Optimal. Leaf size=135 \[ \frac {i a e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {f (c+d x)}{d}\right )}{2 f}+\frac {i a e^{\frac {c f}{d}-e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {f (c+d x)}{d}\right )}{2 f}+\frac {a (c+d x)^{m+1}}{d (m+1)} \]
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Rubi [A] time = 0.15, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3317, 3308, 2181} \[ \frac {i a e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,-\frac {f (c+d x)}{d}\right )}{2 f}+\frac {i a e^{\frac {c f}{d}-e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,\frac {f (c+d x)}{d}\right )}{2 f}+\frac {a (c+d x)^{m+1}}{d (m+1)} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 3308
Rule 3317
Rubi steps
\begin {align*} \int (c+d x)^m (a+i a \sinh (e+f x)) \, dx &=\int \left (a (c+d x)^m+i a (c+d x)^m \sinh (e+f x)\right ) \, dx\\ &=\frac {a (c+d x)^{1+m}}{d (1+m)}+(i a) \int (c+d x)^m \sinh (e+f x) \, dx\\ &=\frac {a (c+d x)^{1+m}}{d (1+m)}+\frac {1}{2} (i a) \int e^{-i (i e+i f x)} (c+d x)^m \, dx-\frac {1}{2} (i a) \int e^{i (i e+i f x)} (c+d x)^m \, dx\\ &=\frac {a (c+d x)^{1+m}}{d (1+m)}+\frac {i a 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 )}{2 f}+\frac {i a 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 )}{2 f}\\ \end {align*}
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Mathematica [A] time = 0.51, size = 207, normalized size = 1.53 \[ -\frac {a e^{-\frac {c f}{d}-e} (c+d x)^m (\sinh (e+f x)-i) \left (-\frac {f^2 (c+d x)^2}{d^2}\right )^{-m} \left (-2 i f (c+d x) e^{\frac {c f}{d}+e} \left (-\frac {f^2 (c+d x)^2}{d^2}\right )^m+d e^{2 e} (m+1) \left (f \left (\frac {c}{d}+x\right )\right )^m \Gamma \left (m+1,-\frac {f (c+d x)}{d}\right )+d (m+1) e^{\frac {2 c f}{d}} \left (-\frac {f (c+d x)}{d}\right )^m \Gamma \left (m+1,\frac {f (c+d x)}{d}\right )\right )}{2 d f (m+1) \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 134, normalized size = 0.99 \[ \frac {{\left (i \, a d m + i \, a d\right )} e^{\left (-\frac {d m \log \left (\frac {f}{d}\right ) + d e - c f}{d}\right )} \Gamma \left (m + 1, \frac {d f x + c f}{d}\right ) + {\left (i \, a d m + i \, a d\right )} e^{\left (-\frac {d m \log \left (-\frac {f}{d}\right ) - d e + c f}{d}\right )} \Gamma \left (m + 1, -\frac {d f x + c f}{d}\right ) + 2 \, {\left (a d f x + a c f\right )} {\left (d x + c\right )}^{m}}{2 \, {\left (d f m + d f\right )}} \]
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 \[ \int {\left (i \, a \sinh \left (f x + e\right ) + a\right )} {\left (d x + c\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right )^{m} \left (a +i a \sinh \left (f x +e \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 101, normalized size = 0.75 \[ \frac {1}{2} i \, {\left (\frac {{\left (d x + c\right )}^{m + 1} e^{\left (-e + \frac {c f}{d}\right )} E_{-m}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{d} - \frac {{\left (d x + c\right )}^{m + 1} e^{\left (e - \frac {c f}{d}\right )} E_{-m}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{d}\right )} a + \frac {{\left (d x + c\right )}^{m + 1} a}{d {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,{\left (c+d\,x\right )}^m \,d x \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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