Integrand size = 23, antiderivative size = 410 \[ \int (c+d x)^m (a+i a \sinh (e+f x))^3 \, dx=\frac {5 a^3 (c+d x)^{1+m}}{2 d (1+m)}-\frac {i 3^{-1-m} a^3 e^{3 e-\frac {3 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {3 f (c+d x)}{d}\right )}{8 f}-\frac {3\ 2^{-3-m} a^3 e^{2 e-\frac {2 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {2 f (c+d x)}{d}\right )}{f}+\frac {15 i a^3 e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )}{8 f}+\frac {15 i a^3 e^{-e+\frac {c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {f (c+d x)}{d}\right )}{8 f}+\frac {3\ 2^{-3-m} a^3 e^{-2 e+\frac {2 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )}{f}-\frac {i 3^{-1-m} a^3 e^{-3 e+\frac {3 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {3 f (c+d x)}{d}\right )}{8 f} \]
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Time = 0.42 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3399, 3393, 3388, 2212, 3389} \[ \int (c+d x)^m (a+i a \sinh (e+f x))^3 \, dx=-\frac {i a^3 3^{-m-1} e^{3 e-\frac {3 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {3 f (c+d x)}{d}\right )}{8 f}-\frac {3 a^3 2^{-m-3} e^{2 e-\frac {2 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {2 f (c+d x)}{d}\right )}{f}+\frac {15 i a^3 e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {f (c+d x)}{d}\right )}{8 f}+\frac {15 i a^3 e^{\frac {c f}{d}-e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {f (c+d x)}{d}\right )}{8 f}+\frac {3 a^3 2^{-m-3} e^{\frac {2 c f}{d}-2 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {2 f (c+d x)}{d}\right )}{f}-\frac {i a^3 3^{-m-1} e^{\frac {3 c f}{d}-3 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {3 f (c+d x)}{d}\right )}{8 f}+\frac {5 a^3 (c+d x)^{m+1}}{2 d (m+1)} \]
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Rule 2212
Rule 3388
Rule 3389
Rule 3393
Rule 3399
Rubi steps \begin{align*} \text {integral}& = \left (8 a^3\right ) \int (c+d x)^m \sin ^6\left (\frac {1}{2} \left (i e+\frac {\pi }{2}\right )+\frac {i f x}{2}\right ) \, dx \\ & = \left (8 a^3\right ) \int \left (\frac {5}{16} (c+d x)^m-\frac {3}{16} (c+d x)^m \cosh (2 e+2 f x)+\frac {15}{32} i (c+d x)^m \sinh (e+f x)-\frac {1}{32} i (c+d x)^m \sinh (3 e+3 f x)\right ) \, dx \\ & = \frac {5 a^3 (c+d x)^{1+m}}{2 d (1+m)}-\frac {1}{4} \left (i a^3\right ) \int (c+d x)^m \sinh (3 e+3 f x) \, dx+\frac {1}{4} \left (15 i a^3\right ) \int (c+d x)^m \sinh (e+f x) \, dx-\frac {1}{2} \left (3 a^3\right ) \int (c+d x)^m \cosh (2 e+2 f x) \, dx \\ & = \frac {5 a^3 (c+d x)^{1+m}}{2 d (1+m)}-\frac {1}{8} \left (i a^3\right ) \int e^{-i (3 i e+3 i f x)} (c+d x)^m \, dx+\frac {1}{8} \left (i a^3\right ) \int e^{i (3 i e+3 i f x)} (c+d x)^m \, dx+\frac {1}{8} \left (15 i a^3\right ) \int e^{-i (i e+i f x)} (c+d x)^m \, dx-\frac {1}{8} \left (15 i a^3\right ) \int e^{i (i e+i f x)} (c+d x)^m \, dx-\frac {1}{4} \left (3 a^3\right ) \int e^{-i (2 i e+2 i f x)} (c+d x)^m \, dx-\frac {1}{4} \left (3 a^3\right ) \int e^{i (2 i e+2 i f x)} (c+d x)^m \, dx \\ & = \frac {5 a^3 (c+d x)^{1+m}}{2 d (1+m)}-\frac {i 3^{-1-m} a^3 e^{3 e-\frac {3 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {3 f (c+d x)}{d}\right )}{8 f}-\frac {3\ 2^{-3-m} a^3 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 )}{f}+\frac {15 i a^3 e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )}{8 f}+\frac {15 i a^3 e^{-e+\frac {c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {f (c+d x)}{d}\right )}{8 f}+\frac {3\ 2^{-3-m} a^3 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 )}{f}-\frac {i 3^{-1-m} a^3 e^{-3 e+\frac {3 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {3 f (c+d x)}{d}\right )}{8 f} \\ \end{align*}
Time = 1.04 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.00 \[ \int (c+d x)^m (a+i a \sinh (e+f x))^3 \, dx=\frac {5 a^3 (c+d x)^{1+m}}{2 d (1+m)}-\frac {i 3^{-1-m} a^3 e^{3 e-\frac {3 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {3 f (c+d x)}{d}\right )}{8 f}-\frac {3\ 2^{-3-m} a^3 e^{2 e-\frac {2 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {2 f (c+d x)}{d}\right )}{f}+\frac {15 i a^3 e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )}{8 f}+\frac {15 i a^3 e^{-e+\frac {c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {f (c+d x)}{d}\right )}{8 f}+\frac {3\ 2^{-3-m} a^3 e^{-2 e+\frac {2 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )}{f}-\frac {i 3^{-1-m} a^3 e^{-3 e+\frac {3 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {3 f (c+d x)}{d}\right )}{8 f} \]
