Optimal. Leaf size=176 \[ -\frac{\left (m^2+2 m+9\right ) (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};-e^{2 a} x^4\right )}{4 e (m+1)}-\frac{e^{-2 a} \left (e^{4 a} (m+5) x^4+e^{2 a} (3-m)\right ) (e x)^{m+1}}{8 e \left (e^{2 a} x^4+1\right )}-\frac{\left (1-e^{2 a} x^4\right )^2 (e x)^{m+1}}{4 e \left (e^{2 a} x^4+1\right )^2}+\frac{(m+3) (m+5) (e x)^{m+1}}{8 e (m+1)} \]
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Rubi [F] time = 0.0739888, 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 = {} \[ \int (e x)^m \tanh ^3(a+2 \log (x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int (e x)^m \tanh ^3(a+2 \log (x)) \, dx &=\int (e x)^m \tanh ^3(a+2 \log (x)) \, dx\\ \end{align*}
Mathematica [A] time = 0.816673, size = 218, normalized size = 1.24 \[ \frac{x (e x)^m \left (\frac{x^8 (\sinh (2 a)+\cosh (2 a)) \left ((m+9) x^4 (\sinh (a)+\cosh (a)) \, _2F_1\left (3,\frac{m+13}{4};\frac{m+17}{4};-x^4 (\cosh (2 a)+\sinh (2 a))\right )-3 (m+13) (\cosh (a)-\sinh (a)) \, _2F_1\left (3,\frac{m+9}{4};\frac{m+13}{4};-x^4 (\cosh (2 a)+\sinh (2 a))\right )\right )}{(m+9) (m+13)}+\frac{3 x^4 (\cosh (a)-\sinh (a)) \, _2F_1\left (3,\frac{m+5}{4};\frac{m+9}{4};-x^4 (\cosh (2 a)+\sinh (2 a))\right )}{m+5}-\frac{(\cosh (a)-\sinh (a))^3 \, _2F_1\left (3,\frac{m+1}{4};\frac{m+5}{4};-x^4 (\cosh (2 a)+\sinh (2 a))\right )}{m+1}\right )}{(\cosh (a)-\sinh (a))^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.033, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( \tanh \left ( a+2\,\ln \left ( x \right ) \right ) \right ) ^{3}\, 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} \tanh \left (a + 2 \, \log \left (x\right )\right )^{3}\,{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} \tanh \left (a + 2 \, \log \left (x\right )\right )^{3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \tanh ^{3}{\left (a + 2 \log{\left (x \right )} \right )}\, dx \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} \tanh \left (a + 2 \, \log \left (x\right )\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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