Optimal. Leaf size=148 \[ -\frac{a e^{i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (-\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{i f (c+d x)}{d}\right )}{2 f}-\frac{a e^{-i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{i f (c+d x)}{d}\right )}{2 f}+\frac{a (c+d x)^{m+1}}{d (m+1)} \]
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Rubi [A] time = 0.144218, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3317, 3308, 2181} \[ -\frac{a e^{i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (-\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{i f (c+d x)}{d}\right )}{2 f}-\frac{a e^{-i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{i f (c+d x)}{d}\right )}{2 f}+\frac{a (c+d x)^{m+1}}{d (m+1)} \]
Antiderivative was successfully verified.
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Rule 3317
Rule 3308
Rule 2181
Rubi steps
\begin{align*} \int (c+d x)^m (a+a \sin (e+f x)) \, dx &=\int \left (a (c+d x)^m+a (c+d x)^m \sin (e+f x)\right ) \, dx\\ &=\frac{a (c+d x)^{1+m}}{d (1+m)}+a \int (c+d x)^m \sin (e+f x) \, dx\\ &=\frac{a (c+d x)^{1+m}}{d (1+m)}+\frac{1}{2} (i a) \int e^{-i (e+f x)} (c+d x)^m \, dx-\frac{1}{2} (i a) \int e^{i (e+f x)} (c+d x)^m \, dx\\ &=\frac{a (c+d x)^{1+m}}{d (1+m)}-\frac{a e^{i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (-\frac{i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac{i f (c+d x)}{d}\right )}{2 f}-\frac{a e^{-i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (\frac{i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac{i f (c+d x)}{d}\right )}{2 f}\\ \end{align*}
Mathematica [A] time = 2.71381, size = 199, normalized size = 1.34 \[ -\frac{a (c+d x)^m (\sin (e+f x)+1) \left (d (m+1) \left (-\frac{i f (c+d x)}{d}\right )^{-m} \left (\cos \left (e-\frac{c f}{d}\right )+i \sin \left (e-\frac{c f}{d}\right )\right ) \text{Gamma}\left (m+1,-\frac{i f (c+d x)}{d}\right )+d (m+1) \left (\frac{i f (c+d x)}{d}\right )^{-m} \left (\cos \left (e-\frac{c f}{d}\right )-i \sin \left (e-\frac{c f}{d}\right )\right ) \text{Gamma}\left (m+1,\frac{i f (c+d x)}{d}\right )-2 c f-2 d (e+f x)+2 d e\right )}{2 d f (m+1) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.093, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) ^{m} \left ( a+a\sin \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 = 1.89878, size = 319, normalized size = 2.16 \begin{align*} -\frac{{\left (a d m + a d\right )} e^{\left (-\frac{d m \log \left (\frac{i \, f}{d}\right ) + i \, d e - i \, c f}{d}\right )} \Gamma \left (m + 1, \frac{i \, d f x + i \, c f}{d}\right ) +{\left (a d m + a d\right )} e^{\left (-\frac{d m \log \left (-\frac{i \, f}{d}\right ) - i \, d e + i \, c f}{d}\right )} \Gamma \left (m + 1, \frac{-i \, d f x - i \, 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] time = 0., size = 0, normalized size = 0. \begin{align*} a \left (\int \left (c + d x\right )^{m} \sin{\left (e + f x \right )}\, dx + \int \left (c + d x\right )^{m}\, dx\right ) \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 (a \sin \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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