3.169 \(\int (e x)^m \coth ^p(a+b \log (x)) \, dx\)

Optimal. Leaf size=99 \[ \frac{(e x)^{m+1} \left (-e^{2 a} x^{2 b}-1\right )^p \left (e^{2 a} x^{2 b}+1\right )^{-p} F_1\left (\frac{m+1}{2 b};p,-p;\frac{m+1}{2 b}+1;e^{2 a} x^{2 b},-e^{2 a} x^{2 b}\right )}{e (m+1)} \]

[Out]

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

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Rubi [F]  time = 0.115831, 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 = {} \[ \int (e x)^m \coth ^p(a+b \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[(e*x)^m*Coth[a + b*Log[x]]^p,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 2.97686, size = 126, normalized size = 1.27 \[ \frac{x (e x)^m \left (1-e^{2 a} x^{2 b}\right )^p \left (e^{2 a} x^{2 b}+1\right )^{-p} \left (\frac{e^{2 a} x^{2 b}+1}{e^{2 a} x^{2 b}-1}\right )^p F_1\left (\frac{m+1}{2 b};p,-p;\frac{m+1}{2 b}+1;e^{2 a} x^{2 b},-e^{2 a} x^{2 b}\right )}{m+1} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.03, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ({\rm coth} \left (a+b\ln \left ( x \right ) \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*coth(a+b*ln(x))^p,x)

[Out]

int((e*x)^m*coth(a+b*ln(x))^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 (b \log \left (x\right ) + a\right )^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*coth(a+b*log(x))^p,x, algorithm="maxima")

[Out]

integrate((e*x)^m*coth(b*log(x) + a)^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 \log \left (x\right ) + a\right )^{p}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*coth(a+b*log(x))^p,x, algorithm="fricas")

[Out]

integral((e*x)^m*coth(b*log(x) + a)^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(a+b*ln(x))**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 (b \log \left (x\right ) + a\right )^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*coth(a+b*log(x))^p,x, algorithm="giac")

[Out]

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