\(\int \frac {\coth ^2(x)}{(a+b \coth ^2(x))^{3/2}} \, dx\) [39]
Optimal result
Integrand size = 17, antiderivative size = 53 \[
\int \frac {\coth ^2(x)}{\left (a+b \coth ^2(x)\right )^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^{3/2}}-\frac {\coth (x)}{(a+b) \sqrt {a+b \coth ^2(x)}}
\]
[Out]
arctanh(coth(x)*(a+b)^(1/2)/(a+b*coth(x)^2)^(1/2))/(a+b)^(3/2)-coth(x)/(a+b)/(a+b*coth(x)^2)^(1/2)
Rubi [A] (verified)
Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of
steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {3751, 482, 385, 212}
\[
\int \frac {\coth ^2(x)}{\left (a+b \coth ^2(x)\right )^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^{3/2}}-\frac {\coth (x)}{(a+b) \sqrt {a+b \coth ^2(x)}}
\]
[In]
Int[Coth[x]^2/(a + b*Coth[x]^2)^(3/2),x]
[Out]
ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]]/(a + b)^(3/2) - Coth[x]/((a + b)*Sqrt[a + b*Coth[x]^2])
Rule 212
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 385
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
Rule 482
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[e^(n - 1
)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(n*(b*c - a*d)*(p + 1))), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Rule 3751
Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[c*(ff/f), Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2
+ ff^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))
Rubi steps \begin{align*}
\text {integral}& = \text {Subst}\left (\int \frac {x^2}{\left (1-x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\coth (x)\right ) \\ & = -\frac {\coth (x)}{(a+b) \sqrt {a+b \coth ^2(x)}}+\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\coth (x)\right )}{a+b} \\ & = -\frac {\coth (x)}{(a+b) \sqrt {a+b \coth ^2(x)}}+\frac {\text {Subst}\left (\int \frac {1}{1-(a+b) x^2} \, dx,x,\frac {\coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{a+b} \\ & = \frac {\text {arctanh}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^{3/2}}-\frac {\coth (x)}{(a+b) \sqrt {a+b \coth ^2(x)}} \\
\end{align*}
Mathematica [B] (verified)
Leaf count is larger than twice the leaf count of optimal. \(109\) vs. \(2(53)=106\).
Time = 1.11 (sec) , antiderivative size = 109, normalized size of antiderivative = 2.06
\[
\int \frac {\coth ^2(x)}{\left (a+b \coth ^2(x)\right )^{3/2}} \, dx=\frac {-2 (a+b) \coth (x)+\frac {\text {arctanh}\left (\frac {\sqrt {\frac {(a+b) \coth ^2(x)}{a}}}{\sqrt {1+\frac {b \coth ^2(x)}{a}}}\right ) (-a+b+(a+b) \cosh (2 x)) \sqrt {\frac {(a+b) \coth ^2(x)}{a}} \text {csch}(x) \text {sech}(x)}{\sqrt {1+\frac {b \coth ^2(x)}{a}}}}{2 (a+b)^2 \sqrt {a+b \coth ^2(x)}}
\]
[In]
Integrate[Coth[x]^2/(a + b*Coth[x]^2)^(3/2),x]
[Out]
(-2*(a + b)*Coth[x] + (ArcTanh[Sqrt[((a + b)*Coth[x]^2)/a]/Sqrt[1 + (b*Coth[x]^2)/a]]*(-a + b + (a + b)*Cosh[2
*x])*Sqrt[((a + b)*Coth[x]^2)/a]*Csch[x]*Sech[x])/Sqrt[1 + (b*Coth[x]^2)/a])/(2*(a + b)^2*Sqrt[a + b*Coth[x]^2
])
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(288\) vs. \(2(45)=90\).
