Optimal. Leaf size=399 \[ \frac{2 b^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{f (a+b) (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right )^2 (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}+\frac{2 d \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 f \left (c^2-d^2\right ) (b c-a d) \sqrt{c+d \sin (e+f x)}}-\frac{2 d \left (-4 a c d+7 b c^2-3 b d^2\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 f \left (c^2-d^2\right )^2 (b c-a d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}} \]
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Rubi [A] time = 1.65294, antiderivative size = 399, 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 = {2802, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ \frac{2 b^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{f (a+b) (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right )^2 (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}+\frac{2 d \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 f \left (c^2-d^2\right ) (b c-a d) \sqrt{c+d \sin (e+f x)}}-\frac{2 d \left (-4 a c d+7 b c^2-3 b d^2\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 f \left (c^2-d^2\right )^2 (b c-a d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}} \]
Antiderivative was successfully verified.
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Rule 2802
Rule 3055
Rule 3059
Rule 2655
Rule 2653
Rule 3002
Rule 2663
Rule 2661
Rule 2807
Rule 2805
Rubi steps
\begin{align*} \int \frac{1}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx &=-\frac{2 d^2 \cos (e+f x)}{3 (b c-a d) \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac{2 \int \frac{-\frac{3}{2} \left (a c d-b \left (c^2-d^2\right )\right )-\frac{1}{2} d (3 b c-a d) \sin (e+f x)+\frac{1}{2} b d^2 \sin ^2(e+f x)}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx}{3 (b c-a d) \left (c^2-d^2\right )}\\ &=-\frac{2 d^2 \cos (e+f x)}{3 (b c-a d) \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}-\frac{2 d^2 \left (7 b c^2-4 a c d-3 b d^2\right ) \cos (e+f x)}{3 (b c-a d)^2 \left (c^2-d^2\right )^2 f \sqrt{c+d \sin (e+f x)}}+\frac{4 \int \frac{\frac{1}{4} \left (3 b^2 \left (c^2-d^2\right )^2-2 a b c d \left (3 c^2-d^2\right )+a^2 d^2 \left (3 c^2+d^2\right )\right )+\frac{1}{2} d \left (2 a^2 c d^2-2 a b d \left (c^2-d^2\right )-b^2 \left (3 c^3-c d^2\right )\right ) \sin (e+f x)-\frac{1}{4} b d^2 \left (7 b c^2-4 a c d-3 b d^2\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{3 (b c-a d)^2 \left (c^2-d^2\right )^2}\\ &=-\frac{2 d^2 \cos (e+f x)}{3 (b c-a d) \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}-\frac{2 d^2 \left (7 b c^2-4 a c d-3 b d^2\right ) \cos (e+f x)}{3 (b c-a d)^2 \left (c^2-d^2\right )^2 f \sqrt{c+d \sin (e+f x)}}-\frac{4 \int \frac{-\frac{1}{4} b d \left (c^2-d^2\right ) \left (a b c d-a^2 d^2+3 b^2 \left (c^2-d^2\right )\right )-\frac{1}{4} b^2 d^2 (b c-a d) \left (c^2-d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{3 b d (b c-a d)^2 \left (c^2-d^2\right )^2}-\frac{\left (d \left (7 b c^2-4 a c d-3 b d^2\right )\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{3 (b c-a d)^2 \left (c^2-d^2\right )^2}\\ &=-\frac{2 d^2 \cos (e+f x)}{3 (b c-a d) \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}-\frac{2 d^2 \left (7 b c^2-4 a c d-3 b d^2\right ) \cos (e+f x)}{3 (b c-a d)^2 \left (c^2-d^2\right )^2 f \sqrt{c+d \sin (e+f x)}}+\frac{b^2 \int \frac{1}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{(b c-a d)^2}+\frac{d \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{3 (b c-a d) \left (c^2-d^2\right )}-\frac{\left (d \left (7 b c^2-4 a c d-3 b 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}{3 (b c-a d)^2 \left (c^2-d^2\right )^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}\\ &=-\frac{2 d^2 \cos (e+f x)}{3 (b c-a d) \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}-\frac{2 d^2 \left (7 b c^2-4 a c d-3 b d^2\right ) \cos (e+f x)}{3 (b c-a d)^2 \left (c^2-d^2\right )^2 f \sqrt{c+d \sin (e+f x)}}-\frac{2 d \left (7 b c^2-4 a c d-3 b 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)}}{3 (b c-a d)^2 \left (c^2-d^2\right )^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left (b^2 \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}{(b c-a d)^2 \sqrt{c+d \sin (e+f x)}}+\frac{\left (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}{3 (b c-a d) \left (c^2-d^2\right ) \sqrt{c+d \sin (e+f x)}}\\ &=-\frac{2 d^2 \cos (e+f x)}{3 (b c-a d) \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}-\frac{2 d^2 \left (7 b c^2-4 a c d-3 b d^2\right ) \cos (e+f x)}{3 (b c-a d)^2 \left (c^2-d^2\right )^2 f \sqrt{c+d \sin (e+f x)}}-\frac{2 d \left (7 b c^2-4 a c d-3 b 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)}}{3 (b c-a d)^2 \left (c^2-d^2\right )^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 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}}}{3 (b c-a d) \left (c^2-d^2\right ) f \sqrt{c+d \sin (e+f x)}}+\frac{2 b^2 \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}}}{(a+b) (b c-a d)^2 f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 7.17685, size = 1079, normalized size = 2.7 \[ \frac{\sqrt{c+d \sin (e+f x)} \left (\frac{2 \left (3 b \cos (e+f x) d^4+4 a c \cos (e+f x) d^3-7 b c^2 \cos (e+f x) d^2\right )}{3 (b c-a d)^2 \left (c^2-d^2\right )^2 (c+d \sin (e+f x))}-\frac{2 d^2 \cos (e+f x)}{3 (b c-a d) \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}\right )}{f}+\frac{-\frac{2 \left (6 b^2 c^4-12 a b d c^3+6 a^2 d^2 c^2-19 b^2 d^2 c^2+8 a b d^3 c+2 a^2 d^4+9 b^2 d^4\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 (8 a b d^4+8 a^2 c d^3+4 b^2 c d^3-8 a b c^2 d^2-12 b^2 c^3 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 (-3 b^2 d^4-4 a b c d^3+7 b^2 c^2 d^2\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}}}}{6 (c-d)^2 (c+d)^2 (b c-a d)^2 f} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 5.758, size = 1072, normalized size = 2.7 \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{1}{{\left (b \sin \left (f x + e\right ) + a\right )}{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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