Optimal. Leaf size=320 \[ -\frac{4 a^2 \left (c^2+5 c d-12 d^2\right ) \cos (e+f x)}{15 d f (c-d) (c+d)^3 \sqrt{c+d \sin (e+f x)}}-\frac{4 a^2 \left (c^2+5 c d-12 d^2\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{15 d^2 f (c-d) (c+d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{4 a^2 (c+5 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{15 d^2 f (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{4 a^2 (c+5 d) \cos (e+f x)}{15 d f (c+d)^2 (c+d \sin (e+f x))^{3/2}}+\frac{2 a^2 (c-d) \cos (e+f x)}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}} \]
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Rubi [A] time = 0.577853, antiderivative size = 320, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {2762, 2754, 2752, 2663, 2661, 2655, 2653} \[ -\frac{4 a^2 \left (c^2+5 c d-12 d^2\right ) \cos (e+f x)}{15 d f (c-d) (c+d)^3 \sqrt{c+d \sin (e+f x)}}-\frac{4 a^2 \left (c^2+5 c d-12 d^2\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{15 d^2 f (c-d) (c+d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{4 a^2 (c+5 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{15 d^2 f (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{4 a^2 (c+5 d) \cos (e+f x)}{15 d f (c+d)^2 (c+d \sin (e+f x))^{3/2}}+\frac{2 a^2 (c-d) \cos (e+f x)}{5 d f (c+d) (c+d \sin (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2762
Rule 2754
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))^{7/2}} \, dx &=\frac{2 a^2 (c-d) \cos (e+f x)}{5 d (c+d) f (c+d \sin (e+f x))^{5/2}}-\frac{(2 a) \int \frac{-5 a d-a (c+4 d) \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx}{5 d (c+d)}\\ &=\frac{2 a^2 (c-d) \cos (e+f x)}{5 d (c+d) f (c+d \sin (e+f x))^{5/2}}-\frac{4 a^2 (c+5 d) \cos (e+f x)}{15 d (c+d)^2 f (c+d \sin (e+f x))^{3/2}}+\frac{(4 a) \int \frac{6 a (c-d) d+\frac{1}{2} a (c-d) (c+5 d) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx}{15 (c-d) d (c+d)^2}\\ &=\frac{2 a^2 (c-d) \cos (e+f x)}{5 d (c+d) f (c+d \sin (e+f x))^{5/2}}-\frac{4 a^2 (c+5 d) \cos (e+f x)}{15 d (c+d)^2 f (c+d \sin (e+f x))^{3/2}}-\frac{4 a^2 \left (c^2+5 c d-12 d^2\right ) \cos (e+f x)}{15 (c-d) d (c+d)^3 f \sqrt{c+d \sin (e+f x)}}-\frac{(8 a) \int \frac{-\frac{1}{4} a (11 c-5 d) (c-d) d+\frac{1}{4} a (c-d) \left (c^2+5 c d-12 d^2\right ) \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{15 (c-d)^2 d (c+d)^3}\\ &=\frac{2 a^2 (c-d) \cos (e+f x)}{5 d (c+d) f (c+d \sin (e+f x))^{5/2}}-\frac{4 a^2 (c+5 d) \cos (e+f x)}{15 d (c+d)^2 f (c+d \sin (e+f x))^{3/2}}-\frac{4 a^2 \left (c^2+5 c d-12 d^2\right ) \cos (e+f x)}{15 (c-d) d (c+d)^3 f \sqrt{c+d \sin (e+f x)}}+\frac{\left (2 a^2 (c+5 d)\right ) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{15 d^2 (c+d)^2}-\frac{\left (2 a^2 \left (c^2+5 c d-12 d^2\right )\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{15 (c-d) d^2 (c+d)^3}\\ &=\frac{2 a^2 (c-d) \cos (e+f x)}{5 d (c+d) f (c+d \sin (e+f x))^{5/2}}-\frac{4 a^2 (c+5 d) \cos (e+f x)}{15 d (c+d)^2 f (c+d \sin (e+f x))^{3/2}}-\frac{4 a^2 \left (c^2+5 c d-12 d^2\right ) \cos (e+f x)}{15 (c-d) d (c+d)^3 f \sqrt{c+d \sin (e+f x)}}-\frac{\left (2 a^2 \left (c^2+5 c d-12 d^2\right ) \sqrt{c+d \sin (e+f x)}\right ) \int \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}} \, dx}{15 (c-d) d^2 (c+d)^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left (2 a^2 (c+5 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}{15 d^2 (c+d)^2 \sqrt{c+d \sin (e+f x)}}\\ &=\frac{2 a^2 (c-d) \cos (e+f x)}{5 d (c+d) f (c+d \sin (e+f x))^{5/2}}-\frac{4 a^2 (c+5 d) \cos (e+f x)}{15 d (c+d)^2 f (c+d \sin (e+f x))^{3/2}}-\frac{4 a^2 \left (c^2+5 c d-12 d^2\right ) \cos (e+f x)}{15 (c-d) d (c+d)^3 f \sqrt{c+d \sin (e+f x)}}-\frac{4 a^2 \left (c^2+5 c d-12 d^2\right ) 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)}}{15 (c-d) d^2 (c+d)^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{4 a^2 (c+5 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}}}{15 d^2 (c+d)^2 f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 2.02668, size = 283, normalized size = 0.88 \[ \frac{2 a^2 (\sin (e+f x)+1)^2 \left (d \cos (e+f x) \left (-2 \left (c^2+5 c d-12 d^2\right ) (c+d \sin (e+f x))^2-2 (c-d) (c+5 d) (c+d) (c+d \sin (e+f x))+3 (c-d)^2 (c+d)^2\right )-2 (c+d \sin (e+f x))^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left (d^2 (11 c-5 d) F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )-\left (c^2+5 c d-12 d^2\right ) \left ((c+d) E\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )-c F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )\right )\right )\right )}{15 d^2 f (c-d) (c+d)^3 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^4 (c+d \sin (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 5.667, size = 1436, normalized size = 4.5 \begin{align*} \text{result too large to display} \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{7}{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^{4} \cos \left (f x + e\right )^{4} + c^{4} + 6 \, c^{2} d^{2} + d^{4} - 2 \,{\left (3 \, c^{2} d^{2} + d^{4}\right )} \cos \left (f x + e\right )^{2} - 4 \,{\left (c d^{3} \cos \left (f x + e\right )^{2} - c^{3} d - c d^{3}\right )} \sin \left (f x + e\right )}, 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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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