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The integral I'm trying to solve is $$ \int{\frac{12\left(c_3-R\ c_2\cos{(\frac{s}{R}})+R\ c_1\sin{(\frac{s}{R})}\right)}{l\ Y\left(\frac{H}{2}+4k\frac{s(s-\pi R)}{(\pi R)^2}\right)^3}ds} $$ but, when given to Wolfram Mathematica, the result is a complex function and I cannot accept this solution.
The code I used is this:

\[Theta][s_] =Integrate[(12 (c3 - R c2 Cos[s/R] + R c1 Sin[s/R]))/(l y (H/2 + (
   4 k s (-\[Pi] R + s))/(\[Pi]^2 R^2))^3),s]

What could be the problem? Is there a problem in the first place or am I getting something wrong? Is there a way to force Mathematica into giving me a real function as a result?
Thanks to anyone in advance

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  • 1
    $\begingroup$ Hi. I think the parameters are a distraction in this question. Also say Integrate[Sin[s]/(1+s^2)^3,s] contains the imaginary unit. $\endgroup$
    – user293787
    Commented Sep 21, 2022 at 11:16
  • 1
    $\begingroup$ An indefinite integral is an anti-derivative. Any function that differs by at most a constant (including complex constants) from the anti-derivative is also a valid anti-derivative. The algorithms that Mathematica uses generally operate in the complex domain and may return results that are or appear complex. Also "many algebraic numbers are real-valued but cannot be expressed in radicals without introducing complex numbers" (casus irreducibilis) $\endgroup$
    – Bob Hanlon
    Commented Sep 21, 2022 at 14:33

1 Answer 1

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the result is a complex function and I cannot accept this solution.

Maple 2022.1 gives anti-derivative with no explicit I in it. Here it is. It is in terms of cos/sin integral special functions. The code is below. I did not verify if it correct or not. You can try to differentiate it and see if you get the integrand back.

