Optimal. Leaf size=100 \[ -\frac {10 a^3 \tan (e+f x)}{c f}-\frac {15 a^3 \tanh ^{-1}(\sin (e+f x))}{2 c f}-\frac {5 a^3 \tan (e+f x) \sec (e+f x)}{2 c f}-\frac {2 a \tan (e+f x) (a \sec (e+f x)+a)^2}{f (c-c \sec (e+f x))} \]
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Rubi [A] time = 0.13, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3957, 3788, 3767, 8, 4046, 3770} \[ -\frac {10 a^3 \tan (e+f x)}{c f}-\frac {15 a^3 \tanh ^{-1}(\sin (e+f x))}{2 c f}-\frac {5 a^3 \tan (e+f x) \sec (e+f x)}{2 c f}-\frac {2 a \tan (e+f x) (a \sec (e+f x)+a)^2}{f (c-c \sec (e+f x))} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 3788
Rule 3957
Rule 4046
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{c-c \sec (e+f x)} \, dx &=-\frac {2 a (a+a \sec (e+f x))^2 \tan (e+f x)}{f (c-c \sec (e+f x))}-\frac {(5 a) \int \sec (e+f x) (a+a \sec (e+f x))^2 \, dx}{c}\\ &=-\frac {2 a (a+a \sec (e+f x))^2 \tan (e+f x)}{f (c-c \sec (e+f x))}-\frac {(5 a) \int \sec (e+f x) \left (a^2+a^2 \sec ^2(e+f x)\right ) \, dx}{c}-\frac {\left (10 a^3\right ) \int \sec ^2(e+f x) \, dx}{c}\\ &=-\frac {5 a^3 \sec (e+f x) \tan (e+f x)}{2 c f}-\frac {2 a (a+a \sec (e+f x))^2 \tan (e+f x)}{f (c-c \sec (e+f x))}-\frac {\left (15 a^3\right ) \int \sec (e+f x) \, dx}{2 c}+\frac {\left (10 a^3\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{c f}\\ &=-\frac {15 a^3 \tanh ^{-1}(\sin (e+f x))}{2 c f}-\frac {10 a^3 \tan (e+f x)}{c f}-\frac {5 a^3 \sec (e+f x) \tan (e+f x)}{2 c f}-\frac {2 a (a+a \sec (e+f x))^2 \tan (e+f x)}{f (c-c \sec (e+f x))}\\ \end {align*}
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Mathematica [B] time = 2.64, size = 287, normalized size = 2.87 \[ \frac {a^3 \cos ^2(e+f x) \tan \left (\frac {1}{2} (e+f x)\right ) \sec ^4\left (\frac {1}{2} (e+f x)\right ) (\sec (e+f x)+1)^3 \left (32 \csc \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right ) \sec \left (\frac {1}{2} (e+f x)\right )+\tan \left (\frac {1}{2} (e+f x)\right ) \left (\frac {16 \sin (f x)}{\left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\sin \left (\frac {e}{2}\right )+\cos \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )}+\frac {1}{\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}-\frac {1}{\left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2}-30 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+30 \log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )\right )\right )}{16 f (c-c \sec (e+f x))} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 125, normalized size = 1.25 \[ -\frac {15 \, a^{3} \cos \left (f x + e\right )^{2} \log \left (\sin \left (f x + e\right ) + 1\right ) \sin \left (f x + e\right ) - 15 \, a^{3} \cos \left (f x + e\right )^{2} \log \left (-\sin \left (f x + e\right ) + 1\right ) \sin \left (f x + e\right ) - 48 \, a^{3} \cos \left (f x + e\right )^{3} - 34 \, a^{3} \cos \left (f x + e\right )^{2} + 16 \, a^{3} \cos \left (f x + e\right ) + 2 \, a^{3}}{4 \, c f \cos \left (f x + e\right )^{2} \sin \left (f x + e\right )} \]
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 = 0.70, size = 166, normalized size = 1.66 \[ \frac {8 a^{3}}{f c \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}-\frac {a^{3}}{2 f c \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )^{2}}+\frac {7 a^{3}}{2 f c \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )}+\frac {15 a^{3} \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )}{2 f c}+\frac {a^{3}}{2 f c \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )^{2}}+\frac {7 a^{3}}{2 f c \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )}-\frac {15 a^{3} \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )}{2 f c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 387, normalized size = 3.87 \[ -\frac {a^{3} {\left (\frac {2 \, {\left (\frac {5 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {2 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - 1\right )}}{\frac {c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {2 \, c \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {c \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}} + \frac {3 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{c} - \frac {3 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{c}\right )} + 6 \, a^{3} {\left (\frac {\frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - 1}{\frac {c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {c \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{c} - \frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{c}\right )} + 6 \, a^{3} {\left (\frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{c} - \frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{c} - \frac {\cos \left (f x + e\right ) + 1}{c \sin \left (f x + e\right )}\right )} - \frac {2 \, a^{3} {\left (\cos \left (f x + e\right ) + 1\right )}}{c \sin \left (f x + e\right )}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.17, size = 105, normalized size = 1.05 \[ \frac {15\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-25\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+8\,a^3}{f\,\left (c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5-2\,c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+c\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}-\frac {15\,a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{c\,f} \]
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 \[ - \frac {a^{3} \left (\int \frac {\sec {\left (e + f x \right )}}{\sec {\left (e + f x \right )} - 1}\, dx + \int \frac {3 \sec ^{2}{\left (e + f x \right )}}{\sec {\left (e + f x \right )} - 1}\, dx + \int \frac {3 \sec ^{3}{\left (e + f x \right )}}{\sec {\left (e + f x \right )} - 1}\, dx + \int \frac {\sec ^{4}{\left (e + f x \right )}}{\sec {\left (e + f x \right )} - 1}\, dx\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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