Optimal. Leaf size=145 \[ \frac{2 a^3 c^4 \cos ^7(e+f x)}{13 f \sqrt{c-c \sin (e+f x)}}+\frac{24 a^3 c^5 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^{3/2}}+\frac{64 a^3 c^6 \cos ^7(e+f x)}{429 f (c-c \sin (e+f x))^{5/2}}+\frac{256 a^3 c^7 \cos ^7(e+f x)}{3003 f (c-c \sin (e+f x))^{7/2}} \]
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Rubi [A] time = 0.330216, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {2736, 2674, 2673} \[ \frac{2 a^3 c^4 \cos ^7(e+f x)}{13 f \sqrt{c-c \sin (e+f x)}}+\frac{24 a^3 c^5 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^{3/2}}+\frac{64 a^3 c^6 \cos ^7(e+f x)}{429 f (c-c \sin (e+f x))^{5/2}}+\frac{256 a^3 c^7 \cos ^7(e+f x)}{3003 f (c-c \sin (e+f x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 2736
Rule 2674
Rule 2673
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2} \, dx &=\left (a^3 c^3\right ) \int \cos ^6(e+f x) \sqrt{c-c \sin (e+f x)} \, dx\\ &=\frac{2 a^3 c^4 \cos ^7(e+f x)}{13 f \sqrt{c-c \sin (e+f x)}}+\frac{1}{13} \left (12 a^3 c^4\right ) \int \frac{\cos ^6(e+f x)}{\sqrt{c-c \sin (e+f x)}} \, dx\\ &=\frac{24 a^3 c^5 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^{3/2}}+\frac{2 a^3 c^4 \cos ^7(e+f x)}{13 f \sqrt{c-c \sin (e+f x)}}+\frac{1}{143} \left (96 a^3 c^5\right ) \int \frac{\cos ^6(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx\\ &=\frac{64 a^3 c^6 \cos ^7(e+f x)}{429 f (c-c \sin (e+f x))^{5/2}}+\frac{24 a^3 c^5 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^{3/2}}+\frac{2 a^3 c^4 \cos ^7(e+f x)}{13 f \sqrt{c-c \sin (e+f x)}}+\frac{1}{429} \left (128 a^3 c^6\right ) \int \frac{\cos ^6(e+f x)}{(c-c \sin (e+f x))^{5/2}} \, dx\\ &=\frac{256 a^3 c^7 \cos ^7(e+f x)}{3003 f (c-c \sin (e+f x))^{7/2}}+\frac{64 a^3 c^6 \cos ^7(e+f x)}{429 f (c-c \sin (e+f x))^{5/2}}+\frac{24 a^3 c^5 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^{3/2}}+\frac{2 a^3 c^4 \cos ^7(e+f x)}{13 f \sqrt{c-c \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 6.18494, size = 112, normalized size = 0.77 \[ \frac{a^3 c^3 \cos ^6(e+f x) \sqrt{c-c \sin (e+f x)} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) (-6377 \sin (e+f x)+231 \sin (3 (e+f x))-1890 \cos (2 (e+f x))+5230)}{6006 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.545, size = 81, normalized size = 0.6 \begin{align*}{\frac{ \left ( -2+2\,\sin \left ( fx+e \right ) \right ){c}^{4} \left ( 1+\sin \left ( fx+e \right ) \right ) ^{4}{a}^{3} \left ( 231\, \left ( \sin \left ( fx+e \right ) \right ) ^{3}-945\, \left ( \sin \left ( fx+e \right ) \right ) ^{2}+1421\,\sin \left ( fx+e \right ) -835 \right ) }{3003\,f\cos \left ( fx+e \right ) }{\frac{1}{\sqrt{c-c\sin \left ( fx+e \right ) }}}} \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{\left (a \sin \left (f x + e\right ) + a\right )}^{3}{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.09878, size = 656, normalized size = 4.52 \begin{align*} \frac{2 \,{\left (231 \, a^{3} c^{3} \cos \left (f x + e\right )^{7} - 21 \, a^{3} c^{3} \cos \left (f x + e\right )^{6} + 28 \, a^{3} c^{3} \cos \left (f x + e\right )^{5} - 40 \, a^{3} c^{3} \cos \left (f x + e\right )^{4} + 64 \, a^{3} c^{3} \cos \left (f x + e\right )^{3} - 128 \, a^{3} c^{3} \cos \left (f x + e\right )^{2} + 512 \, a^{3} c^{3} \cos \left (f x + e\right ) + 1024 \, a^{3} c^{3} +{\left (231 \, a^{3} c^{3} \cos \left (f x + e\right )^{6} + 252 \, a^{3} c^{3} \cos \left (f x + e\right )^{5} + 280 \, a^{3} c^{3} \cos \left (f x + e\right )^{4} + 320 \, a^{3} c^{3} \cos \left (f x + e\right )^{3} + 384 \, a^{3} c^{3} \cos \left (f x + e\right )^{2} + 512 \, a^{3} c^{3} \cos \left (f x + e\right ) + 1024 \, a^{3} c^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt{-c \sin \left (f x + e\right ) + c}}{3003 \,{\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + 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{\left (a \sin \left (f x + e\right ) + a\right )}^{3}{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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