Optimal. Leaf size=449 \[ \frac{d \left (2 a^2 d^2+b^2 \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) \left (c^2-d^2\right ) (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}+\frac{\left (2 a^2 d^2+b^2 \left (c^2-3 d^2\right )\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{f \left (a^2-b^2\right ) \left (c^2-d^2\right ) (b c-a d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}-\frac{b \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{f \left (a^2-b^2\right ) (b c-a d) \sqrt{c+d \sin (e+f x)}}+\frac{b \left (-5 a^2 d+2 a b c+3 b^2 d\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{f (a-b) (a+b)^2 (b c-a d)^2 \sqrt{c+d \sin (e+f x)}} \]
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Rubi [A] time = 1.62744, antiderivative size = 449, 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} \[ \frac{d \left (2 a^2 d^2+b^2 \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) \left (c^2-d^2\right ) (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}+\frac{\left (2 a^2 d^2+b^2 \left (c^2-3 d^2\right )\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{f \left (a^2-b^2\right ) \left (c^2-d^2\right ) (b c-a d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}-\frac{b \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{f \left (a^2-b^2\right ) (b c-a d) \sqrt{c+d \sin (e+f x)}}+\frac{b \left (-5 a^2 d+2 a b c+3 b^2 d\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{f (a-b) (a+b)^2 (b c-a d)^2 \sqrt{c+d \sin (e+f x)}} \]
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))^2 (c+d \sin (e+f x))^{3/2}} \, dx &=\frac{b^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}-\frac{\int \frac{\frac{1}{2} \left (-2 a b c+2 a^2 d-3 b^2 d\right )-a b d \sin (e+f x)+\frac{1}{2} b^2 d \sin ^2(e+f x)}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx}{\left (a^2-b^2\right ) (b c-a d)}\\ &=\frac{d \left (2 a^2 d^2+b^2 \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) f \sqrt{c+d \sin (e+f x)}}+\frac{b^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}-\frac{2 \int \frac{\frac{1}{4} \left (-2 a^3 c d^2-2 a b^2 c \left (c^2-2 d^2\right )+4 a^2 b d \left (c^2-d^2\right )-3 b^3 d \left (c^2-d^2\right )\right )-\frac{1}{2} d \left (a^2 b c d-b^3 c d+a^3 d^2+a b^2 \left (c^2-2 d^2\right )\right ) \sin (e+f x)-\frac{1}{4} b d \left (2 a^2 d^2+b^2 \left (c^2-3 d^2\right )\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{\left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right )}\\ &=\frac{d \left (2 a^2 d^2+b^2 \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) f \sqrt{c+d \sin (e+f x)}}+\frac{b^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}+\frac{2 \int \frac{\frac{1}{4} b^2 d \left (a b c-4 a^2 d+3 b^2 d\right ) \left (c^2-d^2\right )-\frac{1}{4} b^3 d (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}{b \left (a^2-b^2\right ) d (b c-a d)^2 \left (c^2-d^2\right )}+\frac{\left (2 a^2 d^2+b^2 \left (c^2-3 d^2\right )\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{2 \left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right )}\\ &=\frac{d \left (2 a^2 d^2+b^2 \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) f \sqrt{c+d \sin (e+f x)}}+\frac{b^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}-\frac{b \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac{\left (b \left (2 a b c-5 a^2 d+3 b^2 d\right )\right ) \int \frac{1}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{2 \left (a^2-b^2\right ) (b c-a d)^2}+\frac{\left (\left (2 a^2 d^2+b^2 \left (c^2-3 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}{2 \left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}\\ &=\frac{d \left (2 a^2 d^2+b^2 \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) f \sqrt{c+d \sin (e+f x)}}+\frac{b^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}+\frac{\left (2 a^2 d^2+b^2 \left (c^2-3 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)}}{\left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\left (b \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}{2 \left (a^2-b^2\right ) (b c-a d) \sqrt{c+d \sin (e+f x)}}+\frac{\left (b \left (2 a b c-5 a^2 d+3 b^2 d\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}{2 \left (a^2-b^2\right ) (b c-a d)^2 \sqrt{c+d \sin (e+f x)}}\\ &=\frac{d \left (2 a^2 d^2+b^2 \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) f \sqrt{c+d \sin (e+f x)}}+\frac{b^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}+\frac{\left (2 a^2 d^2+b^2 \left (c^2-3 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)}}{\left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{b 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}}}{\left (a^2-b^2\right ) (b c-a d) f \sqrt{c+d \sin (e+f x)}}+\frac{b \left (2 a b c-5 a^2 d+3 b^2 d\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}}}{(a-b) (a+b)^2 (b c-a d)^2 f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 7.63817, size = 1057, normalized size = 2.35 \[ \frac{\sqrt{c+d \sin (e+f x)} \left (\frac{\cos (e+f x) b^3}{\left (a^2-b^2\right ) (a d-b c)^2 (a+b \sin (e+f x))}+\frac{2 d^3 \cos (e+f x)}{(b c-a d)^2 \left (c^2-d^2\right ) (c+d \sin (e+f x))}\right )}{f}+\frac{-\frac{2 \left (4 c d^2 a^3+10 b d^3 a^2-8 b c^2 d a^2+4 b^2 c^3 a-8 b^2 c d^2 a-9 b^3 d^3+7 b^3 c^2 d\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 (-4 c d^2 b^3-8 a d^3 b^2+4 a c^2 d b^2+4 a^2 c d^2 b+4 a^3 d^3\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 d^3 b^3-c^2 d b^3-2 a^2 d^3 b\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}}}}{4 (a-b) (a+b) (c-d) (c+d) (a d-b c)^2 f} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 5.793, size = 1266, normalized size = 2.8 \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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