Optimal. Leaf size=188 \[ -\frac {a^3 c^5 \tan ^7(e+f x)}{7 f}-\frac {a^3 c^5 \tan ^5(e+f x)}{5 f}+\frac {a^3 c^5 \tan ^3(e+f x)}{3 f}-\frac {a^3 c^5 \tan (e+f x)}{f}-\frac {5 a^3 c^5 \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac {a^3 c^5 \tan ^5(e+f x) \sec (e+f x)}{3 f}-\frac {5 a^3 c^5 \tan ^3(e+f x) \sec (e+f x)}{12 f}+\frac {5 a^3 c^5 \tan (e+f x) \sec (e+f x)}{8 f}+a^3 c^5 x \]
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Rubi [A] time = 0.24, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3904, 3886, 3473, 8, 2611, 3770, 2607, 30} \[ -\frac {a^3 c^5 \tan ^7(e+f x)}{7 f}-\frac {a^3 c^5 \tan ^5(e+f x)}{5 f}+\frac {a^3 c^5 \tan ^3(e+f x)}{3 f}-\frac {a^3 c^5 \tan (e+f x)}{f}-\frac {5 a^3 c^5 \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac {a^3 c^5 \tan ^5(e+f x) \sec (e+f x)}{3 f}-\frac {5 a^3 c^5 \tan ^3(e+f x) \sec (e+f x)}{12 f}+\frac {5 a^3 c^5 \tan (e+f x) \sec (e+f x)}{8 f}+a^3 c^5 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2607
Rule 2611
Rule 3473
Rule 3770
Rule 3886
Rule 3904
Rubi steps
\begin {align*} \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5 \, dx &=-\left (\left (a^3 c^3\right ) \int (c-c \sec (e+f x))^2 \tan ^6(e+f x) \, dx\right )\\ &=-\left (\left (a^3 c^3\right ) \int \left (c^2 \tan ^6(e+f x)-2 c^2 \sec (e+f x) \tan ^6(e+f x)+c^2 \sec ^2(e+f x) \tan ^6(e+f x)\right ) \, dx\right )\\ &=-\left (\left (a^3 c^5\right ) \int \tan ^6(e+f x) \, dx\right )-\left (a^3 c^5\right ) \int \sec ^2(e+f x) \tan ^6(e+f x) \, dx+\left (2 a^3 c^5\right ) \int \sec (e+f x) \tan ^6(e+f x) \, dx\\ &=-\frac {a^3 c^5 \tan ^5(e+f x)}{5 f}+\frac {a^3 c^5 \sec (e+f x) \tan ^5(e+f x)}{3 f}+\left (a^3 c^5\right ) \int \tan ^4(e+f x) \, dx-\frac {1}{3} \left (5 a^3 c^5\right ) \int \sec (e+f x) \tan ^4(e+f x) \, dx-\frac {\left (a^3 c^5\right ) \operatorname {Subst}\left (\int x^6 \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {a^3 c^5 \tan ^3(e+f x)}{3 f}-\frac {5 a^3 c^5 \sec (e+f x) \tan ^3(e+f x)}{12 f}-\frac {a^3 c^5 \tan ^5(e+f x)}{5 f}+\frac {a^3 c^5 \sec (e+f x) \tan ^5(e+f x)}{3 f}-\frac {a^3 c^5 \tan ^7(e+f x)}{7 f}-\left (a^3 c^5\right ) \int \tan ^2(e+f x) \, dx+\frac {1}{4} \left (5 a^3 c^5\right ) \int \sec (e+f x) \tan ^2(e+f x) \, dx\\ &=-\frac {a^3 c^5 \tan (e+f x)}{f}+\frac {5 a^3 c^5 \sec (e+f x) \tan (e+f x)}{8 f}+\frac {a^3 c^5 \tan ^3(e+f x)}{3 f}-\frac {5 a^3 c^5 \sec (e+f x) \tan ^3(e+f x)}{12 f}-\frac {a^3 c^5 \tan ^5(e+f x)}{5 f}+\frac {a^3 c^5 \sec (e+f x) \tan ^5(e+f x)}{3 f}-\frac {a^3 c^5 \tan ^7(e+f x)}{7 f}-\frac {1}{8} \left (5 a^3 c^5\right ) \int \sec (e+f x) \, dx+\left (a^3 c^5\right ) \int 1 \, dx\\ &=a^3 c^5 x-\frac {5 a^3 c^5 \tanh ^{-1}(\sin (e+f x))}{8 f}-\frac {a^3 c^5 \tan (e+f x)}{f}+\frac {5 a^3 c^5 \sec (e+f x) \tan (e+f x)}{8 f}+\frac {a^3 c^5 \tan ^3(e+f x)}{3 f}-\frac {5 a^3 c^5 \sec (e+f x) \tan ^3(e+f x)}{12 f}-\frac {a^3 c^5 \tan ^5(e+f x)}{5 f}+\frac {a^3 c^5 \sec (e+f x) \tan ^5(e+f x)}{3 f}-\frac {a^3 c^5 \tan ^7(e+f x)}{7 f}\\ \end {align*}
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Mathematica [A] time = 2.24, size = 189, normalized size = 1.01 \[ \frac {a^3 c^5 \sec ^7(e+f x) \left (-4200 \sin (e+f x)+2975 \sin (2 (e+f x))-2184 \sin (3 (e+f x))+980 \sin (4 (e+f x))-2408 \sin (5 (e+f x))+1155 \sin (6 (e+f x))-584 \sin (7 (e+f x))+14700 (e+f x) \cos (e+f x)+8820 e \cos (3 (e+f x))+8820 f x \cos (3 (e+f x))+2940 e \cos (5 (e+f x))+2940 f x \cos (5 (e+f x))+420 e \cos (7 (e+f x))+420 f x \cos (7 (e+f x))-16800 \cos ^7(e+f x) \tanh ^{-1}(\sin (e+f x))\right )}{26880 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 195, normalized size = 1.04 \[ \frac {1680 \, a^{3} c^{5} f x \cos \left (f x + e\right )^{7} - 525 \, a^{3} c^{5} \cos \left (f x + e\right )^{7} \log \left (\sin \left (f x + e\right ) + 1\right ) + 525 \, a^{3} c^{5} \cos \left (f x + e\right )^{7} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (1168 \, a^{3} c^{5} \cos \left (f x + e\right )^{6} - 1155 \, a^{3} c^{5} \cos \left (f x + e\right )^{5} - 256 \, a^{3} c^{5} \cos \left (f x + e\right )^{4} + 910 \, a^{3} c^{5} \cos \left (f x + e\right )^{3} - 192 \, a^{3} c^{5} \cos \left (f x + e\right )^{2} - 280 \, a^{3} c^{5} \cos \left (f x + e\right ) + 120 \, a^{3} c^{5}\right )} \sin \left (f x + e\right )}{1680 \, f \cos \left (f x + e\right )^{7}} \]
