Optimal. Leaf size=80 \[ -\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{63 c f (c-c \sec (e+f x))^4}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{9 f (c-c \sec (e+f x))^5} \]
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Rubi [A] time = 0.15, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {3951, 3950} \[ -\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{63 c f (c-c \sec (e+f x))^4}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^3}{9 f (c-c \sec (e+f x))^5} \]
Antiderivative was successfully verified.
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Rule 3950
Rule 3951
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^5} \, dx &=-\frac {(a+a \sec (e+f x))^3 \tan (e+f x)}{9 f (c-c \sec (e+f x))^5}+\frac {\int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^4} \, dx}{9 c}\\ &=-\frac {(a+a \sec (e+f x))^3 \tan (e+f x)}{9 f (c-c \sec (e+f x))^5}-\frac {(a+a \sec (e+f x))^3 \tan (e+f x)}{63 c f (c-c \sec (e+f x))^4}\\ \end {align*}
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Mathematica [A] time = 0.40, size = 141, normalized size = 1.76 \[ -\frac {a^3 \csc \left (\frac {e}{2}\right ) \left (315 \sin \left (e+\frac {f x}{2}\right )-189 \sin \left (e+\frac {3 f x}{2}\right )-483 \sin \left (2 e+\frac {3 f x}{2}\right )+225 \sin \left (2 e+\frac {5 f x}{2}\right )+63 \sin \left (3 e+\frac {5 f x}{2}\right )-9 \sin \left (3 e+\frac {7 f x}{2}\right )-63 \sin \left (4 e+\frac {7 f x}{2}\right )+8 \sin \left (4 e+\frac {9 f x}{2}\right )+693 \sin \left (\frac {f x}{2}\right )\right ) \csc ^9\left (\frac {1}{2} (e+f x)\right )}{16128 c^5 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 140, normalized size = 1.75 \[ \frac {8 \, a^{3} \cos \left (f x + e\right )^{5} + 31 \, a^{3} \cos \left (f x + e\right )^{4} + 44 \, a^{3} \cos \left (f x + e\right )^{3} + 26 \, a^{3} \cos \left (f x + e\right )^{2} + 4 \, a^{3} \cos \left (f x + e\right ) - a^{3}}{63 \, {\left (c^{5} f \cos \left (f x + e\right )^{4} - 4 \, c^{5} f \cos \left (f x + e\right )^{3} + 6 \, c^{5} f \cos \left (f x + e\right )^{2} - 4 \, c^{5} f \cos \left (f x + e\right ) + c^{5} f\right )} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.89, size = 43, normalized size = 0.54 \[ -\frac {9 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 7 \, a^{3}}{126 \, c^{5} f \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.95, size = 39, normalized size = 0.49 \[ \frac {a^{3} \left (-\frac {1}{7 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{7}}+\frac {1}{9 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{9}}\right )}{2 f \,c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.37, size = 357, normalized size = 4.46 \[ -\frac {\frac {a^{3} {\left (\frac {180 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {378 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {420 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {315 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - 35\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{c^{5} \sin \left (f x + e\right )^{9}} + \frac {15 \, a^{3} {\left (\frac {18 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {42 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {63 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - 7\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{c^{5} \sin \left (f x + e\right )^{9}} - \frac {5 \, a^{3} {\left (\frac {18 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {42 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {63 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + 7\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{c^{5} \sin \left (f x + e\right )^{9}} + \frac {21 \, a^{3} {\left (\frac {18 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {45 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - 5\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{c^{5} \sin \left (f x + e\right )^{9}}}{5040 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.62, size = 37, normalized size = 0.46 \[ \frac {a^3\,{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (7\,{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-9\right )}{126\,c^5\,f} \]
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 \[ - \frac {a^{3} \left (\int \frac {\sec {\left (e + f x \right )}}{\sec ^{5}{\left (e + f x \right )} - 5 \sec ^{4}{\left (e + f x \right )} + 10 \sec ^{3}{\left (e + f x \right )} - 10 \sec ^{2}{\left (e + f x \right )} + 5 \sec {\left (e + f x \right )} - 1}\, dx + \int \frac {3 \sec ^{2}{\left (e + f x \right )}}{\sec ^{5}{\left (e + f x \right )} - 5 \sec ^{4}{\left (e + f x \right )} + 10 \sec ^{3}{\left (e + f x \right )} - 10 \sec ^{2}{\left (e + f x \right )} + 5 \sec {\left (e + f x \right )} - 1}\, dx + \int \frac {3 \sec ^{3}{\left (e + f x \right )}}{\sec ^{5}{\left (e + f x \right )} - 5 \sec ^{4}{\left (e + f x \right )} + 10 \sec ^{3}{\left (e + f x \right )} - 10 \sec ^{2}{\left (e + f x \right )} + 5 \sec {\left (e + f x \right )} - 1}\, dx + \int \frac {\sec ^{4}{\left (e + f x \right )}}{\sec ^{5}{\left (e + f x \right )} - 5 \sec ^{4}{\left (e + f x \right )} + 10 \sec ^{3}{\left (e + f x \right )} - 10 \sec ^{2}{\left (e + f x \right )} + 5 \sec {\left (e + f x \right )} - 1}\, dx\right )}{c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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