Optimal. Leaf size=518 \[ -\frac{d \left (111 c^2 d^2-27 c^3 d+4 c^4+579 c d^3+357 d^4\right ) \cos (e+f x)}{30 a^3 f (c-d)^5 (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{d \left (-27 c^2 d+4 c^3+114 c d^2+165 d^3\right ) \cos (e+f x)}{30 a^3 f (c-d)^4 (c+d) (c+d \sin (e+f x))^{3/2}}-\frac{\left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{30 f (c-d)^3 \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}+\frac{\left (-27 c^2 d+4 c^3+114 c d^2+165 d^3\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{30 a^3 f (c-d)^4 (c+d) \sqrt{c+d \sin (e+f x)}}-\frac{\left (111 c^2 d^2-27 c^3 d+4 c^4+579 c d^3+357 d^4\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{30 a^3 f (c-d)^5 (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 (c-5 d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}} \]
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Rubi [A] time = 1.21486, antiderivative size = 518, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {2766, 2978, 2754, 2752, 2663, 2661, 2655, 2653} \[ -\frac{d \left (111 c^2 d^2-27 c^3 d+4 c^4+579 c d^3+357 d^4\right ) \cos (e+f x)}{30 a^3 f (c-d)^5 (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{d \left (-27 c^2 d+4 c^3+114 c d^2+165 d^3\right ) \cos (e+f x)}{30 a^3 f (c-d)^4 (c+d) (c+d \sin (e+f x))^{3/2}}-\frac{\left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{30 f (c-d)^3 \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}+\frac{\left (-27 c^2 d+4 c^3+114 c d^2+165 d^3\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{30 a^3 f (c-d)^4 (c+d) \sqrt{c+d \sin (e+f x)}}-\frac{\left (111 c^2 d^2-27 c^3 d+4 c^4+579 c d^3+357 d^4\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{30 a^3 f (c-d)^5 (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 (c-5 d) \cos (e+f x)}{15 a f (c-d)^2 (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{5 f (c-d) (a \sin (e+f x)+a)^3 (c+d \sin (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2766
Rule 2978
Rule 2754
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{1}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2}} \, dx &=-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}}-\frac{\int \frac{-\frac{1}{2} a (4 c-13 d)-\frac{7}{2} a d \sin (e+f x)}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx}{5 a^2 (c-d)}\\ &=-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}}-\frac{2 (c-5 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}}+\frac{\int \frac{\frac{1}{2} a^2 \left (4 c^2-17 c d+69 d^2\right )+5 a^2 (c-5 d) d \sin (e+f x)}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx}{15 a^4 (c-d)^2}\\ &=-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}}-\frac{2 (c-5 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}}-\frac{\left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{30 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}}-\frac{\int \frac{-\frac{15}{4} a^3 (c-33 d) d^2-\frac{3}{4} a^3 d \left (4 c^2-27 c d+119 d^2\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx}{15 a^6 (c-d)^3}\\ &=-\frac{d \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}}-\frac{2 (c-5 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}}-\frac{\left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{30 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}}+\frac{2 \int \frac{\frac{9}{8} a^3 d^2 \left (c^2-138 c d-119 d^2\right )+\frac{3}{8} a^3 d \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx}{45 a^6 (c-d)^4 (c+d)}\\ &=-\frac{d \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}}-\frac{2 (c-5 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}}-\frac{\left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{30 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}}-\frac{d \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \cos (e+f x)}{30 a^3 (c-d)^5 (c+d)^2 f \sqrt{c+d \sin (e+f x)}}-\frac{4 \int \frac{\frac{3}{16} a^3 d^2 \left (c^3+387 c^2 d+471 c d^2+165 d^3\right )+\frac{3}{16} a^3 d \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{45 a^6 (c-d)^5 (c+d)^2}\\ &=-\frac{d \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}}-\frac{2 (c-5 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}}-\frac{\left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{30 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}}-\frac{d \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \cos (e+f x)}{30 a^3 (c-d)^5 (c+d)^2 f \sqrt{c+d \sin (e+f x)}}+\frac{\left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{60 a^3 (c-d)^4 (c+d)}-\frac{\left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{60 a^3 (c-d)^5 (c+d)^2}\\ &=-\frac{d \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}}-\frac{2 (c-5 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}}-\frac{\left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{30 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}}-\frac{d \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \cos (e+f x)}{30 a^3 (c-d)^5 (c+d)^2 f \sqrt{c+d \sin (e+f x)}}-\frac{\left (\left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\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 (c-d)^5 (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left (\left (4 c^3-27 c^2 d+114 c d^2+165 d^3\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 (c-d)^4 (c+d) \sqrt{c+d \sin (e+f x)}}\\ &=-\frac{d \left (4 c^3-27 c^2 d+114 c d^2+165 d^3\right ) \cos (e+f x)}{30 a^3 (c-d)^4 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{5 (c-d) f (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}}-\frac{2 (c-5 d) \cos (e+f x)}{15 a (c-d)^2 f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}}-\frac{\left (4 c^2-27 c d+119 d^2\right ) \cos (e+f x)}{30 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}}-\frac{d \left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\right ) \cos (e+f x)}{30 a^3 (c-d)^5 (c+d)^2 f \sqrt{c+d \sin (e+f x)}}-\frac{\left (4 c^4-27 c^3 d+111 c^2 d^2+579 c d^3+357 d^4\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 (c-d)^5 (c+d)^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left (4 c^3-27 c^2 d+114 c d^2+165 d^3\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 (c-d)^4 (c+d) f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 6.83506, size = 828, normalized size = 1.6 \[ \frac{\sqrt{c+d \sin (e+f x)} \left (-\frac{2 \cos (e+f x) d^4}{3 (c-d)^4 (c+d) (c+d \sin (e+f x))^2}-\frac{4 c^4-27 d c^3+111 d^2 c^2+449 d^3 c+267 d^4}{15 (c-d)^5 (c+d)^2}+\frac{4 \left (c \sin \left (\frac{1}{2} (e+f x)\right )-8 d \sin \left (\frac{1}{2} (e+f x)\right )\right )}{15 (c-d)^4 \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^3}+\frac{4 \sin \left (\frac{1}{2} (e+f x)\right ) c^2-35 d \sin \left (\frac{1}{2} (e+f x)\right ) c+177 d^2 \sin \left (\frac{1}{2} (e+f x)\right )}{15 (c-d)^5 \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )}-\frac{2 \left (9 \cos (e+f x) d^5+13 c \cos (e+f x) d^4\right )}{3 (c-d)^5 (c+d)^2 (c+d \sin (e+f x))}-\frac{2 (c-8 d)}{15 (c-d)^4 \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^2}-\frac{1}{5 (c-d)^3 \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^4}+\frac{2 \sin \left (\frac{1}{2} (e+f x)\right )}{5 (c-d)^3 \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^5}\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^6}{f (\sin (e+f x) a+a)^3}+\frac{d \left (\frac{2 \left (4 c^4-27 d c^3+111 d^2 c^2+579 d^3 c+357 d^4\right ) \sqrt{c+d \sin (e+f x)} \cos ^2(e+f x)}{d \left (1-\sin ^2(e+f x)\right )}-\frac{\left (-4 c^4+27 d c^3-111 d^2 c^2-579 d^3 c-357 d^4\right ) \left (\frac{2 (c+d) E\left (\frac{1}{2} \left (-e-f x+\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{\sqrt{c+d \sin (e+f x)}}-\frac{2 c F\left (\frac{1}{2} \left (-e-f x+\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{\sqrt{c+d \sin (e+f x)}}\right )}{d}-\frac{2 \left (-165 d^4-471 c d^3-387 c^2 d^2-c^3 d\right ) F\left (\frac{1}{2} \left (-e-f x+\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{\sqrt{c+d \sin (e+f x)}}\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^6}{60 (c-d)^5 (c+d)^2 f (\sin (e+f x) a+a)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 9.667, size = 2311, normalized size = 4.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{d \sin \left (f x + e\right ) + c}}{a^{3} d^{3} \cos \left (f x + e\right )^{6} - 4 \, a^{3} c^{3} - 12 \, a^{3} c^{2} d - 12 \, a^{3} c d^{2} - 4 \, a^{3} d^{3} - 3 \,{\left (a^{3} c^{2} d + 3 \, a^{3} c d^{2} + 2 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )^{4} + 3 \,{\left (a^{3} c^{3} + 5 \, a^{3} c^{2} d + 7 \, a^{3} c d^{2} + 3 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )^{2} -{\left (4 \, a^{3} c^{3} + 12 \, a^{3} c^{2} d + 12 \, a^{3} c d^{2} + 4 \, a^{3} d^{3} + 3 \,{\left (a^{3} c d^{2} + a^{3} d^{3}\right )} \cos \left (f x + e\right )^{4} -{\left (a^{3} c^{3} + 9 \, a^{3} c^{2} d + 15 \, a^{3} c d^{2} + 7 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{3}{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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