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 = 1.62, size = 211, normalized size = 1.12 \[ -\frac {13 c^{5} a^{3} \tan \left (f x +e \right ) \left (\sec ^{3}\left (f x +e \right )\right )}{12 f}+\frac {11 a^{3} c^{5} \sec \left (f x +e \right ) \tan \left (f x +e \right )}{8 f}-\frac {5 c^{5} a^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8 f}-\frac {146 a^{3} c^{5} \tan \left (f x +e \right )}{105 f}+a^{3} c^{5} x +\frac {a^{3} c^{5} e}{f}+\frac {8 c^{5} a^{3} \tan \left (f x +e \right ) \left (\sec ^{4}\left (f x +e \right )\right )}{35 f}+\frac {32 c^{5} a^{3} \tan \left (f x +e \right ) \left (\sec ^{2}\left (f x +e \right )\right )}{105 f}+\frac {c^{5} a^{3} \tan \left (f x +e \right ) \left (\sec ^{5}\left (f x +e \right )\right )}{3 f}-\frac {c^{5} a^{3} \tan \left (f x +e \right ) \left (\sec ^{6}\left (f x +e \right )\right )}{7 f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 356, normalized size = 1.89 \[ -\frac {48 \, {\left (5 \, \tan \left (f x + e\right )^{7} + 21 \, \tan \left (f x + e\right )^{5} + 35 \, \tan \left (f x + e\right )^{3} + 35 \, \tan \left (f x + e\right )\right )} a^{3} c^{5} - 224 \, {\left (3 \, \tan \left (f x + e\right )^{5} + 10 \, \tan \left (f x + e\right )^{3} + 15 \, \tan \left (f x + e\right )\right )} a^{3} c^{5} - 1680 \, {\left (f x + e\right )} a^{3} c^{5} + 35 \, a^{3} c^{5} {\left (\frac {2 \, {\left (15 \, \sin \left (f x + e\right )^{5} - 40 \, \sin \left (f x + e\right )^{3} + 33 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{6} - 3 \, \sin \left (f x + e\right )^{4} + 3 \, \sin \left (f x + e\right )^{2} - 1} - 15 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 15 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 630 \, a^{3} c^{5} {\left (\frac {2 \, {\left (3 \, \sin \left (f x + e\right )^{3} - 5 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{4} - 2 \, \sin \left (f x + e\right )^{2} + 1} - 3 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 3 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 2520 \, a^{3} c^{5} {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 3360 \, a^{3} c^{5} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) + 3360 \, a^{3} c^{5} \tan \left (f x + e\right )}{1680 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.62, size = 259, normalized size = 1.38 \[ \frac {\frac {13\,a^3\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{13}}{4}-23\,a^3\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}+\frac {1413\,a^3\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9}{20}-\frac {1768\,a^3\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{35}+\frac {1409\,a^3\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{60}-\frac {19\,a^3\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3}+\frac {3\,a^3\,c^5\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{4}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}-7\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )}+a^3\,c^5\,x-\frac {5\,a^3\,c^5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{4\,f} \]
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 \[ - a^{3} c^{5} \left (\int \left (-1\right )\, dx + \int 2 \sec {\left (e + f x \right )}\, dx + \int 2 \sec ^{2}{\left (e + f x \right )}\, dx + \int \left (- 6 \sec ^{3}{\left (e + f x \right )}\right )\, dx + \int 6 \sec ^{5}{\left (e + f x \right )}\, dx + \int \left (- 2 \sec ^{6}{\left (e + f x \right )}\right )\, dx + \int \left (- 2 \sec ^{7}{\left (e + f x \right )}\right )\, dx + \int \sec ^{8}{\left (e + f x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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