3.163 \(\int \text{csch}^p(a+\frac{\log (c x^n)}{n (-2+p)}) \, dx\)

Optimal. Leaf size=90 \[ -\frac{e^{2 a} (2-p) x \left (c x^n\right )^{-\frac{2}{n (2-p)}} \left (1-e^{-2 a} \left (c x^n\right )^{\frac{2}{n (2-p)}}\right ) \text{csch}^p\left (a-\frac{\log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \]

[Out]

-(E^(2*a)*(2 - p)*x*(1 - (c*x^n)^(2/(n*(2 - p)))/E^(2*a))*Csch[a - Log[c*x^n]/(n*(2 - p))]^p)/(2*(1 - p)*(c*x^
n)^(2/(n*(2 - p))))

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Rubi [A]  time = 0.0872534, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {5546, 5550, 261} \[ -\frac{e^{2 a} (2-p) x \left (c x^n\right )^{-\frac{2}{n (2-p)}} \left (1-e^{-2 a} \left (c x^n\right )^{\frac{2}{n (2-p)}}\right ) \text{csch}^p\left (a-\frac{\log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \]

Antiderivative was successfully verified.

[In]

Int[Csch[a + Log[c*x^n]/(n*(-2 + p))]^p,x]

[Out]

-(E^(2*a)*(2 - p)*x*(1 - (c*x^n)^(2/(n*(2 - p)))/E^(2*a))*Csch[a - Log[c*x^n]/(n*(2 - p))]^p)/(2*(1 - p)*(c*x^
n)^(2/(n*(2 - p))))

Rule 5546

Int[Csch[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[
x^(1/n - 1)*Csch[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n
, 1])

Rule 5550

Int[Csch[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[(Csch[d*(a + b*Log[x])]^p*
(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p)/x^(-(b*d*p)), Int[(e*x)^m/(x^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p), x], x]
/; FreeQ[{a, b, d, e, m, p}, x] &&  !IntegerQ[p]

Rule 261

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

Rubi steps

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

Mathematica [A]  time = 5.95475, size = 115, normalized size = 1.28 \[ \frac{2^{p-1} (p-2) x \left (\frac{e^a \left (c x^n\right )^{\frac{1}{n (p-2)}}}{e^{2 a} \left (c x^n\right )^{\frac{2}{n (p-2)}}-1}\right )^p \left (e^{2 a} \left (c x^n\right )^{\frac{2}{n (p-2)}} \left (\left (1-e^{-2 a} \left (c x^n\right )^{-\frac{2}{n (p-2)}}\right )^p-1\right )+1\right )}{p-1} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Csch[a + Log[c*x^n]/(n*(-2 + p))]^p,x]

[Out]

(2^(-1 + p)*(-2 + p)*x*((E^a*(c*x^n)^(1/(n*(-2 + p))))/(-1 + E^(2*a)*(c*x^n)^(2/(n*(-2 + p)))))^p*(1 + E^(2*a)
*(c*x^n)^(2/(n*(-2 + p)))*(-1 + (1 - 1/(E^(2*a)*(c*x^n)^(2/(n*(-2 + p)))))^p)))/(-1 + p)

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Maple [F]  time = 0.105, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm csch} \left (a+{\frac{\ln \left ( c{x}^{n} \right ) }{n \left ( -2+p \right ) }}\right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(csch(a + log(c*x^n)/(n*(p - 2)))^p, x)

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Fricas [B]  time = 1.81932, size = 1323, normalized size = 14.7 \begin{align*} -\frac{{\left (p - 2\right )} x \cosh \left (p \log \left (\frac{2 \,{\left (\cosh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) + \sinh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )\right )}}{\cosh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )^{2} + 2 \, \cosh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) \sinh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) + \sinh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )^{2} - 1}\right )\right ) \sinh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) +{\left (p - 2\right )} x \sinh \left (p \log \left (\frac{2 \,{\left (\cosh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) + \sinh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )\right )}}{\cosh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )^{2} + 2 \, \cosh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) \sinh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) + \sinh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )^{2} - 1}\right )\right ) \sinh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )}{{\left (p - 1\right )} \cosh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) -{\left (p - 1\right )} \sinh \left (\frac{a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((p - 2)*x*cosh(p*log(2*(cosh((a*n*p - 2*a*n + n*log(x) + log(c))/(n*p - 2*n)) + sinh((a*n*p - 2*a*n + n*log(
x) + log(c))/(n*p - 2*n)))/(cosh((a*n*p - 2*a*n + n*log(x) + log(c))/(n*p - 2*n))^2 + 2*cosh((a*n*p - 2*a*n +
n*log(x) + log(c))/(n*p - 2*n))*sinh((a*n*p - 2*a*n + n*log(x) + log(c))/(n*p - 2*n)) + sinh((a*n*p - 2*a*n +
n*log(x) + log(c))/(n*p - 2*n))^2 - 1)))*sinh((a*n*p - 2*a*n + n*log(x) + log(c))/(n*p - 2*n)) + (p - 2)*x*sin
h(p*log(2*(cosh((a*n*p - 2*a*n + n*log(x) + log(c))/(n*p - 2*n)) + sinh((a*n*p - 2*a*n + n*log(x) + log(c))/(n
*p - 2*n)))/(cosh((a*n*p - 2*a*n + n*log(x) + log(c))/(n*p - 2*n))^2 + 2*cosh((a*n*p - 2*a*n + n*log(x) + log(
c))/(n*p - 2*n))*sinh((a*n*p - 2*a*n + n*log(x) + log(c))/(n*p - 2*n)) + sinh((a*n*p - 2*a*n + n*log(x) + log(
c))/(n*p - 2*n))^2 - 1)))*sinh((a*n*p - 2*a*n + n*log(x) + log(c))/(n*p - 2*n)))/((p - 1)*cosh((a*n*p - 2*a*n
+ n*log(x) + log(c))/(n*p - 2*n)) - (p - 1)*sinh((a*n*p - 2*a*n + n*log(x) + log(c))/(n*p - 2*n)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(csch(a + log(c*x^n)/(n*(p - 2)))^p, x)