Integrand size = 31, antiderivative size = 288 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3 \, dx=\frac {a^3 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \text {arctanh}(\sin (e+f x))}{16 f}+\frac {a^3 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \tan (e+f x)}{16 f}+\frac {\left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{48 f}+\frac {a (3 c+8 d) (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2 \tan (e+f x)}{30 f}+\frac {a (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^3 \tan (e+f x)}{6 f}+\frac {a (a+a \sec (e+f x))^2 \left (2 \left (4 c^3+74 c^2 d+66 c d^2+21 d^3\right )+d \left (6 c^2+62 c d+31 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{120 f} \]
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Time = 0.55 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4072, 102, 152, 52, 65, 223, 209} \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3 \, dx=\frac {a^4 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \tan (e+f x) \arctan \left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a (\sec (e+f x)+1)}}\right )}{8 f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {a^3 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \tan (e+f x)}{16 f}+\frac {\left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \tan (e+f x) \left (a^3 \sec (e+f x)+a^3\right )}{48 f}+\frac {d \tan (e+f x) (a \sec (e+f x)+a)^3 \left (70 c^2+4 d (8 c+3 d) \sec (e+f x)+54 c d+19 d^2\right )}{120 f}+\frac {a \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \tan (e+f x) (a \sec (e+f x)+a)^2}{120 f}+\frac {d \tan (e+f x) (a \sec (e+f x)+a)^3 (c+d \sec (e+f x))^2}{6 f} \]
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Rule 52
Rule 65
Rule 102
Rule 152
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)^3}{\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))^2 \tan (e+f x)}{6 f}+\frac {\tan (e+f x) \text {Subst}\left (\int \frac {(a+a x)^{5/2} (c+d x) \left (-a^2 \left (6 c^2+3 c d+2 d^2\right )-a^2 d (8 c+3 d) x\right )}{\sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{6 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))^2 \tan (e+f x)}{6 f}+\frac {d (a+a \sec (e+f x))^3 \left (70 c^2+54 c d+19 d^2+4 d (8 c+3 d) \sec (e+f x)\right ) \tan (e+f x)}{120 f}-\frac {\left (a^2 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\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 )}{40 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {a \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{120 f}+\frac {d (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \tan (e+f x)}{6 f}+\frac {d (a+a \sec (e+f x))^3 \left (70 c^2+54 c d+19 d^2+4 d (8 c+3 d) \sec (e+f x)\right ) \tan (e+f x)}{120 f}-\frac {\left (a^3 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\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 )}{24 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {a \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{120 f}+\frac {\left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{48 f}+\frac {d (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \tan (e+f x)}{6 f}+\frac {d (a+a \sec (e+f x))^3 \left (70 c^2+54 c d+19 d^2+4 d (8 c+3 d) \sec (e+f x)\right ) \tan (e+f x)}{120 f}-\frac {\left (a^4 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\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 )}{16 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {a^3 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \tan (e+f x)}{16 f}+\frac {a \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{120 f}+\frac {\left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{48 f}+\frac {d (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \tan (e+f x)}{6 f}+\frac {d (a+a \sec (e+f x))^3 \left (70 c^2+54 c d+19 d^2+4 d (8 c+3 d) \sec (e+f x)\right ) \tan (e+f x)}{120 f}-\frac {\left (a^5 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\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 )}{16 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {a^3 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \tan (e+f x)}{16 f}+\frac {a \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{120 f}+\frac {\left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{48 f}+\frac {d (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \tan (e+f x)}{6 f}+\frac {d (a+a \sec (e+f x))^3 \left (70 c^2+54 c d+19 d^2+4 d (8 c+3 d) \sec (e+f x)\right ) \tan (e+f x)}{120 f}+\frac {\left (a^4 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\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 )}{8 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {a^3 