Optimal. Leaf size=268 \[ \frac {3 a^2 (c+d x)^{1+m}}{2 d (1+m)}-\frac {2^{-3-m} a^2 e^{2 e-\frac {2 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {2 f (c+d x)}{d}\right )}{f}+\frac {i a^2 e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )}{f}+\frac {i a^2 e^{-e+\frac {c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {f (c+d x)}{d}\right )}{f}+\frac {2^{-3-m} a^2 e^{-2 e+\frac {2 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )}{f} \]
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Rubi [A]
time = 0.27, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps
used = 9, 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 {a^2 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 {i a^2 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 )}{f}+\frac {i a^2 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 )}{f}+\frac {a^2 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 {3 a^2 (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))^2 \, dx &=\left (4 a^2\right ) \int (c+d x)^m \sin ^4\left (\frac {1}{2} \left (i e+\frac {\pi }{2}\right )+\frac {i f x}{2}\right ) \, dx\\ &=\left (4 a^2\right ) \int \left (\frac {3}{8} (c+d x)^m-\frac {1}{8} (c+d x)^m \cosh (2 e+2 f x)+\frac {1}{2} i (c+d x)^m \sinh (e+f x)\right ) \, dx\\ &=\frac {3 a^2 (c+d x)^{1+m}}{2 d (1+m)}+\left (2 i a^2\right ) \int (c+d x)^m \sinh (e+f x) \, dx-\frac {1}{2} a^2 \int (c+d x)^m \cosh (2 e+2 f x) \, dx\\ &=\frac {3 a^2 (c+d x)^{1+m}}{2 d (1+m)}+\left (i a^2\right ) \int e^{-i (i e+i f x)} (c+d x)^m \, dx-\left (i a^2\right ) \int e^{i (i e+i f x)} (c+d x)^m \, dx-\frac {1}{4} a^2 \int e^{-i (2 i e+2 i f x)} (c+d x)^m \, dx-\frac {1}{4} a^2 \int e^{i (2 i e+2 i f x)} (c+d x)^m \, dx\\ &=\frac {3 a^2 (c+d x)^{1+m}}{2 d (1+m)}-\frac {2^{-3-m} a^2 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 a^2 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 )}{f}+\frac {i a^2 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 )}{f}+\frac {2^{-3-m} a^2 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}\\ \end {align*}
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Mathematica [A]
time = 7.85, size = 383, normalized size = 1.43 \begin {gather*} \frac {(c+d x)^m \left (8 i e^{-e-f x}+e^{-2 (e+f x)}+8 i e^{e+f x}-e^{2 (e+f x)}+12 (e+f x)+\frac {2^{-m} e^{-2 \left (e+\frac {c f}{d}\right )} \left (-\frac {f^2 (c+d x)^2}{d^2}\right )^{-m} \left (-3 2^{2+m} e^{2 \left (e+\frac {c f}{d}\right )} \left (-\frac {f^2 (c+d x)^2}{d^2}\right )^m (-c f+d (e+e m+f m x))+i 2^{3+m} d e^{e+\frac {3 c f}{d}} m (1+m) \left (-\frac {f (c+d x)}{d}\right )^m \Gamma \left (m,f \left (\frac {c}{d}+x\right )\right )-d e^{4 e} m (1+m) \left (f \left (\frac {c}{d}+x\right )\right )^m \Gamma \left (m,-\frac {2 f (c+d x)}{d}\right )+i 2^{3+m} d e^{3 e+\frac {c f}{d}} m (1+m) \left (f \left (\frac {c}{d}+x\right )\right )^m \Gamma \left (m,-\frac {f (c+d x)}{d}\right )+d e^{\frac {4 c f}{d}} m (1+m) \left (-\frac {f (c+d x)}{d}\right )^m \Gamma \left (m,\frac {2 f (c+d x)}{d}\right )\right )}{d (1+m)}\right ) (a+i a \sinh (e+f x))^2}{8 f \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (d x +c \right )^{m} \left (a +i a \sinh \left (f x +e \right )\right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.09, size = 214, normalized size = 0.80 \begin {gather*} \frac {1}{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^{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^{2} + \frac {{\left (d x + c\right )}^{m + 1} a^{2}}{d {\left (m + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.09, size = 263, normalized size = 0.98 \begin {gather*} \frac {{\left (a^{2} d m + a^{2} 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 ) - 8 \, {\left (-i \, a^{2} d m - i \, a^{2} 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 ) - 8 \, {\left (-i \, a^{2} d m - i \, a^{2} 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 ) - {\left (a^{2} d m + a^{2} 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 ) + 12 \, {\left (a^{2} d f x + a^{2} c f\right )} {\left (d x + c\right )}^{m}}{8 \, {\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 )}^2\,{\left (c+d\,x\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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