Optimal. Leaf size=164 \[ \frac {7 c^4 \tan (e+f x)}{a^3 f}-\frac {7 c^4 \tanh ^{-1}(\sin (e+f x))}{a^3 f}+\frac {14 \tan (e+f x) \left (c^4-c^4 \sec (e+f x)\right )}{3 f \left (a^3 \sec (e+f x)+a^3\right )}-\frac {14 \tan (e+f x) \left (c^2-c^2 \sec (e+f x)\right )^2}{15 a f (a \sec (e+f x)+a)^2}+\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{5 f (a \sec (e+f x)+a)^3} \]
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Rubi [A] time = 0.28, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {3957, 3787, 3770, 3767, 8} \[ \frac {7 c^4 \tan (e+f x)}{a^3 f}-\frac {7 c^4 \tanh ^{-1}(\sin (e+f x))}{a^3 f}+\frac {14 \tan (e+f x) \left (c^4-c^4 \sec (e+f x)\right )}{3 f \left (a^3 \sec (e+f x)+a^3\right )}-\frac {14 \tan (e+f x) \left (c^2-c^2 \sec (e+f x)\right )^2}{15 a f (a \sec (e+f x)+a)^2}+\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^3}{5 f (a \sec (e+f x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 3787
Rule 3957
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^3} \, dx &=\frac {2 c (c-c \sec (e+f x))^3 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}-\frac {(7 c) \int \frac {\sec (e+f x) (c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^2} \, dx}{5 a}\\ &=\frac {2 c (c-c \sec (e+f x))^3 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}-\frac {14 \left (c^2-c^2 \sec (e+f x)\right )^2 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {\left (7 c^2\right ) \int \frac {\sec (e+f x) (c-c \sec (e+f x))^2}{a+a \sec (e+f x)} \, dx}{3 a^2}\\ &=\frac {2 c (c-c \sec (e+f x))^3 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}-\frac {14 \left (c^2-c^2 \sec (e+f x)\right )^2 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {14 \left (c^4-c^4 \sec (e+f x)\right ) \tan (e+f x)}{3 f \left (a^3+a^3 \sec (e+f x)\right )}-\frac {\left (7 c^3\right ) \int \sec (e+f x) (c-c \sec (e+f x)) \, dx}{a^3}\\ &=\frac {2 c (c-c \sec (e+f x))^3 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}-\frac {14 \left (c^2-c^2 \sec (e+f x)\right )^2 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {14 \left (c^4-c^4 \sec (e+f x)\right ) \tan (e+f x)}{3 f \left (a^3+a^3 \sec (e+f x)\right )}-\frac {\left (7 c^4\right ) \int \sec (e+f x) \, dx}{a^3}+\frac {\left (7 c^4\right ) \int \sec ^2(e+f x) \, dx}{a^3}\\ &=-\frac {7 c^4 \tanh ^{-1}(\sin (e+f x))}{a^3 f}+\frac {2 c (c-c \sec (e+f x))^3 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}-\frac {14 \left (c^2-c^2 \sec (e+f x)\right )^2 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {14 \left (c^4-c^4 \sec (e+f x)\right ) \tan (e+f x)}{3 f \left (a^3+a^3 \sec (e+f x)\right )}-\frac {\left (7 c^4\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{a^3 f}\\ &=-\frac {7 c^4 \tanh ^{-1}(\sin (e+f x))}{a^3 f}+\frac {7 c^4 \tan (e+f x)}{a^3 f}+\frac {2 c (c-c \sec (e+f x))^3 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}-\frac {14 \left (c^2-c^2 \sec (e+f x)\right )^2 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac {14 \left (c^4-c^4 \sec (e+f x)\right ) \tan (e+f x)}{3 f \left (a^3+a^3 \sec (e+f x)\right )}\\ \end {align*}
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Mathematica [B] time = 6.32, size = 826, normalized size = 5.04 \[ \frac {2 \cos (e+f x) \cot \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec \left (\frac {e}{2}\right ) (c-c \sec (e+f x))^4 \sin \left (\frac {f x}{2}\right ) \csc ^7\left (\frac {e}{2}+\frac {f x}{2}\right )}{5 f (\sec (e+f x) a+a)^3}+\frac {2 \cos (e+f x) \cot ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) (c-c \sec (e+f x))^4 \tan \left (\frac {e}{2}\right ) \csc ^6\left (\frac {e}{2}+\frac {f x}{2}\right )}{5 f (\sec (e+f x) a+a)^3}+\frac {8 \cos (e+f x) \cot ^3\left (\frac {e}{2}+\frac {f x}{2}\right ) \sec \left (\frac {e}{2}\right ) (c-c \sec (e+f x))^4 \sin \left (\frac {f x}{2}\right ) \csc ^5\left (\frac {e}{2}+\frac {f x}{2}\right )}{15 f (\sec (e+f x) a+a)^3}+\frac {8 \cos (e+f x) \cot ^4\left (\frac {e}{2}+\frac {f x}{2}\right ) (c-c \sec (e+f x))^4 \tan \left (\frac {e}{2}\right ) \csc ^4\left (\frac {e}{2}+\frac {f x}{2}\right )}{15 f (\sec (e+f x) a+a)^3}+\frac {76 \cos (e+f x) \cot ^5\left (\frac {e}{2}+\frac {f x}{2}\right ) \sec \left (\frac {e}{2}\right ) (c-c \sec (e+f x))^4 \sin \left (\frac {f x}{2}\right ) \csc ^3\left (\frac {e}{2}+\frac {f x}{2}\right )}{15 f (\sec (e+f x) a+a)^3}+\frac {7 \cos (e+f x) \cot ^6\left (\frac {e}{2}+\frac {f x}{2}\right ) \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )-\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right ) (c-c \sec (e+f x))^4 \csc ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 f (\sec (e+f x) a+a)^3}-\frac {7 \cos (e+f x) \cot ^6\left (\frac {e}{2}+\frac {f x}{2}\right ) \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right ) (c-c \sec (e+f x))^4 \csc ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 f (\sec (e+f x) a+a)^3}+\frac {\cos (e+f x) \cot ^6\left (\frac {e}{2}+\frac {f x}{2}\right ) (c-c \sec (e+f x))^4 \sin \left (\frac {f x}{2}\right ) \csc ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 f (\sec (e+f x) a+a)^3 \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )-\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}+\frac {\cos (e+f x) \cot ^6\left (\frac {e}{2}+\frac {f x}{2}\right ) (c-c \sec (e+f x))^4 \sin \left (\frac {f x}{2}\right ) \csc ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 f (\sec (e+f x) a+a)^3 \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 231, normalized size = 1.41 \[ -\frac {105 \, {\left (c^{4} \cos \left (f x + e\right )^{4} + 3 \, c^{4} \cos \left (f x + e\right )^{3} + 3 \, c^{4} \cos \left (f x + e\right )^{2} + c^{4} \cos \left (f x + e\right )\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 105 \, {\left (c^{4} \cos \left (f x + e\right )^{4} + 3 \, c^{4} \cos \left (f x + e\right )^{3} + 3 \, c^{4} \cos \left (f x + e\right )^{2} + c^{4} \cos \left (f x + e\right )\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (167 \, c^{4} \cos \left (f x + e\right )^{3} + 381 \, c^{4} \cos \left (f x + e\right )^{2} + 277 \, c^{4} \cos \left (f x + e\right ) + 15 \, c^{4}\right )} \sin \left (f x + e\right )}{30 \, {\left (a^{3} f \cos \left (f x + e\right )^{4} + 3 \, a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} + a^{3} f \cos \left (f x + e\right )\right )}} \]
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 = 0.68, size = 160, normalized size = 0.98 \[ \frac {4 c^{4} \left (\tan ^{5}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{5 f \,a^{3}}+\frac {8 c^{4} \left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{3 f \,a^{3}}+\frac {12 c^{4} \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{f \,a^{3}}-\frac {c^{4}}{f \,a^{3} \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )}+\frac {7 c^{4} \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )}{f \,a^{3}}-\frac {c^{4}}{f \,a^{3} \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )}-\frac {7 c^{4} \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )}{f \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 470, normalized size = 2.87 \[ \frac {3 \, c^{4} {\left (\frac {40 \, \sin \left (f x + e\right )}{{\left (a^{3} - \frac {a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (f x + e\right ) + 1\right )}} + \frac {\frac {85 \, \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}}}{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 )} + 4 \, 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 )} + \frac {6 \, 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 {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 {12 \, 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 = 1.63, size = 126, normalized size = 0.77 \[ \frac {12\,c^4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{a^3\,f}+\frac {8\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3\,a^3\,f}+\frac {4\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{5\,a^3\,f}-\frac {14\,c^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{a^3\,f}-\frac {2\,c^4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left (a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-a^3\right )} \]
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 \[ \frac {c^{4} \left (\int \frac {\sec {\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {4 \sec ^{2}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {6 \sec ^{3}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {4 \sec ^{4}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {\sec ^{5}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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