3.27 \(\int \frac{1}{(-\text{csch}^2(x))^{5/2}} \, dx\)

Optimal. Leaf size=49 \[ \frac{8 \coth (x)}{15 \sqrt{-\text{csch}^2(x)}}+\frac{4 \coth (x)}{15 \left (-\text{csch}^2(x)\right )^{3/2}}+\frac{\coth (x)}{5 \left (-\text{csch}^2(x)\right )^{5/2}} \]

[Out]

Coth[x]/(5*(-Csch[x]^2)^(5/2)) + (4*Coth[x])/(15*(-Csch[x]^2)^(3/2)) + (8*Coth[x])/(15*Sqrt[-Csch[x]^2])

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Rubi [A]  time = 0.0196418, antiderivative size = 49, 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, 192, 191} \[ \frac{8 \coth (x)}{15 \sqrt{-\text{csch}^2(x)}}+\frac{4 \coth (x)}{15 \left (-\text{csch}^2(x)\right )^{3/2}}+\frac{\coth (x)}{5 \left (-\text{csch}^2(x)\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Coth[x]/(5*(-Csch[x]^2)^(5/2)) + (4*Coth[x])/(15*(-Csch[x]^2)^(3/2)) + (8*Coth[x])/(15*Sqrt[-Csch[x]^2])

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 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 0] && NeQ[p, -1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{1}{\left (-\text{csch}^2(x)\right )^{5/2}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^{7/2}} \, dx,x,\coth (x)\right )\\ &=\frac{\coth (x)}{5 \left (-\text{csch}^2(x)\right )^{5/2}}+\frac{4}{5} \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^{5/2}} \, dx,x,\coth (x)\right )\\ &=\frac{\coth (x)}{5 \left (-\text{csch}^2(x)\right )^{5/2}}+\frac{4 \coth (x)}{15 \left (-\text{csch}^2(x)\right )^{3/2}}+\frac{8}{15} \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^{3/2}} \, dx,x,\coth (x)\right )\\ &=\frac{\coth (x)}{5 \left (-\text{csch}^2(x)\right )^{5/2}}+\frac{4 \coth (x)}{15 \left (-\text{csch}^2(x)\right )^{3/2}}+\frac{8 \coth (x)}{15 \sqrt{-\text{csch}^2(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0249027, size = 33, normalized size = 0.67 \[ \frac{(150 \cosh (x)-25 \cosh (3 x)+3 \cosh (5 x)) \text{csch}(x)}{240 \sqrt{-\text{csch}^2(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((150*Cosh[x] - 25*Cosh[3*x] + 3*Cosh[5*x])*Csch[x])/(240*Sqrt[-Csch[x]^2])

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Maple [B]  time = 0.04, size = 178, normalized size = 3.6 \begin{align*}{\frac{{{\rm e}^{6\,x}}}{160\,{{\rm e}^{2\,x}}-160}{\frac{1}{\sqrt{-{\frac{{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}-{\frac{5\,{{\rm e}^{4\,x}}}{96\,{{\rm e}^{2\,x}}-96}{\frac{1}{\sqrt{-{\frac{{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}+{\frac{5\,{{\rm e}^{2\,x}}}{16\,{{\rm e}^{2\,x}}-16}{\frac{1}{\sqrt{-{\frac{{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}+{\frac{5}{16\,{{\rm e}^{2\,x}}-16}{\frac{1}{\sqrt{-{\frac{{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}-{\frac{5\,{{\rm e}^{-2\,x}}}{96\,{{\rm e}^{2\,x}}-96}{\frac{1}{\sqrt{-{\frac{{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}+{\frac{{{\rm e}^{-4\,x}}}{160\,{{\rm e}^{2\,x}}-160}{\frac{1}{\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(1/(-csch(x)^2)^(5/2),x)

[Out]

1/160*exp(6*x)/(exp(2*x)-1)/(-exp(2*x)/(exp(2*x)-1)^2)^(1/2)-5/96*exp(4*x)/(exp(2*x)-1)/(-exp(2*x)/(exp(2*x)-1
)^2)^(1/2)+5/16/(-exp(2*x)/(exp(2*x)-1)^2)^(1/2)/(exp(2*x)-1)*exp(2*x)+5/16/(-exp(2*x)/(exp(2*x)-1)^2)^(1/2)/(
exp(2*x)-1)-5/96*exp(-2*x)/(exp(2*x)-1)/(-exp(2*x)/(exp(2*x)-1)^2)^(1/2)+1/160*exp(-4*x)/(exp(2*x)-1)/(-exp(2*
x)/(exp(2*x)-1)^2)^(1/2)

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Maxima [C]  time = 1.57892, size = 47, normalized size = 0.96 \begin{align*} \frac{1}{160} i \, e^{\left (5 \, x\right )} - \frac{5}{96} i \, e^{\left (3 \, x\right )} + \frac{5}{16} i \, e^{\left (-x\right )} - \frac{5}{96} i \, e^{\left (-3 \, x\right )} + \frac{1}{160} i \, e^{\left (-5 \, x\right )} + \frac{5}{16} i \, e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-csch(x)^2)^(5/2),x, algorithm="maxima")

[Out]

1/160*I*e^(5*x) - 5/96*I*e^(3*x) + 5/16*I*e^(-x) - 5/96*I*e^(-3*x) + 1/160*I*e^(-5*x) + 5/16*I*e^x

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Fricas [C]  time = 1.40485, size = 135, normalized size = 2.76 \begin{align*} \frac{1}{480} \,{\left (-3 i \, e^{\left (10 \, x\right )} + 25 i \, e^{\left (8 \, x\right )} - 150 i \, e^{\left (6 \, x\right )} - 150 i \, e^{\left (4 \, x\right )} + 25 i \, e^{\left (2 \, x\right )} - 3 i\right )} e^{\left (-5 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-csch(x)^2)^(5/2),x, algorithm="fricas")

[Out]

1/480*(-3*I*e^(10*x) + 25*I*e^(8*x) - 150*I*e^(6*x) - 150*I*e^(4*x) + 25*I*e^(2*x) - 3*I)*e^(-5*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/(-csch(x)**2)**(5/2),x)

[Out]

Integral((-csch(x)**2)**(-5/2), x)

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Giac [C]  time = 1.17908, size = 86, normalized size = 1.76 \begin{align*} \frac{i \,{\left (150 \, e^{\left (4 \, x\right )} - 25 \, e^{\left (2 \, x\right )} + 3\right )} e^{\left (-5 \, x\right )}}{480 \, \mathrm{sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right )} + \frac{i \,{\left (3 \, e^{\left (5 \, x\right )} - 25 \, e^{\left (3 \, x\right )} + 150 \, e^{x}\right )}}{480 \, \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(1/(-csch(x)^2)^(5/2),x, algorithm="giac")

[Out]

1/480*I*(150*e^(4*x) - 25*e^(2*x) + 3)*e^(-5*x)/sgn(-e^(3*x) + e^x) + 1/480*I*(3*e^(5*x) - 25*e^(3*x) + 150*e^
x)/sgn(-e^(3*x) + e^x)