Integrand size = 31, antiderivative size = 257 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \, dx=\frac {a^3 \left (20 c^2+30 c d+13 d^2\right ) \text {arctanh}(\sin (e+f x))}{8 f}+\frac {a^3 \left (2 c^4-15 c^3 d+72 c^2 d^2+180 c d^3+76 d^4\right ) \tan (e+f x)}{30 d^2 f}+\frac {a^3 \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right ) \sec (e+f x) \tan (e+f x)}{120 d f}+\frac {a^3 \left (2 c^2-15 c d+76 d^2\right ) (c+d \sec (e+f x))^2 \tan (e+f x)}{60 d^2 f}-\frac {a^3 (2 c-11 d) (c+d \sec (e+f x))^3 \tan (e+f x)}{20 d^2 f}+\frac {\left (a^3+a^3 \sec (e+f x)\right ) (c+d \sec (e+f x))^3 \tan (e+f x)}{5 d f} \]
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Time = 0.36 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4072, 92, 81, 52, 65, 223, 209} \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \, dx=\frac {a^4 \left (20 c^2+30 c d+13 d^2\right ) \tan (e+f x) \arctan \left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a (\sec (e+f x)+1)}}\right )}{4 f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {a^3 \left (20 c^2+30 c d+13 d^2\right ) \tan (e+f x)}{8 f}+\frac {\left (20 c^2+30 c d+13 d^2\right ) \tan (e+f x) \left (a^3 \sec (e+f x)+a^3\right )}{24 f}+\frac {a \left (20 c^2+30 c d+13 d^2\right ) \tan (e+f x) (a \sec (e+f x)+a)^2}{60 f}+\frac {3 d (2 c+d) \tan (e+f x) (a \sec (e+f x)+a)^3}{20 f}+\frac {d \tan (e+f x) (a \sec (e+f x)+a)^3 (c+d \sec (e+f x))}{5 f} \]
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Rule 52
Rule 65
Rule 81
Rule 92
Rule 209
Rule 223
Rule 4072
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a^2 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {(a+a x)^{5/2} (c+d x)^2}{\sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {d (a+a \sec (e+f x))^3 (c+d \sec (e+f x)) \tan (e+f x)}{5 f}+\frac {\tan (e+f x) \text {Subst}\left (\int \frac {(a+a x)^{5/2} \left (-a^2 \left (5 c^2+3 c d+d^2\right )-3 a^2 d (2 c+d) x\right )}{\sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{5 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {3 d (2 c+d) (a+a \sec (e+f x))^3 \tan (e+f x)}{20 f}+\frac {d (a+a \sec (e+f x))^3 (c+d \sec (e+f x)) \tan (e+f x)}{5 f}-\frac {\left (a^2 \left (20 c^2+30 c d+13 d^2\right ) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {(a+a x)^{5/2}}{\sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{20 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {a \left (20 c^2+30 c d+13 d^2\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{60 f}+\frac {3 d (2 c+d) (a+a \sec (e+f x))^3 \tan (e+f x)}{20 f}+\frac {d (a+a \sec (e+f x))^3 (c+d \sec (e+f x)) \tan (e+f x)}{5 f}-\frac {\left (a^3 \left (20 c^2+30 c d+13 d^2\right ) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {(a+a x)^{3/2}}{\sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{12 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {a \left (20 c^2+30 c d+13 d^2\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{60 f}+\frac {3 d (2 c+d) (a+a \sec (e+f x))^3 \tan (e+f x)}{20 f}+\frac {\left (20 c^2+30 c d+13 d^2\right ) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{24 f}+\frac {d (a+a \sec (e+f x))^3 (c+d \sec (e+f x)) \tan (e+f x)}{5 f}-\frac {\left (a^4 \left (20 c^2+30 c d+13 d^2\right ) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {\sqrt {a+a x}}{\sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{8 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {a^3 \left (20 c^2+30 c d+13 d^2\right ) \tan (e+f x)}{8 f}+\frac {a \left (20 c^2+30 c d+13 d^2\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{60 f}+\frac {3 d (2 c+d) (a+a \sec (e+f x))^3 \tan (e+f x)}{20 f}+\frac {\left (20 c^2+30 c d+13 d^2\right ) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{24 f}+\frac {d (a+a \sec (e+f x))^3 (c+d \sec (e+f x)) \tan (e+f x)}{5 f}-\frac {\left (a^5 \left (20 c^2+30 c d+13 d^2\right ) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-a x} \sqrt {a+a x}} \, dx,x,\sec (e+f x)\right )}{8 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {a^3 \left (20 c^2+30 c d+13 d^2\right ) \tan (e+f x)}{8 f}+\frac {a \left (20 c^2+30 c d+13 d^2\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{60 f}+\frac {3 d (2 c+d) (a+a \sec (e+f x))^3 \tan (e+f x)}{20 f}+\frac {\left (20 c^2+30 c d+13 d^2\right ) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{24 f}+\frac {d (a+a \sec (e+f x))^3 (c+d \sec (e+f x)) \tan (e+f x)}{5 f}+\frac {\left (a^4 \left (20 c^2+30 c d+13 d^2\right ) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2 a-x^2}} \, dx,x,\sqrt {a-a \sec (e+f x)}\right )}{4 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {a^3 \left (20 c^2+30 c d+13 d^2\right ) \tan (e+f x)}{8 f}+\frac {a \left (20 c^2+30 c d+13 d^2\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{60 f}+\frac {3 d (2 c+d) (a+a \sec (e+f x))^3 \tan (e+f x)}{20 f}+\frac {\left (20 c^2+30 c d+13 d^2\right ) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{24 f}+\frac {d (a+a \sec (e+f x))^3 (c+d \sec (e+f x)) \tan (e+f x)}{5 f}+\frac {\left (a^4 \left (20 c^2+30 c d+13 d^2\right ) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}}\right )}{4 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {a^3 \left (20 c^2+30 c d+13 d^2\right ) \tan (e+f x)}{8 f}+\frac {a^4 \left (20 c^2+30 c d+13 d^2\right ) \arctan \left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}}\right ) \tan (e+f x)}{4 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {a \left (20 c^2+30 c d+13 d^2\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{60 f}+\frac {3 d (2 c+d) (a+a \sec (e+f x))^3 \tan (e+f x)}{20 f}+\frac {\left (20 c^2+30 c d+13 d^2\right ) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{24 f}+\frac {d (a+a \sec (e+f x))^3 (c+d \sec (e+f x)) \tan (e+f x)}{5 f} \\ \end{align*}
Time = 4.43 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.51 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \, dx=\frac {a^3 \left (15 \left (20 c^2+30 c d+13 d^2\right ) \text {arctanh}(\sin (e+f x))+\tan (e+f x) \left (15 \left (12 c^2+30 c d+13 d^2\right ) \sec (e+f x)+30 d (2 c+3 d) \sec ^3(e+f x)+8 \left (60 (c+d)^2+5 \left (c^2+6 c d+5 d^2\right ) \tan ^2(e+f x)+3 d^2 \tan ^4(e+f x)\right )\right )\right )}{120 f} \]
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Time = 4.54 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00
method | result | size |
norman | \(\frac {-\frac {32 a^{3} \left (20 c^{2}+30 c d +13 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{15 f}+\frac {7 a^{3} \left (20 c^{2}+30 c d +13 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{6 f}-\frac {a^{3} \left (20 c^{2}+30 c d +13 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{4 f}-\frac {a^{3} \left (44 c^{2}+98 c d +51 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}+\frac {a^{3} \left (212 c^{2}+366 c d +133 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{6 f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{5}}-\frac {a^{3} \left (20 c^{2}+30 c d +13 d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{8 f}+\frac {a^{3} \left (20 c^{2}+30 c d +13 d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{8 f}\) | \(257\) |
parts | \(\frac {\left (2 a^{3} c d +3 a^{3} d^{2}\right ) \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )}{f}+\frac {\left (3 c^{2} a^{3}+2 a^{3} c d \right ) \tan \left (f x +e \right )}{f}-\frac {\left (c^{2} a^{3}+6 a^{3} c d +3 a^{3} d^{2}\right ) \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )}{f}+\frac {\left (3 c^{2} a^{3}+6 a^{3} c d +a^{3} d^{2}\right ) \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )}{f}-\frac {a^{3} d^{2} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )}{f}+\frac {c^{2} a^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{f}\) | \(259\) |
parallelrisch | \(\frac {26 a^{3} \left (-\frac {75 \left (\frac {\cos \left (5 f x +5 e \right )}{10}+\frac {\cos \left (3 f x +3 e \right )}{2}+\cos \left (f x +e \right )\right ) \left (c^{2}+\frac {3}{2} c d +\frac {13}{20} d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{26}+\frac {75 \left (\frac {\cos \left (5 f x +5 e \right )}{10}+\frac {\cos \left (3 f x +3 e \right )}{2}+\cos \left (f x +e \right )\right ) \left (c^{2}+\frac {3}{2} c d +\frac {13}{20} d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{26}+\left (\frac {57}{26} c d +\frac {75}{52} d^{2}+\frac {9}{13} c^{2}\right ) \sin \left (2 f x +2 e \right )+\left (3 c d +\frac {19}{13} d^{2}+\frac {37}{26} c^{2}\right ) \sin \left (3 f x +3 e \right )+\left (\frac {45}{52} c d +\frac {3}{8} d^{2}+\frac {9}{26} c^{2}\right ) \sin \left (4 f x +4 e \right )+\left (\frac {9}{13} c d +\frac {19}{65} d^{2}+\frac {11}{26} c^{2}\right ) \sin \left (5 f x +5 e \right )+\sin \left (f x +e \right ) \left (\frac {30}{13} c d +\frac {20}{13} d^{2}+c^{2}\right )\right )}{3 f \left (\cos \left (5 f x +5 e \right )+5 \cos \left (3 f x +3 e \right )+10 \cos \left (f x +e \right )\right )}\) | \(273\) |
