Optimal. Leaf size=361 \[ \frac{2 b \left (-3 a^2 d^2+6 a b c d+b^2 \left (-\left (4 c^2-d^2\right )\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 d^2 f \left (c^2-d^2\right )}-\frac{2 b \left (-9 a^2 d^2+18 a b c d+b^2 \left (-\left (8 c^2+d^2\right )\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 d^3 f \sqrt{c+d \sin (e+f x)}}-\frac{2 \left (9 a^2 b c d^2-3 a^3 d^3-9 a b^2 d \left (2 c^2-d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 d^3 f \left (c^2-d^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d f \left (c^2-d^2\right ) \sqrt{c+d \sin (e+f x)}} \]
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Rubi [A] time = 0.621814, antiderivative size = 361, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {2792, 3023, 2752, 2663, 2661, 2655, 2653} \[ \frac{2 b \left (-3 a^2 d^2+6 a b c d+b^2 \left (-\left (4 c^2-d^2\right )\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 d^2 f \left (c^2-d^2\right )}-\frac{2 b \left (-9 a^2 d^2+18 a b c d+b^2 \left (-\left (8 c^2+d^2\right )\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 d^3 f \sqrt{c+d \sin (e+f x)}}-\frac{2 \left (9 a^2 b c d^2-3 a^3 d^3-9 a b^2 d \left (2 c^2-d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 d^3 f \left (c^2-d^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d f \left (c^2-d^2\right ) \sqrt{c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2792
Rule 3023
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx &=\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d \left (c^2-d^2\right ) f \sqrt{c+d \sin (e+f x)}}-\frac{2 \int \frac{\frac{1}{2} \left (2 b (b c-a d)^2-a d \left (\left (a^2+b^2\right ) c-2 a b d\right )\right )+\frac{1}{2} \left (a^2 b c d-b^3 c d-a^3 d^2-a b^2 \left (2 c^2-3 d^2\right )\right ) \sin (e+f x)+\frac{1}{2} b \left (6 a b c d-3 a^2 d^2-b^2 \left (4 c^2-d^2\right )\right ) \sin ^2(e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{d \left (c^2-d^2\right )}\\ &=\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d \left (c^2-d^2\right ) f \sqrt{c+d \sin (e+f x)}}+\frac{2 b \left (6 a b c d-3 a^2 d^2-b^2 \left (4 c^2-d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 d^2 \left (c^2-d^2\right ) f}-\frac{4 \int \frac{-\frac{1}{4} d \left (3 a^3 c d+9 a b^2 c d-9 a^2 b d^2-b^3 \left (2 c^2+d^2\right )\right )+\frac{1}{4} \left (9 a^2 b c d^2-3 a^3 d^3-9 a b^2 d \left (2 c^2-d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{3 d^2 \left (c^2-d^2\right )}\\ &=\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d \left (c^2-d^2\right ) f \sqrt{c+d \sin (e+f x)}}+\frac{2 b \left (6 a b c d-3 a^2 d^2-b^2 \left (4 c^2-d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 d^2 \left (c^2-d^2\right ) f}-\frac{\left (b \left (18 a b c d-9 a^2 d^2-b^2 \left (8 c^2+d^2\right )\right )\right ) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{3 d^3}-\frac{\left (9 a^2 b c d^2-3 a^3 d^3-9 a b^2 d \left (2 c^2-d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{3 d^3 \left (c^2-d^2\right )}\\ &=\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d \left (c^2-d^2\right ) f \sqrt{c+d \sin (e+f x)}}+\frac{2 b \left (6 a b c d-3 a^2 d^2-b^2 \left (4 c^2-d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 d^2 \left (c^2-d^2\right ) f}-\frac{\left (\left (9 a^2 b c d^2-3 a^3 d^3-9 a b^2 d \left (2 c^2-d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) \sqrt{c+d \sin (e+f x)}\right ) \int \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}} \, dx}{3 d^3 \left (c^2-d^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\left (b \left (18 a b c d-9 a^2 d^2-b^2 \left (8 c^2+d^2\right )\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}{3 d^3 \sqrt{c+d \sin (e+f x)}}\\ &=\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d \left (c^2-d^2\right ) f \sqrt{c+d \sin (e+f x)}}+\frac{2 b \left (6 a b c d-3 a^2 d^2-b^2 \left (4 c^2-d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 d^2 \left (c^2-d^2\right ) f}-\frac{2 \left (9 a^2 b c d^2-3 a^3 d^3-9 a b^2 d \left (2 c^2-d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\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)}}{3 d^3 \left (c^2-d^2\right ) f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 b \left (18 a b c d-9 a^2 d^2-b^2 \left (8 c^2+d^2\right )\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}}}{3 d^3 f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 2.05759, size = 311, normalized size = 0.86 \[ \frac{2 \left (\frac{\sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left (d^2 \left (9 a^2 b d^2-3 a^3 c d-9 a b^2 c d+b^3 \left (2 c^2+d^2\right )\right ) F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )+\left (9 a^2 b c d^2-3 a^3 d^3+9 a b^2 d \left (d^2-2 c^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) \left ((c+d) E\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )-c F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )\right )\right )}{(c-d) (c+d)}-\frac{d \cos (e+f x) \left (-9 a^2 b c d^2+3 a^3 d^3+9 a b^2 c^2 d+b^3 d \left (d^2-c^2\right ) \sin (e+f x)+b^3 \left (c d^2-4 c^3\right )\right )}{d^2-c^2}\right )}{3 d^3 f \sqrt{c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 4.096, size = 1398, normalized size = 3.9 \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 (b \sin \left (f x + e\right ) + a\right )}^{3}}{{\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 (3 \, a b^{2} \cos \left (f x + e\right )^{2} - a^{3} - 3 \, a b^{2} +{\left (b^{3} \cos \left (f x + e\right )^{2} - 3 \, a^{2} b - b^{3}\right )} \sin \left (f x + e\right )\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 (b \sin \left (f x + e\right ) + a\right )}^{3}}{{\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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