3.165 \(\int (e x)^m \coth (a+2 \log (x)) \, dx\)

Optimal. Leaf size=59 \[ \frac{(e x)^{m+1}}{e (m+1)}-\frac{2 (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};e^{2 a} x^4\right )}{e (m+1)} \]

[Out]

(e*x)^(1 + m)/(e*(1 + m)) - (2*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, E^(2*a)*x^4])/(e*(1 +
m))

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Rubi [F]  time = 0.0387854, 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 = {} \[ \int (e x)^m \coth (a+2 \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[(e*x)^m*Coth[a + 2*Log[x]],x]

[Out]

Defer[Int][(e*x)^m*Coth[a + 2*Log[x]], x]

Rubi steps

\begin{align*} \int (e x)^m \coth (a+2 \log (x)) \, dx &=\int (e x)^m \coth (a+2 \log (x)) \, dx\\ \end{align*}

Mathematica [A]  time = 0.0933579, size = 46, normalized size = 0.78 \[ -\frac{x (e x)^m \left (2 \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};x^4 (\cosh (2 a)+\sinh (2 a))\right )-1\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m*Coth[a + 2*Log[x]],x]

[Out]

-((x*(e*x)^m*(-1 + 2*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, x^4*(Cosh[2*a] + Sinh[2*a])]))/(1 + m))

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m}{\rm coth} \left (a+2\,\ln \left ( x \right ) \right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*coth(a+2*ln(x)),x)

[Out]

int((e*x)^m*coth(a+2*ln(x)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \coth \left (a + 2 \, \log \left (x\right )\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*coth(a+2*log(x)),x, algorithm="maxima")

[Out]

integrate((e*x)^m*coth(a + 2*log(x)), 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 (a + 2 \, \log \left (x\right )\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*coth(a+2*log(x)),x, algorithm="fricas")

[Out]

integral((e*x)^m*coth(a + 2*log(x)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \coth{\left (a + 2 \log{\left (x \right )} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*coth(a+2*ln(x)),x)

[Out]

Integral((e*x)**m*coth(a + 2*log(x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \coth \left (a + 2 \, \log \left (x\right )\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*coth(a+2*log(x)),x, algorithm="giac")

[Out]

integrate((e*x)^m*coth(a + 2*log(x)), x)