Optimal. Leaf size=333 \[ -\frac {d \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{3 a f (c-d)^3 (c+d)^2 \sqrt {c+d \sin (e+f x)}}-\frac {\left (3 c^2+20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3 a f (c-d)^3 (c+d)^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {d (3 c+5 d) \cos (e+f x)}{3 a f (c-d)^2 (c+d) (c+d \sin (e+f x))^{3/2}}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}+\frac {(3 c+5 d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3 a f (c-d)^2 (c+d) \sqrt {c+d \sin (e+f x)}} \]
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Rubi [A] time = 0.49, antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2768, 2754, 2752, 2663, 2661, 2655, 2653} \[ -\frac {d \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{3 a f (c-d)^3 (c+d)^2 \sqrt {c+d \sin (e+f x)}}-\frac {\left (3 c^2+20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3 a f (c-d)^3 (c+d)^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {d (3 c+5 d) \cos (e+f x)}{3 a f (c-d)^2 (c+d) (c+d \sin (e+f x))^{3/2}}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}+\frac {(3 c+5 d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3 a f (c-d)^2 (c+d) \sqrt {c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2754
Rule 2768
Rubi steps
\begin {align*} \int \frac {1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx &=-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}+\frac {d \int \frac {-\frac {5 a}{2}+\frac {3}{2} a \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx}{a^2 (c-d)}\\ &=-\frac {d (3 c+5 d) \cos (e+f x)}{3 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac {(2 d) \int \frac {\frac {3}{4} a (5 c+3 d)-\frac {1}{4} a (3 c+5 d) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx}{3 a^2 (c-d)^2 (c+d)}\\ &=-\frac {d (3 c+5 d) \cos (e+f x)}{3 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac {d \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{3 a (c-d)^3 (c+d)^2 f \sqrt {c+d \sin (e+f x)}}+\frac {(4 d) \int \frac {-\frac {1}{8} a \left (15 c^2+12 c d+5 d^2\right )-\frac {1}{8} a \left (3 c^2+20 c d+9 d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{3 a^2 (c-d)^3 (c+d)^2}\\ &=-\frac {d (3 c+5 d) \cos (e+f x)}{3 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac {d \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{3 a (c-d)^3 (c+d)^2 f \sqrt {c+d \sin (e+f x)}}+\frac {(3 c+5 d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{6 a (c-d)^2 (c+d)}-\frac {\left (3 c^2+20 c d+9 d^2\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{6 a (c-d)^3 (c+d)^2}\\ &=-\frac {d (3 c+5 d) \cos (e+f x)}{3 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac {d \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{3 a (c-d)^3 (c+d)^2 f \sqrt {c+d \sin (e+f x)}}-\frac {\left (\left (3 c^2+20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{6 a (c-d)^3 (c+d)^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left ((3 c+5 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}{6 a (c-d)^2 (c+d) \sqrt {c+d \sin (e+f x)}}\\ &=-\frac {d (3 c+5 d) \cos (e+f x)}{3 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac {d \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{3 a (c-d)^3 (c+d)^2 f \sqrt {c+d \sin (e+f x)}}-\frac {\left (3 c^2+20 c d+9 d^2\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 a (c-d)^3 (c+d)^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {(3 c+5 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 a (c-d)^2 (c+d) f \sqrt {c+d \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 4.26, size = 367, normalized size = 1.10 \[ \frac {\left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2 \left (2 (c+d \sin (e+f x)) \left (\frac {3 \sin \left (\frac {1}{2} (e+f x)\right )}{\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )}-\frac {\frac {d^2 \cos (e+f x) \left (8 c^2+d (7 c+3 d) \sin (e+f x)+3 c d-d^2\right )}{(c+d \sin (e+f x))^2}+3 c^2+13 c d+6 d^2}{(c+d)^2}\right )+\frac {\left (3 c^2+20 c d+9 d^2\right ) (c+d \sin (e+f x))+d \left (15 c^2+12 c d+5 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 d}{c+d}\right )+\left (3 c^2+20 c d+9 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \left ((c+d) E\left (\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 d}{c+d}\right )-c F\left (\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 d}{c+d}\right )\right )}{(c+d)^2}\right )}{3 a f (c-d)^3 (\sin (e+f x)+1) \sqrt {c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d \sin \left (f x + e\right ) + c}}{a d^{3} \cos \left (f x + e\right )^{4} + a c^{3} + 3 \, a c^{2} d + 3 \, a c d^{2} + a d^{3} - {\left (3 \, a c^{2} d + 3 \, a c d^{2} + 2 \, a d^{3}\right )} \cos \left (f x + e\right )^{2} + {\left (a c^{3} + 3 \, a c^{2} d + 3 \, a c d^{2} + a d^{3} - {\left (3 \, a c d^{2} + a d^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}, x\right ) \]
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 \[ \int \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 6.55, size = 1291, normalized size = 3.88 \[ \text {result too large to display} \]
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 \[ \int \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]
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 \[ \frac {\int \frac {1}{c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )} + c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} + 2 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )} + 2 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )} + d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )} + d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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