Optimal. Leaf size=186 \[ \frac {\left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{a f \sqrt {c+d \sin (e+f x)}}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}-\frac {(c-3 d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{a f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \]
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Rubi [A] time = 0.26, antiderivative size = 186, 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 = {2767, 2752, 2663, 2661, 2655, 2653} \[ \frac {\left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{a f \sqrt {c+d \sin (e+f x)}}-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}-\frac {(c-3 d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{a f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2767
Rubi steps
\begin {align*} \int \frac {(c+d \sin (e+f x))^{3/2}}{a+a \sin (e+f x)} \, dx &=-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a+a \sin (e+f x))}-\frac {d \int \frac {-\frac {1}{2} a (3 c-d)+\frac {1}{2} a (c-3 d) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{a^2}\\ &=-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a+a \sin (e+f x))}-\frac {(c-3 d) \int \sqrt {c+d \sin (e+f x)} \, dx}{2 a}+\frac {\left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{2 a}\\ &=-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a+a \sin (e+f x))}-\frac {\left ((c-3 d) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{2 a \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (\left (c^2-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}{2 a \sqrt {c+d \sin (e+f x)}}\\ &=-\frac {(c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a+a \sin (e+f x))}-\frac {(c-3 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)}}{a f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (c^2-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}}}{a f \sqrt {c+d \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 1.53, size = 223, normalized size = 1.20 \[ \frac {\left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) \left (2 (c-d) \sin \left (\frac {1}{2} (e+f x)\right ) (c+d \sin (e+f x))-\left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) \left ((c-d) \left ((c+d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 d}{c+d}\right )+c+d \sin (e+f x)\right )-\left (c^2-2 c d-3 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} E\left (\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 d}{c+d}\right )\right )\right )}{a f (\sin (e+f x)+1) \sqrt {c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{a \sin \left (f x + e\right ) + a}, x\right ) \]
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 \[ \int \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{a \sin \left (f x + e\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.60, size = 925, normalized size = 4.97 \[ \frac {\sqrt {\left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) d +c \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (2 \sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, \EllipticF \left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{2} d -2 \sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, \EllipticF \left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{3}+\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, \EllipticE \left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{3}-3 \sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, \EllipticE \left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{2} d -\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, \EllipticE \left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c \,d^{2}+3 \sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, \EllipticE \left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{3}-c \left (\cos ^{2}\left (f x +e \right )\right ) d^{2}+\left (\cos ^{2}\left (f x +e \right )\right ) d^{3}+c^{2} d \sin \left (f x +e \right )-2 c \,d^{2} \sin \left (f x +e \right )+d^{3} \sin \left (f x +e \right )-c^{2} d +2 c \,d^{2}-d^{3}\right )}{d \sqrt {-\left (c +d \sin \left (f x +e \right )\right ) \left (\sin \left (f x +e \right )-1\right ) \left (1+\sin \left (f x +e \right )\right )}\, a \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 \[ \int \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{a \sin \left (f x + e\right ) + a}\,{d x} \]
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 \[ \int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}}{a+a\,\sin \left (e+f\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {c \sqrt {c + d \sin {\left (e + f x \right )}}}{\sin {\left (e + f x \right )} + 1}\, dx + \int \frac {d \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}}{\sin {\left (e + f x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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