Optimal. Leaf size=277 \[ -\frac {i e^{i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (-\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (m+1,-\frac {i d (e+f x)}{f}\right )}{2 a d}+\frac {2^{-m-3} e^{2 i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (-\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (m+1,-\frac {2 i d (e+f x)}{f}\right )}{a d}+\frac {i e^{-i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (m+1,\frac {i d (e+f x)}{f}\right )}{2 a d}+\frac {2^{-m-3} e^{-2 i \left (c-\frac {d e}{f}\right )} (e+f x)^m \left (\frac {i d (e+f x)}{f}\right )^{-m} \Gamma \left (m+1,\frac {2 i d (e+f x)}{f}\right )}{a d} \]
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Rubi [A] time = 0.32, antiderivative size = 277, normalized size of antiderivative = 1.00, 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 12
Rule 2181
Rule 3307
Rule 3308
Rule 4406
Rule 4523
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*}
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Mathematica [A] time = 2.55, size = 253, normalized size = 0.91 \[ \frac {2^{-m-3} e^{-\frac {2 i (c f+d e)}{f}} (e+f x)^m \left (\frac {d^2 (e+f x)^2}{f^2}\right )^{-m} \left (i 2^{m+2} e^{i \left (c+\frac {3 d e}{f}\right )} \left (-\frac {i d (e+f x)}{f}\right )^m \Gamma \left (m+1,\frac {i d (e+f x)}{f}\right )-i 2^{m+2} e^{i \left (3 c+\frac {d e}{f}\right )} \left (\frac {i d (e+f x)}{f}\right )^m \Gamma \left (m+1,-\frac {i d (e+f x)}{f}\right )+e^{4 i c} \left (\frac {i d (e+f x)}{f}\right )^m \Gamma \left (m+1,-\frac {2 i d (e+f x)}{f}\right )+e^{\frac {4 i d e}{f}} \left (-\frac {i d (e+f x)}{f}\right )^m \Gamma \left (m+1,\frac {2 i d (e+f x)}{f}\right )\right )}{a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 187, normalized size = 0.68 \[ \frac {e^{\left (-\frac {f m \log \left (\frac {2 i \, d}{f}\right ) - 2 i \, d e + 2 i \, c f}{f}\right )} \Gamma \left (m + 1, \frac {2 i \, d f x + 2 i \, d e}{f}\right ) + 4 i \, e^{\left (-\frac {f m \log \left (\frac {i \, d}{f}\right ) - i \, d e + i \, c f}{f}\right )} \Gamma \left (m + 1, \frac {i \, d f x + i \, d e}{f}\right ) - 4 i \, e^{\left (-\frac {f m \log \left (-\frac {i \, d}{f}\right ) + i \, d e - i \, c f}{f}\right )} \Gamma \left (m + 1, \frac {-i \, d f x - i \, d e}{f}\right ) + e^{\left (-\frac {f m \log \left (-\frac {2 i \, d}{f}\right ) + 2 i \, d e - 2 i \, c f}{f}\right )} \Gamma \left (m + 1, \frac {-2 i \, d f x - 2 i \, d e}{f}\right )}{8 \, a d} \]
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 \[ \int \frac {{\left (f x + e\right )}^{m} \cos \left (d x + c\right )^{3}}{a \sin \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{m} \left (\cos ^{3}\left (d x +c \right )\right )}{a +a \sin \left (d x +c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (f x + e\right )}^{m} \cos \left (d x + c\right )^{3}}{a \sin \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^m}{a+a\,\sin \left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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