Optimal. Leaf size=88 \[ -\frac {8 c \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)}-\frac {3 c \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)^2}-\frac {2 c \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)^3}+\frac {c x}{a^3} \]
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Rubi [A] time = 0.20, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3903, 3777, 3922, 3919, 3794, 3796} \[ -\frac {8 c \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)}-\frac {3 c \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)^2}-\frac {2 c \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)^3}+\frac {c x}{a^3} \]
Antiderivative was successfully verified.
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Rule 3777
Rule 3794
Rule 3796
Rule 3903
Rule 3919
Rule 3922
Rubi steps
\begin {align*} \int \frac {c-c \sec (e+f x)}{(a+a \sec (e+f x))^3} \, dx &=\frac {\int \left (\frac {c}{(1+\sec (e+f x))^3}-\frac {c \sec (e+f x)}{(1+\sec (e+f x))^3}\right ) \, dx}{a^3}\\ &=\frac {c \int \frac {1}{(1+\sec (e+f x))^3} \, dx}{a^3}-\frac {c \int \frac {\sec (e+f x)}{(1+\sec (e+f x))^3} \, dx}{a^3}\\ &=-\frac {2 c \tan (e+f x)}{5 a^3 f (1+\sec (e+f x))^3}-\frac {c \int \frac {-5+2 \sec (e+f x)}{(1+\sec (e+f x))^2} \, dx}{5 a^3}-\frac {(2 c) \int \frac {\sec (e+f x)}{(1+\sec (e+f x))^2} \, dx}{5 a^3}\\ &=-\frac {2 c \tan (e+f x)}{5 a^3 f (1+\sec (e+f x))^3}-\frac {3 c \tan (e+f x)}{5 a^3 f (1+\sec (e+f x))^2}+\frac {c \int \frac {15-7 \sec (e+f x)}{1+\sec (e+f x)} \, dx}{15 a^3}-\frac {(2 c) \int \frac {\sec (e+f x)}{1+\sec (e+f x)} \, dx}{15 a^3}\\ &=\frac {c x}{a^3}-\frac {2 c \tan (e+f x)}{5 a^3 f (1+\sec (e+f x))^3}-\frac {3 c \tan (e+f x)}{5 a^3 f (1+\sec (e+f x))^2}-\frac {2 c \tan (e+f x)}{15 a^3 f (1+\sec (e+f x))}-\frac {(22 c) \int \frac {\sec (e+f x)}{1+\sec (e+f x)} \, dx}{15 a^3}\\ &=\frac {c x}{a^3}-\frac {2 c \tan (e+f x)}{5 a^3 f (1+\sec (e+f x))^3}-\frac {3 c \tan (e+f x)}{5 a^3 f (1+\sec (e+f x))^2}-\frac {8 c \tan (e+f x)}{5 a^3 f (1+\sec (e+f x))}\\ \end {align*}
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Mathematica [A] time = 0.46, size = 169, normalized size = 1.92 \[ \frac {c \sec \left (\frac {e}{2}\right ) \sec ^5\left (\frac {1}{2} (e+f x)\right ) \left (110 \sin \left (e+\frac {f x}{2}\right )-90 \sin \left (e+\frac {3 f x}{2}\right )+40 \sin \left (2 e+\frac {3 f x}{2}\right )-26 \sin \left (2 e+\frac {5 f x}{2}\right )+50 f x \cos \left (e+\frac {f x}{2}\right )+25 f x \cos \left (e+\frac {3 f x}{2}\right )+25 f x \cos \left (2 e+\frac {3 f x}{2}\right )+5 f x \cos \left (2 e+\frac {5 f x}{2}\right )+5 f x \cos \left (3 e+\frac {5 f x}{2}\right )-150 \sin \left (\frac {f x}{2}\right )+50 f x \cos \left (\frac {f x}{2}\right )\right )}{160 a^3 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 124, normalized size = 1.41 \[ \frac {5 \, c f x \cos \left (f x + e\right )^{3} + 15 \, c f x \cos \left (f x + e\right )^{2} + 15 \, c f x \cos \left (f x + e\right ) + 5 \, c f x - {\left (13 \, c \cos \left (f x + e\right )^{2} + 19 \, c \cos \left (f x + e\right ) + 8 \, c\right )} \sin \left (f x + e\right )}{5 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} + 3 \, a^{3} f \cos \left (f x + e\right ) + a^{3} f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.19, size = 75, normalized size = 0.85 \[ \frac {\frac {10 \, {\left (f x + e\right )} c}{a^{3}} - \frac {a^{12} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 5 \, a^{12} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 20 \, a^{12} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a^{15}}}{10 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.75, size = 79, normalized size = 0.90 \[ -\frac {c \left (\tan ^{5}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{10 f \,a^{3}}+\frac {c \left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{2 f \,a^{3}}-\frac {2 c \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{f \,a^{3}}+\frac {2 c \arctan \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{f \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 159, normalized size = 1.81 \[ -\frac {c {\left (\frac {\frac {105 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {20 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{3}}\right )} + \frac {c {\left (\frac {15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {10 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}}}{60 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.38, size = 85, normalized size = 0.97 \[ \frac {c\,x}{a^3}-\frac {\frac {13\,c\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4}{5}-\frac {7\,c\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2}{10}+\frac {c\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}{10}}{a^3\,f\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5} \]
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 \[ - \frac {c \left (\int \frac {\sec {\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {1}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\right )\, dx\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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