Optimal. Leaf size=277 \[ -\frac{i e^{i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (-\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,-\frac{i d (e+f x)}{f}\right )}{2 a d}+\frac{2^{-m-3} e^{2 i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (-\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 i d (e+f x)}{f}\right )}{a d}+\frac{i e^{-i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,\frac{i d (e+f x)}{f}\right )}{2 a d}+\frac{2^{-m-3} e^{-2 i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,\frac{2 i d (e+f x)}{f}\right )}{a d} \]
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Rubi [A] time = 0.319053, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {4523, 3307, 2181, 4406, 12, 3308} \[ -\frac{i e^{i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (-\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,-\frac{i d (e+f x)}{f}\right )}{2 a d}+\frac{2^{-m-3} e^{2 i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (-\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 i d (e+f x)}{f}\right )}{a d}+\frac{i e^{-i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,\frac{i d (e+f x)}{f}\right )}{2 a d}+\frac{2^{-m-3} e^{-2 i \left (c-\frac{d e}{f}\right )} (e+f x)^m \left (\frac{i d (e+f x)}{f}\right )^{-m} \text{Gamma}\left (m+1,\frac{2 i d (e+f x)}{f}\right )}{a d} \]
Antiderivative was successfully verified.
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Rule 4523
Rule 3307
Rule 2181
Rule 4406
Rule 12
Rule 3308
Rubi steps
\begin{align*} \int \frac{(e+f x)^m \cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int (e+f x)^m \cos (c+d x) \, dx}{a}-\frac{\int (e+f x)^m \cos (c+d x) \sin (c+d x) \, dx}{a}\\ &=\frac{\int e^{-i (c+d x)} (e+f x)^m \, dx}{2 a}+\frac{\int e^{i (c+d x)} (e+f x)^m \, dx}{2 a}-\frac{\int \frac{1}{2} (e+f x)^m \sin (2 c+2 d x) \, dx}{a}\\ &=-\frac{i 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{i 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{\int (e+f x)^m \sin (2 c+2 d x) \, dx}{2 a}\\ &=-\frac{i 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{i 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{i \int e^{-i (2 c+2 d x)} (e+f x)^m \, dx}{4 a}+\frac{i \int e^{i (2 c+2 d x)} (e+f x)^m \, dx}{4 a}\\ &=-\frac{i 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{i 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{2^{-3-m} e^{2 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{2 i d (e+f x)}{f}\right )}{a d}+\frac{2^{-3-m} e^{-2 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{2 i d (e+f x)}{f}\right )}{a d}\\ \end{align*}
Mathematica [A] time = 2.51116, size = 253, normalized size = 0.91 \[ \frac{2^{-m-3} e^{-\frac{2 i (c f+d e)}{f}} (e+f x)^m \left (\frac{d^2 (e+f x)^2}{f^2}\right )^{-m} \left (i 2^{m+2} e^{i \left (c+\frac{3 d e}{f}\right )} \left (-\frac{i d (e+f x)}{f}\right )^m \text{Gamma}\left (m+1,\frac{i d (e+f x)}{f}\right )-i 2^{m+2} e^{i \left (3 c+\frac{d e}{f}\right )} \left (\frac{i d (e+f x)}{f}\right )^m \text{Gamma}\left (m+1,-\frac{i d (e+f x)}{f}\right )+e^{4 i c} \left (\frac{i d (e+f x)}{f}\right )^m \text{Gamma}\left (m+1,-\frac{2 i d (e+f x)}{f}\right )+e^{\frac{4 i d e}{f}} \left (-\frac{i d (e+f x)}{f}\right )^m \text{Gamma}\left (m+1,\frac{2 i d (e+f x)}{f}\right )\right )}{a d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.19, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{m} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{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 )^{3}}{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 = 1.9995, size = 468, normalized size = 1.69 \begin{align*} \frac{e^{\left (-\frac{f m \log \left (\frac{2 i \, d}{f}\right ) - 2 i \, d e + 2 i \, c f}{f}\right )} \Gamma \left (m + 1, \frac{2 i \, d f x + 2 i \, d e}{f}\right ) + 4 i \, 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 ) - 4 i \, 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 ) + e^{\left (-\frac{f m \log \left (-\frac{2 i \, d}{f}\right ) + 2 i \, d e - 2 i \, c f}{f}\right )} \Gamma \left (m + 1, \frac{-2 i \, d f x - 2 i \, d e}{f}\right )}{8 \, a d} \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 )^{3}}{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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