Optimal. Leaf size=200 \[ -\frac{2 a^3 c^2 (A+B) \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}-\frac{4 a^3 c (A+B) \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}-\frac{8 a^3 (A+B) \cos (e+f x)}{f \sqrt{c-c \sin (e+f x)}}+\frac{8 \sqrt{2} a^3 (A+B) \tanh ^{-1}\left (\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right )}{\sqrt{c} f}-\frac{2 a^3 B c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}} \]
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Rubi [A] time = 0.521297, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.132, Rules used = {2967, 2860, 2679, 2649, 206} \[ -\frac{2 a^3 c^2 (A+B) \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}-\frac{4 a^3 c (A+B) \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}-\frac{8 a^3 (A+B) \cos (e+f x)}{f \sqrt{c-c \sin (e+f x)}}+\frac{8 \sqrt{2} a^3 (A+B) \tanh ^{-1}\left (\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right )}{\sqrt{c} f}-\frac{2 a^3 B c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 2967
Rule 2860
Rule 2679
Rule 2649
Rule 206
Rubi steps
\begin{align*} \int \frac{(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{\sqrt{c-c \sin (e+f x)}} \, dx &=\left (a^3 c^3\right ) \int \frac{\cos ^6(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx\\ &=-\frac{2 a^3 B c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}}+\left (a^3 (A+B) c^3\right ) \int \frac{\cos ^6(e+f x)}{(c-c \sin (e+f x))^{7/2}} \, dx\\ &=-\frac{2 a^3 B c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}}-\frac{2 a^3 (A+B) c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}+\left (2 a^3 (A+B) c^2\right ) \int \frac{\cos ^4(e+f x)}{(c-c \sin (e+f x))^{5/2}} \, dx\\ &=-\frac{2 a^3 B c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}}-\frac{2 a^3 (A+B) c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}-\frac{4 a^3 (A+B) c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}+\left (4 a^3 (A+B) c\right ) \int \frac{\cos ^2(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx\\ &=-\frac{2 a^3 B c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}}-\frac{2 a^3 (A+B) c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}-\frac{4 a^3 (A+B) c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}-\frac{8 a^3 (A+B) \cos (e+f x)}{f \sqrt{c-c \sin (e+f x)}}+\left (8 a^3 (A+B)\right ) \int \frac{1}{\sqrt{c-c \sin (e+f x)}} \, dx\\ &=-\frac{2 a^3 B c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}}-\frac{2 a^3 (A+B) c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}-\frac{4 a^3 (A+B) c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}-\frac{8 a^3 (A+B) \cos (e+f x)}{f \sqrt{c-c \sin (e+f x)}}-\frac{\left (16 a^3 (A+B)\right ) \operatorname{Subst}\left (\int \frac{1}{2 c-x^2} \, dx,x,-\frac{c \cos (e+f x)}{\sqrt{c-c \sin (e+f x)}}\right )}{f}\\ &=\frac{8 \sqrt{2} a^3 (A+B) \tanh ^{-1}\left (\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right )}{\sqrt{c} f}-\frac{2 a^3 B c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}}-\frac{2 a^3 (A+B) c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}-\frac{4 a^3 (A+B) c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}-\frac{8 a^3 (A+B) \cos (e+f x)}{f \sqrt{c-c \sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 1.41376, size = 193, normalized size = 0.96 \[ -\frac{a^3 (\sin (e+f x)+1)^3 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (-2 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) (-(448 A+673 B) \sin (e+f x)+6 (7 A+22 B) \cos (2 (e+f x))-2086 A+15 B \sin (3 (e+f x))-2236 B)+(6720+6720 i) \sqrt [4]{-1} (A+B) \tan ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt [4]{-1} \left (\tan \left (\frac{1}{4} (e+f x)\right )+1\right )\right )\right )}{420 f \sqrt{c-c \sin (e+f x)} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.292, size = 233, normalized size = 1.2 \begin{align*} -{\frac{ \left ( -2+2\,\sin \left ( fx+e \right ) \right ){a}^{3}}{105\,{c}^{4}\cos \left ( fx+e \right ) f}\sqrt{c \left ( 1+\sin \left ( fx+e \right ) \right ) } \left ( 420\,{c}^{7/2}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( 1+\sin \left ( fx+e \right ) \right ) }\sqrt{2}}{\sqrt{c}}} \right ) A+420\,{c}^{7/2}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( 1+\sin \left ( fx+e \right ) \right ) }\sqrt{2}}{\sqrt{c}}} \right ) B-15\,B \left ( c \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{7/2}-21\,A \left ( c \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{5/2}c-21\,B \left ( c \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{5/2}c-70\,A \left ( c \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{3/2}{c}^{2}-70\,B \left ( c \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{3/2}{c}^{2}-420\,A{c}^{3}\sqrt{c \left ( 1+\sin \left ( fx+e \right ) \right ) }-420\,B{c}^{3}\sqrt{c \left ( 1+\sin \left ( fx+e \right ) \right ) } \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 \frac{{\left (B \sin \left (f x + e\right ) + A\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{3}}{\sqrt{-c \sin \left (f x + e\right ) + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48203, size = 944, normalized size = 4.72 \begin{align*} \frac{2 \,{\left (\frac{210 \, \sqrt{2}{\left ({\left (A + B\right )} a^{3} c \cos \left (f x + e\right ) -{\left (A + B\right )} a^{3} c \sin \left (f x + e\right ) +{\left (A + B\right )} a^{3} c\right )} \log \left (-\frac{\cos \left (f x + e\right )^{2} +{\left (\cos \left (f x + e\right ) - 2\right )} \sin \left (f x + e\right ) + \frac{2 \, \sqrt{2} \sqrt{-c \sin \left (f x + e\right ) + c}{\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )}}{\sqrt{c}} + 3 \, \cos \left (f x + e\right ) + 2}{\cos \left (f x + e\right )^{2} +{\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right )}{\sqrt{c}} -{\left (15 \, B a^{3} \cos \left (f x + e\right )^{4} - 3 \,{\left (7 \, A + 22 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} -{\left (133 \, A + 253 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 4 \,{\left (133 \, A + 148 \, B\right )} a^{3} \cos \left (f x + e\right ) + 4 \,{\left (161 \, A + 191 \, B\right )} a^{3} -{\left (15 \, B a^{3} \cos \left (f x + e\right )^{3} + 3 \,{\left (7 \, A + 27 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 4 \,{\left (28 \, A + 43 \, B\right )} a^{3} \cos \left (f x + e\right ) - 4 \,{\left (161 \, A + 191 \, B\right )} a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt{-c \sin \left (f x + e\right ) + c}\right )}}{105 \,{\left (c f \cos \left (f x + e\right ) - c f \sin \left (f x + e\right ) + c 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 [B] time = 2.04635, size = 938, normalized size = 4.69 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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