Time = 0.10 (sec) , antiderivative size = 289, normalized size of antiderivative = 5.45
| | |
| method | result | size |
| | |
| derivativedivides |
\(-\frac {\coth \left (x \right )}{a \sqrt {a +b \coth \left (x \right )^{2}}}-\frac {1}{2 \left (a +b \right ) \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}+\frac {b \left (2 b \left (\coth \left (x \right )-1\right )+2 b \right )}{\left (a +b \right ) \left (4 \left (a +b \right ) b -4 b^{2}\right ) \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}+\frac {\ln \left (\frac {2 a +2 b +2 b \left (\coth \left (x \right )-1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}{\coth \left (x \right )-1}\right )}{2 \left (a +b \right )^{\frac {3}{2}}}+\frac {1}{2 \left (a +b \right ) \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}+\frac {b \left (2 b \left (1+\coth \left (x \right )\right )-2 b \right )}{\left (a +b \right ) \left (4 \left (a +b \right ) b -4 b^{2}\right ) \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}-\frac {\ln \left (\frac {2 a +2 b -2 b \left (1+\coth \left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}{1+\coth \left (x \right )}\right )}{2 \left (a +b \right )^{\frac {3}{2}}}\) |
\(289\) |
| default |
\(-\frac {\coth \left (x \right )}{a \sqrt {a +b \coth \left (x \right )^{2}}}-\frac {1}{2 \left (a +b \right ) \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}+\frac {b \left (2 b \left (\coth \left (x \right )-1\right )+2 b \right )}{\left (a +b \right ) \left (4 \left (a +b \right ) b -4 b^{2}\right ) \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}+\frac {\ln \left (\frac {2 a +2 b +2 b \left (\coth \left (x \right )-1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}{\coth \left (x \right )-1}\right )}{2 \left (a +b \right )^{\frac {3}{2}}}+\frac {1}{2 \left (a +b \right ) \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}+\frac {b \left (2 b \left (1+\coth \left (x \right )\right )-2 b \right )}{\left (a +b \right ) \left (4 \left (a +b \right ) b -4 b^{2}\right ) \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}-\frac {\ln \left (\frac {2 a +2 b -2 b \left (1+\coth \left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}{1+\coth \left (x \right )}\right )}{2 \left (a +b \right )^{\frac {3}{2}}}\) |
\(289\) |
| | |
|
|
|
|
[In]
int(coth(x)^2/(a+b*coth(x)^2)^(3/2),x,method=_RETURNVERBOSE)
[Out]
-coth(x)/a/(a+b*coth(x)^2)^(1/2)-1/2/(a+b)/(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(1/2)+b/(a+b)*(2*b*(coth(x)-1
)+2*b)/(4*(a+b)*b-4*b^2)/(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(1/2)+1/2/(a+b)^(3/2)*ln((2*a+2*b+2*b*(coth(x)-
1)+2*(a+b)^(1/2)*(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(1/2))/(coth(x)-1))+1/2/(a+b)/(b*(1+coth(x))^2-2*b*(1+c
oth(x))+a+b)^(1/2)+b/(a+b)*(2*b*(1+coth(x))-2*b)/(4*(a+b)*b-4*b^2)/(b*(1+coth(x))^2-2*b*(1+coth(x))+a+b)^(1/2)
-1/2/(a+b)^(3/2)*ln((2*a+2*b-2*b*(1+coth(x))+2*(a+b)^(1/2)*(b*(1+coth(x))^2-2*b*(1+coth(x))+a+b)^(1/2))/(1+cot
h(x)))
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 860 vs. \(2 (45) = 90\).
