Optimal. Leaf size=189 \[ -\frac {4 a^2 (c-d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d^2 f \sqrt {c+d \sin (e+f x)}}+\frac {4 a^2 c \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d^2 f (c+d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 a^2 (c-d) \cos (e+f x)}{d f (c+d) \sqrt {c+d \sin (e+f x)}} \]
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Rubi [A] time = 0.24, antiderivative size = 189, normalized size of antiderivative = 1.00, 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 = {2762, 2752, 2663, 2661, 2655, 2653} \[ -\frac {4 a^2 (c-d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d^2 f \sqrt {c+d \sin (e+f x)}}+\frac {4 a^2 c \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d^2 f (c+d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 a^2 (c-d) \cos (e+f x)}{d f (c+d) \sqrt {c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2762
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^2}{(c+d \sin (e+f x))^{3/2}} \, dx &=\frac {2 a^2 (c-d) \cos (e+f x)}{d (c+d) f \sqrt {c+d \sin (e+f x)}}-\frac {(2 a) \int \frac {-a d-a c \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{d (c+d)}\\ &=\frac {2 a^2 (c-d) \cos (e+f x)}{d (c+d) f \sqrt {c+d \sin (e+f x)}}-\frac {\left (2 a^2 (c-d)\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{d^2}+\frac {\left (2 a^2 c\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{d^2 (c+d)}\\ &=\frac {2 a^2 (c-d) \cos (e+f x)}{d (c+d) f \sqrt {c+d \sin (e+f x)}}+\frac {\left (2 a^2 c \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{d^2 (c+d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (2 a^2 (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}{d^2 \sqrt {c+d \sin (e+f x)}}\\ &=\frac {2 a^2 (c-d) \cos (e+f x)}{d (c+d) f \sqrt {c+d \sin (e+f x)}}+\frac {4 a^2 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)}}{d^2 (c+d) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {4 a^2 (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}}}{d^2 f \sqrt {c+d \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.91, size = 175, normalized size = 0.93 \[ -\frac {2 a^2 (\sin (e+f x)+1)^2 \left (2 c (c+d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} E\left (\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 d}{c+d}\right )-(c-d) \left (2 (c+d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 d}{c+d}\right )+d \cos (e+f x)\right )\right )}{d^2 f (c+d) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^4 \sqrt {c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2} \sin \left (f x + e\right ) - 2 \, a^{2}\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{2}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.40, size = 463, normalized size = 2.45 \[ -\frac {2 \left (2 \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \EllipticE \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{3}-2 \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \EllipticE \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c \,d^{2}-2 \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \EllipticF \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{2} d +2 \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \EllipticF \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{3}+c \,d^{2} \left (\sin ^{2}\left (f x +e \right )\right )-d^{3} \left (\sin ^{2}\left (f x +e \right )\right )-c \,d^{2}+d^{3}\right ) a^{2}}{d^{3} \left (c +d \right ) \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{2}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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