Integrand size = 20, antiderivative size = 224 \[ \int \frac {(c+d x)^m}{(a+a \tanh (e+f x))^3} \, dx=\frac {(c+d x)^{1+m}}{8 a^3 d (1+m)}-\frac {3\ 2^{-4-m} e^{-2 e+\frac {2 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )}{a^3 f}-\frac {3\ 2^{-5-2 m} e^{-4 e+\frac {4 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {4 f (c+d x)}{d}\right )}{a^3 f}-\frac {2^{-4-m} 3^{-1-m} e^{-6 e+\frac {6 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {6 f (c+d x)}{d}\right )}{a^3 f} \]
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Time = 0.17 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3810, 2212} \[ \int \frac {(c+d x)^m}{(a+a \tanh (e+f x))^3} \, dx=-\frac {3\ 2^{-m-4} e^{\frac {2 c f}{d}-2 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {2 f (c+d x)}{d}\right )}{a^3 f}-\frac {3\ 2^{-2 m-5} e^{\frac {4 c f}{d}-4 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {4 f (c+d x)}{d}\right )}{a^3 f}-\frac {2^{-m-4} 3^{-m-1} e^{\frac {6 c f}{d}-6 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {6 f (c+d x)}{d}\right )}{a^3 f}+\frac {(c+d x)^{m+1}}{8 a^3 d (m+1)} \]
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Rule 2212
Rule 3810
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 1.39 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.03 \[ \int \frac {(c+d x)^m}{(a+a \tanh (e+f x))^3} \, dx=\frac {2^{-5-2 m} 3^{-1-m} e^{-3 e} \left (f \left (\frac {c}{d}+x\right )\right )^{-m} (c+d x)^m \left (12^{1+m} e^{6 e} f \left (f \left (\frac {c}{d}+x\right )\right )^m (c+d x)-2^{1+m} 3^{2+m} d e^{4 e+\frac {2 c f}{d}} (1+m) \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )-3^{2+m} d e^{2 e+\frac {4 c f}{d}} (1+m) \Gamma \left (1+m,\frac {4 f (c+d x)}{d}\right )-2^{1+m} d e^{\frac {6 c f}{d}} (1+m) \Gamma \left (1+m,\frac {6 f (c+d x)}{d}\right )\right ) \text {sech}^3(e+f x) (\cosh (f x)+\sinh (f x))^3}{a^3 d f (1+m) (1+\tanh (e+f x))^3} \]
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\[\int \frac {\left (d x +c \right )^{m}}{\left (a +a \tanh \left (f x +e \right )\right )^{3}}d x\]
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none
Time = 0.10 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.54 \[ \int \frac {(c+d x)^m}{(a+a \tanh (e+f x))^3} \, dx=-\frac {2 \, {\left (d m + d\right )} \cosh \left (\frac {d m \log \left (\frac {6 \, f}{d}\right ) + 6 \, d e - 6 \, c f}{d}\right ) \Gamma \left (m + 1, \frac {6 \, {\left (d f x + c f\right )}}{d}\right ) + 9 \, {\left (d m + d\right )} \cosh \left (\frac {d m \log \left (\frac {4 \, f}{d}\right ) + 4 \, d e - 4 \, c f}{d}\right ) \Gamma \left (m + 1, \frac {4 \, {\left (d f x + c f\right )}}{d}\right ) + 18 \, {\left (d m + d\right )} \cosh \left (\frac {d m \log \left (\frac {2 \, f}{d}\right ) + 2 \, d e - 2 \, c f}{d}\right ) \Gamma \left (m + 1, \frac {2 \, {\left (d f x + c f\right )}}{d}\right ) - 2 \, {\left (d m + d\right )} \Gamma \left (m + 1, \frac {6 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {6 \, f}{d}\right ) + 6 \, d e - 6 \, c f}{d}\right ) - 9 \, {\left (d m + d\right )} \Gamma \left (m + 1, \frac {4 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {4 \, f}{d}\right ) + 4 \, d e - 4 \, c f}{d}\right ) - 18 \, {\left (d m + d\right )} \Gamma \left (m + 1, \frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {2 \, f}{d}\right ) + 2 \, d e - 2 \, c f}{d}\right ) - 12 \, {\left (d f x + c f\right )} \cosh \left (m \log \left (d x + c\right )\right ) - 12 \, {\left (d f x + c f\right )} \sinh \left (m \log \left (d x + c\right )\right )}{96 \, {\left (a^{3} d f m + a^{3} d f\right )}} \]
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\[ \int \frac {(c+d x)^m}{(a+a \tanh (e+f x))^3} \, dx=\frac {\int \frac {\left (c + d x\right )^{m}}{\tanh ^{3}{\left (e + f x \right )} + 3 \tanh ^{2}{\left (e + f x \right )} + 3 \tanh {\left (e + f x \right )} + 1}\, dx}{a^{3}} \]
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\[ \int \frac {(c+d x)^m}{(a+a \tanh (e+f x))^3} \, dx=\int { \frac {{\left (d x + c\right )}^{m}}{{\left (a \tanh \left (f x + e\right ) + a\right )}^{3}} \,d x } \]
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\[ \int \frac {(c+d x)^m}{(a+a \tanh (e+f x))^3} \, dx=\int { \frac {{\left (d x + c\right )}^{m}}{{\left (a \tanh \left (f x + e\right ) + a\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(c+d x)^m}{(a+a \tanh (e+f x))^3} \, dx=\int \frac {{\left (c+d\,x\right )}^m}{{\left (a+a\,\mathrm {tanh}\left (e+f\,x\right )\right )}^3} \,d x \]
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