3.166 \(\int \frac{\text{csch}^2(a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=19 \[ -\frac{\coth \left (a+b \log \left (c x^n\right )\right )}{b n} \]

[Out]

-(Coth[a + b*Log[c*x^n]]/(b*n))

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Rubi [A]  time = 0.0271739, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3767, 8} \[ -\frac{\coth \left (a+b \log \left (c x^n\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[Csch[a + b*Log[c*x^n]]^2/x,x]

[Out]

-(Coth[a + b*Log[c*x^n]]/(b*n))

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\text{csch}^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \text{csch}^2(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{i \operatorname{Subst}\left (\int 1 \, dx,x,-i \coth \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ &=-\frac{\coth \left (a+b \log \left (c x^n\right )\right )}{b n}\\ \end{align*}

Mathematica [A]  time = 0.0663661, size = 19, normalized size = 1. \[ -\frac{\coth \left (a+b \log \left (c x^n\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[a + b*Log[c*x^n]]^2/x,x]

[Out]

-(Coth[a + b*Log[c*x^n]]/(b*n))

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Maple [A]  time = 0.012, size = 20, normalized size = 1.1 \begin{align*} -{\frac{{\rm coth} \left (a+b\ln \left ( c{x}^{n} \right ) \right )}{bn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(a+b*ln(c*x^n))^2/x,x)

[Out]

-coth(a+b*ln(c*x^n))/b/n

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Maxima [A]  time = 1.12217, size = 39, normalized size = 2.05 \begin{align*} -\frac{2}{b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(a+b*log(c*x^n))^2/x,x, algorithm="maxima")

[Out]

-2/(b*c^(2*b)*n*e^(2*b*log(x^n) + 2*a) - b*n)

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Fricas [B]  time = 1.92079, size = 219, normalized size = 11.53 \begin{align*} -\frac{2}{b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + b n \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(a+b*log(c*x^n))^2/x,x, algorithm="fricas")

[Out]

-2/(b*n*cosh(b*n*log(x) + b*log(c) + a)^2 + 2*b*n*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) +
a) + b*n*sinh(b*n*log(x) + b*log(c) + a)^2 - b*n)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}^{2}{\left (a + b \log{\left (c x^{n} \right )} \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(a+b*ln(c*x**n))**2/x,x)

[Out]

Integral(csch(a + b*log(c*x**n))**2/x, x)

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Giac [A]  time = 1.14547, size = 38, normalized size = 2. \begin{align*} -\frac{2}{{\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} - 1\right )} b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(a+b*log(c*x^n))^2/x,x, algorithm="giac")

[Out]

-2/((c^(2*b)*x^(2*b*n)*e^(2*a) - 1)*b*n)