Time = 0.35 (sec) , antiderivative size = 2279, normalized size of antiderivative = 43.00
\[
\int \frac {\coth ^2(x)}{\left (a+b \coth ^2(x)\right )^{3/2}} \, dx=\text {Too large to display}
\]
[In]
integrate(coth(x)^2/(a+b*coth(x)^2)^(3/2),x, algorithm="fricas")
[Out]
[1/4*(((a + b)*cosh(x)^4 + 4*(a + b)*cosh(x)*sinh(x)^3 + (a + b)*sinh(x)^4 - 2*(a - b)*cosh(x)^2 + 2*(3*(a + b
)*cosh(x)^2 - a + b)*sinh(x)^2 + 4*((a + b)*cosh(x)^3 - (a - b)*cosh(x))*sinh(x) + a + b)*sqrt(a + b)*log(((a*
b^2 + b^3)*cosh(x)^8 + 8*(a*b^2 + b^3)*cosh(x)*sinh(x)^7 + (a*b^2 + b^3)*sinh(x)^8 + 2*(a*b^2 + 2*b^3)*cosh(x)
^6 + 2*(a*b^2 + 2*b^3 + 14*(a*b^2 + b^3)*cosh(x)^2)*sinh(x)^6 + 4*(14*(a*b^2 + b^3)*cosh(x)^3 + 3*(a*b^2 + 2*b
^3)*cosh(x))*sinh(x)^5 + (a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x)^4 + (70*(a*b^2 + b^3)*cosh(x)^4 + a^3 - a^2*b
+ 4*a*b^2 + 6*b^3 + 30*(a*b^2 + 2*b^3)*cosh(x)^2)*sinh(x)^4 + 4*(14*(a*b^2 + b^3)*cosh(x)^5 + 10*(a*b^2 + 2*b
^3)*cosh(x)^3 + (a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x))*sinh(x)^3 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 - 2*(a^3 -
3*a*b^2 - 2*b^3)*cosh(x)^2 + 2*(14*(a*b^2 + b^3)*cosh(x)^6 + 15*(a*b^2 + 2*b^3)*cosh(x)^4 - a^3 + 3*a*b^2 + 2*
b^3 + 3*(a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x)^2)*sinh(x)^2 + sqrt(2)*(b^2*cosh(x)^6 + 6*b^2*cosh(x)*sinh(x)^
5 + b^2*sinh(x)^6 + 3*b^2*cosh(x)^4 + 3*(5*b^2*cosh(x)^2 + b^2)*sinh(x)^4 + 4*(5*b^2*cosh(x)^3 + 3*b^2*cosh(x)
)*sinh(x)^3 - (a^2 - 2*a*b - 3*b^2)*cosh(x)^2 + (15*b^2*cosh(x)^4 + 18*b^2*cosh(x)^2 - a^2 + 2*a*b + 3*b^2)*si
nh(x)^2 + a^2 + 2*a*b + b^2 + 2*(3*b^2*cosh(x)^5 + 6*b^2*cosh(x)^3 - (a^2 - 2*a*b - 3*b^2)*cosh(x))*sinh(x))*s
qrt(a + b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 - a + b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) +
4*(2*(a*b^2 + b^3)*cosh(x)^7 + 3*(a*b^2 + 2*b^3)*cosh(x)^5 + (a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x)^3 - (a^3
- 3*a*b^2 - 2*b^3)*cosh(x))*sinh(x))/(cosh(x)^6 + 6*cosh(x)^5*sinh(x) + 15*cosh(x)^4*sinh(x)^2 + 20*cosh(x)^3
*sinh(x)^3 + 15*cosh(x)^2*sinh(x)^4 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6)) + ((a + b)*cosh(x)^4 + 4*(a + b)*cosh(
x)*sinh(x)^3 + (a + b)*sinh(x)^4 - 2*(a - b)*cosh(x)^2 + 2*(3*(a + b)*cosh(x)^2 - a + b)*sinh(x)^2 + 4*((a + b
)*cosh(x)^3 - (a - b)*cosh(x))*sinh(x) + a + b)*sqrt(a + b)*log(-((a + b)*cosh(x)^4 + 4*(a + b)*cosh(x)*sinh(x
)^3 + (a + b)*sinh(x)^4 - 2*a*cosh(x)^2 + 2*(3*(a + b)*cosh(x)^2 - a)*sinh(x)^2 + sqrt(2)*(cosh(x)^2 + 2*cosh(
x)*sinh(x) + sinh(x)^2 - 1)*sqrt(a + b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 - a + b)/(cosh(x)^2 - 2*co
sh(x)*sinh(x) + sinh(x)^2)) + 4*((a + b)*cosh(x)^3 - a*cosh(x))*sinh(x) + a + b)/(cosh(x)^2 + 2*cosh(x)*sinh(x
) + sinh(x)^2)) - 4*sqrt(2)*((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 + a + b)*sqrt((
(a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 - a + b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/((a^3 + 3*a^2*b +
3*a*b^2 + b^3)*cosh(x)^4 + 4*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)*sinh(x)^3 + (a^3 + 3*a^2*b + 3*a*b^2 + b
