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

Optimal. Leaf size=68 \[ \frac{4 e^{2 a} x \left (c x^n\right )^{2 b} \, _2F_1\left (2,\frac{1}{2} \left (2+\frac{1}{b n}\right );\frac{1}{2} \left (4+\frac{1}{b n}\right );e^{2 a} \left (c x^n\right )^{2 b}\right )}{2 b n+1} \]

[Out]

(4*E^(2*a)*x*(c*x^n)^(2*b)*Hypergeometric2F1[2, (2 + 1/(b*n))/2, (4 + 1/(b*n))/2, E^(2*a)*(c*x^n)^(2*b)])/(1 +
2*b*n)

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Rubi [A]  time = 0.0653747, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {5546, 5548, 263, 364} \[ \frac{4 e^{2 a} x \left (c x^n\right )^{2 b} \, _2F_1\left (2,\frac{1}{2} \left (2+\frac{1}{b n}\right );\frac{1}{2} \left (4+\frac{1}{b n}\right );e^{2 a} \left (c x^n\right )^{2 b}\right )}{2 b n+1} \]

Antiderivative was successfully verified.

[In]

Int[Csch[a + b*Log[c*x^n]]^2,x]

[Out]

(4*E^(2*a)*x*(c*x^n)^(2*b)*Hypergeometric2F1[2, (2 + 1/(b*n))/2, (4 + 1/(b*n))/2, E^(2*a)*(c*x^n)^(2*b)])/(1 +
2*b*n)

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 5548

Int[Csch[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[2^p/E^(a*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}, x] && IntegerQ[p]

Rule 263

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

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \text{csch}^2\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1}{n}} \text{csch}^2(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (4 e^{-2 a} x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{x^{-1-2 b+\frac{1}{n}}}{\left (1-e^{-2 a} x^{-2 b}\right )^2} \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (4 e^{-2 a} x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{x^{-1+2 b+\frac{1}{n}}}{\left (-e^{-2 a}+x^{2 b}\right )^2} \, dx,x,c x^n\right )}{n}\\ &=\frac{4 e^{2 a} x \left (c x^n\right )^{2 b} \, _2F_1\left (2,\frac{1}{2} \left (2+\frac{1}{b n}\right );\frac{1}{2} \left (4+\frac{1}{b n}\right );e^{2 a} \left (c x^n\right )^{2 b}\right )}{1+2 b n}\\ \end{align*}

Mathematica [A]  time = 4.24291, size = 126, normalized size = 1.85 \[ \frac{x \left (-\frac{e^{2 a} \left (c x^n\right )^{2 b} \, _2F_1\left (1,1+\frac{1}{2 b n};2+\frac{1}{2 b n};e^{2 \left (a+b \log \left (c x^n\right )\right )}\right )}{2 b n+1}-\, _2F_1\left (1,\frac{1}{2 b n};1+\frac{1}{2 b n};e^{2 \left (a+b \log \left (c x^n\right )\right )}\right )-\coth \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Csch[a + b*Log[c*x^n]]^2,x]

[Out]

(x*(-Coth[a + b*Log[c*x^n]] - (E^(2*a)*(c*x^n)^(2*b)*Hypergeometric2F1[1, 1 + 1/(2*b*n), 2 + 1/(2*b*n), E^(2*(
a + b*Log[c*x^n]))])/(1 + 2*b*n) - Hypergeometric2F1[1, 1/(2*b*n), 1 + 1/(2*b*n), E^(2*(a + b*Log[c*x^n]))]))/
(b*n)

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Maple [F]  time = 1.119, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm csch} \left (a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2 \, x}{b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - b n} - 4 \, \int \frac{1}{4 \,{\left (b c^{b} n e^{\left (b \log \left (x^{n}\right ) + a\right )} + b n\right )}}\,{d x} + 4 \, \int \frac{1}{4 \,{\left (b c^{b} n e^{\left (b \log \left (x^{n}\right ) + a\right )} - b n\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x/(b*c^(2*b)*n*e^(2*b*log(x^n) + 2*a) - b*n) - 4*integrate(1/4/(b*c^b*n*e^(b*log(x^n) + a) + b*n), x) + 4*i
ntegrate(1/4/(b*c^b*n*e^(b*log(x^n) + a) - b*n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(csch(b*log(c*x^n) + a)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(csch(b*log(c*x^n) + a)^2, x)