Optimal. Leaf size=285 \[ \frac {2 \left (4 a c d-b \left (c^2+3 d^2\right )\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right )^2 \sqrt {c+d \sin (e+f x)}}-\frac {2 (b c-a d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}+\frac {2 (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3 d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (4 a c d-b \left (c^2+3 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3 d f \left (c^2-d^2\right )^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \]
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Rubi [A] time = 0.39, antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2754, 2752, 2663, 2661, 2655, 2653} \[ \frac {2 \left (4 a c d-b \left (c^2+3 d^2\right )\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right )^2 \sqrt {c+d \sin (e+f x)}}-\frac {2 (b c-a d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}+\frac {2 (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3 d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (4 a c d-b \left (c^2+3 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3 d f \left (c^2-d^2\right )^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2754
Rubi steps
\begin {align*} \int \frac {a+b \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx &=-\frac {2 (b c-a d) \cos (e+f x)}{3 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}-\frac {2 \int \frac {-\frac {3}{2} (a c-b d)-\frac {1}{2} (b c-a d) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx}{3 \left (c^2-d^2\right )}\\ &=-\frac {2 (b c-a d) \cos (e+f x)}{3 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {2 \left (4 a c d-b \left (c^2+3 d^2\right )\right ) \cos (e+f x)}{3 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}}+\frac {4 \int \frac {\frac {1}{4} \left (-4 b c d+a \left (3 c^2+d^2\right )\right )+\frac {1}{4} \left (4 a c d-b \left (c^2+3 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{3 \left (c^2-d^2\right )^2}\\ &=-\frac {2 (b c-a d) \cos (e+f x)}{3 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {2 \left (4 a c d-b \left (c^2+3 d^2\right )\right ) \cos (e+f x)}{3 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}}+\frac {(b c-a d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{3 d \left (c^2-d^2\right )}+\frac {\left (4 a c d-b \left (c^2+3 d^2\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{3 d \left (c^2-d^2\right )^2}\\ &=-\frac {2 (b c-a d) \cos (e+f x)}{3 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {2 \left (4 a c d-b \left (c^2+3 d^2\right )\right ) \cos (e+f x)}{3 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}}+\frac {\left (\left (4 a c d-b \left (c^2+3 d^2\right )\right ) \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 \left (c^2-d^2\right )^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left ((b c-a 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}{3 d \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}\\ &=-\frac {2 (b c-a d) \cos (e+f x)}{3 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {2 \left (4 a c d-b \left (c^2+3 d^2\right )\right ) \cos (e+f x)}{3 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (4 a c d-b \left (c^2+3 d^2\right )\right ) 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 \left (c^2-d^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 (b c-a 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 d \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 1.55, size = 199, normalized size = 0.70 \[ \frac {2 \left (\frac {\left (\frac {c+d \sin (e+f x)}{c+d}\right )^{3/2} \left (\left (b \left (c^2+3 d^2\right )-4 a c d\right ) E\left (\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 d}{c+d}\right )-(c-d) (b c-a d) F\left (\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 d}{c+d}\right )\right )}{d (c-d)^2}-\frac {\cos (e+f x) \left (d \left (b \left (c^2+3 d^2\right )-4 a c d\right ) \sin (e+f x)+a d \left (d^2-5 c^2\right )+2 b c \left (c^2+d^2\right )\right )}{\left (c^2-d^2\right )^2}\right )}{3 f (c+d \sin (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (b \sin \left (f x + e\right ) + a\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{3 \, c d^{2} \cos \left (f x + e\right )^{2} - c^{3} - 3 \, c d^{2} + {\left (d^{3} \cos \left (f x + e\right )^{2} - 3 \, c^{2} d - d^{3}\right )} \sin \left (f x + e\right )}, x\right ) \]
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 \[ \int \frac {b \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 5.13, size = 887, normalized size = 3.11 \[ \frac {\sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (\frac {\left (d a -c b \right ) \left (\frac {2 \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}{3 \left (c^{2}-d^{2}\right ) d \left (\sin \left (f x +e \right )+\frac {c}{d}\right )^{2}}+\frac {8 d \left (\cos ^{2}\left (f x +e \right )\right ) c}{3 \left (c^{2}-d^{2}\right )^{2} \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}+\frac {2 \left (3 c^{2}+d^{2}\right ) \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 (3 c^{4}-6 c^{2} d^{2}+3 d^{4}\right ) \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}+\frac {8 c 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 )}{3 \left (c^{2}-d^{2}\right )^{2} \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}\right )}{d}+\frac {b \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}\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 \[ \int \frac {b \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {a+b\,\sin \left (e+f\,x\right )}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]
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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