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I'd like to obtain an analytical form for the following definite integral, in which $f$ and $b$ are positive quantities.

Integrate[1/( Sqrt[f + b*x^2]), {x, 1, a}]

Mathematica is unable to provide an answer directly. Is there another way to obtain an analytical answer?

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4 Answers 4

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You can use Rubi

Get["Rubi`"]

Assuming[f > 0 && b > 0, Int[1/(Sqrt[f + b*x^2]), {x, 1, a}]]

rubi

Edit: Without Rubi and giving an extra assumption

FullSimplify@
 Assuming[f > 0 && b > 0 && a > 1, 
  Integrate[1/(Sqrt[f + b*x^2]), {x, 1, a}]]

res2

Edit 3: Noteworthy

FullSimplify@
   Assuming[f > 0 && b > 0, Int[1/(Sqrt[f + b*x^2]), {x, 1, a}]] // 
  TrigToExp // Simplify

rubiagain

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  • $\begingroup$ Note to OP: In case you're unfamiliar with Rubi, it's an add-on package developed by Albert Rich. Installation instructions can be found here: rulebasedintegration.org $\endgroup$
    – theorist
    Commented Apr 13, 2022 at 6:21
  • $\begingroup$ @theorist many thanks. I have added the link to the answer $\endgroup$
    – bmf
    Commented Apr 13, 2022 at 14:16
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Expanding the integrand into a series

coef[n_] = SeriesCoefficient[1/(Sqrt[f + b*x^2]), {x, 0, n}]

enter image description here

Verifying the series expansion,

1/Sqrt[f + b*x^2] == Sum[coef[n] x^n, {n, 0, Infinity, 2}] // 
 Simplify[#, f > 0] &

(* True *)

Exchanging the order of integration and summation, the integral is

int = Assuming[{a > 1, f > 0}, 
  Sum[Integrate[coef[n] x^n, {x, 1, a}], {n, 0, Infinity, 2}]]

(* (-ArcSinh[Sqrt[b/f]] + ArcSinh[a Sqrt[b/f]])/Sqrt[b] *)

The Rubi result provided by bmf is

int2 = -ArcTanh[Sqrt[b/(b + f)]]/Sqrt[b] + ArcTanh[a/Sqrt[a^2 + f/b]]/Sqrt[b];

Numerically, comparing the results

And @@ Table[int == int2 /. {
    a -> RandomReal[{11/10, 100}, WorkingPrecision -> 15],
    b -> RandomReal[{1/10, 100}, WorkingPrecision -> 15],
    f -> RandomReal[{1/10, 100}, WorkingPrecision -> 15]}, {10000}]

(* True *)
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  • $\begingroup$ Very clever approach! $\endgroup$
    – bmf
    Commented Apr 12, 2022 at 22:02
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Extending a trick I picked up from one of Daniel Lichtblau's answers, we can try substituting symbolic mathematical constants for the parameters f and b, and for the upper integration limit a, and then then back-substitute:

expr = (1/Sqrt[f + b*x^2]) /. {f -> Khinchin, b -> GoldenRatio};
int = Integrate[expr, {x, 1, Catalan}]
int1 = int /. {Khinchin -> f, GoldenRatio -> b, Catalan -> a}

enter image description here

enter image description here

Note that since these mathematical constants are all specific real-valued positive numbers, the answer may not be general. Nevertheless, in this case, it gives an answer identical to Bob Hanlon's, whose only restrictions were $a > 1, f > 0$.

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  • 1
    $\begingroup$ Integrate[1/(Sqrt[f + b*x^2]), {x, 1, a}, GenerateConditions -> False] yields this result without tricks, while with assumptions in the questions Integrate[1/(Sqrt[f + b*x^2]), {x, 1, a}, Assumptions -> b > 0 && f > 0 && a \[Element] Reals] it returns that with condition a>1. $\endgroup$
    – Artes
    Commented Apr 13, 2022 at 6:57
  • $\begingroup$ @Artes Ah--I assumed the other posters had already tried those more basic approaches before giving their answers, but I should have checked those myself. $\endgroup$
    – theorist
    Commented Apr 13, 2022 at 17:01
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The older version 8.0 for Microsoft Windows (32-bit) has no problems with that integral at all.

