Optimal. Leaf size=419 \[ -\frac{4 a^3 \left (21 c^2 d+4 c^3+62 c d^2-147 d^3\right ) \cos (e+f x)}{105 d^2 f (c-d) (c+d)^4 \sqrt{c+d \sin (e+f x)}}-\frac{4 a^3 \left (4 c^2+21 c d+65 d^2\right ) \cos (e+f x)}{105 d^2 f (c+d)^3 (c+d \sin (e+f x))^{3/2}}+\frac{4 a^3 \left (4 c^2+21 c d+65 d^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{105 d^3 f (c+d)^3 \sqrt{c+d \sin (e+f x)}}-\frac{4 a^3 \left (21 c^2 d+4 c^3+62 c d^2-147 d^3\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{105 d^3 f (c-d) (c+d)^4 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 a^3 (c-d) (c+4 d) \cos (e+f x)}{35 d^2 f (c+d)^2 (c+d \sin (e+f x))^{5/2}}+\frac{2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{7 d f (c+d) (c+d \sin (e+f x))^{7/2}} \]
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Rubi [A] time = 0.928286, antiderivative size = 419, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2762, 2968, 3021, 2754, 2752, 2663, 2661, 2655, 2653} \[ -\frac{4 a^3 \left (21 c^2 d+4 c^3+62 c d^2-147 d^3\right ) \cos (e+f x)}{105 d^2 f (c-d) (c+d)^4 \sqrt{c+d \sin (e+f x)}}-\frac{4 a^3 \left (4 c^2+21 c d+65 d^2\right ) \cos (e+f x)}{105 d^2 f (c+d)^3 (c+d \sin (e+f x))^{3/2}}+\frac{4 a^3 \left (4 c^2+21 c d+65 d^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{105 d^3 f (c+d)^3 \sqrt{c+d \sin (e+f x)}}-\frac{4 a^3 \left (21 c^2 d+4 c^3+62 c d^2-147 d^3\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{105 d^3 f (c-d) (c+d)^4 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 a^3 (c-d) (c+4 d) \cos (e+f x)}{35 d^2 f (c+d)^2 (c+d \sin (e+f x))^{5/2}}+\frac{2 (c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{7 d f (c+d) (c+d \sin (e+f x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 2762
Rule 2968
Rule 3021
Rule 2754
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^{9/2}} \, dx &=\frac{2 (c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{7 d (c+d) f (c+d \sin (e+f x))^{7/2}}-\frac{(2 a) \int \frac{(a+a \sin (e+f x)) (a (c-8 d)-a (2 c+5 d) \sin (e+f x))}{(c+d \sin (e+f x))^{7/2}} \, dx}{7 d (c+d)}\\ &=\frac{2 (c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{7 d (c+d) f (c+d \sin (e+f x))^{7/2}}-\frac{(2 a) \int \frac{a^2 (c-8 d)+\left (a^2 (c-8 d)-a^2 (2 c+5 d)\right ) \sin (e+f x)-a^2 (2 c+5 d) \sin ^2(e+f x)}{(c+d \sin (e+f x))^{7/2}} \, dx}{7 d (c+d)}\\ &=\frac{2 (c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{7 d (c+d) f (c+d \sin (e+f x))^{7/2}}+\frac{8 a^3 (c-d) (c+4 d) \cos (e+f x)}{35 d^2 (c+d)^2 f (c+d \sin (e+f x))^{5/2}}+\frac{(4 a) \int \frac{\frac{5}{2} a^2 (c-d) d (c+13 d)+\frac{1}{2} a^2 (c-d) \left (4 c^2+17 c d+49 d^2\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx}{35 (c-d) d^2 (c+d)^2}\\ &=\frac{2 (c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{7 d (c+d) f (c+d \sin (e+f x))^{7/2}}+\frac{8 a^3 (c-d) (c+4 d) \cos (e+f x)}{35 d^2 (c+d)^2 f (c+d \sin (e+f x))^{5/2}}-\frac{4 a^3 \left (4 c^2+21 c d+65 d^2\right ) \cos (e+f x)}{105 d^2 (c+d)^3 f (c+d \sin (e+f x))^{3/2}}-\frac{(8 a) \int \frac{-\frac{3}{4} a^2 (c-d)^2 d (c+49 d)-\frac{1}{4} a^2 (c-d)^2 \left (4 c^2+21 c d+65 d^2\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx}{105 (c-d)^2 d^2 (c+d)^3}\\ &=\frac{2 (c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{7 d (c+d) f (c+d \sin (e+f x))^{7/2}}+\frac{8 a^3 (c-d) (c+4 d) \cos (e+f x)}{35 d^2 (c+d)^2 f (c+d \sin (e+f x))^{5/2}}-\frac{4 a^3 \left (4 c^2+21 c d+65 d^2\right ) \cos (e+f x)}{105 d^2 (c+d)^3 f (c+d \sin (e+f x))^{3/2}}-\frac{4 a^3 \left (4 c^3+21 c^2 d+62 c d^2-147 d^3\right ) \cos (e+f x)}{105 (c-d) d^2 (c+d)^4 f \sqrt{c+d \sin (e+f x)}}+\frac{(16 a) \int \frac{-\frac{1}{8} a^2 (c-d)^2 d \left (c^2-126 c d+65 d^2\right )-\frac{1}{8} a^2 (c-d)^2 \left (4 c^3+21 c^2 d+62 c d^2-147 d^3\right ) \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{105 (c-d)^3 d^2 (c+d)^4}\\ &=\frac{2 (c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{7 d (c+d) f (c+d \sin (e+f x))^{7/2}}+\frac{8 a^3 (c-d) (c+4 d) \cos (e+f x)}{35 d^2 (c+d)^2 f (c+d \sin (e+f x))^{5/2}}-\frac{4 a^3 \left (4 c^2+21 c d+65 d^2\right ) \cos (e+f x)}{105 d^2 (c+d)^3 f (c+d \sin (e+f x))^{3/2}}-\frac{4 a^3 \left (4 c^3+21 c^2 d+62 c d^2-147 d^3\right ) \cos (e+f x)}{105 (c-d) d^2 (c+d)^4 f \sqrt{c+d \sin (e+f x)}}+\frac{\left (2 a^3 \left (4 c^2+21 c d+65 d^2\right )\right ) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{105 d^3 (c+d)^3}-\frac{\left (2 a^3 \left (4 c^3+21 c^2 d+62 c d^2-147 d^3\right )\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{105 (c-d) d^3 (c+d)^4}\\ &=\frac{2 (c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{7 d (c+d) f (c+d \sin (e+f x))^{7/2}}+\frac{8 a^3 (c-d) (c+4 d) \cos (e+f x)}{35 d^2 (c+d)^2 f (c+d \sin (e+f x))^{5/2}}-\frac{4 a^3 \left (4 c^2+21 c d+65 d^2\right ) \cos (e+f x)}{105 d^2 (c+d)^3 f (c+d \sin (e+f x))^{3/2}}-\frac{4 a^3 \left (4 c^3+21 c^2 d+62 c d^2-147 d^3\right ) \cos (e+f x)}{105 (c-d) d^2 (c+d)^4 f \sqrt{c+d \sin (e+f x)}}-\frac{\left (2 a^3 \left (4 c^3+21 c^2 d+62 c d^2-147 d^3\right ) \sqrt{c+d \sin (e+f x)}\right ) \int \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}} \, dx}{105 (c-d) d^3 (c+d)^4 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left (2 a^3 \left (4 c^2+21 c d+65 d^2\right ) \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}{105 d^3 (c+d)^3 \sqrt{c+d \sin (e+f x)}}\\ &=\frac{2 (c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{7 d (c+d) f (c+d \sin (e+f x))^{7/2}}+\frac{8 a^3 (c-d) (c+4 d) \cos (e+f x)}{35 d^2 (c+d)^2 f (c+d \sin (e+f x))^{5/2}}-\frac{4 a^3 \left (4 c^2+21 c d+65 d^2\right ) \cos (e+f x)}{105 d^2 (c+d)^3 f (c+d \sin (e+f x))^{3/2}}-\frac{4 a^3 \left (4 c^3+21 c^2 d+62 c d^2-147 d^3\right ) \cos (e+f x)}{105 (c-d) d^2 (c+d)^4 f \sqrt{c+d \sin (e+f x)}}-\frac{4 a^3 \left (4 c^3+21 c^2 d+62 c d^2-147 d^3\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)}}{105 (c-d) d^3 (c+d)^4 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{4 a^3 \left (4 c^2+21 c d+65 d^2\right ) 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}}}{105 d^3 (c+d)^3 f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 3.56264, size = 351, normalized size = 0.84 \[ -\frac{2 a^3 (\sin (e+f x)+1)^3 \left (d \cos (e+f x) \left (2 (c-d) \left (4 c^2+21 c d+65 d^2\right ) (c+d) (c+d \sin (e+f x))^2+2 \left (21 c^2 d+4 c^3+62 c d^2-147 d^3\right ) (c+d \sin (e+f x))^3-9 (c-d)^2 (3 c+7 d) (c+d)^2 (c+d \sin (e+f x))+15 (c-d)^3 (c+d)^3\right )-2 (c+d \sin (e+f x))^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left (d^2 \left (c^2-126 c d+65 d^2\right ) F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )+\left (21 c^2 d+4 c^3+62 c d^2-147 d^3\right ) \left ((c+d) E\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )-c F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )\right )\right )\right )}{105 d^3 f (c-d) (c+d)^4 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^6 (c+d \sin (e+f x))^{7/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 8.848, size = 2079, normalized size = 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 )}^{3}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{9}{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 (3 \, a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3} +{\left (a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt{d \sin \left (f x + e\right ) + c}}{5 \, c d^{4} \cos \left (f x + e\right )^{4} + c^{5} + 10 \, c^{3} d^{2} + 5 \, c d^{4} - 10 \,{\left (c^{3} d^{2} + c d^{4}\right )} \cos \left (f x + e\right )^{2} +{\left (d^{5} \cos \left (f x + e\right )^{4} + 5 \, c^{4} d + 10 \, c^{2} d^{3} + d^{5} - 2 \,{\left (5 \, c^{2} d^{3} + d^{5}\right )} \cos \left (f x + e\right )^{2}\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 )}^{3}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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