Optimal. Leaf size=410 \[ \frac {5 a^3 (c+d x)^{1+m}}{2 d (1+m)}-\frac {i 3^{-1-m} a^3 e^{3 e-\frac {3 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {3 f (c+d x)}{d}\right )}{8 f}-\frac {3\ 2^{-3-m} a^3 e^{2 e-\frac {2 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {2 f (c+d x)}{d}\right )}{f}+\frac {15 i a^3 e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )}{8 f}+\frac {15 i a^3 e^{-e+\frac {c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {f (c+d x)}{d}\right )}{8 f}+\frac {3\ 2^{-3-m} a^3 e^{-2 e+\frac {2 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )}{f}-\frac {i 3^{-1-m} a^3 e^{-3 e+\frac {3 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {3 f (c+d x)}{d}\right )}{8 f} \]
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Rubi [A]
time = 0.42, antiderivative size = 410, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3399, 3393,
3388, 2212, 3389} \begin {gather*} -\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} \text {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} \text {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} \text {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} \text {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} \text {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} \text {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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 3388
Rule 3389
Rule 3393
Rule 3399
Rubi steps
\begin {align*} \int (c+d x)^m (a+i a \sinh (e+f x))^3 \, dx &=\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*}
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Mathematica [A]
time = 4.37, size = 452, normalized size = 1.10 \begin {gather*} \frac {2^{-3-m} 3^{-1-m} a^3 e^{-3 \left (e+\frac {c f}{d}\right )} (c+d x)^m \left (-\frac {f^2 (c+d x)^2}{d^2}\right )^{-m} \left (5\ 2^m 3^{2+m} d e^{2 e+\frac {4 c f}{d}} (1+m) \left (-\frac {f (c+d x)}{d}\right )^m \Gamma \left (1+m,f \left (\frac {c}{d}+x\right )\right )-2^m d e^{6 e} (1+m) \left (\frac {f (c+d x)}{d}\right )^m \Gamma \left (1+m,-\frac {3 f (c+d x)}{d}\right )+i 3^{2+m} d e^{5 e+\frac {c f}{d}} (1+m) \left (f \left (\frac {c}{d}+x\right )\right )^m \Gamma \left (1+m,-\frac {2 f (c+d x)}{d}\right )+5\ 2^m 3^{2+m} d e^{4 e+\frac {2 c f}{d}} (1+m) \left (\frac {f (c+d x)}{d}\right )^m \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )-i 3^{2+m} d e^{e+\frac {5 c f}{d}} (1+m) \left (-\frac {f (c+d x)}{d}\right )^m \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )-2^m e^{\frac {3 c f}{d}} \left (20 i 3^{1+m} e^{3 e} f (c+d x) \left (-\frac {f^2 (c+d x)^2}{d^2}\right )^m+d e^{\frac {3 c f}{d}} (1+m) \left (-\frac {f (c+d x)}{d}\right )^m \Gamma \left (1+m,\frac {3 f (c+d x)}{d}\right )\right )\right ) (-i+\sinh (e+f x))^3}{d f (1+m) \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (d x +c \right )^{m} \left (a +i a \sinh \left (f x +e \right )\right )^{3}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.12, size = 383, normalized size = 0.93 \begin {gather*} -\frac {1}{8} i \, {\left (\frac {{\left (d x + c\right )}^{m + 1} e^{\left (\frac {3 \, c f}{d} - 3 \, e\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 (\frac {c f}{d} - e\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 (-\frac {c f}{d} + e\right )} E_{-m}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{d} - \frac {{\left (d x + c\right )}^{m + 1} e^{\left (-\frac {3 \, c f}{d} + 3 \, e\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 (\frac {2 \, c f}{d} - 2 \, e\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 (-\frac {2 \, c f}{d} + 2 \, e\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 (\frac {c f}{d} - e\right )} E_{-m}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{d} - \frac {{\left (d x + c\right )}^{m + 1} e^{\left (-\frac {c f}{d} + e\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.10, size = 380, normalized size = 0.93 \begin {gather*} \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 \, c f + 3 \, d e}{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 \, c f + 2 \, d e}{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 ) - c f + d e}{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 ) + c f - d e}{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 \, c f - 2 \, d e}{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 \, c f - 3 \, d e}{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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^3\,{\left (c+d\,x\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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