^3)*sinh(x)^4 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 - 2*(a^3 + a^2*b - a*b^2 - b^3)*cosh(x)^2 - 2*(a^3 + a^2*b - a*b
^2 - b^3 - 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^2)*sinh(x)^2 + 4*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)
^3 - (a^3 + a^2*b - a*b^2 - b^3)*cosh(x))*sinh(x)), -1/2*(((a + b)*cosh(x)^4 + 4*(a + b)*cosh(x)*sinh(x)^3 + (
a + b)*sinh(x)^4 - 2*(a - b)*cosh(x)^2 + 2*(3*(a + b)*cosh(x)^2 - a + b)*sinh(x)^2 + 4*((a + b)*cosh(x)^3 - (a
- b)*cosh(x))*sinh(x) + a + b)*sqrt(-a - b)*arctan(sqrt(2)*(b*cosh(x)^2 + 2*b*cosh(x)*sinh(x) + b*sinh(x)^2 +
a + b)*sqrt(-a - b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 - a + b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sin
h(x)^2))/((a*b + b^2)*cosh(x)^4 + 4*(a*b + b^2)*cosh(x)*sinh(x)^3 + (a*b + b^2)*sinh(x)^4 - (a^2 - a*b - 2*b^2
)*cosh(x)^2 + (6*(a*b + b^2)*cosh(x)^2 - a^2 + a*b + 2*b^2)*sinh(x)^2 + a^2 + 2*a*b + b^2 + 2*(2*(a*b + b^2)*c
osh(x)^3 - (a^2 - a*b - 2*b^2)*cosh(x))*sinh(x))) + ((a + b)*cosh(x)^4 + 4*(a + b)*cosh(x)*sinh(x)^3 + (a + b)
*sinh(x)^4 - 2*(a - b)*cosh(x)^2 + 2*(3*(a + b)*cosh(x)^2 - a + b)*sinh(x)^2 + 4*((a + b)*cosh(x)^3 - (a - b)*
cosh(x))*sinh(x) + a + b)*sqrt(-a - b)*arctan(sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*sqrt(-a
- b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 - a + b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/((a + b
)*cosh(x)^4 + 4*(a + b)*cosh(x)*sinh(x)^3 + (a + b)*sinh(x)^4 - 2*(a - b)*cosh(x)^2 + 2*(3*(a + b)*cosh(x)^2 -
a + b)*sinh(x)^2 + 4*((a + b)*cosh(x)^3 - (a - b)*cosh(x))*sinh(x) + a + b)) + 2*sqrt(2)*((a + b)*cosh(x)^2 +
2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 + a + b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 - a + b)/(
cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^4 + 4*(a^3 + 3*a^2*b + 3
*a*b^2 + b^3)*cosh(x)*sinh(x)^3 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sinh(x)^4 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 -
2*(a^3 + a^2*b - a*b^2 - b^3)*cosh(x)^2 - 2*(a^3 + a^2*b - a*b^2 - b^3 - 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos
h(x)^2)*sinh(x)^2 + 4*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^3 - (a^3 + a^2*b - a*b^2 - b^3)*cosh(x))*sinh(x
))]
Sympy [F]
\[
\int \frac {\coth ^2(x)}{\left (a+b \coth ^2(x)\right )^{3/2}} \, dx=\int \frac {\coth ^{2}{\left (x \right )}}{\left (a + b \coth ^{2}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx
\]
[In]
integrate(coth(x)**2/(a+b*coth(x)**2)**(3/2),x)
[Out]
Integral(coth(x)**2/(a + b*coth(x)**2)**(3/2), x)
Maxima [F]
\[
\int \frac {\coth ^2(x)}{\left (a+b \coth ^2(x)\right )^{3/2}} \, dx=\int { \frac {\coth \left (x\right )^{2}}{{\left (b \coth \left (x\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x }
\]
[In]
integrate(coth(x)^2/(a+b*coth(x)^2)^(3/2),x, algorithm="maxima")
[Out]
integrate(coth(x)^2/(b*coth(x)^2 + a)^(3/2), x)
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 363 vs. \(2 (45) = 90\).
