Integrand size = 26, antiderivative size = 140 \[ \int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^4 \, dx=a^2 c^4 x-\frac {3 a^2 c^4 \text {arctanh}(\sin (e+f x))}{4 f}-\frac {a^2 c^4 \tan (e+f x)}{f}+\frac {3 a^2 c^4 \sec (e+f x) \tan (e+f x)}{4 f}+\frac {a^2 c^4 \tan ^3(e+f x)}{3 f}-\frac {a^2 c^4 \sec (e+f x) \tan ^3(e+f x)}{2 f}+\frac {a^2 c^4 \tan ^5(e+f x)}{5 f} \]
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Time = 0.26 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3989, 3971, 3554, 8, 2691, 3855, 2687, 30} \[ \int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^4 \, dx=-\frac {3 a^2 c^4 \text {arctanh}(\sin (e+f x))}{4 f}+\frac {a^2 c^4 \tan ^5(e+f x)}{5 f}+\frac {a^2 c^4 \tan ^3(e+f x)}{3 f}-\frac {a^2 c^4 \tan (e+f x)}{f}-\frac {a^2 c^4 \tan ^3(e+f x) \sec (e+f x)}{2 f}+\frac {3 a^2 c^4 \tan (e+f x) \sec (e+f x)}{4 f}+a^2 c^4 x \]
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Rule 8
Rule 30
Rule 2687
Rule 2691
Rule 3554
Rule 3855
Rule 3971
Rule 3989
Rubi steps \begin{align*} \text {integral}& = \left (a^2 c^2\right ) \int (c-c \sec (e+f x))^2 \tan ^4(e+f x) \, dx \\ & = \left (a^2 c^2\right ) \int \left (c^2 \tan ^4(e+f x)-2 c^2 \sec (e+f x) \tan ^4(e+f x)+c^2 \sec ^2(e+f x) \tan ^4(e+f x)\right ) \, dx \\ & = \left (a^2 c^4\right ) \int \tan ^4(e+f x) \, dx+\left (a^2 c^4\right ) \int \sec ^2(e+f x) \tan ^4(e+f x) \, dx-\left (2 a^2 c^4\right ) \int \sec (e+f x) \tan ^4(e+f x) \, dx \\ & = \frac {a^2 c^4 \tan ^3(e+f x)}{3 f}-\frac {a^2 c^4 \sec (e+f x) \tan ^3(e+f x)}{2 f}-\left (a^2 c^4\right ) \int \tan ^2(e+f x) \, dx+\frac {1}{2} \left (3 a^2 c^4\right ) \int \sec (e+f x) \tan ^2(e+f x) \, dx+\frac {\left (a^2 c^4\right ) \text {Subst}\left (\int x^4 \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {a^2 c^4 \tan (e+f x)}{f}+\frac {3 a^2 c^4 \sec (e+f x) \tan (e+f x)}{4 f}+\frac {a^2 c^4 \tan ^3(e+f x)}{3 f}-\frac {a^2 c^4 \sec (e+f x) \tan ^3(e+f x)}{2 f}+\frac {a^2 c^4 \tan ^5(e+f x)}{5 f}-\frac {1}{4} \left (3 a^2 c^4\right ) \int \sec (e+f x) \, dx+\left (a^2 c^4\right ) \int 1 \, dx \\ & = a^2 c^4 x-\frac {3 a^2 c^4 \text {arctanh}(\sin (e+f x))}{4 f}-\frac {a^2 c^4 \tan (e+f x)}{f}+\frac {3 a^2 c^4 \sec (e+f x) \tan (e+f x)}{4 f}+\frac {a^2 c^4 \tan ^3(e+f x)}{3 f}-\frac {a^2 c^4 \sec (e+f x) \tan ^3(e+f x)}{2 f}+\frac {a^2 c^4 \tan ^5(e+f x)}{5 f} \\ \end{align*}
Time = 1.16 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.04 \[ \int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^4 \, dx=\frac {a^2 c^4 \sec ^5(e+f x) \left (600 (e+f x) \cos (e+f x)-720 \text {arctanh}(\sin (e+f x)) \cos ^5(e+f x)+300 e \cos (3 (e+f x))+300 f x \cos (3 (e+f x))+60 e \cos (5 (e+f x))+60 f x \cos (5 (e+f x))+40 \sin (e+f x)+60 \sin (2 (e+f x))-220 \sin (3 (e+f x))+150 \sin (4 (e+f x))-68 \sin (5 (e+f x))\right )}{960 f} \]
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Time = 3.13 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.23
method | result | size |
parts | \(a^{2} c^{4} x -\frac {c^{4} a^{2} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )}{f}-\frac {a^{2} c^{4} \tan \left (f x +e \right )}{f}+\frac {2 a^{2} c^{4} \sec \left (f x +e \right ) \tan \left (f x +e \right )}{f}+\frac {c^{4} a^{2} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )}{f}-\frac {2 c^{4} a^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )}{f}\) | \(172\) |
risch | \(a^{2} c^{4} x -\frac {i c^{4} a^{2} \left (75 \,{\mathrm e}^{9 i \left (f x +e \right )}+60 \,{\mathrm e}^{8 i \left (f x +e \right )}+30 \,{\mathrm e}^{7 i \left (f x +e \right )}+360 \,{\mathrm e}^{6 i \left (f x +e \right )}+320 \,{\mathrm e}^{4 i \left (f x +e \right )}-30 \,{\mathrm e}^{3 i \left (f x +e \right )}+280 \,{\mathrm e}^{2 i \left (f x +e \right )}-75 \,{\mathrm e}^{i \left (f x +e \right )}+68\right )}{30 f \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{5}}-\frac {3 c^{4} a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{4 f}+\frac {3 c^{4} a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{4 f}\) | \(173\) |
derivativedivides | \(\frac {-c^{4} a^{2} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )-2 c^{4} a^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )+c^{4} a^{2} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+4 c^{4} a^{2} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-c^{4} a^{2} \tan \left (f x +e \right )-2 c^{4} a^{2} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+c^{4} a^{2} \left (f x +e \right )}{f}\) | \(206\) |
default | \(\frac {-c^{4} a^{2} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )-2 c^{4} a^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )+c^{4} a^{2} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+4 c^{4} a^{2} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-c^{4} a^{2} \tan \left (f x +e \right )-2 c^{4} a^{2} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+c^{4} a^{2} \left (f x +e \right )}{f}\) | \(206\) |
