Optimal. Leaf size=65 \[ \frac{(c+2 d) \tan (e+f x)}{3 f \left (a^2 \sec (e+f x)+a^2\right )}+\frac{(c-d) \tan (e+f x)}{3 f (a \sec (e+f x)+a)^2} \]
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Rubi [A] time = 0.0865492, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {4000, 3794} \[ \frac{(c+2 d) \tan (e+f x)}{3 f \left (a^2 \sec (e+f x)+a^2\right )}+\frac{(c-d) \tan (e+f x)}{3 f (a \sec (e+f x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 4000
Rule 3794
Rubi steps
\begin{align*} \int \frac{\sec (e+f x) (c+d \sec (e+f x))}{(a+a \sec (e+f x))^2} \, dx &=\frac{(c-d) \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}+\frac{(c+2 d) \int \frac{\sec (e+f x)}{a+a \sec (e+f x)} \, dx}{3 a}\\ &=\frac{(c-d) \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}+\frac{(c+2 d) \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )}\\ \end{align*}
Mathematica [A] time = 0.210091, size = 76, normalized size = 1.17 \[ \frac{\sec \left (\frac{e}{2}\right ) \cos \left (\frac{1}{2} (e+f x)\right ) \left ((2 c+d) \sin \left (e+\frac{3 f x}{2}\right )+3 (c+d) \sin \left (\frac{f x}{2}\right )-3 c \sin \left (e+\frac{f x}{2}\right )\right )}{3 a^2 f (\cos (e+f x)+1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 60, normalized size = 0.9 \begin{align*}{\frac{1}{2\,f{a}^{2}} \left ( -{\frac{c}{3} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}+{\frac{d}{3} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}+c\tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) +\tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) d \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.97264, size = 126, normalized size = 1.94 \begin{align*} \frac{\frac{d{\left (\frac{3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2}} + \frac{c{\left (\frac{3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac{\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2}}}{6 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.435586, size = 144, normalized size = 2.22 \begin{align*} \frac{{\left ({\left (2 \, c + d\right )} \cos \left (f x + e\right ) + c + 2 \, d\right )} \sin \left (f x + e\right )}{3 \,{\left (a^{2} f \cos \left (f x + e\right )^{2} + 2 \, a^{2} f \cos \left (f x + e\right ) + a^{2} 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{\int \frac{c \sec{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec{\left (e + f x \right )} + 1}\, dx + \int \frac{d \sec ^{2}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec{\left (e + f x \right )} + 1}\, dx}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14865, size = 86, normalized size = 1.32 \begin{align*} -\frac{c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - d \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 3 \, c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 3 \, d \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}{6 \, a^{2} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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