Optimal. Leaf size=176 \[ \frac{a^2 c^2 (A+B) \cos ^5(e+f x)}{2 f (c-c \sin (e+f x))^{7/2}}-\frac{\sqrt{2} a^2 (3 A+7 B) \tanh ^{-1}\left (\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right )}{c^{3/2} f}+\frac{a^2 (3 A+7 B) \cos ^3(e+f x)}{6 f (c-c \sin (e+f x))^{3/2}}+\frac{a^2 (3 A+7 B) \cos (e+f x)}{c f \sqrt{c-c \sin (e+f x)}} \]
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Rubi [A] time = 0.480917, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{a^2 c^2 (A+B) \cos ^5(e+f x)}{2 f (c-c \sin (e+f x))^{7/2}}-\frac{\sqrt{2} a^2 (3 A+7 B) \tanh ^{-1}\left (\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right )}{c^{3/2} f}+\frac{a^2 (3 A+7 B) \cos ^3(e+f x)}{6 f (c-c \sin (e+f x))^{3/2}}+\frac{a^2 (3 A+7 B) \cos (e+f x)}{c f \sqrt{c-c \sin (e+f x)}} \]
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))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx &=\left (a^2 c^2\right ) \int \frac{\cos ^4(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx\\ &=\frac{a^2 (A+B) c^2 \cos ^5(e+f x)}{2 f (c-c \sin (e+f x))^{7/2}}-\frac{1}{4} \left (a^2 (3 A+7 B) c\right ) \int \frac{\cos ^4(e+f x)}{(c-c \sin (e+f x))^{5/2}} \, dx\\ &=\frac{a^2 (A+B) c^2 \cos ^5(e+f x)}{2 f (c-c \sin (e+f x))^{7/2}}+\frac{a^2 (3 A+7 B) \cos ^3(e+f x)}{6 f (c-c \sin (e+f x))^{3/2}}-\frac{1}{2} \left (a^2 (3 A+7 B)\right ) \int \frac{\cos ^2(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx\\ &=\frac{a^2 (A+B) c^2 \cos ^5(e+f x)}{2 f (c-c \sin (e+f x))^{7/2}}+\frac{a^2 (3 A+7 B) \cos ^3(e+f x)}{6 f (c-c \sin (e+f x))^{3/2}}+\frac{a^2 (3 A+7 B) \cos (e+f x)}{c f \sqrt{c-c \sin (e+f x)}}-\frac{\left (a^2 (3 A+7 B)\right ) \int \frac{1}{\sqrt{c-c \sin (e+f x)}} \, dx}{c}\\ &=\frac{a^2 (A+B) c^2 \cos ^5(e+f x)}{2 f (c-c \sin (e+f x))^{7/2}}+\frac{a^2 (3 A+7 B) \cos ^3(e+f x)}{6 f (c-c \sin (e+f x))^{3/2}}+\frac{a^2 (3 A+7 B) \cos (e+f x)}{c f \sqrt{c-c \sin (e+f x)}}+\frac{\left (2 a^2 (3 A+7 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{\sqrt{2} a^2 (3 A+7 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^2 (A+B) c^2 \cos ^5(e+f x)}{2 f (c-c \sin (e+f x))^{7/2}}+\frac{a^2 (3 A+7 B) \cos ^3(e+f x)}{6 f (c-c \sin (e+f x))^{3/2}}+\frac{a^2 (3 A+7 B) \cos (e+f x)}{c f \sqrt{c-c \sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 0.916248, size = 355, normalized size = 2.02 \[ \frac{a^2 (\sin (e+f x)+1)^2 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (12 (A+B) \sin \left (\frac{1}{2} (e+f x)\right )+3 (2 A+7 B) \cos \left (\frac{1}{2} (e+f x)\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^2+3 (2 A+7 B) \sin \left (\frac{1}{2} (e+f x)\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^2+6 (A+B) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )+(6+6 i) \sqrt [4]{-1} (3 A+7 B) \tan ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt [4]{-1} \left (\tan \left (\frac{1}{4} (e+f x)\right )+1\right )\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^2-B \cos \left (\frac{3}{2} (e+f x)\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^2+B \sin \left (\frac{3}{2} (e+f x)\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^2\right )}{3 f (c-c \sin (e+f x))^{3/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.065, size = 282, normalized size = 1.6 \begin{align*}{\frac{{a}^{2}}{3\,f\cos \left ( fx+e \right ) } \left ( \sin \left ( fx+e \right ) \left ( 9\,A\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c+c\sin \left ( fx+e \right ) }\sqrt{2}}{\sqrt{c}}} \right ){c}^{2}-6\,A\sqrt{c+c\sin \left ( fx+e \right ) }{c}^{3/2}+21\,B\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c+c\sin \left ( fx+e \right ) }\sqrt{2}}{\sqrt{c}}} \right ){c}^{2}-18\,B\sqrt{c+c\sin \left ( fx+e \right ) }{c}^{3/2}-2\,B \left ( c+c\sin \left ( fx+e \right ) \right ) ^{3/2}\sqrt{c} \right ) -9\,A\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c+c\sin \left ( fx+e \right ) }\sqrt{2}}{\sqrt{c}}} \right ){c}^{2}+12\,A\sqrt{c+c\sin \left ( fx+e \right ) }{c}^{3/2}-21\,B\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c+c\sin \left ( fx+e \right ) }\sqrt{2}}{\sqrt{c}}} \right ){c}^{2}+24\,B\sqrt{c+c\sin \left ( fx+e \right ) }{c}^{3/2}+2\,B \left ( c+c\sin \left ( fx+e \right ) \right ) ^{3/2}\sqrt{c} \right ) \sqrt{c \left ( 1+\sin \left ( fx+e \right ) \right ) }{c}^{-{\frac{7}{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 )}^{2}}{{\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 = 1.52835, size = 988, normalized size = 5.61 \begin{align*} \frac{\frac{3 \, \sqrt{2}{\left ({\left (3 \, A + 7 \, B\right )} a^{2} c \cos \left (f x + e\right )^{2} -{\left (3 \, A + 7 \, B\right )} a^{2} c \cos \left (f x + e\right ) - 2 \,{\left (3 \, A + 7 \, B\right )} a^{2} c +{\left ({\left (3 \, A + 7 \, B\right )} a^{2} c \cos \left (f x + e\right ) + 2 \,{\left (3 \, A + 7 \, B\right )} a^{2} 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}} - 4 \,{\left (B a^{2} \cos \left (f x + e\right )^{3} +{\left (3 \, A + 10 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} + 6 \,{\left (A + 2 \, B\right )} a^{2} \cos \left (f x + e\right ) + 3 \,{\left (A + B\right )} a^{2} +{\left (B a^{2} \cos \left (f x + e\right )^{2} - 3 \,{\left (A + 3 \, B\right )} a^{2} \cos \left (f x + e\right ) + 3 \,{\left (A + B\right )} a^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt{-c \sin \left (f x + e\right ) + c}}{6 \,{\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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