Optimal. Leaf size=189 \[ -\frac{4 a^2 (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{d^2 f \sqrt{c+d \sin (e+f x)}}+\frac{4 a^2 c \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{d^2 f (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 a^2 (c-d) \cos (e+f x)}{d f (c+d) \sqrt{c+d \sin (e+f x)}} \]
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Rubi [A] time = 0.239003, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2762, 2752, 2663, 2661, 2655, 2653} \[ -\frac{4 a^2 (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{d^2 f \sqrt{c+d \sin (e+f x)}}+\frac{4 a^2 c \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{d^2 f (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 a^2 (c-d) \cos (e+f x)}{d f (c+d) \sqrt{c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2762
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{(a+a \sin (e+f x))^2}{(c+d \sin (e+f x))^{3/2}} \, dx &=\frac{2 a^2 (c-d) \cos (e+f x)}{d (c+d) f \sqrt{c+d \sin (e+f x)}}-\frac{(2 a) \int \frac{-a d-a c \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{d (c+d)}\\ &=\frac{2 a^2 (c-d) \cos (e+f x)}{d (c+d) f \sqrt{c+d \sin (e+f x)}}-\frac{\left (2 a^2 (c-d)\right ) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{d^2}+\frac{\left (2 a^2 c\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{d^2 (c+d)}\\ &=\frac{2 a^2 (c-d) \cos (e+f x)}{d (c+d) f \sqrt{c+d \sin (e+f x)}}+\frac{\left (2 a^2 c \sqrt{c+d \sin (e+f x)}\right ) \int \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}} \, dx}{d^2 (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\left (2 a^2 (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}\right ) \int \frac{1}{\sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}}} \, dx}{d^2 \sqrt{c+d \sin (e+f x)}}\\ &=\frac{2 a^2 (c-d) \cos (e+f x)}{d (c+d) f \sqrt{c+d \sin (e+f x)}}+\frac{4 a^2 c E\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{c+d \sin (e+f x)}}{d^2 (c+d) f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{4 a^2 (c-d) F\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{d^2 f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.92012, size = 175, normalized size = 0.93 \[ -\frac{2 a^2 (\sin (e+f x)+1)^2 \left (2 c (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} E\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )-(c-d) \left (2 (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )+d \cos (e+f x)\right )\right )}{d^2 f (c+d) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^4 \sqrt{c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.826, size = 463, normalized size = 2.5 \begin{align*} -2\,{\frac{{a}^{2}}{{d}^{3} \left ( c+d \right ) \cos \left ( fx+e \right ) \sqrt{c+d\sin \left ( fx+e \right ) }f} \left ( 2\,\sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}}\sqrt{-{\frac{ \left ( -1+\sin \left ( fx+e \right ) \right ) d}{c+d}}}\sqrt{-{\frac{d \left ( 1+\sin \left ( fx+e \right ) \right ) }{c-d}}}{\it EllipticE} \left ( \sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}},\sqrt{{\frac{c-d}{c+d}}} \right ){c}^{3}-2\,\sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}}\sqrt{-{\frac{ \left ( -1+\sin \left ( fx+e \right ) \right ) d}{c+d}}}\sqrt{-{\frac{d \left ( 1+\sin \left ( fx+e \right ) \right ) }{c-d}}}{\it EllipticE} \left ( \sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}},\sqrt{{\frac{c-d}{c+d}}} \right ) c{d}^{2}-2\,\sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}}\sqrt{-{\frac{ \left ( -1+\sin \left ( fx+e \right ) \right ) d}{c+d}}}\sqrt{-{\frac{d \left ( 1+\sin \left ( fx+e \right ) \right ) }{c-d}}}{\it EllipticF} \left ( \sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}},\sqrt{{\frac{c-d}{c+d}}} \right ){c}^{2}d+2\,\sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}}\sqrt{-{\frac{ \left ( -1+\sin \left ( fx+e \right ) \right ) d}{c+d}}}\sqrt{-{\frac{d \left ( 1+\sin \left ( fx+e \right ) \right ) }{c-d}}}{\it EllipticF} \left ( \sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}},\sqrt{{\frac{c-d}{c+d}}} \right ){d}^{3}+c{d}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{2}-{d}^{3} \left ( \sin \left ( fx+e \right ) \right ) ^{2}-c{d}^{2}+{d}^{3} \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 (a \sin \left (f x + e\right ) + a\right )}^{2}}{{\left (d \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2} \sin \left (f x + e\right ) - 2 \, a^{2}\right )} \sqrt{d \sin \left (f x + e\right ) + c}}{d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2}}, x\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 \frac{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}}{{\left (d \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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