Optimal. Leaf size=223 \[ \frac {3\ 2^{-m-4} e^{\frac {2 c f}{d}-2 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {2 f (c+d x)}{d}\right )}{a^3 f}-\frac {3\ 2^{-2 m-5} e^{\frac {4 c f}{d}-4 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {4 f (c+d x)}{d}\right )}{a^3 f}+\frac {2^{-m-4} 3^{-m-1} e^{\frac {6 c f}{d}-6 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {6 f (c+d x)}{d}\right )}{a^3 f}+\frac {(c+d x)^{m+1}}{8 a^3 d (m+1)} \]
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Rubi [A] time = 0.23, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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 2181
Rule 3729
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*}
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Mathematica [F] time = 180.00, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]
Verification is Not applicable to the result.
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fricas [A] time = 0.41, size = 345, normalized size = 1.55 \[ \frac {2 \, {\left (d m + d\right )} \cosh \left (\frac {d m \log \left (\frac {6 \, f}{d}\right ) + 6 \, d e - 6 \, c f}{d}\right ) \Gamma \left (m + 1, \frac {6 \, {\left (d f x + c f\right )}}{d}\right ) - 9 \, {\left (d m + d\right )} \cosh \left (\frac {d m \log \left (\frac {4 \, f}{d}\right ) + 4 \, d e - 4 \, c f}{d}\right ) \Gamma \left (m + 1, \frac {4 \, {\left (d f x + c f\right )}}{d}\right ) + 18 \, {\left (d m + d\right )} \cosh \left (\frac {d m \log \left (\frac {2 \, f}{d}\right ) + 2 \, d e - 2 \, c f}{d}\right ) \Gamma \left (m + 1, \frac {2 \, {\left (d f x + c f\right )}}{d}\right ) - 2 \, {\left (d m + d\right )} \Gamma \left (m + 1, \frac {6 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {6 \, f}{d}\right ) + 6 \, d e - 6 \, c f}{d}\right ) + 9 \, {\left (d m + d\right )} \Gamma \left (m + 1, \frac {4 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {4 \, f}{d}\right ) + 4 \, d e - 4 \, c f}{d}\right ) - 18 \, {\left (d m + d\right )} \Gamma \left (m + 1, \frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {2 \, f}{d}\right ) + 2 \, d e - 2 \, c f}{d}\right ) + 12 \, {\left (d f x + c f\right )} \cosh \left (m \log \left (d x + c\right )\right ) + 12 \, {\left (d f x + c f\right )} \sinh \left (m \log \left (d x + c\right )\right )}{96 \, {\left (a^{3} d f m + a^{3} d f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x + c\right )}^{m}}{{\left (a \coth \left (f x + e\right ) + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.55, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x +c \right )^{m}}{\left (a +a \coth \left (f x +e \right )\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x + c\right )}^{m}}{{\left (a \coth \left (f x + e\right ) + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c+d\,x\right )}^m}{{\left (a+a\,\mathrm {coth}\left (e+f\,x\right )\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\left (c + d x\right )^{m}}{\coth ^{3}{\left (e + f x \right )} + 3 \coth ^{2}{\left (e + f x \right )} + 3 \coth {\left (e + f x \right )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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