3.26 \(\int \frac{1}{(-\text{csch}^2(x))^{3/2}} \, dx\)

Optimal. Leaf size=33 \[ \frac{2 \coth (x)}{3 \sqrt{-\text{csch}^2(x)}}+\frac{\coth (x)}{3 \left (-\text{csch}^2(x)\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.014503, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4122, 192, 191} \[ \frac{2 \coth (x)}{3 \sqrt{-\text{csch}^2(x)}}+\frac{\coth (x)}{3 \left (-\text{csch}^2(x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(-Csch[x]^2)^(-3/2),x]

[Out]

Coth[x]/(3*(-Csch[x]^2)^(3/2)) + (2*Coth[x])/(3*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 )^{3/2}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^{5/2}} \, dx,x,\coth (x)\right )\\ &=\frac{\coth (x)}{3 \left (-\text{csch}^2(x)\right )^{3/2}}+\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^{3/2}} \, dx,x,\coth (x)\right )\\ &=\frac{\coth (x)}{3 \left (-\text{csch}^2(x)\right )^{3/2}}+\frac{2 \coth (x)}{3 \sqrt{-\text{csch}^2(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0144007, size = 27, normalized size = 0.82 \[ \frac{9 \coth (x)-\cosh (3 x) \text{csch}(x)}{12 \sqrt{-\text{csch}^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-Csch[x]^2)^(-3/2),x]

[Out]

(9*Coth[x] - Cosh[3*x]*Csch[x])/(12*Sqrt[-Csch[x]^2])

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Maple [B]  time = 0.039, size = 118, normalized size = 3.6 \begin{align*} -{\frac{{{\rm e}^{4\,x}}}{24\,{{\rm e}^{2\,x}}-24}{\frac{1}{\sqrt{-{\frac{{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}+{\frac{3\,{{\rm e}^{2\,x}}}{8\,{{\rm e}^{2\,x}}-8}{\frac{1}{\sqrt{-{\frac{{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}+{\frac{3}{8\,{{\rm e}^{2\,x}}-8}{\frac{1}{\sqrt{-{\frac{{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}-{\frac{{{\rm e}^{-2\,x}}}{24\,{{\rm e}^{2\,x}}-24}{\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)^(3/2),x)

[Out]

-1/24*exp(4*x)/(exp(2*x)-1)/(-exp(2*x)/(exp(2*x)-1)^2)^(1/2)+3/8/(-exp(2*x)/(exp(2*x)-1)^2)^(1/2)/(exp(2*x)-1)
*exp(2*x)+3/8/(-exp(2*x)/(exp(2*x)-1)^2)^(1/2)/(exp(2*x)-1)-1/24*exp(-2*x)/(exp(2*x)-1)/(-exp(2*x)/(exp(2*x)-1
)^2)^(1/2)

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Maxima [C]  time = 1.64072, size = 31, normalized size = 0.94 \begin{align*} -\frac{1}{24} i \, e^{\left (3 \, x\right )} + \frac{3}{8} i \, e^{\left (-x\right )} - \frac{1}{24} i \, e^{\left (-3 \, x\right )} + \frac{3}{8} i \, e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-csch(x)^2)^(3/2),x, algorithm="maxima")

[Out]

-1/24*I*e^(3*x) + 3/8*I*e^(-x) - 1/24*I*e^(-3*x) + 3/8*I*e^x

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Fricas [C]  time = 1.55686, size = 80, normalized size = 2.42 \begin{align*} \frac{1}{24} \,{\left (i \, e^{\left (6 \, x\right )} - 9 i \, e^{\left (4 \, x\right )} - 9 i \, e^{\left (2 \, x\right )} + i\right )} e^{\left (-3 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-csch(x)^2)^(3/2),x, algorithm="fricas")

[Out]

1/24*(I*e^(6*x) - 9*I*e^(4*x) - 9*I*e^(2*x) + I)*e^(-3*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{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-csch(x)**2)**(3/2),x)

[Out]

Integral((-csch(x)**2)**(-3/2), x)

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Giac [C]  time = 1.1529, size = 68, normalized size = 2.06 \begin{align*} \frac{i \,{\left (9 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-3 \, x\right )}}{24 \, \mathrm{sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right )} - \frac{i \,{\left (e^{\left (3 \, x\right )} - 9 \, e^{x}\right )}}{24 \, \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)^(3/2),x, algorithm="giac")

[Out]

1/24*I*(9*e^(2*x) - 1)*e^(-3*x)/sgn(-e^(3*x) + e^x) - 1/24*I*(e^(3*x) - 9*e^x)/sgn(-e^(3*x) + e^x)