Optimal. Leaf size=333 \[ -\frac{d \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{3 a f (c-d)^3 (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{\left (3 c^2+20 c d+9 d^2\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 a f (c-d)^3 (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{d (3 c+5 d) \cos (e+f x)}{3 a f (c-d)^2 (c+d) (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}+\frac{(3 c+5 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 a f (c-d)^2 (c+d) \sqrt{c+d \sin (e+f x)}} \]
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Rubi [A] time = 0.488061, antiderivative size = 333, 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 = {2768, 2754, 2752, 2663, 2661, 2655, 2653} \[ -\frac{d \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{3 a f (c-d)^3 (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{\left (3 c^2+20 c d+9 d^2\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 a f (c-d)^3 (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{d (3 c+5 d) \cos (e+f x)}{3 a f (c-d)^2 (c+d) (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}+\frac{(3 c+5 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 a f (c-d)^2 (c+d) \sqrt{c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2768
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)) (c+d \sin (e+f x))^{5/2}} \, dx &=-\frac{\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}+\frac{d \int \frac{-\frac{5 a}{2}+\frac{3}{2} a \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx}{a^2 (c-d)}\\ &=-\frac{d (3 c+5 d) \cos (e+f x)}{3 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac{(2 d) \int \frac{\frac{3}{4} a (5 c+3 d)-\frac{1}{4} a (3 c+5 d) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx}{3 a^2 (c-d)^2 (c+d)}\\ &=-\frac{d (3 c+5 d) \cos (e+f x)}{3 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac{d \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{3 a (c-d)^3 (c+d)^2 f \sqrt{c+d \sin (e+f x)}}+\frac{(4 d) \int \frac{-\frac{1}{8} a \left (15 c^2+12 c d+5 d^2\right )-\frac{1}{8} a \left (3 c^2+20 c d+9 d^2\right ) \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{3 a^2 (c-d)^3 (c+d)^2}\\ &=-\frac{d (3 c+5 d) \cos (e+f x)}{3 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac{d \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{3 a (c-d)^3 (c+d)^2 f \sqrt{c+d \sin (e+f x)}}+\frac{(3 c+5 d) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{6 a (c-d)^2 (c+d)}-\frac{\left (3 c^2+20 c d+9 d^2\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{6 a (c-d)^3 (c+d)^2}\\ &=-\frac{d (3 c+5 d) \cos (e+f x)}{3 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac{d \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{3 a (c-d)^3 (c+d)^2 f \sqrt{c+d \sin (e+f x)}}-\frac{\left (\left (3 c^2+20 c d+9 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}{6 a (c-d)^3 (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left ((3 c+5 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}{6 a (c-d)^2 (c+d) \sqrt{c+d \sin (e+f x)}}\\ &=-\frac{d (3 c+5 d) \cos (e+f x)}{3 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac{d \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{3 a (c-d)^3 (c+d)^2 f \sqrt{c+d \sin (e+f x)}}-\frac{\left (3 c^2+20 c d+9 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 a (c-d)^3 (c+d)^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{(3 c+5 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 a (c-d)^2 (c+d) f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 4.22967, size = 367, normalized size = 1.1 \[ \frac{\left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2 \left (2 (c+d \sin (e+f x)) \left (\frac{3 \sin \left (\frac{1}{2} (e+f x)\right )}{\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )}-\frac{\frac{d^2 \cos (e+f x) \left (8 c^2+d (7 c+3 d) \sin (e+f x)+3 c d-d^2\right )}{(c+d \sin (e+f x))^2}+3 c^2+13 c d+6 d^2}{(c+d)^2}\right )+\frac{\left (3 c^2+20 c d+9 d^2\right ) (c+d \sin (e+f x))+d \left (15 c^2+12 c d+5 d^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )+\left (3 c^2+20 c d+9 d^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left ((c+d) E\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )-c F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )\right )}{(c+d)^2}\right )}{3 a f (c-d)^3 (\sin (e+f x)+1) \sqrt{c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 5.365, size = 1291, normalized size = 3.9 \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}}{a d^{3} \cos \left (f x + e\right )^{4} + a c^{3} + 3 \, a c^{2} d + 3 \, a c d^{2} + a d^{3} -{\left (3 \, a c^{2} d + 3 \, a c d^{2} + 2 \, a d^{3}\right )} \cos \left (f x + e\right )^{2} +{\left (a c^{3} + 3 \, a c^{2} d + 3 \, a c d^{2} + a d^{3} -{\left (3 \, a c d^{2} + a d^{3}\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] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{c^{2} \sqrt{c + d \sin{\left (e + f x \right )}} \sin{\left (e + f x \right )} + c^{2} \sqrt{c + d \sin{\left (e + f x \right )}} + 2 c d \sqrt{c + d \sin{\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )} + 2 c d \sqrt{c + d \sin{\left (e + f x \right )}} \sin{\left (e + f x \right )} + d^{2} \sqrt{c + d \sin{\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )} + d^{2} \sqrt{c + d \sin{\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}}\, dx}{a} \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 )}{\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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