Optimal. Leaf size=203 \[ \frac{2 \left (d^2 \left (3 a^2+b^2\right )+2 b c (b c-3 a d)\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 d^2 f \sqrt{c+d \sin (e+f x)}}-\frac{4 b (b c-3 a d) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 d f} \]
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Rubi [A] time = 0.28484, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2791, 2752, 2663, 2661, 2655, 2653} \[ \frac{2 \left (d^2 \left (3 a^2+b^2\right )+2 b c (b c-3 a d)\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 d^2 f \sqrt{c+d \sin (e+f x)}}-\frac{4 b (b c-3 a d) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 d f} \]
Antiderivative was successfully verified.
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Rule 2791
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{(a+b \sin (e+f x))^2}{\sqrt{c+d \sin (e+f x)}} \, dx &=-\frac{2 b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 d f}+\frac{2 \int \frac{\frac{1}{2} \left (3 a^2+b^2\right ) d-b (b c-3 a d) \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{3 d}\\ &=-\frac{2 b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 d f}-\frac{(2 b (b c-3 a d)) \int \sqrt{c+d \sin (e+f x)} \, dx}{3 d^2}+\frac{1}{3} \left (3 a^2+b^2+\frac{2 b c (b c-3 a d)}{d^2}\right ) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx\\ &=-\frac{2 b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 d f}-\frac{\left (2 b (b c-3 a d) \sqrt{c+d \sin (e+f x)}\right ) \int \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}} \, dx}{3 d^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left (\left (3 a^2+b^2+\frac{2 b c (b c-3 a d)}{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}{3 \sqrt{c+d \sin (e+f x)}}\\ &=-\frac{2 b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 d f}-\frac{4 b (b c-3 a d) 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 d^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 \left (3 a^2+b^2+\frac{2 b c (b c-3 a d)}{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}}}{3 f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.905828, size = 173, normalized size = 0.85 \[ -\frac{2 \left (\left (3 a^2 d^2-6 a b c d+b^2 \left (2 c^2+d^2\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )-2 b (c+d) (b c-3 a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} E\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )+b^2 d \cos (e+f x) (c+d \sin (e+f x))\right )}{3 d^2 f \sqrt{c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 2.608, size = 695, normalized size = 3.4 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sin \left (f x + e\right ) + a\right )}^{2}}{\sqrt{d \sin \left (f x + e\right ) + c}}\,{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{b^{2} \cos \left (f x + e\right )^{2} - 2 \, a b \sin \left (f x + e\right ) - a^{2} - b^{2}}{\sqrt{d \sin \left (f x + e\right ) + c}}, 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*} \int \frac{\left (a + b \sin{\left (e + f x \right )}\right )^{2}}{\sqrt{c + d \sin{\left (e + f x \right )}}}\, dx \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{{\left (b \sin \left (f x + e\right ) + a\right )}^{2}}{\sqrt{d \sin \left (f x + e\right ) + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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