Optimal. Leaf size=148 \[ \frac {c^4 \tanh ^{-1}(\sin (e+f x))}{a^3 f}-\frac {c^4 \tan (e+f x) \sec ^2(e+f x)}{5 a^3 f (\sec (e+f x)+1)^3}-\frac {23 c^4 \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)}+\frac {14 c^4 \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)^2}-\frac {3 c^4 \tan (e+f x)}{a^3 f (\sec (e+f x)+1)^3}+\frac {c^4 x}{a^3} \]
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Rubi [A] time = 0.61, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 13, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3903, 3777, 3922, 3919, 3794, 3796, 3797, 3799, 4000, 3816, 4008, 3998, 3770} \[ \frac {c^4 \tanh ^{-1}(\sin (e+f x))}{a^3 f}-\frac {c^4 \tan (e+f x) \sec ^2(e+f x)}{5 a^3 f (\sec (e+f x)+1)^3}-\frac {23 c^4 \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)}+\frac {14 c^4 \tan (e+f x)}{5 a^3 f (\sec (e+f x)+1)^2}-\frac {3 c^4 \tan (e+f x)}{a^3 f (\sec (e+f x)+1)^3}+\frac {c^4 x}{a^3} \]
Antiderivative was successfully verified.
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Rule 3770
Rule 3777
Rule 3794
Rule 3796
Rule 3797
Rule 3799
Rule 3816
Rule 3903
Rule 3919
Rule 3922
Rule 3998
Rule 4000
Rule 4008
Rubi steps
\begin {align*} \int \frac {(c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^3} \, dx &=\frac {\int \left (\frac {c^4}{(1+\sec (e+f x))^3}-\frac {4 c^4 \sec (e+f x)}{(1+\sec (e+f x))^3}+\frac {6 c^4 \sec ^2(e+f x)}{(1+\sec (e+f x))^3}-\frac {4 c^4 \sec ^3(e+f x)}{(1+\sec (e+f x))^3}+\frac {c^4 \sec ^4(e+f x)}{(1+\sec (e+f x))^3}\right ) \, dx}{a^3}\\ &=\frac {c^4 \int \frac {1}{(1+\sec (e+f x))^3} \, dx}{a^3}+\frac {c^4 \int \frac {\sec ^4(e+f x)}{(1+\sec (e+f x))^3} \, dx}{a^3}-\frac {\left (4 c^4\right ) \int \frac {\sec (e+f x)}{(1+\sec (e+f x))^3} \, dx}{a^3}-\frac {\left (4 c^4\right ) \int \frac {\sec ^3(e+f x)}{(1+\sec (e+f x))^3} \, dx}{a^3}+\frac {\left (6 c^4\right ) \int \frac {\sec ^2(e+f x)}{(1+\sec (e+f x))^3} \, dx}{a^3}\\ &=-\frac {3 c^4 \tan (e+f x)}{a^3 f (1+\sec (e+f x))^3}-\frac {c^4 \sec ^2(e+f x) \tan (e+f x)}{5 a^3 f (1+\sec (e+f x))^3}-\frac {c^4 \int \frac {(2-5 \sec (e+f x)) \sec ^2(e+f x)}{(1+\sec (e+f x))^2} \, dx}{5 a^3}-\frac {c^4 \int \frac {-5+2 \sec (e+f x)}{(1+\sec (e+f x))^2} \, dx}{5 a^3}-\frac {\left (4 c^4\right ) \int \frac {\sec (e+f x) (-3+5 \sec (e+f x))}{(1+\sec (e+f x))^2} \, dx}{5 a^3}-\frac {\left (8 c^4\right ) \int \frac {\sec (e+f x)}{(1+\sec (e+f x))^2} \, dx}{5 a^3}+\frac {\left (18 c^4\right ) \int \frac {\sec (e+f x)}{(1+\sec (e+f x))^2} \, dx}{5 a^3}\\ &=-\frac {3 c^4 \tan (e+f x)}{a^3 f (1+\sec (e+f x))^3}-\frac {c^4 \sec ^2(e+f x) \tan (e+f x)}{5 a^3 f (1+\sec (e+f x))^3}+\frac {14 c^4 \tan (e+f x)}{5 a^3 f (1+\sec (e+f x))^2}+\frac {c^4 \int \frac {15-7 \sec (e+f x)}{1+\sec (e+f x)} \, dx}{15 a^3}+\frac {c^4 \int \frac {\sec (e+f x) (-14+15 \sec (e+f x))}{1+\sec (e+f x)} \, dx}{15 a^3}-\frac {\left (8 c^4\right ) \int \frac {\sec (e+f x)}{1+\sec (e+f x)} \, dx}{15 a^3}+\frac {\left (6 c^4\right ) \int \frac {\sec (e+f x)}{1+\sec (e+f x)} \, dx}{5 a^3}-\frac {\left (28 c^4\right ) \int \frac {\sec (e+f x)}{1+\sec (e+f x)} \, dx}{15 a^3}\\ &=\frac {c^4 x}{a^3}-\frac {3 c^4 \tan (e+f x)}{a^3 f (1+\sec (e+f x))^3}-\frac {c^4 \sec ^2(e+f x) \tan (e+f x)}{5 a^3 f (1+\sec (e+f x))^3}+\frac {14 c^4 \tan (e+f x)}{5 a^3 f (1+\sec (e+f x))^2}-\frac {6 c^4 \tan (e+f x)}{5 a^3 f (1+\sec (e+f x))}+\frac {c^4 \int \sec (e+f x) \, dx}{a^3}-\frac {\left (22 c^4\right ) \int \frac {\sec (e+f x)}{1+\sec (e+f x)} \, dx}{15 a^3}-\frac {\left (29 c^4\right ) \int \frac {\sec (e+f x)}{1+\sec (e+f x)} \, dx}{15 a^3}\\ &=\frac {c^4 x}{a^3}+\frac {c^4 \tanh ^{-1}(\sin (e+f x))}{a^3 f}-\frac {3 c^4 \tan (e+f x)}{a^3 f (1+\sec (e+f x))^3}-\frac {c^4 \sec ^2(e+f x) \tan (e+f x)}{5 a^3 f (1+\sec (e+f x))^3}+\frac {14 c^4 \tan (e+f x)}{5 a^3 f (1+\sec (e+f x))^2}-\frac {23 c^4 \tan (e+f x)}{5 a^3 f (1+\sec (e+f x))}\\ \end {align*}
