\(\int \frac {\coth ^2(x)}{(a+b \coth ^2(x))^{5/2}} \, dx\) [44]

Optimal result

Integrand size = 17, antiderivative size = 88 \[ \int \frac {\coth ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^{5/2}}-\frac {\coth (x)}{3 (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}-\frac {(2 a-b) \coth (x)}{3 a (a+b)^2 \sqrt {a+b \coth ^2(x)}} \]

[Out]

arctanh(coth(x)*(a+b)^(1/2)/(a+b*coth(x)^2)^(1/2))/(a+b)^(5/2)-1/3*coth(x)/(a+b)/(a+b*coth(x)^2)^(3/2)-1/3*(2*
a-b)*coth(x)/a/(a+b)^2/(a+b*coth(x)^2)^(1/2)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3751, 482, 541, 12, 385, 212} \[ \int \frac {\coth ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^{5/2}}-\frac {(2 a-b) \coth (x)}{3 a (a+b)^2 \sqrt {a+b \coth ^2(x)}}-\frac {\coth (x)}{3 (a+b) \left (a+b \coth ^2(x)\right )^{3/2}} \]

[In]

Int[Coth[x]^2/(a + b*Coth[x]^2)^(5/2),x]

[Out]

ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]]/(a + b)^(5/2) - Coth[x]/(3*(a + b)*(a + b*Coth[x]^2)^(3/2
)) - ((2*a - b)*Coth[x])/(3*a*(a + b)^2*Sqrt[a + b*Coth[x]^2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 541

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

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 )^{5/2}} \, dx,x,\coth (x)\right ) \\ & = -\frac {\coth (x)}{3 (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {1+2 x^2}{\left (1-x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\coth (x)\right )}{3 (a+b)} \\ & = -\frac {\coth (x)}{3 (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}-\frac {(2 a-b) \coth (x)}{3 a (a+b)^2 \sqrt {a+b \coth ^2(x)}}-\frac {\text {Subst}\left (\int -\frac {3 a}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\coth (x)\right )}{3 a (a+b)^2} \\ & = -\frac {\coth (x)}{3 (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}-\frac {(2 a-b) \coth (x)}{3 a (a+b)^2 \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)^2} \\ & = -\frac {\coth (x)}{3 (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}-\frac {(2 a-b) \coth (x)}{3 a (a+b)^2 \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)^2} \\ & = \frac {\text {arctanh}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^{5/2}}-\frac {\coth (x)}{3 (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}-\frac {(2 a-b) \coth (x)}{3 a (a+b)^2 \sqrt {a+b \coth ^2(x)}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 7.07 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.44 \[ \int \frac {\coth ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=-\frac {\cosh ^2(x) \coth (x) \left (\frac {4 (a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \operatorname {Hypergeometric2F1}\left (2,2,\frac {9}{2},\frac {(a+b) \cosh ^2(x)}{a}\right )}{35 a^2}-\frac {\left (-5 a-2 b \coth ^2(x)\right ) \left (3 \arcsin \left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \left (a+b \coth ^2(x)\right )^2-a \left (-4 b \coth ^2(x)+a \left (-3-\coth ^2(x)\right )\right ) \text {csch}^2(x) \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}\right ) \tanh ^4(x)}{3 a (a+b)^2 \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}}\right )}{3 a^2 \sqrt {a+b \coth ^2(x)} \left (1+\frac {b \coth ^2(x)}{a}\right )} \]

[In]

Integrate[Coth[x]^2/(a + b*Coth[x]^2)^(5/2),x]

[Out]

-1/3*(Cosh[x]^2*Coth[x]*((4*(a + b)*Cosh[x]^2*(a + b*Coth[x]^2)*Hypergeometric2F1[2, 2, 9/2, ((a + b)*Cosh[x]^
2)/a])/(35*a^2) - ((-5*a - 2*b*Coth[x]^2)*(3*ArcSin[Sqrt[((a + b)*Cosh[x]^2)/a]]*(a + b*Coth[x]^2)^2 - a*(-4*b
*Coth[x]^2 + a*(-3 - Coth[x]^2))*Csch[x]^2*Sqrt[-(((a + b)*Cosh[x]^2*(a + b*Coth[x]^2)*Sinh[x]^2)/a^2)])*Tanh[
x]^4)/(3*a*(a + b)^2*Sqrt[-(((a + b)*Cosh[x]^2*(a + b*Coth[x]^2)*Sinh[x]^2)/a^2)])))/(a^2*Sqrt[a + b*Coth[x]^2
]*(1 + (b*Coth[x]^2)/a))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(453\) vs. \(2(74)=148\).

