Optimal. Leaf size=334 \[ -\frac{\left (4 c^2-5 c d-3 d^2\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{30 f (c-d)^2 \left (a^3 \sin (e+f x)+a^3\right )}-\frac{\left (4 c^2-5 c d-3 d^2\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{30 a^3 f (c-d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{(4 c-5 d) (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{30 a^3 f (c-d) \sqrt{c+d \sin (e+f x)}}-\frac{(2 c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 a f (c-d) (a \sin (e+f x)+a)^2}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3} \]
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Rubi [A] time = 0.767721, antiderivative size = 334, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {2764, 2978, 2752, 2663, 2661, 2655, 2653} \[ -\frac{\left (4 c^2-5 c d-3 d^2\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{30 f (c-d)^2 \left (a^3 \sin (e+f x)+a^3\right )}-\frac{\left (4 c^2-5 c d-3 d^2\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{30 a^3 f (c-d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{(4 c-5 d) (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{30 a^3 f (c-d) \sqrt{c+d \sin (e+f x)}}-\frac{(2 c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 a f (c-d) (a \sin (e+f x)+a)^2}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{5 f (a \sin (e+f x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 2764
Rule 2978
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{\sqrt{c+d \sin (e+f x)}}{(a+a \sin (e+f x))^3} \, dx &=-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{5 f (a+a \sin (e+f x))^3}+\frac{\int \frac{\frac{1}{2} a (4 c+d)+\frac{3}{2} a d \sin (e+f x)}{(a+a \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)}} \, dx}{5 a^2}\\ &=-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{5 f (a+a \sin (e+f x))^3}-\frac{(2 c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 a (c-d) f (a+a \sin (e+f x))^2}-\frac{\int \frac{-\frac{1}{2} a^2 \left (4 c^2-3 c d-4 d^2\right )-\frac{1}{2} a^2 (2 c-d) d \sin (e+f x)}{(a+a \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{15 a^4 (c-d)}\\ &=-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{5 f (a+a \sin (e+f x))^3}-\frac{(2 c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 a (c-d) f (a+a \sin (e+f x))^2}-\frac{\left (4 c^2-5 c d-3 d^2\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{30 (c-d)^2 f \left (a^3+a^3 \sin (e+f x)\right )}+\frac{\int \frac{-\frac{1}{4} a^3 (c-5 d) d^2-\frac{1}{4} a^3 d \left (4 c^2-5 c d-3 d^2\right ) \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{15 a^6 (c-d)^2}\\ &=-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{5 f (a+a \sin (e+f x))^3}-\frac{(2 c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 a (c-d) f (a+a \sin (e+f x))^2}-\frac{\left (4 c^2-5 c d-3 d^2\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{30 (c-d)^2 f \left (a^3+a^3 \sin (e+f x)\right )}+\frac{((4 c-5 d) (c+d)) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{60 a^3 (c-d)}-\frac{\left (4 c^2-5 c d-3 d^2\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{60 a^3 (c-d)^2}\\ &=-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{5 f (a+a \sin (e+f x))^3}-\frac{(2 c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 a (c-d) f (a+a \sin (e+f x))^2}-\frac{\left (4 c^2-5 c d-3 d^2\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{30 (c-d)^2 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac{\left (\left (4 c^2-5 c d-3 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}{60 a^3 (c-d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left ((4 c-5 d) (c+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}{60 a^3 (c-d) \sqrt{c+d \sin (e+f x)}}\\ &=-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{5 f (a+a \sin (e+f x))^3}-\frac{(2 c-d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 a (c-d) f (a+a \sin (e+f x))^2}-\frac{\left (4 c^2-5 c d-3 d^2\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{30 (c-d)^2 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac{\left (4 c^2-5 c d-3 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)}}{30 a^3 (c-d)^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{(4 c-5 d) (c+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}}}{30 a^3 (c-d) f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 5.95959, size = 449, normalized size = 1.34 \[ \frac{\left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^6 \left (-\left (4 c^2-5 c d-3 d^2\right ) (c+d \sin (e+f x))+\frac{2 (c+d \sin (e+f x)) \left (\left (4 c^2-5 c d-3 d^2\right ) \sin \left (\frac{1}{2} (e+f x)\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^4+6 (c-d)^2 \sin \left (\frac{1}{2} (e+f x)\right )-(c-d) (2 c-d) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3+2 (c-d) (2 c-d) \sin \left (\frac{1}{2} (e+f x)\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2-3 (c-d)^2 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )\right )}{\left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^5}+\left (4 c^2-5 c d-3 d^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left ((c+d) E\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )-c F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )\right )+d^2 (c-5 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )\right )}{30 a^3 f (c-d)^2 (\sin (e+f x)+1)^3 \sqrt{c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 5.708, size = 1056, normalized size = 3.2 \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{\sqrt{d \sin \left (f x + e\right ) + c}}{{\left (a \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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{d \sin \left (f x + e\right ) + c}}{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 )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\sqrt{c + d \sin{\left (e + f x \right )}}}{\sin ^{3}{\left (e + f x \right )} + 3 \sin ^{2}{\left (e + f x \right )} + 3 \sin{\left (e + f x \right )} + 1}\, dx}{a^{3}} \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{\sqrt{d \sin \left (f x + e\right ) + c}}{{\left (a \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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