3.22 \(\int (-\text{csch}^2(x))^{5/2} \, dx\)

Optimal. Leaf size=40 \[ \frac{3}{8} \sin ^{-1}(\coth (x))+\frac{1}{4} \coth (x) \left (-\text{csch}^2(x)\right )^{3/2}+\frac{3}{8} \coth (x) \sqrt{-\text{csch}^2(x)} \]

[Out]

(3*ArcSin[Coth[x]])/8 + (3*Coth[x]*Sqrt[-Csch[x]^2])/8 + (Coth[x]*(-Csch[x]^2)^(3/2))/4

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Rubi [A]  time = 0.0170618, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4122, 195, 216} \[ \frac{3}{8} \sin ^{-1}(\coth (x))+\frac{1}{4} \coth (x) \left (-\text{csch}^2(x)\right )^{3/2}+\frac{3}{8} \coth (x) \sqrt{-\text{csch}^2(x)} \]

Antiderivative was successfully verified.

[In]

Int[(-Csch[x]^2)^(5/2),x]

[Out]

(3*ArcSin[Coth[x]])/8 + (3*Coth[x]*Sqrt[-Csch[x]^2])/8 + (Coth[x]*(-Csch[x]^2)^(3/2))/4

Rule 4122

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(b*ff)
/f, Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p
]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \left (-\text{csch}^2(x)\right )^{5/2} \, dx &=\operatorname{Subst}\left (\int \left (1-x^2\right )^{3/2} \, dx,x,\coth (x)\right )\\ &=\frac{1}{4} \coth (x) \left (-\text{csch}^2(x)\right )^{3/2}+\frac{3}{4} \operatorname{Subst}\left (\int \sqrt{1-x^2} \, dx,x,\coth (x)\right )\\ &=\frac{3}{8} \coth (x) \sqrt{-\text{csch}^2(x)}+\frac{1}{4} \coth (x) \left (-\text{csch}^2(x)\right )^{3/2}+\frac{3}{8} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,\coth (x)\right )\\ &=\frac{3}{8} \sin ^{-1}(\coth (x))+\frac{3}{8} \coth (x) \sqrt{-\text{csch}^2(x)}+\frac{1}{4} \coth (x) \left (-\text{csch}^2(x)\right )^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.118668, size = 41, normalized size = 1.02 \[ \frac{1}{64} \sinh (x) \left (-\text{csch}^2(x)\right )^{5/2} \left (6 \left (\cosh (3 x)+4 \sinh ^4(x) \log \left (\tanh \left (\frac{x}{2}\right )\right )\right )-22 \cosh (x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(-Csch[x]^2)^(5/2),x]

[Out]

((-Csch[x]^2)^(5/2)*Sinh[x]*(-22*Cosh[x] + 6*(Cosh[3*x] + 4*Log[Tanh[x/2]]*Sinh[x]^4)))/64

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Maple [B]  time = 0.053, size = 114, normalized size = 2.9 \begin{align*}{\frac{3\,{{\rm e}^{6\,x}}-11\,{{\rm e}^{4\,x}}-11\,{{\rm e}^{2\,x}}+3}{4\, \left ({{\rm e}^{2\,x}}-1 \right ) ^{3}}\sqrt{-{\frac{{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}+{\frac{3\,{{\rm e}^{-x}} \left ({{\rm e}^{2\,x}}-1 \right ) \ln \left ({{\rm e}^{x}}-1 \right ) }{8}\sqrt{-{\frac{{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}-{\frac{3\,{{\rm e}^{-x}} \left ({{\rm e}^{2\,x}}-1 \right ) \ln \left ({{\rm e}^{x}}+1 \right ) }{8}\sqrt{-{\frac{{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-csch(x)^2)^(5/2),x)

[Out]

