Optimal. Leaf size=79 \[ -\frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};e^{2 a} x^4\right )}{e}+\frac{(e x)^{m+1}}{e \left (1-e^{2 a} x^4\right )}+\frac{(e x)^{m+1}}{e (m+1)} \]
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Rubi [F] time = 0.0679443, 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 = {} \[ \int (e x)^m \coth ^2(a+2 \log (x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int (e x)^m \coth ^2(a+2 \log (x)) \, dx &=\int (e x)^m \coth ^2(a+2 \log (x)) \, dx\\ \end{align*}
Mathematica [B] time = 0.369473, size = 165, normalized size = 2.09 \[ \frac{x (e x)^m \left (\frac{x^4 (\sinh (a)+\cosh (a)) \left ((m+5) x^4 (\sinh (a)+\cosh (a)) \, _2F_1\left (2,\frac{m+9}{4};\frac{m+13}{4};x^4 (\cosh (2 a)+\sinh (2 a))\right )+2 (m+9) (\cosh (a)-\sinh (a)) \, _2F_1\left (2,\frac{m+5}{4};\frac{m+9}{4};x^4 (\cosh (2 a)+\sinh (2 a))\right )\right )}{(m+5) (m+9)}+\frac{(\cosh (2 a)-\sinh (2 a)) \, _2F_1\left (2,\frac{m+1}{4};\frac{m+5}{4};x^4 (\cosh (2 a)+\sinh (2 a))\right )}{m+1}\right )}{(\cosh (a)-\sinh (a))^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.027, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ({\rm coth} \left (a+2\,\ln \left ( x \right ) \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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 )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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 )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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 )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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