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\[\int \left (d x +c \right )^{m} \left (a +i a \sinh \left (f x +e \right )\right )^{3}d x\]
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Time = 0.10 (sec) , antiderivative size = 374, normalized size of antiderivative = 0.91 \[ \int (c+d x)^m (a+i a \sinh (e+f x))^3 \, dx=\frac {{\left (-i \, a^{3} d m - i \, a^{3} d\right )} e^{\left (-\frac {d m \log \left (\frac {3 \, f}{d}\right ) + 3 \, d e - 3 \, c f}{d}\right )} \Gamma \left (m + 1, \frac {3 \, {\left (d f x + c f\right )}}{d}\right ) + 9 \, {\left (a^{3} d m + a^{3} d\right )} e^{\left (-\frac {d m \log \left (\frac {2 \, f}{d}\right ) + 2 \, d e - 2 \, c f}{d}\right )} \Gamma \left (m + 1, \frac {2 \, {\left (d f x + c f\right )}}{d}\right ) - 45 \, {\left (-i \, a^{3} d m - i \, a^{3} d\right )} e^{\left (-\frac {d m \log \left (\frac {f}{d}\right ) + d e - c f}{d}\right )} \Gamma \left (m + 1, \frac {d f x + c f}{d}\right ) - 45 \, {\left (-i \, a^{3} d m - i \, a^{3} d\right )} e^{\left (-\frac {d m \log \left (-\frac {f}{d}\right ) - d e + c f}{d}\right )} \Gamma \left (m + 1, -\frac {d f x + c f}{d}\right ) - 9 \, {\left (a^{3} d m + a^{3} d\right )} e^{\left (-\frac {d m \log \left (-\frac {2 \, f}{d}\right ) - 2 \, d e + 2 \, c f}{d}\right )} \Gamma \left (m + 1, -\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + {\left (-i \, a^{3} d m - i \, a^{3} d\right )} e^{\left (-\frac {d m \log \left (-\frac {3 \, f}{d}\right ) - 3 \, d e + 3 \, c f}{d}\right )} \Gamma \left (m + 1, -\frac {3 \, {\left (d f x + c f\right )}}{d}\right ) + 60 \, {\left (a^{3} d f x + a^{3} c f\right )} {\left (d x + c\right )}^{m}}{24 \, {\left (d f m + d f\right )}} \]
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Exception generated. \[ \int (c+d x)^m (a+i a \sinh (e+f x))^3 \, dx=\text {Exception raised: TypeError} \]
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Time = 0.14 (sec) , antiderivative size = 375, normalized size of antiderivative = 0.91 \[ \int (c+d x)^m (a+i a \sinh (e+f x))^3 \, dx=-\frac {1}{8} i \, {\left (\frac {{\left (d x + c\right )}^{m + 1} e^{\left (-3 \, e + \frac {3 \, c f}{d}\right )} E_{-m}\left (\frac {3 \, {\left (d x + c\right )} f}{d}\right )}{d} - \frac {3 \, {\left (d x + c\right )}^{m + 1} e^{\left (-e + \frac {c f}{d}\right )} E_{-m}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{d} + \frac {3 \, {\left (d x + c\right )}^{m + 1} e^{\left (e - \frac {c f}{d}\right )} E_{-m}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{d} - \frac {{\left (d x + c\right )}^{m + 1} e^{\left (3 \, e - \frac {3 \, c f}{d}\right )} E_{-m}\left (-\frac {3 \, {\left (d x + c\right )} f}{d}\right )}{d}\right )} a^{3} + \frac {3}{4} \, {\left (\frac {{\left (d x + c\right )}^{m + 1} e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{-m}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{d} + \frac {{\left (d x + c\right )}^{m + 1} e^{\left (2 \, e - \frac {2 \, c f}{d}\right )} E_{-m}\left (-\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{d} + \frac {2 \, {\left (d x + c\right )}^{m + 1}}{d {\left (m + 1\right )}}\right )} a^{3} + \frac {3}{2} i \, {\left (\frac {{\left (d x + c\right )}^{m + 1} e^{\left (-e + \frac {c f}{d}\right )} E_{-m}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{d} - \frac {{\left (d x + c\right )}^{m + 1} e^{\left (e - \frac {c f}{d}\right )} E_{-m}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{d}\right )} a^{3} + \frac {{\left (d x + c\right )}^{m + 1} a^{3}}{d {\left (m + 1\right )}} \]
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\[ \int (c+d x)^m (a+i a \sinh (e+f x))^3 \, dx=\int { {\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{3} {\left (d x + c\right )}^{m} \,d x } \]
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Timed out. \[ \int (c+d x)^m (a+i a \sinh (e+f x))^3 \, dx=\int {\left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^3\,{\left (c+d\,x\right )}^m \,d x \]
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