derivativedivides | \(\frac {-c^{2} a^{3} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+2 a^{3} c d \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )-a^{3} d^{2} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )+3 c^{2} a^{3} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-6 a^{3} c d \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+3 a^{3} d^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )+3 c^{2} a^{3} \tan \left (f x +e \right )+6 a^{3} c d \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-3 a^{3} d^{2} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+c^{2} a^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+2 a^{3} c d \tan \left (f x +e \right )+a^{3} d^{2} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )}{f}\) | \(385\) |
default | \(\frac {-c^{2} a^{3} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+2 a^{3} c d \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )-a^{3} d^{2} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )+3 c^{2} a^{3} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-6 a^{3} c d \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+3 a^{3} d^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )+3 c^{2} a^{3} \tan \left (f x +e \right )+6 a^{3} c d \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-3 a^{3} d^{2} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+c^{2} a^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+2 a^{3} c d \tan \left (f x +e \right )+a^{3} d^{2} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )}{f}\) | \(385\) |
risch | \(-\frac {i a^{3} \left (-440 c^{2}-4800 c d \,{\mathrm e}^{4 i \left (f x +e \right )}-3360 c d \,{\mathrm e}^{2 i \left (f x +e \right )}-450 d \,{\mathrm e}^{i \left (f x +e \right )} c -720 c d -304 d^{2}-1520 d^{2} {\mathrm e}^{2 i \left (f x +e \right )}-195 d^{2} {\mathrm e}^{i \left (f x +e \right )}-1140 c d \,{\mathrm e}^{3 i \left (f x +e \right )}+450 c d \,{\mathrm e}^{9 i \left (f x +e \right )}-2400 c d \,{\mathrm e}^{6 i \left (f x +e \right )}-240 c d \,{\mathrm e}^{8 i \left (f x +e \right )}+1140 c d \,{\mathrm e}^{7 i \left (f x +e \right )}-2720 c^{2} {\mathrm e}^{4 i \left (f x +e \right )}-1840 c^{2} {\mathrm e}^{2 i \left (f x +e \right )}-360 c^{2} {\mathrm e}^{3 i \left (f x +e \right )}-750 d^{2} {\mathrm e}^{3 i \left (f x +e \right )}+360 c^{2} {\mathrm e}^{7 i \left (f x +e \right )}+180 c^{2} {\mathrm e}^{9 i \left (f x +e \right )}+750 d^{2} {\mathrm e}^{7 i \left (f x +e \right )}-180 c^{2} {\mathrm e}^{i \left (f x +e \right )}-360 c^{2} {\mathrm e}^{8 i \left (f x +e \right )}+195 d^{2} {\mathrm e}^{9 i \left (f x +e \right )}-2320 d^{2} {\mathrm e}^{4 i \left (f x +e \right )}-1680 c^{2} {\mathrm e}^{6 i \left (f x +e \right )}-720 d^{2} {\mathrm e}^{6 i \left (f x +e \right )}\right )}{60 f \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{5}}+\frac {5 c^{2} a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{2 f}+\frac {15 a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) c d}{4 f}+\frac {13 a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) d^{2}}{8 f}-\frac {5 c^{2} a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{2 f}-\frac {15 a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) c d}{4 f}-\frac {13 a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) d^{2}}{8 f}\) | \(495\) |
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Time = 0.28 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.95 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \, dx=\frac {15 \, {\left (20 \, a^{3} c^{2} + 30 \, a^{3} c d + 13 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )^{5} \log \left (\sin \left (f x + e\right ) + 1\right ) - 15 \, {\left (20 \, a^{3} c^{2} + 30 \, a^{3} c d + 13 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )^{5} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (24 \, a^{3} d^{2} + 8 \, {\left (55 \, a^{3} c^{2} + 90 \, a^{3} c d + 38 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )^{4} + 15 \, {\left (12 \, a^{3} c^{2} + 30 \, a^{3} c d + 13 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )^{3} + 8 \, {\left (5 \, a^{3} c^{2} + 30 \, a^{3} c d + 19 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )^{2} + 30 \, {\left (2 \, a^{3} c d + 3 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{240 \, f \cos \left (f x + e\right )^{5}} \]