Time = 0.10 (sec) , antiderivative size = 454, normalized size of antiderivative = 5.16

method	result	size
derivativedivides	\(-\frac {\coth \left (x \right )}{3 a \left (a +b \coth \left (x \right )^{2}\right )^{\frac {3}{2}}}-\frac {2 \coth \left (x \right )}{3 a^{2} \sqrt {a +b \coth \left (x \right )^{2}}}-\frac {1}{6 \left (a +b \right ) \left (b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \coth \left (x \right )}{6 \left (a +b \right ) a \left (b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \coth \left (x \right )}{3 \left (a +b \right ) a^{2} \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}-\frac {1}{2 \left (a +b \right )^{2} \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}+\frac {b \coth \left (x \right )}{2 \left (a +b \right )^{2} a \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 {5}{2}}}+\frac {1}{6 \left (a +b \right ) \left (b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \coth \left (x \right )}{6 \left (a +b \right ) a \left (b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \coth \left (x \right )}{3 \left (a +b \right ) a^{2} \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}+\frac {1}{2 \left (a +b \right )^{2} \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}+\frac {b \coth \left (x \right )}{2 \left (a +b \right )^{2} a \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 {5}{2}}}\)	\(454\)
default	\(-\frac {\coth \left (x \right )}{3 a \left (a +b \coth \left (x \right )^{2}\right )^{\frac {3}{2}}}-\frac {2 \coth \left (x \right )}{3 a^{2} \sqrt {a +b \coth \left (x \right )^{2}}}-\frac {1}{6 \left (a +b \right ) \left (b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \coth \left (x \right )}{6 \left (a +b \right ) a \left (b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \coth \left (x \right )}{3 \left (a +b \right ) a^{2} \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}-\frac {1}{2 \left (a +b \right )^{2} \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}+\frac {b \coth \left (x \right )}{2 \left (a +b \right )^{2} a \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 {5}{2}}}+\frac {1}{6 \left (a +b \right ) \left (b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \coth \left (x \right )}{6 \left (a +b \right ) a \left (b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \coth \left (x \right )}{3 \left (a +b \right ) a^{2} \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}+\frac {1}{2 \left (a +b \right )^{2} \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}+\frac {b \coth \left (x \right )}{2 \left (a +b \right )^{2} a \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 {5}{2}}}\)	\(454\)

[In]

int(coth(x)^2/(a+b*coth(x)^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*coth(x)/a/(a+b*coth(x)^2)^(3/2)-2/3/a^2*coth(x)/(a+b*coth(x)^2)^(1/2)-1/6/(a+b)/(b*(coth(x)-1)^2+2*b*(cot
h(x)-1)+a+b)^(3/2)+1/6*b/(a+b)/a/(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(3/2)*coth(x)+1/3*b/(a+b)/a^2/(b*(coth(
x)-1)^2+2*b*(coth(x)-1)+a+b)^(1/2)*coth(x)-1/2/(a+b)^2/(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(1/2)+1/2/(a+b)^2
/a/(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(1/2)*b*coth(x)+1/2/(a+b)^(5/2)*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))+1/6/(a+b)/(b*(1+coth(x))^2-2*b*(1+coth(x))+a+b)
^(3/2)+1/6*b/(a+b)/a/(b*(1+coth(x))^2-2*b*(1+coth(x))+a+b)^(3/2)*coth(x)+1/3*b/(a+b)/a^2/(b*(1+coth(x))^2-2*b*
(1+coth(x))+a+b)^(1/2)*coth(x)+1/2/(a+b)^2/(b*(1+coth(x))^2-2*b*(1+coth(x))+a+b)^(1/2)+1/2/(a+b)^2/a/(b*(1+cot
h(x))^2-2*b*(1+coth(x))+a+b)^(1/2)*b*coth(x)-1/2/(a+b)^(5/2)*ln((2*a+2*b-2*b*(1+coth(x))+2*(a+b)^(1/2)*(b*(1+c
oth(x))^2-2*b*(1+coth(x))+a+b)^(1/2))/(1+coth(x)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3016 vs. \(2 (74) = 148\).

