Optimal. Leaf size=237 \[ -\frac {2 a (c-3 d) \cos (e+f x)}{3 f (c-d) (c+d)^2 \sqrt {c+d \sin (e+f x)}}-\frac {2 a \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}+\frac {2 a \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3 d f (c+d) \sqrt {c+d \sin (e+f x)}}-\frac {2 a (c-3 d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3 d f (c-d) (c+d)^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \]
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Rubi [A] time = 0.34, antiderivative size = 237, 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} \[ -\frac {2 a (c-3 d) \cos (e+f x)}{3 f (c-d) (c+d)^2 \sqrt {c+d \sin (e+f x)}}-\frac {2 a \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}+\frac {2 a \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3 d f (c+d) \sqrt {c+d \sin (e+f x)}}-\frac {2 a (c-3 d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3 d f (c-d) (c+d)^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \]
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+a \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx &=-\frac {2 a \cos (e+f x)}{3 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac {2 \int \frac {-\frac {3}{2} a (c-d)-\frac {1}{2} a (c-d) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx}{3 \left (c^2-d^2\right )}\\ &=-\frac {2 a \cos (e+f x)}{3 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac {2 a (c-3 d) \cos (e+f x)}{3 (c-d) (c+d)^2 f \sqrt {c+d \sin (e+f x)}}+\frac {4 \int \frac {\frac {1}{4} a (c-d) (3 c-d)-\frac {1}{4} a (c-3 d) (c-d) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{3 \left (c^2-d^2\right )^2}\\ &=-\frac {2 a \cos (e+f x)}{3 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac {2 a (c-3 d) \cos (e+f x)}{3 (c-d) (c+d)^2 f \sqrt {c+d \sin (e+f x)}}-\frac {(a (c-3 d)) \int \sqrt {c+d \sin (e+f x)} \, dx}{3 (c-d) d (c+d)^2}+\frac {a \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{3 d (c+d)}\\ &=-\frac {2 a \cos (e+f x)}{3 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac {2 a (c-3 d) \cos (e+f x)}{3 (c-d) (c+d)^2 f \sqrt {c+d \sin (e+f x)}}-\frac {\left (a (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}{3 (c-d) d (c+d)^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (a \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 (c+d) \sqrt {c+d \sin (e+f x)}}\\ &=-\frac {2 a \cos (e+f x)}{3 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac {2 a (c-3 d) \cos (e+f x)}{3 (c-d) (c+d)^2 f \sqrt {c+d \sin (e+f x)}}-\frac {2 a (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)}}{3 (c-d) d (c+d)^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 a 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 (c+d) f \sqrt {c+d \sin (e+f x)}}\\ \end {align*}
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Mathematica [C] time = 6.77, size = 1870, normalized size = 7.89 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a \sin \left (f x + e\right ) + a\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{3 \, c d^{2} \cos \left (f x + e\right )^{2} - c^{3} - 3 \, c d^{2} + {\left (d^{3} \cos \left (f x + e\right )^{2} - 3 \, c^{2} d - d^{3}\right )} \sin \left (f x + e\right )}, x\right ) \]
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 \[ \int \frac {a \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 4.68, size = 884, normalized size = 3.73 \[ \frac {\sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, a \left (\frac {\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 )}}}{d}+\frac {\left (-c +d \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}\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 \[ \int \frac {a \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {a+a\,\sin \left (e+f\,x\right )}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]
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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