Optimal. Leaf size=472 \[ -\frac{\left (-2 a^2 b^2 \left (4 c^2+5 d^2\right )+4 a^3 b c d+a^4 d^2+20 a b^3 c d-b^4 \left (4 c^2+3 d^2\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{4 b^2 f (a-b)^2 (a+b)^3 \sqrt{c+d \sin (e+f x)}}+\frac{\left (a^2 (-d)+6 a b c-5 b^2 d\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f \left (a^2-b^2\right )^2 (a+b \sin (e+f x))}+\frac{(b c-a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac{(b c-a d) \left (a^2 d+6 a b c-7 b^2 d\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{4 b^2 f \left (a^2-b^2\right )^2 \sqrt{c+d \sin (e+f x)}}+\frac{\left (a^2 (-d)+6 a b c-5 b^2 d\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{4 b f \left (a^2-b^2\right )^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}} \]
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Rubi [A] time = 1.80938, antiderivative size = 472, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.37, Rules used = {2799, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ -\frac{\left (-2 a^2 b^2 \left (4 c^2+5 d^2\right )+4 a^3 b c d+a^4 d^2+20 a b^3 c d-b^4 \left (4 c^2+3 d^2\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{4 b^2 f (a-b)^2 (a+b)^3 \sqrt{c+d \sin (e+f x)}}+\frac{\left (a^2 (-d)+6 a b c-5 b^2 d\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 f \left (a^2-b^2\right )^2 (a+b \sin (e+f x))}+\frac{(b c-a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac{(b c-a d) \left (a^2 d+6 a b c-7 b^2 d\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{4 b^2 f \left (a^2-b^2\right )^2 \sqrt{c+d \sin (e+f x)}}+\frac{\left (a^2 (-d)+6 a b c-5 b^2 d\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{4 b f \left (a^2-b^2\right )^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}} \]
Antiderivative was successfully verified.
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Rule 2799
Rule 3055
Rule 3059
Rule 2655
Rule 2653
Rule 3002
Rule 2663
Rule 2661
Rule 2807
Rule 2805
Rubi steps
\begin{align*} \int \frac{(c+d \sin (e+f x))^{3/2}}{(a+b \sin (e+f x))^3} \, dx &=\frac{(b c-a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac{\int \frac{\frac{1}{2} \left (5 b c d-a \left (4 c^2+d^2\right )\right )-\left (3 a c d-b \left (c^2+2 d^2\right )\right ) \sin (e+f x)+\frac{1}{2} d (b c-a d) \sin ^2(e+f x)}{(a+b \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)}} \, dx}{2 \left (a^2-b^2\right )}\\ &=\frac{(b c-a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac{\left (6 a b c-a^2 d-5 b^2 d\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac{\int \frac{\frac{1}{4} (b c-a d) \left (8 a^2 c^2+4 b^2 c^2-18 a b c d+3 a^2 d^2+3 b^2 d^2\right )+\frac{1}{2} d (b c-a d) \left (5 a^2 c+b^2 c-6 a b d\right ) \sin (e+f x)+\frac{1}{4} d (b c-a d) \left (6 a b c-a^2 d-5 b^2 d\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{2 \left (a^2-b^2\right )^2 (b c-a d)}\\ &=\frac{(b c-a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac{\left (6 a b c-a^2 d-5 b^2 d\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}-\frac{\int \frac{-\frac{1}{4} d (b c-a d) \left (a^3 c d-13 a b^2 c d+a^2 b \left (2 c^2+3 d^2\right )+b^3 \left (4 c^2+3 d^2\right )\right )+\frac{1}{4} d (b c-a d)^2 \left (6 a b c+a^2 d-7 b^2 d\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{2 b \left (a^2-b^2\right )^2 d (b c-a d)}+\frac{\left (6 a b c-a^2 d-5 b^2 d\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{8 b \left (a^2-b^2\right )^2}\\ &=\frac{(b c-a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac{\left (6 a b c-a^2 d-5 b^2 d\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}-\frac{\left ((b c-a d) \left (6 a b c+a^2 d-7 b^2 d\right )\right ) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{8 b^2 \left (a^2-b^2\right )^2}-\frac{\left (4 a^3 b c d+20 a b^3 c d+a^4 d^2-b^4 \left (4 c^2+3 d^2\right )-2 a^2 b^2 \left (4 c^2+5 d^2\right )\right ) \int \frac{1}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{8 b^2 \left (a^2-b^2\right )^2}+\frac{\left (\left (6 a b c-a^2 d-5 b^2 d\right ) \sqrt{c+d \sin (e+f x)}\right ) \int \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}} \, dx}{8 b \left (a^2-b^2\right )^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}\\ &=\frac{(b c-a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac{\left (6 a b c-a^2 d-5 b^2 d\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac{\left (6 a b c-a^2 d-5 b^2 d\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)}}{4 b \left (a^2-b^2\right )^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\left ((b c-a d) \left (6 a b c+a^2 d-7 b^2 d\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}{8 b^2 \left (a^2-b^2\right )^2 \sqrt{c+d \sin (e+f x)}}-\frac{\left (\left (4 a^3 b c d+20 a b^3 c d+a^4 d^2-b^4 \left (4 c^2+3 d^2\right )-2 a^2 b^2 \left (4 c^2+5 d^2\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}\right ) \int \frac{1}{(a+b \sin (e+f x)) \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}}} \, dx}{8 b^2 \left (a^2-b^2\right )^2 \sqrt{c+d \sin (e+f x)}}\\ &=\frac{(b c-a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{2 \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac{\left (6 a b c-a^2 d-5 b^2 d\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{4 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac{\left (6 a b c-a^2 d-5 b^2 d\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)}}{4 b \left (a^2-b^2\right )^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{(b c-a d) \left (6 a b c+a^2 d-7 b^2 d\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}}}{4 b^2 \left (a^2-b^2\right )^2 f \sqrt{c+d \sin (e+f x)}}-\frac{\left (4 a^3 b c d+20 a b^3 c d+a^4 d^2-b^4 \left (4 c^2+3 d^2\right )-2 a^2 b^2 \left (4 c^2+5 d^2\right )\right ) \Pi \left (\frac{2 b}{a+b};\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}}}{4 (a-b)^2 b^2 (a+b)^3 f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 7.51627, size = 1001, normalized size = 2.12 \[ \frac{\sqrt{c+d \sin (e+f x)} \left (\frac{b c \cos (e+f x)-a d \cos (e+f x)}{2 \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}+\frac{-d \cos (e+f x) a^2+6 b c \cos (e+f x) a-5 b^2 d \cos (e+f x)}{4 \left (a^2-b^2\right )^2 (a+b \sin (e+f x))}\right )}{f}+\frac{-\frac{2 \left (16 a^2 c^2+8 b^2 c^2-30 a b d c+5 a^2 d^2+b^2 d^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (-e-f x+\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{(a+b) \sqrt{c+d \sin (e+f x)}}-\frac{2 i \left (20 c d a^2-24 b d^2 a+4 b^2 c d\right ) \cos (e+f x) \left ((b c-a d) F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )+a d \Pi \left (\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )\right ) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b d^2 \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}-\frac{2 i \left (a^2 d^2+5 b^2 d^2-6 a b c d\right ) \cos (e+f x) \cos (2 (e+f x)) \left (2 b (c-d) (b c-a d) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )+d \left (\left (2 a^2-b^2\right ) d \Pi \left (\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )-2 (a+b) (a d-b c) F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )\right )\right ) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b^2 d \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \left (-2 c^2+4 (c+d \sin (e+f x)) c+d^2-2 (c+d \sin (e+f x))^2\right ) \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}}{16 (a-b)^2 (a+b)^2 f} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 8.563, size = 1718, normalized size = 3.6 \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(-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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Fricas [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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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 (d \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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