Optimal. Leaf size=121 \[ -\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^3}{693 c^2 f (c-c \sec (e+f x))^4}-\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^3}{99 c f (c-c \sec (e+f x))^5}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^3}{11 f (c-c \sec (e+f x))^6} \]
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Rubi [A] time = 0.230325, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {3951, 3950} \[ -\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^3}{693 c^2 f (c-c \sec (e+f x))^4}-\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^3}{99 c f (c-c \sec (e+f x))^5}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^3}{11 f (c-c \sec (e+f x))^6} \]
Antiderivative was successfully verified.
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Rule 3951
Rule 3950
Rubi steps
\begin{align*} \int \frac{\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^6} \, dx &=-\frac{(a+a \sec (e+f x))^3 \tan (e+f x)}{11 f (c-c \sec (e+f x))^6}+\frac{2 \int \frac{\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^5} \, dx}{11 c}\\ &=-\frac{(a+a \sec (e+f x))^3 \tan (e+f x)}{11 f (c-c \sec (e+f x))^6}-\frac{2 (a+a \sec (e+f x))^3 \tan (e+f x)}{99 c f (c-c \sec (e+f x))^5}+\frac{2 \int \frac{\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^4} \, dx}{99 c^2}\\ &=-\frac{(a+a \sec (e+f x))^3 \tan (e+f x)}{11 f (c-c \sec (e+f x))^6}-\frac{2 (a+a \sec (e+f x))^3 \tan (e+f x)}{99 c f (c-c \sec (e+f x))^5}-\frac{2 (a+a \sec (e+f x))^3 \tan (e+f x)}{693 c^2 f (c-c \sec (e+f x))^4}\\ \end{align*}
Mathematica [A] time = 0.677084, size = 167, normalized size = 1.38 \[ -\frac{a^3 \csc \left (\frac{e}{2}\right ) \left (21252 \sin \left (e+\frac{f x}{2}\right )-15444 \sin \left (e+\frac{3 f x}{2}\right )-10626 \sin \left (2 e+\frac{3 f x}{2}\right )+4950 \sin \left (2 e+\frac{5 f x}{2}\right )+8085 \sin \left (3 e+\frac{5 f x}{2}\right )-2959 \sin \left (3 e+\frac{7 f x}{2}\right )-1386 \sin \left (4 e+\frac{7 f x}{2}\right )+176 \sin \left (4 e+\frac{9 f x}{2}\right )+693 \sin \left (5 e+\frac{9 f x}{2}\right )-79 \sin \left (5 e+\frac{11 f x}{2}\right )+15246 \sin \left (\frac{f x}{2}\right )\right ) \csc ^{11}\left (\frac{1}{2} (e+f x)\right )}{709632 c^6 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.125, size = 52, normalized size = 0.4 \begin{align*}{\frac{{a}^{3}}{4\,f{c}^{6}} \left ( -{\frac{1}{11} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-11}}-{\frac{1}{7} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-7}}+{\frac{2}{9} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-9}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.09299, size = 699, normalized size = 5.78 \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 = 0.466873, size = 413, normalized size = 3.41 \begin{align*} \frac{79 \, a^{3} \cos \left (f x + e\right )^{6} + 298 \, a^{3} \cos \left (f x + e\right )^{5} + 404 \, a^{3} \cos \left (f x + e\right )^{4} + 216 \, a^{3} \cos \left (f x + e\right )^{3} + 19 \, a^{3} \cos \left (f x + e\right )^{2} - 10 \, a^{3} \cos \left (f x + e\right ) + 2 \, a^{3}}{693 \,{\left (c^{6} f \cos \left (f x + e\right )^{5} - 5 \, c^{6} f \cos \left (f x + e\right )^{4} + 10 \, c^{6} f \cos \left (f x + e\right )^{3} - 10 \, c^{6} f \cos \left (f x + e\right )^{2} + 5 \, c^{6} f \cos \left (f x + e\right ) - c^{6} 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 = 1.29698, size = 81, normalized size = 0.67 \begin{align*} -\frac{99 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 154 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 63 \, a^{3}}{2772 \, c^{6} f \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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