Optimal. Leaf size=61 \[ -\frac{5 c \tan (e+f x)}{3 a^2 f (\sec (e+f x)+1)}-\frac{2 c \tan (e+f x)}{3 a^2 f (\sec (e+f x)+1)^2}+\frac{c x}{a^2} \]
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Rubi [A] time = 0.145132, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {3903, 3777, 3919, 3794, 3796} \[ -\frac{5 c \tan (e+f x)}{3 a^2 f (\sec (e+f x)+1)}-\frac{2 c \tan (e+f x)}{3 a^2 f (\sec (e+f x)+1)^2}+\frac{c x}{a^2} \]
Antiderivative was successfully verified.
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Rule 3903
Rule 3777
Rule 3919
Rule 3794
Rule 3796
Rubi steps
\begin{align*} \int \frac{c-c \sec (e+f x)}{(a+a \sec (e+f x))^2} \, dx &=\frac{\int \left (\frac{c}{(1+\sec (e+f x))^2}-\frac{c \sec (e+f x)}{(1+\sec (e+f x))^2}\right ) \, dx}{a^2}\\ &=\frac{c \int \frac{1}{(1+\sec (e+f x))^2} \, dx}{a^2}-\frac{c \int \frac{\sec (e+f x)}{(1+\sec (e+f x))^2} \, dx}{a^2}\\ &=-\frac{2 c \tan (e+f x)}{3 a^2 f (1+\sec (e+f x))^2}-\frac{c \int \frac{-3+\sec (e+f x)}{1+\sec (e+f x)} \, dx}{3 a^2}-\frac{c \int \frac{\sec (e+f x)}{1+\sec (e+f x)} \, dx}{3 a^2}\\ &=\frac{c x}{a^2}-\frac{2 c \tan (e+f x)}{3 a^2 f (1+\sec (e+f x))^2}-\frac{c \tan (e+f x)}{3 a^2 f (1+\sec (e+f x))}-\frac{(4 c) \int \frac{\sec (e+f x)}{1+\sec (e+f x)} \, dx}{3 a^2}\\ &=\frac{c x}{a^2}-\frac{2 c \tan (e+f x)}{3 a^2 f (1+\sec (e+f x))^2}-\frac{5 c \tan (e+f x)}{3 a^2 f (1+\sec (e+f x))}\\ \end{align*}
Mathematica [A] time = 0.313902, size = 113, normalized size = 1.85 \[ \frac{c \sec \left (\frac{e}{2}\right ) \sec ^3\left (\frac{1}{2} (e+f x)\right ) \left (18 \sin \left (e+\frac{f x}{2}\right )-14 \sin \left (e+\frac{3 f x}{2}\right )+9 f x \cos \left (e+\frac{f x}{2}\right )+3 f x \cos \left (e+\frac{3 f x}{2}\right )+3 f x \cos \left (2 e+\frac{3 f x}{2}\right )-24 \sin \left (\frac{f x}{2}\right )+9 f x \cos \left (\frac{f x}{2}\right )\right )}{24 a^2 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.08, size = 59, normalized size = 1. \begin{align*}{\frac{c}{3\,f{a}^{2}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}-2\,{\frac{c\tan \left ( 1/2\,fx+e/2 \right ) }{f{a}^{2}}}+2\,{\frac{c\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) }{f{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.54762, size = 161, normalized size = 2.64 \begin{align*} -\frac{c{\left (\frac{\frac{9 \, \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}}}{a^{2}} - \frac{12 \, \arctan \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{2}}\right )} + \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 = 1.02543, size = 212, normalized size = 3.48 \begin{align*} \frac{3 \, c f x \cos \left (f x + e\right )^{2} + 6 \, c f x \cos \left (f x + e\right ) + 3 \, c f x -{\left (7 \, c \cos \left (f x + e\right ) + 5 \, c\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{c \left (\int \frac{\sec{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec{\left (e + f x \right )} + 1}\, dx + \int - \frac{1}{\sec ^{2}{\left (e + f x \right )} + 2 \sec{\left (e + f x \right )} + 1}\, dx\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.42365, size = 76, normalized size = 1.25 \begin{align*} \frac{\frac{3 \,{\left (f x + e\right )} c}{a^{2}} + \frac{a^{4} c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 6 \, a^{4} c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}{a^{6}}}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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