enter image description here

Gives

-((18432*c3*k^2*Pi^7*
      R)/(L*(32*H*k*Pi^2 - 64*k^2*Pi^2)^2*(H*Pi^2 - (8*k*Pi*s)/
         R + (8*k*s^2)/R^2)*y)) + (36864*c3*k^2*Pi^6*
    s)/(L*(32*H*k*Pi^2 - 64*k^2*Pi^2)^2*(H*Pi^2 - (8*k*Pi*s)/
       R + (8*k*s^2)/R^2)*y) + 
   (18432*c3*k^2*Pi^6*R*
    ArcTan[(-8*k*Pi + (16*k*s)/R)/(4*
        Sqrt[2*H*k*Pi^2 - 4*k^2*Pi^2])])/(L*(32*H*k*Pi^2 - 
       64*k^2*Pi^2)^2*Sqrt[2*H*k*Pi^2 - 4*k^2*Pi^2]*y) - 
   (9*c1*Pi^2*R^2*
    Cos[(2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)]*
    CosIntegral[-((2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)) + 
      s/R])/(4*(H - 2*k)^2*k*L*y) + 
   (9*c2*Pi^2*R^2*
    Cos[(2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)]*
    CosIntegral[-((2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)) + 
      s/R])/((H - 2*k)^2*L*Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2]*y) - 
   (3*c2*Pi^4*R^2*
    Cos[(2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)]*
    CosIntegral[-((2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)) + 
      s/R])/(4*(H - 2*k)^2*L*Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2]*y) + 
   (3*c2*H*Pi^4*R^2*
    Cos[(2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)]*
    CosIntegral[-((2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)) + 
      s/R])/(8*(H - 2*k)^2*k*L*Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2]*y) - 
   (9*c1*Pi^2*R^2*
    Cos[(2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)]*
    CosIntegral[-((2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)) + 
      s/R])/(4*(H - 2*k)^2*k*L*y) - 
   (9*c2*Pi^2*R^2*
    Cos[(2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)]*
    CosIntegral[-((2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)) + 
      s/R])/((H - 2*k)^2*L*Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2]*y) + 
   (3*c2*Pi^4*R^2*
    Cos[(2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)]*
    CosIntegral[-((2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)) + 
      s/R])/(4*(H - 2*k)^2*L*Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2]*y) - 
   (3*c2*H*Pi^4*R^2*
    Cos[(2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)]*
    CosIntegral[-((2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)) + 
      s/R])/(8*(H - 2*k)^2*k*L*Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2]*y) - 
   (288*c2*k*Pi^2*s^3*
    Cos[s/R])/(L*(H*Pi^2 - (8*k*Pi*s)/R + (8*k*s^2)/R^2)^2*y*
    R[H^2 - 4*H*k + 4*k^2]) - 
   (9*c2*Pi^2*R^2*
    CosIntegral[-((2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)) + 
      s/R]*Sin[(2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*
        k)])/(4*(H - 2*k)^2*k*L*y) - 
   (9*c1*Pi^2*R^2*
    CosIntegral[-((2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)) + 
      s/R]*Sin[(2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*
        k)])/((H - 2*k)^2*L*Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2]*y) + 
   (3*c1*Pi^4*R^2*
    CosIntegral[-((2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)) + 
      s/R]*Sin[(2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*
        k)])/(4*(H - 2*k)^2*L*Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2]*y) - 
   (3*c1*H*Pi^4*R^2*
    CosIntegral[-((2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)) + 
      s/R]*Sin[(2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*
        k)])/(8*(H - 2*k)^2*k*L*Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2]*y) - 
   (9*c2*Pi^2*R^2*
    CosIntegral[-((2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)) + 
      s/R]*Sin[(2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*
        k)])/(4*(H - 2*k)^2*k*L*y) + 
   (9*c1*Pi^2*R^2*
    CosIntegral[-((2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)) + 
      s/R]*Sin[(2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*
        k)])/((H - 2*k)^2*L*Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2]*y) - 
   (3*c1*Pi^4*R^2*
    CosIntegral[-((2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)) + 
      s/R]*Sin[(2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*
        k)])/(4*(H - 2*k)^2*L*Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2]*y) + 
   (3*c1*H*Pi^4*R^2*
    CosIntegral[-((2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)) + 
      s/R]*Sin[(2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*
        k)])/(8*(H - 2*k)^2*k*L*Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2]*y) + 
   (288*c1*k*Pi^2*s^3*
    Sin[s/R])/(L*(H*Pi^2 - (8*k*Pi*s)/R + (8*k*s^2)/R^2)^2*y*
    R[H^2 - 4*H*k + 4*k^2]) + 
   (9*c2*Pi^2*R^2*
    Cos[(2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)]*
    SinIntegral[(2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k) - 
      s/R])/(4*(H - 2*k)^2*k*L*y) + 
   (9*c1*Pi^2*R^2*
    Cos[(2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)]*
    SinIntegral[(2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k) - 
      s/R])/((H - 2*k)^2*L*Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2]*y) - 
