3.197 \(\int \coth ^p(d (a+b \log (c x^n))) \, dx\)

Optimal. Leaf size=115 \[ x \left (-e^{2 a d} \left (c x^n\right )^{2 b d}-1\right )^p \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )^{-p} F_1\left (\frac{1}{2 b d n};p,-p;1+\frac{1}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right ) \]

[Out]

(x*(-1 - E^(2*a*d)*(c*x^n)^(2*b*d))^p*AppellF1[1/(2*b*d*n), p, -p, 1 + 1/(2*b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d),
-(E^(2*a*d)*(c*x^n)^(2*b*d))])/(1 + E^(2*a*d)*(c*x^n)^(2*b*d))^p

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Rubi [F]  time = 0.0150993, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \coth ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Coth[d*(a + b*Log[c*x^n])]^p,x]

[Out]

Defer[Int][Coth[d*(a + b*Log[c*x^n])]^p, x]

Rubi steps

\begin{align*} \int \coth ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int \coth ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end{align*}

Mathematica [B]  time = 3.6288, size = 387, normalized size = 3.37 \[ \frac{x (2 b d n+1) \left (\frac{e^{2 a d} \left (c x^n\right )^{2 b d}+1}{e^{2 a d} \left (c x^n\right )^{2 b d}-1}\right )^p F_1\left (\frac{1}{2 b d n};p,-p;1+\frac{1}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{2 b d n p e^{2 a d} \left (c x^n\right )^{2 b d} F_1\left (1+\frac{1}{2 b d n};p,1-p;2+\frac{1}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )+2 b d n p e^{2 a d} \left (c x^n\right )^{2 b d} F_1\left (1+\frac{1}{2 b d n};p+1,-p;2+\frac{1}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )+(2 b d n+1) F_1\left (\frac{1}{2 b d n};p,-p;1+\frac{1}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Coth[d*(a + b*Log[c*x^n])]^p,x]

[Out]

((1 + 2*b*d*n)*x*((1 + E^(2*a*d)*(c*x^n)^(2*b*d))/(-1 + E^(2*a*d)*(c*x^n)^(2*b*d)))^p*AppellF1[1/(2*b*d*n), p,
-p, 1 + 1/(2*b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d), -(E^(2*a*d)*(c*x^n)^(2*b*d))])/(2*b*d*E^(2*a*d)*n*p*(c*x^n)^(
2*b*d)*AppellF1[1 + 1/(2*b*d*n), p, 1 - p, 2 + 1/(2*b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d), -(E^(2*a*d)*(c*x^n)^(2*
b*d))] + 2*b*d*E^(2*a*d)*n*p*(c*x^n)^(2*b*d)*AppellF1[1 + 1/(2*b*d*n), 1 + p, -p, 2 + 1/(2*b*d*n), E^(2*a*d)*(
c*x^n)^(2*b*d), -(E^(2*a*d)*(c*x^n)^(2*b*d))] + (1 + 2*b*d*n)*AppellF1[1/(2*b*d*n), p, -p, 1 + 1/(2*b*d*n), E^
(2*a*d)*(c*x^n)^(2*b*d), -(E^(2*a*d)*(c*x^n)^(2*b*d))])

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Maple [F]  time = 0.081, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm coth} \left (d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(d*(a+b*ln(c*x^n)))^p,x)

[Out]

int(coth(d*(a+b*ln(c*x^n)))^p,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(coth((b*log(c*x^n) + a)*d)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(coth(b*d*log(c*x^n) + a*d)^p, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*(a+b*ln(c*x**n)))**p,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(coth((b*log(c*x^n) + a)*d)^p, x)