Optimal. Leaf size=220 \[ -\frac {2 d^2 \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {2 d \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 b \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \Pi \left (\frac {2 b}{a+b};\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f (a+b) (b c-a d) \sqrt {c+d \sin (e+f x)}} \]
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Rubi [A] time = 0.63, antiderivative size = 220, normalized size of antiderivative = 1.00, 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 = {2802, 3059, 2655, 2653, 12, 2807, 2805} \[ -\frac {2 d^2 \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {2 d \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \left (c^2-d^2\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 b \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \Pi \left (\frac {2 b}{a+b};\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f (a+b) (b c-a d) \sqrt {c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2653
Rule 2655
Rule 2802
Rule 2805
Rule 2807
Rule 3059
Rubi steps
\begin {align*} \int \frac {1}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx &=-\frac {2 d^2 \cos (e+f x)}{(b c-a d) \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {2 \int \frac {\frac {1}{2} \left (-a c d+b \left (c^2-d^2\right )\right )-\frac {1}{2} d (b c+a d) \sin (e+f x)-\frac {1}{2} b d^2 \sin ^2(e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{(b c-a d) \left (c^2-d^2\right )}\\ &=-\frac {2 d^2 \cos (e+f x)}{(b c-a d) \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}-\frac {2 \int -\frac {b^2 d \left (c^2-d^2\right )}{2 (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{b d (b c-a d) \left (c^2-d^2\right )}-\frac {d \int \sqrt {c+d \sin (e+f x)} \, dx}{(b c-a d) \left (c^2-d^2\right )}\\ &=-\frac {2 d^2 \cos (e+f x)}{(b c-a d) \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {b \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{b c-a d}-\frac {\left (d \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{(b c-a d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\\ &=-\frac {2 d^2 \cos (e+f x)}{(b c-a d) \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}-\frac {2 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)}}{(b c-a d) \left (c^2-d^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (b \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}} \, dx}{(b c-a d) \sqrt {c+d \sin (e+f x)}}\\ &=-\frac {2 d^2 \cos (e+f x)}{(b c-a d) \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}-\frac {2 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)}}{(b c-a d) \left (c^2-d^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 b \Pi \left (\frac {2 b}{a+b};\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+b) (b c-a d) f \sqrt {c+d \sin (e+f x)}}\\ \end {align*}
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Mathematica [C] time = 6.89, size = 617, normalized size = 2.80 \[ -\frac {\frac {4 d^2 \cos (e+f x)}{\left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}+\frac {-\frac {2 i \sec (e+f x) \sqrt {-\frac {d (\sin (e+f x)-1)}{c+d}} \sqrt {\frac {d (\sin (e+f x)+1)}{d-c}} \left (d \left (d \left (b^2-2 a^2\right ) \Pi \left (\frac {b (c+d)}{b c-a d};i \sinh ^{-1}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )+2 (a+b) (a d-b c) F\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )\right )-2 b (c-d) (b c-a d) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )\right )}{b \sqrt {-\frac {1}{c+d}} (b c-a d)}+\frac {2 \left (-2 a c d+2 b c^2-3 b d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \Pi \left (\frac {2 b}{a+b};\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 d}{c+d}\right )}{(a+b) \sqrt {c+d \sin (e+f x)}}+\frac {4 i (a d+b c) \sec (e+f x) \sqrt {-\frac {d (\sin (e+f x)-1)}{c+d}} \sqrt {\frac {d (\sin (e+f x)+1)}{d-c}} \left ((a d-b c) F\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )-a d \Pi \left (\frac {b (c+d)}{b c-a d};i \sinh ^{-1}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )\right )}{b \sqrt {-\frac {1}{c+d}} (b c-a d)}}{(c-d) (c+d)}}{2 f (b c-a d)} \]
Antiderivative was successfully verified.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \sin \left (f x + e\right ) + a\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 3.77, size = 610, normalized size = 2.77 \[ \frac {\sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (-\frac {2 \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, \EllipticPi \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \frac {-\frac {c}{d}+1}{-\frac {c}{d}+\frac {a}{b}}, \sqrt {\frac {c -d}{c +d}}\right )}{\left (d a -c b \right ) \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (-\frac {c}{d}+\frac {a}{b}\right )}+\frac {d \left (\frac {2 d \left (\cos ^{2}\left (f x +e \right )\right )}{\left (c^{2}-d^{2}\right ) \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}+\frac {2 c \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, \EllipticF \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )}{\left (c^{2}-d^{2}\right ) \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}+\frac {2 d \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, \left (\left (-\frac {c}{d}-1\right ) \EllipticE \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )+\EllipticF \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )\right )}{\left (c^{2}-d^{2}\right ) \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}\right )}{d a -c b}\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 \[ \int \frac {1}{{\left (b \sin \left (f x + e\right ) + a\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{\left (a+b\,\sin \left (e+f\,x\right )\right )\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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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