Time = 0.68 (sec) , antiderivative size = 6591, normalized size of antiderivative = 74.90 \[ \int \frac {\coth ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate(coth(x)^2/(a+b*coth(x)^2)^(5/2),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F]

\[ \int \frac {\coth ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\int \frac {\coth ^{2}{\left (x \right )}}{\left (a + b \coth ^{2}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(coth(x)**2/(a+b*coth(x)**2)**(5/2),x)

[Out]

Integral(coth(x)**2/(a + b*coth(x)**2)**(5/2), x)

Maxima [F]

\[ \int \frac {\coth ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\int { \frac {\coth \left (x\right )^{2}}{{\left (b \coth \left (x\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(coth(x)^2/(a+b*coth(x)^2)^(5/2),x, algorithm="maxima")

[Out]

integrate(coth(x)^2/(b*coth(x)^2 + a)^(5/2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 952 vs. \(2 (74) = 148\).

Time = 0.55 (sec) , antiderivative size = 952, normalized size of antiderivative = 10.82 \[ \int \frac {\coth ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate(coth(x)^2/(a+b*coth(x)^2)^(5/2),x, algorithm="giac")

[Out]

-1/3*((((3*a^7*b^2*sgn(e^(2*x) - 1) + 14*a^6*b^3*sgn(e^(2*x) - 1) + 25*a^5*b^4*sgn(e^(2*x) - 1) + 20*a^4*b^5*s
gn(e^(2*x) - 1) + 5*a^3*b^6*sgn(e^(2*x) - 1) - 2*a^2*b^7*sgn(e^(2*x) - 1) - a*b^8*sgn(e^(2*x) - 1))*e^(2*x)/(a
^8*b^2 + 6*a^7*b^3 + 15*a^6*b^4 + 20*a^5*b^5 + 15*a^4*b^6 + 6*a^3*b^7 + a^2*b^8) - 3*(a^7*b^2*sgn(e^(2*x) - 1)
+ 2*a^6*b^3*sgn(e^(2*x) - 1) - a^5*b^4*sgn(e^(2*x) - 1) - 4*a^4*b^5*sgn(e^(2*x) - 1) - a^3*b^6*sgn(e^(2*x) -
1) + 2*a^2*b^7*sgn(e^(2*x) - 1) + a*b^8*sgn(e^(2*x) - 1))/(a^8*b^2 + 6*a^7*b^3 + 15*a^6*b^4 + 20*a^5*b^5 + 15*
a^4*b^6 + 6*a^3*b^7 + a^2*b^8))*e^(2*x) - 3*(a^7*b^2*sgn(e^(2*x) - 1) + 2*a^6*b^3*sgn(e^(2*x) - 1) - a^5*b^4*s
gn(e^(2*x) - 1) - 4*a^4*b^5*sgn(e^(2*x) - 1) - a^3*b^6*sgn(e^(2*x) - 1) + 2*a^2*b^7*sgn(e^(2*x) - 1) + a*b^8*s
gn(e^(2*x) - 1))/(a^8*b^2 + 6*a^7*b^3 + 15*a^6*b^4 + 20*a^5*b^5 + 15*a^4*b^6 + 6*a^3*b^7 + a^2*b^8))*e^(2*x) +
(3*a^7*b^2*sgn(e^(2*x) - 1) + 14*a^6*b^3*sgn(e^(2*x) - 1) + 25*a^5*b^4*sgn(e^(2*x) - 1) + 20*a^4*b^5*sgn(e^(2
*x) - 1) + 5*a^3*b^6*sgn(e^(2*x) - 1) - 2*a^2*b^7*sgn(e^(2*x) - 1) - a*b^8*sgn(e^(2*x) - 1))/(a^8*b^2 + 6*a^7*
b^3 + 15*a^6*b^4 + 20*a^5*b^5 + 15*a^4*b^6 + 6*a^3*b^7 + a^2*b^8))/(a*e^(4*x) + b*e^(4*x) - 2*a*e^(2*x) + 2*b*
e^(2*x) + a + b)^(3/2) - 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^2 + 2*a*b + b^2)*sqrt(a + b)*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))/((a^2 +
2*a*b + b^2)*sqrt(a + b)*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^2 + 2*a*b + b^2)*sqrt(a + b)*sgn(e^(2*x) - 1))

Mupad [F(-1)]

Timed out. \[ \int \frac {\coth ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\int \frac {{\mathrm {coth}\left (x\right )}^2}{{\left (b\,{\mathrm {coth}\left (x\right )}^2+a\right )}^{5/2}} \,d x \]

[In]

int(coth(x)^2/(a + b*coth(x)^2)^(5/2),x)

[Out]

int(coth(x)^2/(a + b*coth(x)^2)^(5/2), x)