Let me show, how it works with different assumptions:

int1[a_, b_, f_] = 
 Integrate[1/(Sqrt[f + b*x^2]), {x, 1, a}, 
  Assumptions -> {f > 0, b > 0, a > 1}]

(*   Log[(a b + Sqrt[b (a^2 b + f)])/(b + Sqrt[b (b + f)])]/Sqrt[b]   *)

int2[a_, b_, f_] = 
 Integrate[1/(Sqrt[f + b*x^2]), {x, 1, a}, 
  Assumptions -> {f > 0, b > 0}]

(*   ConditionalExpression[(-ArcSinh[Sqrt[b/f]] + 
  ArcSinh[a Sqrt[b/f]])/Sqrt[b], a > 1]   *)

int1[a, b, f] == int2[a, b, f] // TrigToExp // 
 FullSimplify[#, Assumptions -> {f > 0, b > 0, a > 1}] &

(*   True   *)

int3[a_, b_, f_] = 
 Integrate[1/(Sqrt[f + b*x^2]), {x, 1, a}, 
  Assumptions -> {f > 0, b \[Element] Reals}]

(*   ConditionalExpression[-((
  I Log[(a + Sqrt[a^2 + f/b])/(1 + Sqrt[(b + f)/b])])/Sqrt[-b]), 
 b + f < 0 && a > 1]   *)

int4[a_, b_, f_] = 
 Integrate[1/(Sqrt[f + b*x^2]), {x, 1, a}, 
  Assumptions -> Element[{b, f}, Reals]]

(*   ConditionalExpression[-((
  I Log[(a + Sqrt[a^2 + f/b])/(1 + Sqrt[(b + f)/b])])/Sqrt[-b]), 
 f > 0 && b + f < 0 && a > 1]   *)

int5[a_, b_, f_] = 
 Integrate[1/(Sqrt[f + b*x^2]), {x, 1, a}, 
  Assumptions -> {Element[{b, f}, Reals], a < 1}]

(*   ConditionalExpression[-((
  I Log[(a + Sqrt[a^2 + f/b])/(1 + Sqrt[(b + f)/b])])/Sqrt[-b]), 
 f > 0 && a > 0 && a^2 b + f < 0]   *)

int6[a_, b_, f_] = 
 Integrate[1/(Sqrt[f + b*x^2]), {x, 1, a}, 
  Assumptions -> {Element[{b, f}, Reals], a < 0}]

(*   ConditionalExpression[(-ArcSinh[Sqrt[b/f]] + 
  ArcSinh[a Sqrt[b/f]])/Sqrt[b], (b >= 0 || a + Sqrt[-(f/b)] > 0) && 
  f > 0 && b + f > 0]   *)

nint[a_, b_, f_] := 
 NIntegrate[1/(Sqrt[f + b*x^2]), {x, 1, a}, MaxRecursion -> 50]

Comparison with numerical integration indicates, that int1 is valid for all parameter values. (Didn't check it intensivly)

tab = (Table[
       With[{a = RandomReal[{-3, 3}] // Rationalize[#, 10^-3] &, 
         b = RandomReal[{-3, 3}] // Rationalize[#, 10^-3] &, 
         f = RandomReal[{-3, 3}] // Rationalize[#, 10^-3] &}, {{a, b, 
          f}, nint[a, b, 
          f], {int1[a, b, f] // N}, {int2[a, b, f] // 
           N}, {int3[a, b, f] // N}, {int4[a, b, f] // 
           N}, {int5[a, b, f] // N}, {int6[a, b, f] // N}}], {10}] /. 
      Undefined -> {}) // Quiet // Chop // MatrixForm
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