Optimal. Leaf size=164 \[ -\frac{496 a^3 \tan (e+f x)}{315 c^5 f (1-\sec (e+f x))}-\frac{181 a^3 \tan (e+f x)}{315 c^5 f (1-\sec (e+f x))^2}-\frac{38 a^3 \tan (e+f x)}{105 c^5 f (1-\sec (e+f x))^3}+\frac{4 a^3 \tan (e+f x)}{63 c^5 f (1-\sec (e+f x))^4}-\frac{8 a^3 \tan (e+f x)}{9 c^5 f (1-\sec (e+f x))^5}+\frac{a^3 x}{c^5} \]
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Rubi [A] time = 0.7334, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {3903, 3777, 3922, 3919, 3794, 3796, 3797, 3799, 4000} \[ -\frac{496 a^3 \tan (e+f x)}{315 c^5 f (1-\sec (e+f x))}-\frac{181 a^3 \tan (e+f x)}{315 c^5 f (1-\sec (e+f x))^2}-\frac{38 a^3 \tan (e+f x)}{105 c^5 f (1-\sec (e+f x))^3}+\frac{4 a^3 \tan (e+f x)}{63 c^5 f (1-\sec (e+f x))^4}-\frac{8 a^3 \tan (e+f x)}{9 c^5 f (1-\sec (e+f x))^5}+\frac{a^3 x}{c^5} \]
Antiderivative was successfully verified.
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Rule 3903
Rule 3777
Rule 3922
Rule 3919
Rule 3794
Rule 3796
Rule 3797
Rule 3799
Rule 4000
Rubi steps
\begin{align*} \int \frac{(a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^5} \, dx &=\frac{\int \left (\frac{a^3}{(1-\sec (e+f x))^5}+\frac{3 a^3 \sec (e+f x)}{(1-\sec (e+f x))^5}+\frac{3 a^3 \sec ^2(e+f x)}{(1-\sec (e+f x))^5}+\frac{a^3 \sec ^3(e+f x)}{(1-\sec (e+f x))^5}\right ) \, dx}{c^5}\\ &=\frac{a^3 \int \frac{1}{(1-\sec (e+f x))^5} \, dx}{c^5}+\frac{a^3 \int \frac{\sec ^3(e+f x)}{(1-\sec (e+f x))^5} \, dx}{c^5}+\frac{\left (3 a^3\right ) \int \frac{\sec (e+f x)}{(1-\sec (e+f x))^5} \, dx}{c^5}+\frac{\left (3 a^3\right ) \int \frac{\sec ^2(e+f x)}{(1-\sec (e+f x))^5} \, dx}{c^5}\\ &=-\frac{8 a^3 \tan (e+f x)}{9 c^5 f (1-\sec (e+f x))^5}-\frac{a^3 \int \frac{-9-4 \sec (e+f x)}{(1-\sec (e+f x))^4} \, dx}{9 c^5}+\frac{a^3 \int \frac{(-5-9 \sec (e+f x)) \sec (e+f x)}{(1-\sec (e+f x))^4} \, dx}{9 c^5}+\frac{\left (4 a^3\right ) \int \frac{\sec (e+f x)}{(1-\sec (e+f x))^4} \, dx}{3 c^5}-\frac{\left (5 a^3\right ) \int \frac{\sec (e+f x)}{(1-\sec (e+f x))^4} \, dx}{3 c^5}\\ &=-\frac{8 a^3 \tan (e+f x)}{9 c^5 f (1-\sec (e+f x))^5}+\frac{4 a^3 \tan (e+f x)}{63 c^5 f (1-\sec (e+f x))^4}+\frac{a^3 \int \frac{63+39 \sec (e+f x)}{(1-\sec (e+f x))^3} \, dx}{63 c^5}+\frac{a^3 \int \frac{\sec (e+f x)}{(1-\sec (e+f x))^3} \, dx}{3 c^5}+\frac{\left (4 a^3\right ) \int \frac{\sec (e+f x)}{(1-\sec (e+f x))^3} \, dx}{7 c^5}-\frac{\left (5 a^3\right ) \int \frac{\sec (e+f x)}{(1-\sec (e+f x))^3} \, dx}{7 c^5}\\ &=-\frac{8 a^3 \tan (e+f x)}{9 c^5 f (1-\sec (e+f x))^5}+\frac{4 a^3 \tan (e+f x)}{63 c^5 f (1-\sec (e+f x))^4}-\frac{38 a^3 \tan (e+f x)}{105 c^5 f (1-\sec (e+f x))^3}-\frac{a^3 \int \frac{-315-204 \sec (e+f x)}{(1-\sec (e+f x))^2} \, dx}{315 c^5}+\frac{\left (2 a^3\right ) \int \frac{\sec (e+f x)}{(1-\sec (e+f x))^2} \, dx}{15 c^5}+\frac{\left (8 a^3\right ) \int \frac{\sec (e+f x)}{(1-\sec (e+f x))^2} \, dx}{35 c^5}-\frac{\left (2 a^3\right ) \int \frac{\sec (e+f x)}{(1-\sec (e+f x))^2} \, dx}{7 c^5}\\ &=-\frac{8 a^3 \tan (e+f x)}{9 c^5 f (1-\sec (e+f x))^5}+\frac{4 a^3 \tan (e+f x)}{63 c^5 f (1-\sec (e+f x))^4}-\frac{38 a^3 \tan (e+f x)}{105 c^5 f (1-\sec (e+f x))^3}-\frac{181 a^3 \tan (e+f x)}{315 c^5 f (1-\sec (e+f x))^2}+\frac{a^3 \int \frac{945+519 \sec (e+f x)}{1-\sec (e+f x)} \, dx}{945 c^5}+\frac{\left (2 a^3\right ) \int \frac{\sec (e+f x)}{1-\sec (e+f x)} \, dx}{45 c^5}+\frac{\left (8 a^3\right ) \int \frac{\sec (e+f x)}{1-\sec (e+f x)} \, dx}{105 c^5}-\frac{\left (2 a^3\right ) \int \frac{\sec (e+f x)}{1-\sec (e+f x)} \, dx}{21 c^5}\\ &=\frac{a^3 x}{c^5}-\frac{8 a^3 \tan (e+f x)}{9 c^5 f (1-\sec (e+f x))^5}+\frac{4 a^3 \tan (e+f x)}{63 c^5 f (1-\sec (e+f x))^4}-\frac{38 a^3 \tan (e+f x)}{105 c^5 f (1-\sec (e+f x))^3}-\frac{181 a^3 \tan (e+f x)}{315 c^5 f (1-\sec (e+f x))^2}-\frac{8 a^3 \tan (e+f x)}{315 c^5 f (1-\sec (e+f x))}+\frac{\left (488 a^3\right ) \int \frac{\sec (e+f x)}{1-\sec (e+f x)} \, dx}{315 c^5}\\ &=\frac{a^3 x}{c^5}-\frac{8 a^3 \tan (e+f x)}{9 c^5 f (1-\sec (e+f x))^5}+\frac{4 a^3 \tan (e+f x)}{63 c^5 f (1-\sec (e+f x))^4}-\frac{38 a^3 \tan (e+f x)}{105 c^5 f (1-\sec (e+f x))^3}-\frac{181 a^3 \tan (e+f x)}{315 c^5 f (1-\sec (e+f x))^2}-\frac{496 a^3 \tan (e+f x)}{315 c^5 f (1-\sec (e+f x))}\\ \end{align*}
Mathematica [A] time = 0.879814, size = 283, normalized size = 1.73 \[ \frac{a^3 \csc \left (\frac{e}{2}\right ) \csc ^9\left (\frac{1}{2} (e+f x)\right ) \left (-122850 \sin \left (e+\frac{f x}{2}\right )+103278 \sin \left (e+\frac{3 f x}{2}\right )+73290 \sin \left (2 e+\frac{3 f x}{2}\right )-51102 \sin \left (2 e+\frac{5 f x}{2}\right )-24570 \sin \left (3 e+\frac{5 f x}{2}\right )+13878 \sin \left (3 e+\frac{7 f x}{2}\right )+5040 \sin \left (4 e+\frac{7 f x}{2}\right )-2102 \sin \left (4 e+\frac{9 f x}{2}\right )-39690 f x \cos \left (e+\frac{f x}{2}\right )-26460 f x \cos \left (e+\frac{3 f x}{2}\right )+26460 f x \cos \left (2 e+\frac{3 f x}{2}\right )+11340 f x \cos \left (2 e+\frac{5 f x}{2}\right )-11340 f x \cos \left (3 e+\frac{5 f x}{2}\right )-2835 f x \cos \left (3 e+\frac{7 f x}{2}\right )+2835 f x \cos \left (4 e+\frac{7 f x}{2}\right )+315 f x \cos \left (4 e+\frac{9 f x}{2}\right )-315 f x \cos \left (5 e+\frac{9 f x}{2}\right )-142002 \sin \left (\frac{f x}{2}\right )+39690 f x \cos \left (\frac{f x}{2}\right )\right )}{161280 c^5 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.128, size = 133, normalized size = 0.8 \begin{align*} 2\,{\frac{{a}^{3}\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) }{f{c}^{5}}}+{\frac{{a}^{3}}{18\,f{c}^{5}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-9}}-{\frac{3\,{a}^{3}}{14\,f{c}^{5}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-7}}+{\frac{2\,{a}^{3}}{5\,f{c}^{5}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-5}}-{\frac{2\,{a}^{3}}{3\,f{c}^{5}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-3}}+2\,{\frac{{a}^{3}}{f{c}^{5}\tan \left ( 1/2\,fx+e/2 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.66351, size = 544, normalized size = 3.32 \begin{align*} \frac{a^{3}{\left (\frac{10080 \, \arctan \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{c^{5}} - \frac{{\left (\frac{270 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{1008 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac{2730 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac{9765 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - 35\right )}{\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{c^{5} \sin \left (f x + e\right )^{9}}\right )} - \frac{3 \, a^{3}{\left (\frac{180 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{378 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac{420 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac{315 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - 35\right )}{\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{c^{5} \sin \left (f x + e\right )^{9}} - \frac{15 \, a^{3}{\left (\frac{18 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{42 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac{63 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - 7\right )}{\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{c^{5} \sin \left (f x + e\right )^{9}} - \frac{7 \, a^{3}{\left (\frac{18 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac{45 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - 5\right )}{\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{c^{5} \sin \left (f x + e\right )^{9}}}{5040 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.11534, size = 535, normalized size = 3.26 \begin{align*} \frac{1051 \, a^{3} \cos \left (f x + e\right )^{5} - 1684 \, a^{3} \cos \left (f x + e\right )^{4} + 898 \, a^{3} \cos \left (f x + e\right )^{3} + 1468 \, a^{3} \cos \left (f x + e\right )^{2} - 1669 \, a^{3} \cos \left (f x + e\right ) + 496 \, a^{3} + 315 \,{\left (a^{3} f x \cos \left (f x + e\right )^{4} - 4 \, a^{3} f x \cos \left (f x + e\right )^{3} + 6 \, a^{3} f x \cos \left (f x + e\right )^{2} - 4 \, a^{3} f x \cos \left (f x + e\right ) + a^{3} f x\right )} \sin \left (f x + e\right )}{315 \,{\left (c^{5} f \cos \left (f x + e\right )^{4} - 4 \, c^{5} f \cos \left (f x + e\right )^{3} + 6 \, c^{5} f \cos \left (f x + e\right )^{2} - 4 \, c^{5} f \cos \left (f x + e\right ) + c^{5} f\right )} \sin \left (f x + e\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 [A] time = 1.50893, size = 149, normalized size = 0.91 \begin{align*} \frac{\frac{630 \,{\left (f x + e\right )} a^{3}}{c^{5}} + \frac{1260 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} - 420 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} + 252 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 135 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 35 \, a^{3}}{c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{9}}}{630 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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