Optimal. Leaf size=69 \[ -\frac{\cot ^3(e+f x) (1-\sec (e+f x))}{3 a^2 c f}+\frac{\cot (e+f x) (3-2 \sec (e+f x))}{3 a^2 c f}+\frac{x}{a^2 c} \]
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Rubi [A] time = 0.113739, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {3904, 3882, 8} \[ -\frac{\cot ^3(e+f x) (1-\sec (e+f x))}{3 a^2 c f}+\frac{\cot (e+f x) (3-2 \sec (e+f x))}{3 a^2 c f}+\frac{x}{a^2 c} \]
Antiderivative was successfully verified.
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Rule 3904
Rule 3882
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))} \, dx &=\frac{\int \cot ^4(e+f x) (c-c \sec (e+f x)) \, dx}{a^2 c^2}\\ &=-\frac{\cot ^3(e+f x) (1-\sec (e+f x))}{3 a^2 c f}+\frac{\int \cot ^2(e+f x) (-3 c+2 c \sec (e+f x)) \, dx}{3 a^2 c^2}\\ &=\frac{\cot (e+f x) (3-2 \sec (e+f x))}{3 a^2 c f}-\frac{\cot ^3(e+f x) (1-\sec (e+f x))}{3 a^2 c f}+\frac{\int 3 c \, dx}{3 a^2 c^2}\\ &=\frac{x}{a^2 c}+\frac{\cot (e+f x) (3-2 \sec (e+f x))}{3 a^2 c f}-\frac{\cot ^3(e+f x) (1-\sec (e+f x))}{3 a^2 c f}\\ \end{align*}
Mathematica [A] time = 0.543341, size = 135, normalized size = 1.96 \[ \frac{\csc \left (\frac{e}{2}\right ) \sec \left (\frac{e}{2}\right ) \csc \left (\frac{1}{2} (e+f x)\right ) \sec ^3\left (\frac{1}{2} (e+f x)\right ) (10 \sin (e+f x)+5 \sin (2 (e+f x))-6 \sin (2 e+f x)-8 \sin (e+2 f x)-6 f x \cos (2 e+f x)+3 f x \cos (e+2 f x)-3 f x \cos (3 e+2 f x)-10 \sin (f x)+6 f x \cos (f x))}{96 a^2 c f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 87, normalized size = 1.3 \begin{align*}{\frac{1}{12\,f{a}^{2}c} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}-{\frac{1}{f{a}^{2}c}\tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) }+2\,{\frac{\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) }{f{a}^{2}c}}+{\frac{1}{4\,f{a}^{2}c} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.51585, size = 138, normalized size = 2. \begin{align*} -\frac{\frac{\frac{12 \, \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} c} - \frac{24 \, \arctan \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{2} c} - \frac{3 \,{\left (\cos \left (f x + e\right ) + 1\right )}}{a^{2} c \sin \left (f x + e\right )}}{12 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.04229, size = 180, normalized size = 2.61 \begin{align*} \frac{4 \, \cos \left (f x + e\right )^{2} + 3 \,{\left (f x \cos \left (f x + e\right ) + f x\right )} \sin \left (f x + e\right ) + \cos \left (f x + e\right ) - 2}{3 \,{\left (a^{2} c f \cos \left (f x + e\right ) + a^{2} c 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{\int \frac{1}{\sec ^{3}{\left (e + f x \right )} + \sec ^{2}{\left (e + f x \right )} - \sec{\left (e + f x \right )} - 1}\, dx}{a^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34305, size = 115, normalized size = 1.67 \begin{align*} \frac{\frac{12 \,{\left (f x + e\right )}}{a^{2} c} + \frac{3}{a^{2} c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )} + \frac{a^{4} c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 12 \, a^{4} c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}{a^{6} c^{3}}}{12 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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