Optimal. Leaf size=223 \[ \frac{3\ 2^{-m-4} e^{\frac{2 c f}{d}-2 e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{2 f (c+d x)}{d}\right )}{a^3 f}-\frac{3\ 2^{-2 m-5} e^{\frac{4 c f}{d}-4 e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{4 f (c+d x)}{d}\right )}{a^3 f}+\frac{2^{-m-4} 3^{-m-1} e^{\frac{6 c f}{d}-6 e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{6 f (c+d x)}{d}\right )}{a^3 f}+\frac{(c+d x)^{m+1}}{8 a^3 d (m+1)} \]
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Rubi [A] time = 0.233118, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3729, 2181} \[ \frac{3\ 2^{-m-4} e^{\frac{2 c f}{d}-2 e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{2 f (c+d x)}{d}\right )}{a^3 f}-\frac{3\ 2^{-2 m-5} e^{\frac{4 c f}{d}-4 e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{4 f (c+d x)}{d}\right )}{a^3 f}+\frac{2^{-m-4} 3^{-m-1} e^{\frac{6 c f}{d}-6 e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{6 f (c+d x)}{d}\right )}{a^3 f}+\frac{(c+d x)^{m+1}}{8 a^3 d (m+1)} \]
Antiderivative was successfully verified.
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Rule 3729
Rule 2181
Rubi steps
\begin{align*} \int \frac{(c+d x)^m}{(a+a \coth (e+f x))^3} \, dx &=\int \left (\frac{(c+d x)^m}{8 a^3}-\frac{e^{-6 e-6 f x} (c+d x)^m}{8 a^3}+\frac{3 e^{-4 e-4 f x} (c+d x)^m}{8 a^3}-\frac{3 e^{-2 e-2 f x} (c+d x)^m}{8 a^3}\right ) \, dx\\ &=\frac{(c+d x)^{1+m}}{8 a^3 d (1+m)}-\frac{\int e^{-6 e-6 f x} (c+d x)^m \, dx}{8 a^3}+\frac{3 \int e^{-4 e-4 f x} (c+d x)^m \, dx}{8 a^3}-\frac{3 \int e^{-2 e-2 f x} (c+d x)^m \, dx}{8 a^3}\\ &=\frac{(c+d x)^{1+m}}{8 a^3 d (1+m)}+\frac{3\ 2^{-4-m} e^{-2 e+\frac{2 c f}{d}} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac{2 f (c+d x)}{d}\right )}{a^3 f}-\frac{3\ 2^{-5-2 m} e^{-4 e+\frac{4 c f}{d}} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac{4 f (c+d x)}{d}\right )}{a^3 f}+\frac{2^{-4-m} 3^{-1-m} e^{-6 e+\frac{6 c f}{d}} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac{6 f (c+d x)}{d}\right )}{a^3 f}\\ \end{align*}
Mathematica [F] time = 180.002, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
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Maple [F] time = 0.161, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx+c \right ) ^{m}}{ \left ( a+a{\rm coth} \left (fx+e\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 \frac{{\left (d x + c\right )}^{m}}{{\left (a \coth \left (f x + e\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.28678, size = 853, normalized size = 3.83 \begin{align*} \frac{2 \,{\left (d m + d\right )} \cosh \left (\frac{d m \log \left (\frac{6 \, f}{d}\right ) + 6 \, d e - 6 \, c f}{d}\right ) \Gamma \left (m + 1, \frac{6 \,{\left (d f x + c f\right )}}{d}\right ) - 9 \,{\left (d m + d\right )} \cosh \left (\frac{d m \log \left (\frac{4 \, f}{d}\right ) + 4 \, d e - 4 \, c f}{d}\right ) \Gamma \left (m + 1, \frac{4 \,{\left (d f x + c f\right )}}{d}\right ) + 18 \,{\left (d m + d\right )} \cosh \left (\frac{d m \log \left (\frac{2 \, f}{d}\right ) + 2 \, d e - 2 \, c f}{d}\right ) \Gamma \left (m + 1, \frac{2 \,{\left (d f x + c f\right )}}{d}\right ) - 2 \,{\left (d m + d\right )} \Gamma \left (m + 1, \frac{6 \,{\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac{d m \log \left (\frac{6 \, f}{d}\right ) + 6 \, d e - 6 \, c f}{d}\right ) + 9 \,{\left (d m + d\right )} \Gamma \left (m + 1, \frac{4 \,{\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac{d m \log \left (\frac{4 \, f}{d}\right ) + 4 \, d e - 4 \, c f}{d}\right ) - 18 \,{\left (d m + d\right )} \Gamma \left (m + 1, \frac{2 \,{\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac{d m \log \left (\frac{2 \, f}{d}\right ) + 2 \, d e - 2 \, c f}{d}\right ) + 12 \,{\left (d f x + c f\right )} \cosh \left (m \log \left (d x + c\right )\right ) + 12 \,{\left (d f x + c f\right )} \sinh \left (m \log \left (d x + c\right )\right )}{96 \,{\left (a^{3} d f m + a^{3} d f\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*} \frac{\int \frac{\left (c + d x\right )^{m}}{\coth ^{3}{\left (e + f x \right )} + 3 \coth ^{2}{\left (e + f x \right )} + 3 \coth{\left (e + f x \right )} + 1}\, dx}{a^{3}} \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 \frac{{\left (d x + c\right )}^{m}}{{\left (a \coth \left (f x + e\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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