Optimal. Leaf size=135 \[ \frac{i a e^{e-\frac{c f}{d}} (c+d x)^m \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{f (c+d x)}{d}\right )}{2 f}+\frac{i a e^{\frac{c f}{d}-e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{f (c+d x)}{d}\right )}{2 f}+\frac{a (c+d x)^{m+1}}{d (m+1)} \]
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Rubi [A] time = 0.153541, antiderivative size = 135, normalized size of antiderivative = 1., 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 3317
Rule 3308
Rule 2181
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*}
Mathematica [A] time = 0.55988, size = 207, normalized size = 1.53 \[ -\frac{a e^{-\frac{c f}{d}-e} (c+d x)^m (\sinh (e+f x)-i) \left (-\frac{f^2 (c+d x)^2}{d^2}\right )^{-m} \left (d e^{2 e} (m+1) \left (f \left (\frac{c}{d}+x\right )\right )^m \text{Gamma}\left (m+1,-\frac{f (c+d x)}{d}\right )+d (m+1) e^{\frac{2 c f}{d}} \left (-\frac{f (c+d x)}{d}\right )^m \text{Gamma}\left (m+1,\frac{f (c+d x)}{d}\right )-2 i f (c+d x) e^{\frac{c f}{d}+e} \left (-\frac{f^2 (c+d x)^2}{d^2}\right )^m\right )}{2 d f (m+1) \left (\cosh \left (\frac{1}{2} (e+f x)\right )+i \sinh \left (\frac{1}{2} (e+f x)\right )\right )^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.056, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) ^{m} \left ( a+ia\sinh \left ( fx+e \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.54164, size = 301, normalized size = 2.23 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \sinh \left (f x + e\right ) + a\right )}{\left (d x + c\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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