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 = 0.11, antiderivative size = 69, normalized size of antiderivative = 1.00, 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 8
Rule 3882
Rule 3904
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*}
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Mathematica [A] time = 0.55, 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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fricas [A] time = 0.43, size = 70, normalized size = 1.01 \[ \frac {4 \, \cos \left (f x + e\right )^{2} + 3 \, {\left (f x \cos \left (f x + e\right ) + f x\right )} \sin \left (f x + e\right ) + \cos \left (f x + e\right ) - 2}{3 \, {\left (a^{2} c f \cos \left (f x + e\right ) + a^{2} c f\right )} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 85, normalized size = 1.23 \[ \frac {\frac {12 \, {\left (f x + e\right )}}{a^{2} c} + \frac {3}{a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )} + \frac {a^{4} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 12 \, a^{4} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a^{6} c^{3}}}{12 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.84, size = 87, normalized size = 1.26 \[ \frac {\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )}{12 f \,a^{2} c}-\frac {\tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{f \,a^{2} c}+\frac {1}{4 f \,a^{2} c \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}+\frac {2 \arctan \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{f \,a^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 102, normalized size = 1.48 \[ -\frac {\frac {\frac {12 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a^{2} c} - \frac {24 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{2} c} - \frac {3 \, {\left (\cos \left (f x + e\right ) + 1\right )}}{a^{2} c \sin \left (f x + e\right )}}{12 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.42, size = 69, normalized size = 1.00 \[ \frac {x}{a^2\,c}+\frac {\frac {4\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4}{3}-\frac {7\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2}{6}+\frac {1}{12}}{a^2\,c\,f\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {1}{\sec ^{3}{\left (e + f x \right )} + \sec ^{2}{\left (e + f x \right )} - \sec {\left (e + f x \right )} - 1}\, dx}{a^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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