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

Optimal. Leaf size=62 \[ -\frac{2 e^a x \left (c x^n\right )^b \, _2F_1\left (1,\frac{b+\frac{1}{n}}{2 b};\frac{1}{2} \left (3+\frac{1}{b n}\right );e^{2 a} \left (c x^n\right )^{2 b}\right )}{b n+1} \]

[Out]

(-2*E^a*x*(c*x^n)^b*Hypergeometric2F1[1, (b + n^(-1))/(2*b), (3 + 1/(b*n))/2, E^(2*a)*(c*x^n)^(2*b)])/(1 + b*n
)

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Rubi [A]  time = 0.0547365, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {5546, 5548, 263, 364} \[ -\frac{2 e^a x \left (c x^n\right )^b \, _2F_1\left (1,\frac{b+\frac{1}{n}}{2 b};\frac{1}{2} \left (3+\frac{1}{b n}\right );e^{2 a} \left (c x^n\right )^{2 b}\right )}{b n+1} \]

Antiderivative was successfully verified.

[In]

Int[Csch[a + b*Log[c*x^n]],x]

[Out]

(-2*E^a*x*(c*x^n)^b*Hypergeometric2F1[1, (b + n^(-1))/(2*b), (3 + 1/(b*n))/2, E^(2*a)*(c*x^n)^(2*b)])/(1 + 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}\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}(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (2 e^{-a} x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{x^{-1-b+\frac{1}{n}}}{1-e^{-2 a} x^{-2 b}} \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (2 e^{-a} x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{x^{-1+b+\frac{1}{n}}}{-e^{-2 a}+x^{2 b}} \, dx,x,c x^n\right )}{n}\\ &=-\frac{2 e^a x \left (c x^n\right )^b \, _2F_1\left (1,\frac{b+\frac{1}{n}}{2 b};\frac{1}{2} \left (3+\frac{1}{b n}\right );e^{2 a} \left (c x^n\right )^{2 b}\right )}{1+b n}\\ \end{align*}

Mathematica [A]  time = 1.15678, size = 62, normalized size = 1. \[ -\frac{2 e^a x \left (c x^n\right )^b \, _2F_1\left (1,\frac{1}{2} \left (1+\frac{1}{b n}\right );\frac{1}{2} \left (3+\frac{1}{b n}\right );e^{2 \left (a+b \log \left (c x^n\right )\right )}\right )}{b n+1} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[a + b*Log[c*x^n]],x]

[Out]

(-2*E^a*x*(c*x^n)^b*Hypergeometric2F1[1, (1 + 1/(b*n))/2, (3 + 1/(b*n))/2, E^(2*(a + b*Log[c*x^n]))])/(1 + b*n
)

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Maple [F]  time = 0.093, size = 0, normalized size = 0. \begin{align*} \int{\rm csch} \left (a+b\ln \left ( c{x}^{n} \right ) \right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(csch(b*log(c*x^n) + a), 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 ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(csch(b*log(c*x^n) + a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{csch}{\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)),x)

[Out]

Integral(csch(a + b*log(c*x**n)), 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 )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(csch(b*log(c*x^n) + a), x)