Optimal. Leaf size=121 \[ -\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^2}{315 c^2 f (c-c \sec (e+f x))^3}-\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^2}{63 c f (c-c \sec (e+f x))^4}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^2}{9 f (c-c \sec (e+f x))^5} \]
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Rubi [A] time = 0.230276, 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} \[ -\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^2}{315 c^2 f (c-c \sec (e+f x))^3}-\frac{2 \tan (e+f x) (a \sec (e+f x)+a)^2}{63 c f (c-c \sec (e+f x))^4}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^2}{9 f (c-c \sec (e+f x))^5} \]
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))^2}{(c-c \sec (e+f x))^5} \, dx &=-\frac{(a+a \sec (e+f x))^2 \tan (e+f x)}{9 f (c-c \sec (e+f x))^5}+\frac{2 \int \frac{\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^4} \, dx}{9 c}\\ &=-\frac{(a+a \sec (e+f x))^2 \tan (e+f x)}{9 f (c-c \sec (e+f x))^5}-\frac{2 (a+a \sec (e+f x))^2 \tan (e+f x)}{63 c f (c-c \sec (e+f x))^4}+\frac{2 \int \frac{\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^3} \, dx}{63 c^2}\\ &=-\frac{(a+a \sec (e+f x))^2 \tan (e+f x)}{9 f (c-c \sec (e+f x))^5}-\frac{2 (a+a \sec (e+f x))^2 \tan (e+f x)}{63 c f (c-c \sec (e+f x))^4}-\frac{2 (a+a \sec (e+f x))^2 \tan (e+f x)}{315 c^2 f (c-c \sec (e+f x))^3}\\ \end{align*}
Mathematica [A] time = 0.420944, size = 141, normalized size = 1.17 \[ -\frac{a^2 \csc \left (\frac{e}{2}\right ) \left (2520 \sin \left (e+\frac{f x}{2}\right )-1638 \sin \left (e+\frac{3 f x}{2}\right )-2310 \sin \left (2 e+\frac{3 f x}{2}\right )+1062 \sin \left (2 e+\frac{5 f x}{2}\right )+630 \sin \left (3 e+\frac{5 f x}{2}\right )-108 \sin \left (3 e+\frac{7 f x}{2}\right )-315 \sin \left (4 e+\frac{7 f x}{2}\right )+47 \sin \left (4 e+\frac{9 f x}{2}\right )+3402 \sin \left (\frac{f x}{2}\right )\right ) \csc ^9\left (\frac{1}{2} (e+f x)\right )}{80640 c^5 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.102, size = 52, normalized size = 0.4 \begin{align*}{\frac{{a}^{2}}{4\,f{c}^{5}} \left ({\frac{1}{5} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-5}}-{\frac{2}{7} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-7}}+{\frac{1}{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 = 1.05711, size = 363, normalized size = 3. \begin{align*} -\frac{\frac{a^{2}{\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{10 \, a^{2}{\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^{2}{\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 = 0.45639, size = 342, normalized size = 2.83 \begin{align*} \frac{47 \, a^{2} \cos \left (f x + e\right )^{5} + 127 \, a^{2} \cos \left (f x + e\right )^{4} + 101 \, a^{2} \cos \left (f x + e\right )^{3} + 11 \, a^{2} \cos \left (f x + e\right )^{2} - 8 \, a^{2} \cos \left (f x + e\right ) + 2 \, a^{2}}{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] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{a^{2} \left (\int \frac{\sec{\left (e + f x \right )}}{\sec ^{5}{\left (e + f x \right )} - 5 \sec ^{4}{\left (e + f x \right )} + 10 \sec ^{3}{\left (e + f x \right )} - 10 \sec ^{2}{\left (e + f x \right )} + 5 \sec{\left (e + f x \right )} - 1}\, dx + \int \frac{2 \sec ^{2}{\left (e + f x \right )}}{\sec ^{5}{\left (e + f x \right )} - 5 \sec ^{4}{\left (e + f x \right )} + 10 \sec ^{3}{\left (e + f x \right )} - 10 \sec ^{2}{\left (e + f x \right )} + 5 \sec{\left (e + f x \right )} - 1}\, dx + \int \frac{\sec ^{3}{\left (e + f x \right )}}{\sec ^{5}{\left (e + f x \right )} - 5 \sec ^{4}{\left (e + f x \right )} + 10 \sec ^{3}{\left (e + f x \right )} - 10 \sec ^{2}{\left (e + f x \right )} + 5 \sec{\left (e + f x \right )} - 1}\, dx\right )}{c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26366, size = 81, normalized size = 0.67 \begin{align*} \frac{63 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 90 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 35 \, a^{2}}{1260 \, c^{5} f \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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