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

Optimal. Leaf size=87 \[ \frac{(e x)^{m+1}}{e (m+1)}-\frac{2 (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2 b d n};\frac{m+1}{2 b d n}+1;e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{e (m+1)} \]

[Out]

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

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Rubi [F]  time = 0.0464916, 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 = {} \[ \int (e x)^m \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 13.2313, size = 158, normalized size = 1.82 \[ \frac{x (e x)^m \left (-\frac{(m+1) e^{2 a d} \left (c x^n\right )^{2 b d} \, _2F_1\left (1,\frac{m+2 b d n+1}{2 b d n};\frac{m+4 b d n+1}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{2 b d n+m+1}-\, _2F_1\left (1,\frac{m+1}{2 b d n};\frac{m+1}{2 b d n}+1;e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(e*x)^m*(-Hypergeometric2F1[1, (1 + m)/(2*b*d*n), 1 + (1 + m)/(2*b*d*n), E^(2*d*(a + b*Log[c*x^n]))] - (E^(
2*a*d)*(1 + m)*(c*x^n)^(2*b*d)*Hypergeometric2F1[1, (1 + m + 2*b*d*n)/(2*b*d*n), (1 + m + 4*b*d*n)/(2*b*d*n),
E^(2*a*d)*(c*x^n)^(2*b*d)])/(1 + m + 2*b*d*n)))/(1 + m)

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Maple [F]  time = 1.436, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m}{\rm coth} \left (d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right )\, 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))),x)

[Out]

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

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

[Out]

e^m*x*x^m/(m + 1) - e^m*integrate(x^m/(c^(b*d)*e^(b*d*log(x^n) + a*d) + 1), x) + e^m*integrate(x^m/(c^(b*d)*e^
(b*d*log(x^n) + a*d) - 1), 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 ), 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))),x, algorithm="fricas")

[Out]

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

[Out]

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