Optimal. Leaf size=105 \[ \frac {a c^4 \tan ^5(e+f x)}{5 f}+\frac {4 a c^4 \tan ^3(e+f x)}{3 f}+\frac {7 a c^4 \tanh ^{-1}(\sin (e+f x))}{8 f}-\frac {3 a c^4 \tan (e+f x) \sec ^3(e+f x)}{4 f}-\frac {a c^4 \tan (e+f x) \sec (e+f x)}{8 f} \]
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Rubi [A] time = 0.20, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {3958, 2611, 3770, 2607, 30, 3768, 14} \[ \frac {a c^4 \tan ^5(e+f x)}{5 f}+\frac {4 a c^4 \tan ^3(e+f x)}{3 f}+\frac {7 a c^4 \tanh ^{-1}(\sin (e+f x))}{8 f}-\frac {3 a c^4 \tan (e+f x) \sec ^3(e+f x)}{4 f}-\frac {a c^4 \tan (e+f x) \sec (e+f x)}{8 f} \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 2607
Rule 2611
Rule 3768
Rule 3770
Rule 3958
Rubi steps
\begin {align*} \int \sec (e+f x) (a+a \sec (e+f x)) (c-c \sec (e+f x))^4 \, dx &=-\left ((a c) \int \left (c^3 \sec (e+f x) \tan ^2(e+f x)-3 c^3 \sec ^2(e+f x) \tan ^2(e+f x)+3 c^3 \sec ^3(e+f x) \tan ^2(e+f x)-c^3 \sec ^4(e+f x) \tan ^2(e+f x)\right ) \, dx\right )\\ &=-\left (\left (a c^4\right ) \int \sec (e+f x) \tan ^2(e+f x) \, dx\right )+\left (a c^4\right ) \int \sec ^4(e+f x) \tan ^2(e+f x) \, dx+\left (3 a c^4\right ) \int \sec ^2(e+f x) \tan ^2(e+f x) \, dx-\left (3 a c^4\right ) \int \sec ^3(e+f x) \tan ^2(e+f x) \, dx\\ &=-\frac {a c^4 \sec (e+f x) \tan (e+f x)}{2 f}-\frac {3 a c^4 \sec ^3(e+f x) \tan (e+f x)}{4 f}+\frac {1}{2} \left (a c^4\right ) \int \sec (e+f x) \, dx+\frac {1}{4} \left (3 a c^4\right ) \int \sec ^3(e+f x) \, dx+\frac {\left (a c^4\right ) \operatorname {Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,\tan (e+f x)\right )}{f}+\frac {\left (3 a c^4\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {a c^4 \tanh ^{-1}(\sin (e+f x))}{2 f}-\frac {a c^4 \sec (e+f x) \tan (e+f x)}{8 f}-\frac {3 a c^4 \sec ^3(e+f x) \tan (e+f x)}{4 f}+\frac {a c^4 \tan ^3(e+f x)}{f}+\frac {1}{8} \left (3 a c^4\right ) \int \sec (e+f x) \, dx+\frac {\left (a c^4\right ) \operatorname {Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {7 a c^4 \tanh ^{-1}(\sin (e+f x))}{8 f}-\frac {a c^4 \sec (e+f x) \tan (e+f x)}{8 f}-\frac {3 a c^4 \sec ^3(e+f x) \tan (e+f x)}{4 f}+\frac {4 a c^4 \tan ^3(e+f x)}{3 f}+\frac {a c^4 \tan ^5(e+f x)}{5 f}\\ \end {align*}
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Mathematica [B] time = 1.70, size = 499, normalized size = 4.75 \[ -\frac {a c^4 \sec (e) \sec ^5(e+f x) \left (-1920 \sin (2 e+f x)+780 \sin (e+2 f x)+780 \sin (3 e+2 f x)+640 \sin (2 e+3 f x)-720 \sin (4 e+3 f x)+30 \sin (3 e+4 f x)+30 \sin (5 e+4 f x)+272 \sin (4 e+5 f x)+525 \cos (2 e+3 f x) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+525 \cos (4 e+3 f x) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+105 \cos (4 e+5 f x) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+105 \cos (6 e+5 f x) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+1050 \cos (f x) \left (\log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )\right )+1050 \cos (2 e+f x) \left (\log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )\right )-525 \cos (2 e+3 f x) \log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )-525 \cos (4 e+3 f x) \log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )-105 \cos (4 e+5 f x) \log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )-105 \cos (6 e+5 f x) \log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )+800 \sin (f x)\right )}{3840 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 131, normalized size = 1.25 \[ \frac {105 \, a c^{4} \cos \left (f x + e\right )^{5} \log \left (\sin \left (f x + e\right ) + 1\right ) - 105 \, a c^{4} \cos \left (f x + e\right )^{5} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (136 \, a c^{4} \cos \left (f x + e\right )^{4} + 15 \, a c^{4} \cos \left (f x + e\right )^{3} - 112 \, a c^{4} \cos \left (f x + e\right )^{2} + 90 \, a c^{4} \cos \left (f x + e\right ) - 24 \, a c^{4}\right )} \sin \left (f x + e\right )}{240 \, f \cos \left (f x + e\right )^{5}} \]
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 = 1.72, size = 130, normalized size = 1.24 \[ -\frac {3 a \,c^{4} \left (\sec ^{3}\left (f x +e \right )\right ) \tan \left (f x +e \right )}{4 f}-\frac {a \,c^{4} \sec \left (f x +e \right ) \tan \left (f x +e \right )}{8 f}+\frac {7 a \,c^{4} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8 f}-\frac {17 a \,c^{4} \tan \left (f x +e \right )}{15 f}+\frac {14 a \,c^{4} \tan \left (f x +e \right ) \left (\sec ^{2}\left (f x +e \right )\right )}{15 f}+\frac {a \,c^{4} \tan \left (f x +e \right ) \left (\sec ^{4}\left (f x +e \right )\right )}{5 f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.65, size = 215, normalized size = 2.05 \[ \frac {16 \, {\left (3 \, \tan \left (f x + e\right )^{5} + 10 \, \tan \left (f x + e\right )^{3} + 15 \, \tan \left (f x + e\right )\right )} a c^{4} + 160 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a c^{4} + 45 \, a c^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (f x + e\right )^{3} - 5 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{4} - 2 \, \sin \left (f x + e\right )^{2} + 1} - 3 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 3 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 120 \, a c^{4} {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 240 \, a c^{4} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) - 720 \, a c^{4} \tan \left (f x + e\right )}{240 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.64, size = 176, normalized size = 1.68 \[ \frac {7\,a\,c^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{4\,f}-\frac {\frac {7\,a\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9}{4}+\frac {79\,a\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{6}-\frac {224\,a\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{15}+\frac {49\,a\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{6}-\frac {7\,a\,c^4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{4}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )} \]
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 \[ a c^{4} \left (\int \sec {\left (e + f x \right )}\, dx + \int \left (- 3 \sec ^{2}{\left (e + f x \right )}\right )\, dx + \int 2 \sec ^{3}{\left (e + f x \right )}\, dx + \int 2 \sec ^{4}{\left (e + f x \right )}\, dx + \int \left (- 3 \sec ^{5}{\left (e + f x \right )}\right )\, dx + \int \sec ^{6}{\left (e + f x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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