Optimal. Leaf size=38 \[ \frac{\tan (e+f x) (c-c \sec (e+f x))^2}{5 f (a \sec (e+f x)+a)^3} \]
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Rubi [A] time = 0.0737364, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.031, Rules used = {3950} \[ \frac{\tan (e+f x) (c-c \sec (e+f x))^2}{5 f (a \sec (e+f x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 3950
Rubi steps
\begin{align*} \int \frac{\sec (e+f x) (c-c \sec (e+f x))^2}{(a+a \sec (e+f x))^3} \, dx &=\frac{(c-c \sec (e+f x))^2 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}\\ \end{align*}
Mathematica [A] time = 0.113791, size = 25, normalized size = 0.66 \[ \frac{c^2 \tan ^5\left (\frac{1}{2} (e+f x)\right )}{5 a^3 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.082, size = 23, normalized size = 0.6 \begin{align*}{\frac{{c}^{2}}{5\,f{a}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.00711, size = 250, normalized size = 6.58 \begin{align*} \frac{\frac{c^{2}{\left (\frac{15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{10 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac{3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}} + \frac{c^{2}{\left (\frac{15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac{10 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac{3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}} - \frac{6 \, c^{2}{\left (\frac{5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac{\sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}}}{60 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.441037, size = 196, normalized size = 5.16 \begin{align*} \frac{{\left (c^{2} \cos \left (f x + e\right )^{2} - 2 \, c^{2} \cos \left (f x + e\right ) + c^{2}\right )} \sin \left (f x + e\right )}{5 \,{\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} + 3 \, a^{3} f \cos \left (f x + e\right ) + a^{3} f\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{c^{2} \left (\int \frac{\sec{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec{\left (e + f x \right )} + 1}\, dx + \int - \frac{2 \sec ^{2}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec{\left (e + f x \right )} + 1}\, dx + \int \frac{\sec ^{3}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec{\left (e + f x \right )} + 1}\, dx\right )}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.35964, size = 31, normalized size = 0.82 \begin{align*} \frac{c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5}}{5 \, a^{3} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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