1/4/(exp(2*x)-1)^3*(-exp(2*x)/(exp(2*x)-1)^2)^(1/2)*(3*exp(6*x)-11*exp(4*x)-11*exp(2*x)+3)+3/8*exp(-x)*(exp(2*
x)-1)*(-exp(2*x)/(exp(2*x)-1)^2)^(1/2)*ln(exp(x)-1)-3/8*exp(-x)*(exp(2*x)-1)*(-exp(2*x)/(exp(2*x)-1)^2)^(1/2)*
ln(exp(x)+1)

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Maxima [C]  time = 1.69723, size = 100, normalized size = 2.5 \begin{align*} \frac{3 i \, e^{\left (-x\right )} - 11 i \, e^{\left (-3 \, x\right )} - 11 i \, e^{\left (-5 \, x\right )} + 3 i \, e^{\left (-7 \, x\right )}}{4 \,{\left (4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1\right )}} + \frac{3}{8} i \, \log \left (e^{\left (-x\right )} + 1\right ) - \frac{3}{8} i \, \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-csch(x)^2)^(5/2),x, algorithm="maxima")

[Out]

1/4*(3*I*e^(-x) - 11*I*e^(-3*x) - 11*I*e^(-5*x) + 3*I*e^(-7*x))/(4*e^(-2*x) - 6*e^(-4*x) + 4*e^(-6*x) - e^(-8*
x) - 1) + 3/8*I*log(e^(-x) + 1) - 3/8*I*log(e^(-x) - 1)

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Fricas [C]  time = 1.60467, size = 365, normalized size = 9.12 \begin{align*} \frac{{\left (-3 i \, e^{\left (8 \, x\right )} + 12 i \, e^{\left (6 \, x\right )} - 18 i \, e^{\left (4 \, x\right )} + 12 i \, e^{\left (2 \, x\right )} - 3 i\right )} \log \left (e^{x} + 1\right ) +{\left (3 i \, e^{\left (8 \, x\right )} - 12 i \, e^{\left (6 \, x\right )} + 18 i \, e^{\left (4 \, x\right )} - 12 i \, e^{\left (2 \, x\right )} + 3 i\right )} \log \left (e^{x} - 1\right ) + 6 i \, e^{\left (7 \, x\right )} - 22 i \, e^{\left (5 \, x\right )} - 22 i \, e^{\left (3 \, x\right )} + 6 i \, e^{x}}{8 \,{\left (e^{\left (8 \, x\right )} - 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} - 4 \, e^{\left (2 \, x\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-csch(x)^2)^(5/2),x, algorithm="fricas")

[Out]

1/8*((-3*I*e^(8*x) + 12*I*e^(6*x) - 18*I*e^(4*x) + 12*I*e^(2*x) - 3*I)*log(e^x + 1) + (3*I*e^(8*x) - 12*I*e^(6
*x) + 18*I*e^(4*x) - 12*I*e^(2*x) + 3*I)*log(e^x - 1) + 6*I*e^(7*x) - 22*I*e^(5*x) - 22*I*e^(3*x) + 6*I*e^x)/(
e^(8*x) - 4*e^(6*x) + 6*e^(4*x) - 4*e^(2*x) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (- \operatorname{csch}^{2}{\left (x \right )}\right )^{\frac{5}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-csch(x)**2)**(5/2),x)

[Out]

Integral((-csch(x)**2)**(5/2), x)

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Giac [C]  time = 1.18677, size = 97, normalized size = 2.42 \begin{align*} -\frac{1}{16} \,{\left (\frac{4 i \,{\left (3 \,{\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 20 \, e^{\left (-x\right )} - 20 \, e^{x}\right )}}{{\left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4\right )}^{2}} - 3 i \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) + 3 i \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right )\right )} \mathrm{sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-csch(x)^2)^(5/2),x, algorithm="giac")

[Out]

-1/16*(4*I*(3*(e^(-x) + e^x)^3 - 20*e^(-x) - 20*e^x)/((e^(-x) + e^x)^2 - 4)^2 - 3*I*log(e^(-x) + e^x + 2) + 3*
I*log(e^(-x) + e^x - 2))*sgn(-e^(3*x) + e^x)