Optimal. Leaf size=322 \[ -\frac {\left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{30 f \left (a^3 \sin (e+f x)+a^3\right )}+\frac {(c+d) \left (4 c^2+11 c d+15 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{30 a^3 f \sqrt {c+d \sin (e+f x)}}-\frac {\left (4 c^2+15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{30 a^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a \sin (e+f x)+a)^3}-\frac {2 (c-d) (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 a f (a \sin (e+f x)+a)^2} \]
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Rubi [A] time = 0.83, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2765, 2977, 2978, 2752, 2663, 2661, 2655, 2653} \[ -\frac {\left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{30 f \left (a^3 \sin (e+f x)+a^3\right )}+\frac {(c+d) \left (4 c^2+11 c d+15 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{30 a^3 f \sqrt {c+d \sin (e+f x)}}-\frac {\left (4 c^2+15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{30 a^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a \sin (e+f x)+a)^3}-\frac {2 (c-d) (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 a f (a \sin (e+f x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2765
Rule 2977
Rule 2978
Rubi steps
\begin {align*} \int \frac {(c+d \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^3} \, dx &=-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a+a \sin (e+f x))^3}-\frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (-\frac {1}{2} a \left (4 c^2+9 c d-3 d^2\right )-\frac {1}{2} a d (c+9 d) \sin (e+f x)\right )}{(a+a \sin (e+f x))^2} \, dx}{5 a^2}\\ &=-\frac {2 (c-d) (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 a f (a+a \sin (e+f x))^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a+a \sin (e+f x))^3}-\frac {\int \frac {-\frac {1}{2} a^2 \left (4 c^3+13 c^2 d+19 c d^2-6 d^3\right )-\frac {1}{2} a^2 d \left (2 c^2+7 c d+21 d^2\right ) \sin (e+f x)}{(a+a \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{15 a^4}\\ &=-\frac {2 (c-d) (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 a f (a+a \sin (e+f x))^2}-\frac {\left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{30 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a+a \sin (e+f x))^3}+\frac {\int \frac {-\frac {1}{4} a^3 (c-15 d) (c-d) d^2-\frac {1}{4} a^3 (c-d) d \left (4 c^2+15 c d+27 d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{15 a^6 (c-d)}\\ &=-\frac {2 (c-d) (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 a f (a+a \sin (e+f x))^2}-\frac {\left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{30 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a+a \sin (e+f x))^3}+\frac {\left ((c+d) \left (4 c^2+11 c d+15 d^2\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{60 a^3}-\frac {\left (4 c^2+15 c d+27 d^2\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{60 a^3}\\ &=-\frac {2 (c-d) (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 a f (a+a \sin (e+f x))^2}-\frac {\left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{30 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a+a \sin (e+f x))^3}-\frac {\left (\left (4 c^2+15 c d+27 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}{60 a^3 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left ((c+d) \left (4 c^2+11 c d+15 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}{60 a^3 \sqrt {c+d \sin (e+f x)}}\\ &=-\frac {2 (c-d) (c+3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 a f (a+a \sin (e+f x))^2}-\frac {\left (4 c^2+15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{30 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f (a+a \sin (e+f x))^3}-\frac {\left (4 c^2+15 c d+27 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)}}{30 a^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {(c+d) \left (4 c^2+11 c d+15 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}}}{30 a^3 f \sqrt {c+d \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 5.81, size = 385, normalized size = 1.20 \[ \frac {\left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^6 \left (-\left (4 c^2+15 c d+27 d^2\right ) (c+d \sin (e+f x))-\frac {(c+d \sin (e+f x)) \left (\left (20 c^2+74 c d+90 d^2\right ) \cos \left (\frac {3}{2} (e+f x)\right )+2 \sin \left (\frac {1}{2} (e+f x)\right ) \left (2 \left (2 c^2+7 c d-9 d^2\right ) \cos (e+f x)+\left (4 c^2+15 c d+27 d^2\right ) \cos (2 (e+f x))-3 \left (6 c^2+11 c d+29 d^2\right )\right )-2 d (35 c+57 d) \cos \left (\frac {1}{2} (e+f x)\right )\right )}{2 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^5}+\left (4 c^2+15 c d+27 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \left ((c+d) E\left (\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 d}{c+d}\right )-c F\left (\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 d}{c+d}\right )\right )+d^2 (c-15 d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 d}{c+d}\right )\right )}{30 a^3 f (\sin (e+f x)+1)^3 \sqrt {c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2}\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{3 \, a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3} + {\left (a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3}\right )} \sin \left (f x + e\right )}, x\right ) \]
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 \[ \int \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 7.63, size = 1615, normalized size = 5.02 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{3}}\,{d x} \]
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 \[ \int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3} \,d x \]
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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