Optimal. Leaf size=150 \[ -\frac{8 a^3 (a \sin (c+d x)+a)^{m-3} (e \cos (c+d x))^{6-2 m}}{d e (5-m) \left (m^2-7 m+12\right )}-\frac{4 a^2 (a \sin (c+d x)+a)^{m-2} (e \cos (c+d x))^{6-2 m}}{d e \left (m^2-9 m+20\right )}-\frac{a (a \sin (c+d x)+a)^{m-1} (e \cos (c+d x))^{6-2 m}}{d e (5-m)} \]
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Rubi [A] time = 0.240436, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {2674, 2673} \[ -\frac{8 a^3 (a \sin (c+d x)+a)^{m-3} (e \cos (c+d x))^{6-2 m}}{d e (5-m) \left (m^2-7 m+12\right )}-\frac{4 a^2 (a \sin (c+d x)+a)^{m-2} (e \cos (c+d x))^{6-2 m}}{d e \left (m^2-9 m+20\right )}-\frac{a (a \sin (c+d x)+a)^{m-1} (e \cos (c+d x))^{6-2 m}}{d e (5-m)} \]
Antiderivative was successfully verified.
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Rule 2674
Rule 2673
Rubi steps
\begin{align*} \int (e \cos (c+d x))^{5-2 m} (a+a \sin (c+d x))^m \, dx &=-\frac{a (e \cos (c+d x))^{6-2 m} (a+a \sin (c+d x))^{-1+m}}{d e (5-m)}+\frac{(4 a) \int (e \cos (c+d x))^{5-2 m} (a+a \sin (c+d x))^{-1+m} \, dx}{5-m}\\ &=-\frac{4 a^2 (e \cos (c+d x))^{6-2 m} (a+a \sin (c+d x))^{-2+m}}{d e \left (20-9 m+m^2\right )}-\frac{a (e \cos (c+d x))^{6-2 m} (a+a \sin (c+d x))^{-1+m}}{d e (5-m)}+\frac{\left (8 a^2\right ) \int (e \cos (c+d x))^{5-2 m} (a+a \sin (c+d x))^{-2+m} \, dx}{20-9 m+m^2}\\ &=-\frac{8 a^3 (e \cos (c+d x))^{6-2 m} (a+a \sin (c+d x))^{-3+m}}{d e (3-m) \left (20-9 m+m^2\right )}-\frac{4 a^2 (e \cos (c+d x))^{6-2 m} (a+a \sin (c+d x))^{-2+m}}{d e \left (20-9 m+m^2\right )}-\frac{a (e \cos (c+d x))^{6-2 m} (a+a \sin (c+d x))^{-1+m}}{d e (5-m)}\\ \end{align*}
Mathematica [A] time = 0.434272, size = 105, normalized size = 0.7 \[ \frac{e^5 \cos ^6(c+d x) \left (\left (m^2-7 m+12\right ) \sin ^2(c+d x)+2 \left (m^2-9 m+18\right ) \sin (c+d x)+m^2-11 m+32\right ) (a (\sin (c+d x)+1))^m (e \cos (c+d x))^{-2 m}}{d (m-5) (m-4) (m-3) (\sin (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.914, size = 0, normalized size = 0. \begin{align*} \int \left ( e\cos \left ( dx+c \right ) \right ) ^{5-2\,m} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.66588, size = 842, normalized size = 5.61 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.46202, size = 813, normalized size = 5.42 \begin{align*} -\frac{{\left ({\left (m^{2} - 7 \, m + 12\right )} \cos \left (d x + c\right )^{3} -{\left (m^{2} - 11 \, m + 24\right )} \cos \left (d x + c\right )^{2} - 2 \,{\left (m^{2} - 9 \, m + 22\right )} \cos \left (d x + c\right ) -{\left ({\left (m^{2} - 7 \, m + 12\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (m^{2} - 9 \, m + 18\right )} \cos \left (d x + c\right ) - 8\right )} \sin \left (d x + c\right ) - 8\right )} \left (e \cos \left (d x + c\right )\right )^{-2 \, m + 5}{\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{4 \, d m^{3} -{\left (d m^{3} - 12 \, d m^{2} + 47 \, d m - 60 \, d\right )} \cos \left (d x + c\right )^{3} - 48 \, d m^{2} - 3 \,{\left (d m^{3} - 12 \, d m^{2} + 47 \, d m - 60 \, d\right )} \cos \left (d x + c\right )^{2} + 188 \, d m + 2 \,{\left (d m^{3} - 12 \, d m^{2} + 47 \, d m - 60 \, d\right )} \cos \left (d x + c\right ) +{\left (4 \, d m^{3} - 48 \, d m^{2} -{\left (d m^{3} - 12 \, d m^{2} + 47 \, d m - 60 \, d\right )} \cos \left (d x + c\right )^{2} + 188 \, d m + 2 \,{\left (d m^{3} - 12 \, d m^{2} + 47 \, d m - 60 \, d\right )} \cos \left (d x + c\right ) - 240 \, d\right )} \sin \left (d x + c\right ) - 240 \, 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 \left (e \cos \left (d x + c\right )\right )^{-2 \, m + 5}{\left (a \sin \left (d x + c\right ) + a\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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