   (3*c1*Pi^4*R^2*
    Cos[(2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)]*
    SinIntegral[(2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k) - 
      s/R])/(4*(H - 2*k)^2*L*Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2]*y) + 
   (3*c1*H*Pi^4*R^2*
    Cos[(2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)]*
    SinIntegral[(2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k) - 
      s/R])/(8*(H - 2*k)^2*k*L*Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2]*y) - 
   (9*c1*Pi^2*R^2*
    Sin[(2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)]*
    SinIntegral[(2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k) - 
      s/R])/(4*(H - 2*k)^2*k*L*y) + 
   (9*c2*Pi^2*R^2*
    Sin[(2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)]*
    SinIntegral[(2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k) - 
      s/R])/((H - 2*k)^2*L*Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2]*y) - 
   (3*c2*Pi^4*R^2*
    Sin[(2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)]*
    SinIntegral[(2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k) - 
      s/R])/(4*(H - 2*k)^2*L*Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2]*y) + 
   (3*c2*H*Pi^4*R^2*
    Sin[(2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)]*
    SinIntegral[(2*k*Pi - Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k) - 
      s/R])/(8*(H - 2*k)^2*k*L*Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2]*y) + 
   (9*c2*Pi^2*R^2*
    Cos[(2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)]*
    SinIntegral[(2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k) - 
      s/R])/(4*(H - 2*k)^2*k*L*y) - 
   (9*c1*Pi^2*R^2*
    Cos[(2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)]*
    SinIntegral[(2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k) - 
      s/R])/((H - 2*k)^2*L*Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2]*y) + 
   (3*c1*Pi^4*R^2*
    Cos[(2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)]*
    SinIntegral[(2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k) - 
      s/R])/(4*(H - 2*k)^2*L*Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2]*y) - 
   (3*c1*H*Pi^4*R^2*
    Cos[(2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)]*
    SinIntegral[(2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k) - 
      s/R])/(8*(H - 2*k)^2*k*L*Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2]*y) - 
   (9*c1*Pi^2*R^2*
    Sin[(2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)]*
    SinIntegral[(2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k) - 
      s/R])/(4*(H - 2*k)^2*k*L*y) - 
   (9*c2*Pi^2*R^2*
    Sin[(2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)]*
    SinIntegral[(2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k) - 
      s/R])/((H - 2*k)^2*L*Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2]*y) + 
   (3*c2*Pi^4*R^2*
    Sin[(2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)]*
    SinIntegral[(2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k) - 
      s/R])/(4*(H - 2*k)^2*L*Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2]*y) - 
   (3*c2*H*Pi^4*R^2*
    Sin[(2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k)]*
    SinIntegral[(2*k*Pi + Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2])/(4*k) - 
      s/R])/(8*(H - 2*k)^2*k*L*Sqrt[-2*H*k*Pi^2 + 4*k^2*Pi^2]*y) + 
   (3*c1*Pi^4*R^2*Cos[s/R])/(2*L*
    k[H*Pi^2 - (8*k*Pi*s)/R + (8*k*s^2)/R^2]*y[H - 2*k]) + (3*c2*Pi^4*
    R^2*Sin[s/R])/(2*L*k[H*Pi^2 - (8*k*Pi*s)/R + (8*k*s^2)/R^2]*
    y[H - 2*k]) + 
   (30*c2*H*Pi^5*R^2*
    Cos[s/R])/(L*(H*Pi^2 - (8*k*Pi*s)/R + (8*k*s^2)/R^2)^2*
    y[H^2 - 4*H*k + 4*k^2]) - (24*c2*k*Pi^5*R^2*
    Cos[s/R])/(L*(H*Pi^2 - (8*k*Pi*s)/R + (8*k*s^2)/R^2)^2*
    y[H^2 - 4*H*k + 4*k^2]) - 
   (60*c2*H*Pi^4*R*s*
    Cos[s/R])/(L*(H*Pi^2 - (8*k*Pi*s)/R + (8*k*s^2)/R^2)^2*
    y[H^2 - 4*H*k + 4*k^2]) - (96*c2*k*Pi^4*R*s*
    Cos[s/R])/(L*(H*Pi^2 - (8*k*Pi*s)/R + (8*k*s^2)/R^2)^2*
    y[H^2 - 4*H*k + 4*k^2]) + 
   (432*c2*k*Pi^3*s^2*
    Cos[s/R])/(L*(H*Pi^2 - (8*k*Pi*s)/R + (8*k*s^2)/R^2)^2*
    y[H^2 - 4*H*k + 4*k^2]) - (30*c1*H*Pi^5*R^2*
    Sin[s/R])/(L*(H*Pi^2 - (8*k*Pi*s)/R + (8*k*s^2)/R^2)^2*
    y[H^2 - 4*H*k + 4*k^2]) + 
   (24*c1*k*Pi^5*R^2*
    Sin[s/R])/(L*(H*Pi^2 - (8*k*Pi*s)/R + (8*k*s^2)/R^2)^2*
    y[H^2 - 4*H*k + 4*k^2]) + (60*c1*H*Pi^4*R*s*
    Sin[s/R])/(L*(H*Pi^2 - (8*k*Pi*s)/R + (8*k*s^2)/R^2)^2*
    y[H^2 - 4*H*k + 4*k^2]) + 
   (96*c1*k*Pi^4*R*s*
    Sin[s/R])/(L*(H*Pi^2 - (8*k*Pi*s)/R + (8*k*s^2)/R^2)^2*
    y[H^2 - 4*H*k + 4*k^2]) - (432*c1*k*Pi^3*s^2*
    Sin[s/R])/(L*(H*Pi^2 - (8*k*Pi*s)/R + (8*k*s^2)/R^2)^2*
    y[H^2 - 4*H*k + 4*k^2]) - 
   (384*c3*k*Pi^7*R)/(L*(H*Pi^2 - (8*k*Pi*s)/R + (8*k*s^2)/R^2)^2*
    y[32*H*k*Pi^2 - 64*k^2*Pi^2]) + (768*c3*k*Pi^6*
    s)/(L*(H*Pi^2 - (8*k*Pi*s)/R + (8*k*s^2)/R^2)^2*
    y[32*H*k*Pi^2 - 64*k^2*Pi^2])

Screen shot

Mathematica graphics

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