Optimal. Leaf size=233 \[ -\frac{c \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 a^2 f (c-d) (\sin (e+f x)+1)}+\frac{(c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 a^2 f \sqrt{c+d \sin (e+f x)}}-\frac{c \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 a^2 f (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2} \]
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Rubi [A] time = 0.410489, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{c \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 a^2 f (c-d) (\sin (e+f x)+1)}+\frac{(c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 a^2 f \sqrt{c+d \sin (e+f x)}}-\frac{c \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 a^2 f (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f (a \sin (e+f x)+a)^2} \]
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))^2} \, dx &=-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f (a+a \sin (e+f x))^2}+\frac{\int \frac{\frac{1}{2} a (2 c+d)+\frac{1}{2} a d \sin (e+f x)}{(a+a \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{3 a^2}\\ &=-\frac{c \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 a^2 (c-d) f (1+\sin (e+f x))}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f (a+a \sin (e+f x))^2}-\frac{\int \frac{\frac{a^2 d^2}{2}+\frac{1}{2} a^2 c d \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{3 a^4 (c-d)}\\ &=-\frac{c \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 a^2 (c-d) f (1+\sin (e+f x))}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f (a+a \sin (e+f x))^2}-\frac{c \int \sqrt{c+d \sin (e+f x)} \, dx}{6 a^2 (c-d)}+\frac{(c+d) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{6 a^2}\\ &=-\frac{c \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 a^2 (c-d) f (1+\sin (e+f x))}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f (a+a \sin (e+f x))^2}-\frac{\left (c \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^2 (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left ((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}{6 a^2 \sqrt{c+d \sin (e+f x)}}\\ &=-\frac{c \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 a^2 (c-d) f (1+\sin (e+f x))}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f (a+a \sin (e+f x))^2}-\frac{c 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^2 (c-d) f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{(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}}}{3 a^2 f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 2.56064, size = 256, normalized size = 1.1 \[ \frac{\left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^4 \left (-\left (c^2-d^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )-c (c+d \sin (e+f x))+\frac{(c+d \sin (e+f x)) \left ((3 c-d) \sin \left (\frac{1}{2} (e+f x)\right )-c \cos \left (\frac{3}{2} (e+f x)\right )+d \cos \left (\frac{1}{2} (e+f x)\right )\right )}{\left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3}+c (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} E\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )\right )}{3 a^2 f (c-d) (\sin (e+f x)+1)^2 \sqrt{c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 4.282, size = 906, 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] 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 )}^{2}}\,{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}}{a^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2} \sin \left (f x + e\right ) - 2 \, a^{2}}, 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 ^{2}{\left (e + f x \right )} + 2 \sin{\left (e + f x \right )} + 1}\, dx}{a^{2}} \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 )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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