Optimal. Leaf size=162 \[ -\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^3}{3003 c^3 f (c-c \sec (e+f x))^4}-\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^3}{429 c^2 f (c-c \sec (e+f x))^5}-\frac{3 \tan (e+f x) (a \sec (e+f x)+a)^3}{143 c f (c-c \sec (e+f x))^6}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^3}{13 f (c-c \sec (e+f x))^7} \]
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Rubi [A] time = 0.316715, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {3951, 3950} \[ -\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^3}{3003 c^3 f (c-c \sec (e+f x))^4}-\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^3}{429 c^2 f (c-c \sec (e+f x))^5}-\frac{3 \tan (e+f x) (a \sec (e+f x)+a)^3}{143 c f (c-c \sec (e+f x))^6}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^3}{13 f (c-c \sec (e+f x))^7} \]
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))^7} \, dx &=-\frac{(a+a \sec (e+f x))^3 \tan (e+f x)}{13 f (c-c \sec (e+f x))^7}+\frac{3 \int \frac{\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^6} \, dx}{13 c}\\ &=-\frac{(a+a \sec (e+f x))^3 \tan (e+f x)}{13 f (c-c \sec (e+f x))^7}-\frac{3 (a+a \sec (e+f x))^3 \tan (e+f x)}{143 c f (c-c \sec (e+f x))^6}+\frac{6 \int \frac{\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^5} \, dx}{143 c^2}\\ &=-\frac{(a+a \sec (e+f x))^3 \tan (e+f x)}{13 f (c-c \sec (e+f x))^7}-\frac{3 (a+a \sec (e+f x))^3 \tan (e+f x)}{143 c f (c-c \sec (e+f x))^6}-\frac{2 (a+a \sec (e+f x))^3 \tan (e+f x)}{429 c^2 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}{429 c^3}\\ &=-\frac{(a+a \sec (e+f x))^3 \tan (e+f x)}{13 f (c-c \sec (e+f x))^7}-\frac{3 (a+a \sec (e+f x))^3 \tan (e+f x)}{143 c f (c-c \sec (e+f x))^6}-\frac{2 (a+a \sec (e+f x))^3 \tan (e+f x)}{429 c^2 f (c-c \sec (e+f x))^5}-\frac{2 (a+a \sec (e+f x))^3 \tan (e+f x)}{3003 c^3 f (c-c \sec (e+f x))^4}\\ \end{align*}
Mathematica [A] time = 0.575218, size = 193, normalized size = 1.19 \[ -\frac{a^3 \csc \left (\frac{e}{2}\right ) \left (246246 \sin \left (e+\frac{f x}{2}\right )-182754 \sin \left (e+\frac{3 f x}{2}\right )-216216 \sin \left (2 e+\frac{3 f x}{2}\right )+122551 \sin \left (2 e+\frac{5 f x}{2}\right )+99099 \sin \left (3 e+\frac{5 f x}{2}\right )-37609 \sin \left (3 e+\frac{7 f x}{2}\right )-51051 \sin \left (4 e+\frac{7 f x}{2}\right )+15171 \sin \left (4 e+\frac{9 f x}{2}\right )+9009 \sin \left (5 e+\frac{9 f x}{2}\right )-1027 \sin \left (5 e+\frac{11 f x}{2}\right )-3003 \sin \left (6 e+\frac{11 f x}{2}\right )+310 \sin \left (6 e+\frac{13 f x}{2}\right )+285714 \sin \left (\frac{f x}{2}\right )\right ) \csc ^{13}\left (\frac{1}{2} (e+f x)\right )}{12300288 c^7 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.135, size = 65, normalized size = 0.4 \begin{align*}{\frac{{a}^{3}}{8\,f{c}^{7}} \left ({\frac{1}{13} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-13}}-{\frac{3}{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{1}{3} \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 = 1.1346, size = 698, normalized size = 4.31 \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 = 0.474635, size = 486, normalized size = 3. \begin{align*} \frac{310 \, a^{3} \cos \left (f x + e\right )^{7} + 1143 \, a^{3} \cos \left (f x + e\right )^{6} + 1492 \, a^{3} \cos \left (f x + e\right )^{5} + 736 \, a^{3} \cos \left (f x + e\right )^{4} + 34 \, a^{3} \cos \left (f x + e\right )^{3} - 29 \, a^{3} \cos \left (f x + e\right )^{2} + 12 \, a^{3} \cos \left (f x + e\right ) - 2 \, a^{3}}{3003 \,{\left (c^{7} f \cos \left (f x + e\right )^{6} - 6 \, c^{7} f \cos \left (f x + e\right )^{5} + 15 \, c^{7} f \cos \left (f x + e\right )^{4} - 20 \, c^{7} f \cos \left (f x + e\right )^{3} + 15 \, c^{7} f \cos \left (f x + e\right )^{2} - 6 \, c^{7} f \cos \left (f x + e\right ) + c^{7} 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 = 1.35317, size = 104, normalized size = 0.64 \begin{align*} -\frac{429 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} - 1001 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 819 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 231 \, a^{3}}{24024 \, c^{7} f \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{13}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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