Optimal. Leaf size=423 \[ -\frac{d \left (-21 c^2 d+4 c^3+62 c d^2+147 d^3\right ) \cos (e+f x)}{30 a^3 f (c-d)^4 (c+d) \sqrt{c+d \sin (e+f x)}}-\frac{\left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x)}{30 f (c-d)^3 \left (a^3 \sin (e+f x)+a^3\right ) \sqrt{c+d \sin (e+f x)}}+\frac{\left (4 c^2-21 c d+65 d^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{30 a^3 f (c-d)^3 \sqrt{c+d \sin (e+f x)}}-\frac{\left (-21 c^2 d+4 c^3+62 c d^2+147 d^3\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{30 a^3 f (c-d)^4 (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 (c-4 d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2 \sqrt{c+d \sin (e+f x)}}-\frac{\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 \sqrt{c+d \sin (e+f x)}} \]
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Rubi [A] time = 1.02531, antiderivative size = 423, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {2766, 2978, 2754, 2752, 2663, 2661, 2655, 2653} \[ -\frac{d \left (-21 c^2 d+4 c^3+62 c d^2+147 d^3\right ) \cos (e+f x)}{30 a^3 f (c-d)^4 (c+d) \sqrt{c+d \sin (e+f x)}}-\frac{\left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x)}{30 f (c-d)^3 \left (a^3 \sin (e+f x)+a^3\right ) \sqrt{c+d \sin (e+f x)}}+\frac{\left (4 c^2-21 c d+65 d^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{30 a^3 f (c-d)^3 \sqrt{c+d \sin (e+f x)}}-\frac{\left (-21 c^2 d+4 c^3+62 c d^2+147 d^3\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{30 a^3 f (c-d)^4 (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 (c-4 d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2 \sqrt{c+d \sin (e+f x)}}-\frac{\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 \sqrt{c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2766
Rule 2978
Rule 2754
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{1}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx &=-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 \sqrt{c+d \sin (e+f x)}}-\frac{\int \frac{-\frac{1}{2} a (4 c-11 d)-\frac{5}{2} a d \sin (e+f x)}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}} \, dx}{5 a^2 (c-d)}\\ &=-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 \sqrt{c+d \sin (e+f x)}}-\frac{2 (c-4 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)}}+\frac{\int \frac{\frac{1}{2} a^2 \left (4 c^2-15 c d+41 d^2\right )+3 a^2 (c-4 d) d \sin (e+f x)}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx}{15 a^4 (c-d)^2}\\ &=-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 \sqrt{c+d \sin (e+f x)}}-\frac{2 (c-4 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)}}-\frac{\left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x)}{30 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) \sqrt{c+d \sin (e+f x)}}-\frac{\int \frac{-\frac{3}{4} a^3 (c-49 d) d^2-\frac{1}{4} a^3 d \left (4 c^2-21 c d+65 d^2\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx}{15 a^6 (c-d)^3}\\ &=-\frac{d \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f \sqrt{c+d \sin (e+f x)}}-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 \sqrt{c+d \sin (e+f x)}}-\frac{2 (c-4 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)}}-\frac{\left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x)}{30 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) \sqrt{c+d \sin (e+f x)}}+\frac{2 \int \frac{-\frac{1}{8} a^3 d^2 \left (c^2+126 c d+65 d^2\right )-\frac{1}{8} a^3 d \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{15 a^6 (c-d)^4 (c+d)}\\ &=-\frac{d \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f \sqrt{c+d \sin (e+f x)}}-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 \sqrt{c+d \sin (e+f x)}}-\frac{2 (c-4 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)}}-\frac{\left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x)}{30 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) \sqrt{c+d \sin (e+f x)}}+\frac{\left (4 c^2-21 c d+65 d^2\right ) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{60 a^3 (c-d)^3}-\frac{\left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{60 a^3 (c-d)^4 (c+d)}\\ &=-\frac{d \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f \sqrt{c+d \sin (e+f x)}}-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 \sqrt{c+d \sin (e+f x)}}-\frac{2 (c-4 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)}}-\frac{\left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x)}{30 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) \sqrt{c+d \sin (e+f x)}}-\frac{\left (\left (4 c^3-21 c^2 d+62 c d^2+147 d^3\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)^4 (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left (\left (4 c^2-21 c d+65 d^2\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}{60 a^3 (c-d)^3 \sqrt{c+d \sin (e+f x)}}\\ &=-\frac{d \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f \sqrt{c+d \sin (e+f x)}}-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 \sqrt{c+d \sin (e+f x)}}-\frac{2 (c-4 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)}}-\frac{\left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x)}{30 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) \sqrt{c+d \sin (e+f x)}}-\frac{\left (4 c^3-21 c^2 d+62 c d^2+147 d^3\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)^4 (c+d) f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left (4 c^2-21 c d+65 d^2\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}}}{30 a^3 (c-d)^3 f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 6.54493, size = 745, normalized size = 1.76 \[ \frac{\left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^6 \sqrt{c+d \sin (e+f x)} \left (\frac{4 c^2 \sin \left (\frac{1}{2} (e+f x)\right )-25 c d \sin \left (\frac{1}{2} (e+f x)\right )+87 d^2 \sin \left (\frac{1}{2} (e+f x)\right )}{15 (c-d)^4 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )}-\frac{-21 c^2 d+4 c^3+62 c d^2+117 d^3}{15 (c-d)^4 (c+d)}-\frac{2 d^4 \cos (e+f x)}{(c-d)^4 (c+d) (c+d \sin (e+f x))}+\frac{2 \left (2 c \sin \left (\frac{1}{2} (e+f x)\right )-11 d \sin \left (\frac{1}{2} (e+f x)\right )\right )}{15 (c-d)^3 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3}+\frac{11 d-2 c}{15 (c-d)^3 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2}-\frac{1}{5 (c-d)^2 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^4}+\frac{2 \sin \left (\frac{1}{2} (e+f x)\right )}{5 (c-d)^2 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^5}\right )}{f (a \sin (e+f x)+a)^3}+\frac{d \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^6 \left (\frac{2 \left (-21 c^2 d+4 c^3+62 c d^2+147 d^3\right ) \cos ^2(e+f x) \sqrt{c+d \sin (e+f x)}}{d \left (1-\sin ^2(e+f x)\right )}-\frac{2 \left (-c^2 d-126 c d^2-65 d^3\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 )}{\sqrt{c+d \sin (e+f x)}}-\frac{\left (21 c^2 d-4 c^3-62 c d^2-147 d^3\right ) \left (\frac{2 (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} E\left (\frac{1}{2} \left (-e-f x+\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{\sqrt{c+d \sin (e+f x)}}-\frac{2 c \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 )}{\sqrt{c+d \sin (e+f x)}}\right )}{d}\right )}{60 f (c-d)^4 (c+d) (a \sin (e+f x)+a)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 7.014, size = 1851, normalized size = 4.4 \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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d \sin \left (f x + e\right ) + c}}{4 \, a^{3} c^{2} + 8 \, a^{3} c d + 4 \, a^{3} d^{2} +{\left (2 \, a^{3} c d + 3 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )^{4} -{\left (3 \, a^{3} c^{2} + 10 \, a^{3} c d + 7 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )^{2} +{\left (a^{3} d^{2} \cos \left (f x + e\right )^{4} + 4 \, a^{3} c^{2} + 8 \, a^{3} c d + 4 \, a^{3} d^{2} -{\left (a^{3} c^{2} + 6 \, a^{3} c d + 5 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )^{2}\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(-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 (a \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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