Optimal. Leaf size=154 \[ \frac{e^{i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (-\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,-\frac{i d (e+f x)}{f}\right )}{2 a d}+\frac{e^{-i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,\frac{i d (e+f x)}{f}\right )}{2 a d}+\frac{(e+f x)^{m+1}}{a f (m+1)} \]
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Rubi [A] time = 0.176591, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4523, 32, 3308, 2181} \[ \frac{e^{i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (-\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,-\frac{i d (e+f x)}{f}\right )}{2 a d}+\frac{e^{-i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,\frac{i d (e+f x)}{f}\right )}{2 a d}+\frac{(e+f x)^{m+1}}{a f (m+1)} \]
Antiderivative was successfully verified.
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Rule 4523
Rule 32
Rule 3308
Rule 2181
Rubi steps
\begin{align*} \int \frac{(e+f x)^m \cos ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int (e+f x)^m \, dx}{a}-\frac{\int (e+f x)^m \sin (c+d x) \, dx}{a}\\ &=\frac{(e+f x)^{1+m}}{a f (1+m)}-\frac{i \int e^{-i (c+d x)} (e+f x)^m \, dx}{2 a}+\frac{i \int e^{i (c+d x)} (e+f x)^m \, dx}{2 a}\\ &=\frac{(e+f x)^{1+m}}{a f (1+m)}+\frac{e^{i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (-\frac{i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,-\frac{i d (e+f x)}{f}\right )}{2 a d}+\frac{e^{-i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (\frac{i d (e+f x)}{f}\right )^{-m} \Gamma \left (1+m,\frac{i d (e+f x)}{f}\right )}{2 a d}\\ \end{align*}
Mathematica [A] time = 0.983611, size = 220, normalized size = 1.43 \[ \frac{e^{i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2 \left (\frac{d^2 (e+f x)^2}{f^2}\right )^{-m} \left (f (m+1) e^{-2 i \left (c-\frac{d e}{f}\right )} \left (-\frac{i d (e+f x)}{f}\right )^m \text{Gamma}\left (m+1,\frac{i d (e+f x)}{f}\right )+f (m+1) \left (\frac{i d (e+f x)}{f}\right )^m \text{Gamma}\left (m+1,-\frac{i d (e+f x)}{f}\right )+2 d (e+f x) e^{-i \left (c-\frac{d e}{f}\right )} \left (\frac{d^2 (e+f x)^2}{f^2}\right )^m\right )}{2 a d f (m+1) (\sin (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.116, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{m} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{a+a\sin \left ( dx+c \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )}^{m} \cos \left (d x + c\right )^{2}}{a \sin \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8099, size = 306, normalized size = 1.99 \begin{align*} \frac{{\left (f m + f\right )} e^{\left (-\frac{f m \log \left (\frac{i \, d}{f}\right ) - i \, d e + i \, c f}{f}\right )} \Gamma \left (m + 1, \frac{i \, d f x + i \, d e}{f}\right ) +{\left (f m + f\right )} e^{\left (-\frac{f m \log \left (-\frac{i \, d}{f}\right ) + i \, d e - i \, c f}{f}\right )} \Gamma \left (m + 1, \frac{-i \, d f x - i \, d e}{f}\right ) + 2 \,{\left (d f x + d e\right )}{\left (f x + e\right )}^{m}}{2 \,{\left (a d f m + a d f\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \frac{{\left (f x + e\right )}^{m} \cos \left (d x + c\right )^{2}}{a \sin \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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