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

Optimal. Leaf size=135 \[ \frac{(e x)^{m+1} \left (-e^{2 a d} \left (c x^n\right )^{2 b d}-1\right )^p \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )^{-p} F_1\left (\frac{m+1}{2 b d n};p,-p;\frac{m+1}{2 b d n}+1;e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{e (m+1)} \]

[Out]

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

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Rubi [F]  time = 0.0973732, 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 = {} \[ \int (e x)^m \coth ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 5.07176, size = 174, normalized size = 1.29 \[ \frac{x (e x)^m \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )^p \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )^{-p} \left (\frac{e^{2 a d} \left (c x^n\right )^{2 b d}+1}{e^{2 a d} \left (c x^n\right )^{2 b d}-1}\right )^p F_1\left (\frac{m+1}{2 b d n};p,-p;\frac{m+1}{2 b d n}+1;e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{m+1} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x*(e*x)^m*(1 - E^(2*a*d)*(c*x^n)^(2*b*d))^p*((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 + m)/(2*b*d*n), p, -p, 1 + (1 + m)/(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 + m)*(1 + E^(2*a*d)*(c*x^n)^(2*b*d))^p)

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Maple [F]  time = 0.069, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \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((e*x)^m*coth(d*(a+b*ln(c*x^n)))^p,x)

[Out]

int((e*x)^m*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 \left (e x\right )^{m} \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((e*x)^m*coth(d*(a+b*log(c*x^n)))^p,x, algorithm="maxima")

[Out]

integrate((e*x)^m*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 (\left (e x\right )^{m} \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((e*x)^m*coth(d*(a+b*log(c*x^n)))^p,x, algorithm="fricas")

[Out]

integral((e*x)^m*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((e*x)**m*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 \left (e x\right )^{m} \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((e*x)^m*coth(d*(a+b*log(c*x^n)))^p,x, algorithm="giac")

[Out]

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