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Mathematica [A] time = 1.20, size = 231, normalized size = 1.56 \[ \frac {c^4 (\cos (e+f x)-1)^4 \cot \left (\frac {1}{2} (e+f x)\right ) \csc ^2\left (\frac {1}{2} (e+f x)\right ) \left (8 \tan \left (\frac {e}{2}\right ) \cot ^3\left (\frac {1}{2} (e+f x)\right ) \csc ^2\left (\frac {1}{2} (e+f x)\right )-4 \tan \left (\frac {e}{2}\right ) \cot \left (\frac {1}{2} (e+f x)\right ) \csc ^4\left (\frac {1}{2} (e+f x)\right )+5 \cot ^5\left (\frac {1}{2} (e+f x)\right ) \left (-\log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+\log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )+f x\right )-\sec \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right ) (8 \cos (e+f x)+3 \cos (2 (e+f x))+9) \csc ^5\left (\frac {1}{2} (e+f x)\right )\right )}{10 a^3 f (\cos (e+f x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 242, normalized size = 1.64 \[ \frac {10 \, c^{4} f x \cos \left (f x + e\right )^{3} + 30 \, c^{4} f x \cos \left (f x + e\right )^{2} + 30 \, c^{4} f x \cos \left (f x + e\right ) + 10 \, c^{4} f x + 5 \, {\left (c^{4} \cos \left (f x + e\right )^{3} + 3 \, c^{4} \cos \left (f x + e\right )^{2} + 3 \, c^{4} \cos \left (f x + e\right ) + c^{4}\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 5 \, {\left (c^{4} \cos \left (f x + e\right )^{3} + 3 \, c^{4} \cos \left (f x + e\right )^{2} + 3 \, c^{4} \cos \left (f x + e\right ) + c^{4}\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 16 \, {\left (3 \, c^{4} \cos \left (f x + e\right )^{2} + 4 \, c^{4} \cos \left (f x + e\right ) + 3 \, c^{4}\right )} \sin \left (f x + e\right )}{10 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} + 3 \, a^{3} f \cos \left (f x + e\right ) + a^{3} f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.77, size = 110, normalized size = 0.74 \[ -\frac {4 c^{4} \left (\tan ^{5}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{5 f \,a^{3}}-\frac {4 c^{4} \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{f \,a^{3}}-\frac {c^{4} \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )}{f \,a^{3}}+\frac {c^{4} \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )}{f \,a^{3}}+\frac {2 c^{4} \arctan \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{f \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 396, normalized size = 2.68 \[ -\frac {c^{4} {\left (\frac {\frac {105 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {20 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}}{a^{3}} - \frac {60 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{3}} + \frac {60 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{3}}\right )} + c^{4} {\left (\frac {\frac {105 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {20 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{3}}\right )} + \frac {4 \, c^{4} {\left (\frac {15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}} + \frac {4 \, c^{4} {\left (\frac {15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {10 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}} - \frac {18 \, c^{4} {\left (\frac {5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}}}{60 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.42, size = 50, normalized size = 0.34 \[ \frac {c^4\,\left (2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )-\frac {4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{5}+f\,x\right )}{a^3\,f} \]
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 \[ \frac {c^{4} \left (\int \left (- \frac {4 \sec {\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {6 \sec ^{2}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {4 \sec ^{3}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {\sec ^{4}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {1}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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