Time = 0.46 (sec) , antiderivative size = 363, normalized size of antiderivative = 6.85
\[
\int \frac {\coth ^2(x)}{\left (a+b \coth ^2(x)\right )^{3/2}} \, dx=-\frac {\frac {{\left (a^{2} b + a b^{2}\right )} e^{\left (2 \, x\right )}}{a^{3} b \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + 2 \, a^{2} b^{2} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + a b^{3} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} + \frac {a^{2} b + a b^{2}}{a^{3} b \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + 2 \, a^{2} b^{2} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + a b^{3} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )}}{\sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + 2 \, b e^{\left (2 \, x\right )} + a + b}} - \frac {\log \left ({\left | {\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + 2 \, b e^{\left (2 \, x\right )} + a + b}\right )} \sqrt {a + b} - a + b \right |}\right )}{2 \, {\left (a + b\right )}^{\frac {3}{2}} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} + \frac {\log \left ({\left | {\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + 2 \, b e^{\left (2 \, x\right )} + a + b}\right )} \sqrt {a + b} - a - b \right |}\right )}{2 \, {\left (a + b\right )}^{\frac {3}{2}} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} - \frac {\log \left ({\left | -\sqrt {a + b} e^{\left (2 \, x\right )} + \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + 2 \, b e^{\left (2 \, x\right )} + a + b} - \sqrt {a + b} \right |}\right )}{2 \, {\left (a + b\right )}^{\frac {3}{2}} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )}
\]
[In]
integrate(coth(x)^2/(a+b*coth(x)^2)^(3/2),x, algorithm="giac")
[Out]
-((a^2*b + a*b^2)*e^(2*x)/(a^3*b*sgn(e^(2*x) - 1) + 2*a^2*b^2*sgn(e^(2*x) - 1) + a*b^3*sgn(e^(2*x) - 1)) + (a^
2*b + a*b^2)/(a^3*b*sgn(e^(2*x) - 1) + 2*a^2*b^2*sgn(e^(2*x) - 1) + a*b^3*sgn(e^(2*x) - 1)))/sqrt(a*e^(4*x) +
b*e^(4*x) - 2*a*e^(2*x) + 2*b*e^(2*x) + a + b) - 1/2*log(abs((sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x)
- 2*a*e^(2*x) + 2*b*e^(2*x) + a + b))*sqrt(a + b) - a + b))/((a + b)^(3/2)*sgn(e^(2*x) - 1)) + 1/2*log(abs((s
qrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) - 2*a*e^(2*x) + 2*b*e^(2*x) + a + b))*sqrt(a + b) - a - b))/((
a + b)^(3/2)*sgn(e^(2*x) - 1)) - 1/2*log(abs(-sqrt(a + b)*e^(2*x) + sqrt(a*e^(4*x) + b*e^(4*x) - 2*a*e^(2*x) +
2*b*e^(2*x) + a + b) - sqrt(a + b)))/((a + b)^(3/2)*sgn(e^(2*x) - 1))
Mupad [F(-1)]
Timed out. \[
\int \frac {\coth ^2(x)}{\left (a+b \coth ^2(x)\right )^{3/2}} \, dx=\int \frac {{\mathrm {coth}\left (x\right )}^2}{{\left (b\,{\mathrm {coth}\left (x\right )}^2+a\right )}^{3/2}} \,d x
\]
[In]
int(coth(x)^2/(a + b*coth(x)^2)^(3/2),x)
[Out]
int(coth(x)^2/(a + b*coth(x)^2)^(3/2), x)