Optimal. Leaf size=100 \[ \frac{\cot ^5(e+f x) (1-\sec (e+f x))}{5 a^3 c^2 f}-\frac{\cot ^3(e+f x) (5-4 \sec (e+f x))}{15 a^3 c^2 f}+\frac{\cot (e+f x) (15-8 \sec (e+f x))}{15 a^3 c^2 f}+\frac{x}{a^3 c^2} \]
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Rubi [A] time = 0.142863, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {3904, 3882, 8} \[ \frac{\cot ^5(e+f x) (1-\sec (e+f x))}{5 a^3 c^2 f}-\frac{\cot ^3(e+f x) (5-4 \sec (e+f x))}{15 a^3 c^2 f}+\frac{\cot (e+f x) (15-8 \sec (e+f x))}{15 a^3 c^2 f}+\frac{x}{a^3 c^2} \]
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))^3 (c-c \sec (e+f x))^2} \, dx &=-\frac{\int \cot ^6(e+f x) (c-c \sec (e+f x)) \, dx}{a^3 c^3}\\ &=\frac{\cot ^5(e+f x) (1-\sec (e+f x))}{5 a^3 c^2 f}-\frac{\int \cot ^4(e+f x) (-5 c+4 c \sec (e+f x)) \, dx}{5 a^3 c^3}\\ &=-\frac{\cot ^3(e+f x) (5-4 \sec (e+f x))}{15 a^3 c^2 f}+\frac{\cot ^5(e+f x) (1-\sec (e+f x))}{5 a^3 c^2 f}-\frac{\int \cot ^2(e+f x) (15 c-8 c \sec (e+f x)) \, dx}{15 a^3 c^3}\\ &=\frac{\cot (e+f x) (15-8 \sec (e+f x))}{15 a^3 c^2 f}-\frac{\cot ^3(e+f x) (5-4 \sec (e+f x))}{15 a^3 c^2 f}+\frac{\cot ^5(e+f x) (1-\sec (e+f x))}{5 a^3 c^2 f}-\frac{\int -15 c \, dx}{15 a^3 c^3}\\ &=\frac{x}{a^3 c^2}+\frac{\cot (e+f x) (15-8 \sec (e+f x))}{15 a^3 c^2 f}-\frac{\cot ^3(e+f x) (5-4 \sec (e+f x))}{15 a^3 c^2 f}+\frac{\cot ^5(e+f x) (1-\sec (e+f x))}{5 a^3 c^2 f}\\ \end{align*}
Mathematica [B] time = 0.91673, size = 257, normalized size = 2.57 \[ \frac{\csc \left (\frac{e}{2}\right ) \sec \left (\frac{e}{2}\right ) \csc ^3\left (\frac{1}{2} (e+f x)\right ) \sec ^5\left (\frac{1}{2} (e+f x)\right ) (534 \sin (e+f x)+178 \sin (2 (e+f x))-178 \sin (3 (e+f x))-89 \sin (4 (e+f x))-520 \sin (2 e+f x)-248 \sin (e+2 f x)-120 \sin (3 e+2 f x)+248 \sin (2 e+3 f x)+120 \sin (4 e+3 f x)+184 \sin (3 e+4 f x)-360 f x \cos (2 e+f x)+120 f x \cos (e+2 f x)-120 f x \cos (3 e+2 f x)-120 f x \cos (2 e+3 f x)+120 f x \cos (4 e+3 f x)-60 f x \cos (3 e+4 f x)+60 f x \cos (5 e+4 f x)-200 \sin (e)-584 \sin (f x)+360 f x \cos (f x))}{30720 a^3 c^2 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.063, size = 131, normalized size = 1.3 \begin{align*} -{\frac{1}{80\,f{a}^{3}{c}^{2}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{5}}+{\frac{1}{8\,f{a}^{3}{c}^{2}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}-{\frac{1}{f{a}^{3}{c}^{2}}\tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) }+2\,{\frac{\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) }{f{a}^{3}{c}^{2}}}-{\frac{1}{48\,f{a}^{3}{c}^{2}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-3}}+{\frac{3}{8\,f{a}^{3}{c}^{2}} \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 = 1.54584, size = 197, normalized size = 1.97 \begin{align*} -\frac{\frac{3 \,{\left (\frac{80 \, \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{\sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3} c^{2}} - \frac{480 \, \arctan \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{3} c^{2}} - \frac{5 \,{\left (\frac{18 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - 1\right )}{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}{a^{3} c^{2} \sin \left (f x + e\right )^{3}}}{240 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.07982, size = 374, normalized size = 3.74 \begin{align*} \frac{23 \, \cos \left (f x + e\right )^{4} + 8 \, \cos \left (f x + e\right )^{3} - 27 \, \cos \left (f x + e\right )^{2} + 15 \,{\left (f x \cos \left (f x + e\right )^{3} + f x \cos \left (f x + e\right )^{2} - f x \cos \left (f x + e\right ) - f x\right )} \sin \left (f x + e\right ) - 7 \, \cos \left (f x + e\right ) + 8}{15 \,{\left (a^{3} c^{2} f \cos \left (f x + e\right )^{3} + a^{3} c^{2} f \cos \left (f x + e\right )^{2} - a^{3} c^{2} f \cos \left (f x + e\right ) - a^{3} c^{2} 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 ^{5}{\left (e + f x \right )} + \sec ^{4}{\left (e + f x \right )} - 2 \sec ^{3}{\left (e + f x \right )} - 2 \sec ^{2}{\left (e + f x \right )} + \sec{\left (e + f x \right )} + 1}\, dx}{a^{3} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.39182, size = 165, normalized size = 1.65 \begin{align*} \frac{\frac{240 \,{\left (f x + e\right )}}{a^{3} c^{2}} + \frac{5 \,{\left (18 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right )}}{a^{3} c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3}} - \frac{3 \,{\left (a^{12} c^{8} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 10 \, a^{12} c^{8} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 80 \, a^{12} c^{8} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{a^{15} c^{10}}}{240 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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