parallelrisch | \(\frac {a^{2} c^{4} \left (\left (\frac {15 \cos \left (f x +e \right )}{2}+\frac {15 \cos \left (3 f x +3 e \right )}{4}+\frac {3 \cos \left (5 f x +5 e \right )}{4}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )+\left (-\frac {15 \cos \left (f x +e \right )}{2}-\frac {15 \cos \left (3 f x +3 e \right )}{4}-\frac {3 \cos \left (5 f x +5 e \right )}{4}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )+10 f x \cos \left (f x +e \right )+5 f x \cos \left (3 f x +3 e \right )+f x \cos \left (5 f x +5 e \right )+\sin \left (2 f x +2 e \right )-\frac {11 \sin \left (3 f x +3 e \right )}{3}+\frac {5 \sin \left (4 f x +4 e \right )}{2}-\frac {17 \sin \left (5 f x +5 e \right )}{15}+\frac {2 \sin \left (f x +e \right )}{3}\right )}{f \left (\cos \left (5 f x +5 e \right )+5 \cos \left (3 f x +3 e \right )+10 \cos \left (f x +e \right )\right )}\) | \(216\) |
norman | \(\frac {a^{2} c^{4} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}-a^{2} c^{4} x +5 a^{2} c^{4} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-10 a^{2} c^{4} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+10 a^{2} c^{4} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-5 a^{2} c^{4} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}+\frac {c^{4} a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 f}-\frac {11 c^{4} a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3 f}+\frac {164 c^{4} a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{15 f}-\frac {53 c^{4} a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{3 f}+\frac {7 c^{4} a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{2 f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{5}}+\frac {3 c^{4} a^{2} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{4 f}-\frac {3 c^{4} a^{2} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{4 f}\) | \(281\) |
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Time = 0.27 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.16 \[ \int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^4 \, dx=\frac {120 \, a^{2} c^{4} f x \cos \left (f x + e\right )^{5} - 45 \, a^{2} c^{4} \cos \left (f x + e\right )^{5} \log \left (\sin \left (f x + e\right ) + 1\right ) + 45 \, a^{2} c^{4} \cos \left (f x + e\right )^{5} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (68 \, a^{2} c^{4} \cos \left (f x + e\right )^{4} - 75 \, a^{2} c^{4} \cos \left (f x + e\right )^{3} + 4 \, a^{2} c^{4} \cos \left (f x + e\right )^{2} + 30 \, a^{2} c^{4} \cos \left (f x + e\right ) - 12 \, a^{2} c^{4}\right )} \sin \left (f x + e\right )}{120 \, f \cos \left (f x + e\right )^{5}} \]
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\[ \int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^4 \, dx=a^{2} c^{4} \left (\int 1\, dx + \int \left (- 2 \sec {\left (e + f x \right )}\right )\, dx + \int \left (- \sec ^{2}{\left (e + f x \right )}\right )\, dx + \int 4 \sec ^{3}{\left (e + f x \right )}\, dx + \int \left (- \sec ^{4}{\left (e + f x \right )}\right )\, dx + \int \left (- 2 \sec ^{5}{\left (e + f x \right )}\right )\, dx + \int \sec ^{6}{\left (e + f x \right )}\, dx\right ) \]
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Time = 0.21 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.71 \[ \int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^4 \, dx=\frac {8 \, {\left (3 \, \tan \left (f x + e\right )^{5} + 10 \, \tan \left (f x + e\right )^{3} + 15 \, \tan \left (f x + e\right )\right )} a^{2} c^{4} - 40 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{2} c^{4} + 120 \, {\left (f x + e\right )} a^{2} c^{4} + 15 \, a^{2} c^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (f x + e\right )^{3} - 5 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{4} - 2 \, \sin \left (f x + e\right )^{2} + 1} - 3 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 3 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 120 \, a^{2} c^{4} {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 240 \, a^{2} c^{4} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) - 120 \, a^{2} c^{4} \tan \left (f x + e\right )}{120 \, f} \]
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Time = 0.37 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.23 \[ \int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^4 \, dx=\frac {60 \, {\left (f x + e\right )} a^{2} c^{4} - 45 \, a^{2} c^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) + 45 \, a^{2} c^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) + \frac {2 \, {\left (105 \, a^{2} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} - 530 \, a^{2} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 328 \, a^{2} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 110 \, a^{2} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15 \, a^{2} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{5}}}{60 \, f} \]
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Time = 15.15 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.39 \[ \int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^4 \, dx=a^2\,c^4\,x+\frac {\frac {7\,a^2\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9}{2}-\frac {53\,a^2\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{3}+\frac {164\,a^2\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{15}-\frac {11\,a^2\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3}+\frac {a^2\,c^4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{2}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )}-\frac {3\,a^2\,c^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{2\,f} \]
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