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\[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \, dx=a^{3} \left (\int c^{2} \sec {\left (e + f x \right )}\, dx + \int 3 c^{2} \sec ^{2}{\left (e + f x \right )}\, dx + \int 3 c^{2} \sec ^{3}{\left (e + f x \right )}\, dx + \int c^{2} \sec ^{4}{\left (e + f x \right )}\, dx + \int d^{2} \sec ^{3}{\left (e + f x \right )}\, dx + \int 3 d^{2} \sec ^{4}{\left (e + f x \right )}\, dx + \int 3 d^{2} \sec ^{5}{\left (e + f x \right )}\, dx + \int d^{2} \sec ^{6}{\left (e + f x \right )}\, dx + \int 2 c d \sec ^{2}{\left (e + f x \right )}\, dx + \int 6 c d \sec ^{3}{\left (e + f x \right )}\, dx + \int 6 c d \sec ^{4}{\left (e + f x \right )}\, dx + \int 2 c d \sec ^{5}{\left (e + f x \right )}\, dx\right ) \]
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Time = 0.24 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.79 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \, dx=\frac {80 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} c^{2} + 480 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} c d + 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^{3} d^{2} + 240 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} d^{2} - 30 \, a^{3} c d {\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 )} - 45 \, a^{3} d^{2} {\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 )} - 180 \, a^{3} c^{2} {\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 )} - 360 \, a^{3} c d {\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 )} - 60 \, a^{3} d^{2} {\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^{3} c^{2} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) + 720 \, a^{3} c^{2} \tan \left (f x + e\right ) + 480 \, a^{3} c d \tan \left (f x + e\right )}{240 \, f} \]
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Time = 0.38 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.46 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \, dx=\frac {15 \, {\left (20 \, a^{3} c^{2} + 30 \, a^{3} c d + 13 \, a^{3} d^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) - 15 \, {\left (20 \, a^{3} c^{2} + 30 \, a^{3} c d + 13 \, a^{3} d^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) - \frac {2 \, {\left (300 \, a^{3} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 450 \, a^{3} c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 195 \, a^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} - 1400 \, a^{3} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 2100 \, a^{3} c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 910 \, a^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 2560 \, a^{3} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3840 \, a^{3} c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 1664 \, a^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 2120 \, a^{3} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3660 \, a^{3} c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 1330 \, a^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 660 \, a^{3} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1470 \, a^{3} c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 765 \, a^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{5}}}{120 \, f} \]
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Time = 17.10 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.12 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \, dx=\frac {a^3\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (20\,c^2+30\,c\,d+13\,d^2\right )}{2\,\left (10\,c^2+15\,c\,d+\frac {13\,d^2}{2}\right )}\right )\,\left (20\,c^2+30\,c\,d+13\,d^2\right )}{4\,f}-\frac {\left (5\,a^3\,c^2+\frac {15\,a^3\,c\,d}{2}+\frac {13\,a^3\,d^2}{4}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9+\left (-\frac {70\,a^3\,c^2}{3}-35\,a^3\,c\,d-\frac {91\,a^3\,d^2}{6}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+\left (\frac {128\,a^3\,c^2}{3}+64\,a^3\,c\,d+\frac {416\,a^3\,d^2}{15}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+\left (-\frac {106\,a^3\,c^2}{3}-61\,a^3\,c\,d-\frac {133\,a^3\,d^2}{6}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+\left (11\,a^3\,c^2+\frac {49\,a^3\,c\,d}{2}+\frac {51\,a^3\,d^2}{4}\right )\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )} \]
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