Optimal. Leaf size=218 \[ \frac{a^3 c^3 (A+B) \cos ^7(e+f x)}{2 f (c-c \sin (e+f x))^{9/2}}-\frac{2 \sqrt{2} a^3 (5 A+9 B) \tanh ^{-1}\left (\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right )}{c^{3/2} f}+\frac{a^3 c (5 A+9 B) \cos ^5(e+f x)}{10 f (c-c \sin (e+f x))^{5/2}}+\frac{a^3 (5 A+9 B) \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}+\frac{2 a^3 (5 A+9 B) \cos (e+f x)}{c f \sqrt{c-c \sin (e+f x)}} \]
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Rubi [A] time = 0.546137, antiderivative size = 218, 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, 2859, 2679, 2649, 206} \[ \frac{a^3 c^3 (A+B) \cos ^7(e+f x)}{2 f (c-c \sin (e+f x))^{9/2}}-\frac{2 \sqrt{2} a^3 (5 A+9 B) \tanh ^{-1}\left (\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right )}{c^{3/2} f}+\frac{a^3 c (5 A+9 B) \cos ^5(e+f x)}{10 f (c-c \sin (e+f x))^{5/2}}+\frac{a^3 (5 A+9 B) \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}+\frac{2 a^3 (5 A+9 B) \cos (e+f x)}{c f \sqrt{c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2967
Rule 2859
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))}{(c-c \sin (e+f x))^{3/2}} \, 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))^{9/2}} \, dx\\ &=\frac{a^3 (A+B) c^3 \cos ^7(e+f x)}{2 f (c-c \sin (e+f x))^{9/2}}-\frac{1}{4} \left (a^3 (5 A+9 B) c^2\right ) \int \frac{\cos ^6(e+f x)}{(c-c \sin (e+f x))^{7/2}} \, dx\\ &=\frac{a^3 (A+B) c^3 \cos ^7(e+f x)}{2 f (c-c \sin (e+f x))^{9/2}}+\frac{a^3 (5 A+9 B) c \cos ^5(e+f x)}{10 f (c-c \sin (e+f x))^{5/2}}-\frac{1}{2} \left (a^3 (5 A+9 B) c\right ) \int \frac{\cos ^4(e+f x)}{(c-c \sin (e+f x))^{5/2}} \, dx\\ &=\frac{a^3 (A+B) c^3 \cos ^7(e+f x)}{2 f (c-c \sin (e+f x))^{9/2}}+\frac{a^3 (5 A+9 B) c \cos ^5(e+f x)}{10 f (c-c \sin (e+f x))^{5/2}}+\frac{a^3 (5 A+9 B) \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}-\left (a^3 (5 A+9 B)\right ) \int \frac{\cos ^2(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx\\ &=\frac{a^3 (A+B) c^3 \cos ^7(e+f x)}{2 f (c-c \sin (e+f x))^{9/2}}+\frac{a^3 (5 A+9 B) c \cos ^5(e+f x)}{10 f (c-c \sin (e+f x))^{5/2}}+\frac{a^3 (5 A+9 B) \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}+\frac{2 a^3 (5 A+9 B) \cos (e+f x)}{c f \sqrt{c-c \sin (e+f x)}}-\frac{\left (2 a^3 (5 A+9 B)\right ) \int \frac{1}{\sqrt{c-c \sin (e+f x)}} \, dx}{c}\\ &=\frac{a^3 (A+B) c^3 \cos ^7(e+f x)}{2 f (c-c \sin (e+f x))^{9/2}}+\frac{a^3 (5 A+9 B) c \cos ^5(e+f x)}{10 f (c-c \sin (e+f x))^{5/2}}+\frac{a^3 (5 A+9 B) \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}+\frac{2 a^3 (5 A+9 B) \cos (e+f x)}{c f \sqrt{c-c \sin (e+f x)}}+\frac{\left (4 a^3 (5 A+9 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 )}{c f}\\ &=-\frac{2 \sqrt{2} a^3 (5 A+9 B) \tanh ^{-1}\left (\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right )}{c^{3/2} f}+\frac{a^3 (A+B) c^3 \cos ^7(e+f x)}{2 f (c-c \sin (e+f x))^{9/2}}+\frac{a^3 (5 A+9 B) c \cos ^5(e+f x)}{10 f (c-c \sin (e+f x))^{5/2}}+\frac{a^3 (5 A+9 B) \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}+\frac{2 a^3 (5 A+9 B) \cos (e+f x)}{c f \sqrt{c-c \sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 1.74262, size = 444, normalized size = 2.04 \[ \frac{a^3 (\sin (e+f x)+1)^3 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (240 (A+B) \sin \left (\frac{1}{2} (e+f x)\right )+30 (9 A+20 B) \cos \left (\frac{1}{2} (e+f x)\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^2-5 (2 A+9 B) \cos \left (\frac{3}{2} (e+f x)\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^2+30 (9 A+20 B) \sin \left (\frac{1}{2} (e+f x)\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^2+5 (2 A+9 B) \sin \left (\frac{3}{2} (e+f x)\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^2+120 (A+B) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )+(120+120 i) \sqrt [4]{-1} (5 A+9 B) \tan ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt [4]{-1} \left (\tan \left (\frac{1}{4} (e+f x)\right )+1\right )\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^2-3 B \cos \left (\frac{5}{2} (e+f x)\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^2-3 B \sin \left (\frac{5}{2} (e+f x)\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^2\right )}{30 f (c-c \sin (e+f x))^{3/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.217, size = 354, normalized size = 1.6 \begin{align*}{\frac{2\,{a}^{3}}{15\,f\cos \left ( fx+e \right ) } \left ( \sin \left ( fx+e \right ) \left ( -60\,A\sqrt{c+c\sin \left ( fx+e \right ) }{c}^{5/2}-5\,A \left ( c+c\sin \left ( fx+e \right ) \right ) ^{3/2}{c}^{3/2}-120\,B\sqrt{c+c\sin \left ( fx+e \right ) }{c}^{5/2}-15\,B \left ( c+c\sin \left ( fx+e \right ) \right ) ^{3/2}{c}^{3/2}-3\,B \left ( c+c\sin \left ( fx+e \right ) \right ) ^{5/2}\sqrt{c}+75\,A\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c+c\sin \left ( fx+e \right ) }\sqrt{2}}{\sqrt{c}}} \right ){c}^{3}+135\,B\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c+c\sin \left ( fx+e \right ) }\sqrt{2}}{\sqrt{c}}} \right ){c}^{3} \right ) +90\,A\sqrt{c+c\sin \left ( fx+e \right ) }{c}^{5/2}+5\,A \left ( c+c\sin \left ( fx+e \right ) \right ) ^{3/2}{c}^{3/2}+150\,B\sqrt{c+c\sin \left ( fx+e \right ) }{c}^{5/2}+15\,B \left ( c+c\sin \left ( fx+e \right ) \right ) ^{3/2}{c}^{3/2}+3\,B \left ( c+c\sin \left ( fx+e \right ) \right ) ^{5/2}\sqrt{c}-75\,A\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c+c\sin \left ( fx+e \right ) }\sqrt{2}}{\sqrt{c}}} \right ){c}^{3}-135\,B\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c+c\sin \left ( fx+e \right ) }\sqrt{2}}{\sqrt{c}}} \right ){c}^{3} \right ) \sqrt{c \left ( 1+\sin \left ( fx+e \right ) \right ) }{c}^{-{\frac{9}{2}}}{\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}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.53495, size = 1099, normalized size = 5.04 \begin{align*} \frac{\frac{15 \, \sqrt{2}{\left ({\left (5 \, A + 9 \, B\right )} a^{3} c \cos \left (f x + e\right )^{2} -{\left (5 \, A + 9 \, B\right )} a^{3} c \cos \left (f x + e\right ) - 2 \,{\left (5 \, A + 9 \, B\right )} a^{3} c +{\left ({\left (5 \, A + 9 \, B\right )} a^{3} c \cos \left (f x + e\right ) + 2 \,{\left (5 \, A + 9 \, B\right )} a^{3} c\right )} \sin \left (f x + e\right )\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}} + 2 \,{\left (3 \, B a^{3} \cos \left (f x + e\right )^{4} -{\left (5 \, A + 18 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} -{\left (65 \, A + 141 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 30 \,{\left (3 \, A + 5 \, B\right )} a^{3} \cos \left (f x + e\right ) - 30 \,{\left (A + B\right )} a^{3} -{\left (3 \, B a^{3} \cos \left (f x + e\right )^{3} +{\left (5 \, A + 21 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 60 \,{\left (A + 2 \, B\right )} a^{3} \cos \left (f x + e\right ) + 30 \,{\left (A + B\right )} a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt{-c \sin \left (f x + e\right ) + c}}{15 \,{\left (c^{2} f \cos \left (f x + e\right )^{2} - c^{2} f \cos \left (f x + e\right ) - 2 \, c^{2} f +{\left (c^{2} f \cos \left (f x + e\right ) + 2 \, c^{2} f\right )} \sin \left (f x + e\right )\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*} \mathit{sage}_{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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