Optimal. Leaf size=162 \[ \frac{2 (7 A-2 B) \tan ^5(e+f x)}{105 a^3 c^5 f}+\frac{4 (7 A-2 B) \tan ^3(e+f x)}{63 a^3 c^5 f}+\frac{2 (7 A-2 B) \tan (e+f x)}{21 a^3 c^5 f}+\frac{(7 A-2 B) \sec ^5(e+f x)}{63 a^3 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac{(A+B) \sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2} \]
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Rubi [A] time = 0.288791, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2967, 2859, 2672, 3767} \[ \frac{2 (7 A-2 B) \tan ^5(e+f x)}{105 a^3 c^5 f}+\frac{4 (7 A-2 B) \tan ^3(e+f x)}{63 a^3 c^5 f}+\frac{2 (7 A-2 B) \tan (e+f x)}{21 a^3 c^5 f}+\frac{(7 A-2 B) \sec ^5(e+f x)}{63 a^3 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac{(A+B) \sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2} \]
Antiderivative was successfully verified.
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Rule 2967
Rule 2859
Rule 2672
Rule 3767
Rubi steps
\begin{align*} \int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5} \, dx &=\frac{\int \frac{\sec ^6(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^2} \, dx}{a^3 c^3}\\ &=\frac{(A+B) \sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2}+\frac{(7 A-2 B) \int \frac{\sec ^6(e+f x)}{c-c \sin (e+f x)} \, dx}{9 a^3 c^4}\\ &=\frac{(A+B) \sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2}+\frac{(7 A-2 B) \sec ^5(e+f x)}{63 a^3 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac{(2 (7 A-2 B)) \int \sec ^6(e+f x) \, dx}{21 a^3 c^5}\\ &=\frac{(A+B) \sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2}+\frac{(7 A-2 B) \sec ^5(e+f x)}{63 a^3 f \left (c^5-c^5 \sin (e+f x)\right )}-\frac{(2 (7 A-2 B)) \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (e+f x)\right )}{21 a^3 c^5 f}\\ &=\frac{(A+B) \sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2}+\frac{(7 A-2 B) \sec ^5(e+f x)}{63 a^3 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac{2 (7 A-2 B) \tan (e+f x)}{21 a^3 c^5 f}+\frac{4 (7 A-2 B) \tan ^3(e+f x)}{63 a^3 c^5 f}+\frac{2 (7 A-2 B) \tan ^5(e+f x)}{105 a^3 c^5 f}\\ \end{align*}
Mathematica [B] time = 1.3207, size = 373, normalized size = 2.3 \[ \frac{\left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) (1125 (49 A+13 B) \cos (e+f x)-20480 (7 A-2 B) \cos (2 (e+f x))-322560 A \sin (e+f x)-24500 A \sin (2 (e+f x))-136192 A \sin (3 (e+f x))-19600 A \sin (4 (e+f x))-7168 A \sin (5 (e+f x))-4900 A \sin (6 (e+f x))+7168 A \sin (7 (e+f x))+23275 A \cos (3 (e+f x))-114688 A \cos (4 (e+f x))+1225 A \cos (5 (e+f x))-28672 A \cos (6 (e+f x))-1225 A \cos (7 (e+f x))+92160 B \sin (e+f x)-6500 B \sin (2 (e+f x))+38912 B \sin (3 (e+f x))-5200 B \sin (4 (e+f x))+2048 B \sin (5 (e+f x))-1300 B \sin (6 (e+f x))-2048 B \sin (7 (e+f x))+6175 B \cos (3 (e+f x))+32768 B \cos (4 (e+f x))+325 B \cos (5 (e+f x))+8192 B \cos (6 (e+f x))-325 B \cos (7 (e+f x))-184320 B)}{1290240 a^3 c^5 f (\sin (e+f x)-1)^5 (\sin (e+f x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \text{hanged} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.18959, size = 1621, normalized size = 10.01 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03789, size = 459, normalized size = 2.83 \begin{align*} -\frac{32 \,{\left (7 \, A - 2 \, B\right )} \cos \left (f x + e\right )^{6} - 16 \,{\left (7 \, A - 2 \, B\right )} \cos \left (f x + e\right )^{4} - 4 \,{\left (7 \, A - 2 \, B\right )} \cos \left (f x + e\right )^{2} -{\left (16 \,{\left (7 \, A - 2 \, B\right )} \cos \left (f x + e\right )^{6} - 24 \,{\left (7 \, A - 2 \, B\right )} \cos \left (f x + e\right )^{4} - 10 \,{\left (7 \, A - 2 \, B\right )} \cos \left (f x + e\right )^{2} - 49 \, A + 14 \, B\right )} \sin \left (f x + e\right ) - 14 \, A + 49 \, B}{315 \,{\left (a^{3} c^{5} f \cos \left (f x + e\right )^{7} + 2 \, a^{3} c^{5} f \cos \left (f x + e\right )^{5} \sin \left (f x + e\right ) - 2 \, a^{3} c^{5} f \cos \left (f x + e\right )^{5}\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 [B] time = 1.25432, size = 560, normalized size = 3.46 \begin{align*} -\frac{\frac{21 \,{\left (435 \, A \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 225 \, B \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 1470 \, A \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 690 \, B \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 2060 \, A \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 940 \, B \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 1330 \, A \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 590 \, B \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 353 \, A - 163 \, B\right )}}{a^{3} c^{5}{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1\right )}^{5}} + \frac{31185 \, A \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} + 4725 \, B \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} - 185220 \, A \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} - 11340 \, B \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} + 546840 \, A \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} + 15120 \, B \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} - 961380 \, A \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 3780 \, B \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 1101618 \, A \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 24318 \, B \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 828492 \, A \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 33852 \, B \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 404208 \, A \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 19368 \, B \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 116172 \, A \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 6732 \, B \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 16373 \, A - 223 \, B}{a^{3} c^{5}{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1\right )}^{9}}}{20160 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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