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \tan (e+f x)}{16 f}+\frac {a \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{120 f}+\frac {\left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{48 f}+\frac {d (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \tan (e+f x)}{6 f}+\frac {d (a+a \sec (e+f x))^3 \left (70 c^2+54 c d+19 d^2+4 d (8 c+3 d) \sec (e+f x)\right ) \tan (e+f x)}{120 f}+\frac {\left (a^4 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\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 )}{8 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {a^3 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \tan (e+f x)}{16 f}+\frac {a^4 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \arctan \left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}}\right ) \tan (e+f x)}{8 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {a \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{120 f}+\frac {\left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{48 f}+\frac {d (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \tan (e+f x)}{6 f}+\frac {d (a+a \sec (e+f x))^3 \left (70 c^2+54 c d+19 d^2+4 d (8 c+3 d) \sec (e+f x)\right ) \tan (e+f x)}{120 f} \\ \end{align*}
Time = 6.09 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.59 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3 \, dx=\frac {a^3 \left (15 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \text {arctanh}(\sin (e+f x))+\tan (e+f x) \left (15 \left (24 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \sec (e+f x)+10 d \left (18 c^2+54 c d+23 d^2\right ) \sec ^3(e+f x)+40 d^3 \sec ^5(e+f x)+16 (c+d) \left (60 (c+d)^2+5 \left (c^2+8 c d+7 d^2\right ) \tan ^2(e+f x)+9 d^2 \tan ^4(e+f x)\right )\right )\right )}{240 f} \]
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Time = 6.19 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.23
method | result | size |
norman | \(\frac {-\frac {33 a^{3} \left (40 c^{3}+90 c^{2} d +78 c \,d^{2}+23 d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{20 f}+\frac {17 a^{3} \left (40 c^{3}+90 c^{2} d +78 c \,d^{2}+23 d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{24 f}-\frac {a^{3} \left (40 c^{3}+90 c^{2} d +78 c \,d^{2}+23 d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{8 f}+\frac {a^{3} \left (88 c^{3}+294 c^{2} d +306 c \,d^{2}+105 d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{8 f}+\frac {3 a^{3} \left (520 c^{3}+1250 c^{2} d +998 c \,d^{2}+323 d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{20 f}-\frac {a^{3} \left (1112 c^{3}+3078 c^{2} d +2514 c \,d^{2}+633 d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{24 f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{6}}-\frac {a^{3} \left (40 c^{3}+90 c^{2} d +78 c \,d^{2}+23 d^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{16 f}+\frac {a^{3} \left (40 c^{3}+90 c^{2} d +78 c \,d^{2}+23 d^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{16 f}\) | \(355\) |
parallelrisch | \(\frac {6 \left (-\frac {25 \left (\frac {2 \cos \left (4 f x +4 e \right )}{5}+\frac {2}{3}+\cos \left (2 f x +2 e \right )+\frac {\cos \left (6 f x +6 e \right )}{15}\right ) \left (c^{3}+\frac {9}{4} c^{2} d +\frac {39}{20} c \,d^{2}+\frac {23}{40} d^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{4}+\frac {25 \left (\frac {2 \cos \left (4 f x +4 e \right )}{5}+\frac {2}{3}+\cos \left (2 f x +2 e \right )+\frac {\cos \left (6 f x +6 e \right )}{15}\right ) \left (c^{3}+\frac {9}{4} c^{2} d +\frac {39}{20} c \,d^{2}+\frac {23}{40} d^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{4}+\left (\frac {23}{2} c^{2} d +13 c \,d^{2}+\frac {7}{2} c^{3}+5 d^{3}\right ) \sin \left (2 f x +2 e \right )+\frac {\left (\frac {53}{4} c^{2} d +\frac {63}{4} c \,d^{2}+3 c^{3}+\frac {391}{72} d^{3}\right ) \sin \left (3 f x +3 e \right )}{2}+\frac {8 \left (c^{2}+2 c d +\frac {17}{20} d^{2}\right ) \left (c +d \right ) \sin \left (4 f x +4 e \right )}{3}+\frac {\left (\frac {13}{4} c \,d^{2}+\frac {15}{4} c^{2} d +c^{3}+\frac {23}{24} d^{3}\right ) \sin \left (5 f x +5 e \right )}{2}+\frac {11 \left (c^{2}+\frac {16}{11} c d +\frac {34}{55} d^{2}\right ) \left (c +d \right ) \sin \left (6 f x +6 e \right )}{18}+\sin \left (f x +e \right ) \left (\frac {19}{4} c^{2} d +\frac {25}{4} c \,d^{2}+\frac {25}{8} d^{3}+c^{3}\right )\right ) a^{3}}{f \left (6 \cos \left (4 f x +4 e \right )+10+15 \cos \left (2 f x +2 e \right )+\cos \left (6 f x +6 e \right )\right )}\) | \(362\) |
parts | \(-\frac {\left (3 a^{3} c \,d^{2}+3 a^{3} d^{3}\right ) \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )}{f}+\frac {\left (3 c^{3} a^{3}+3 a^{3} c^{2} d \right ) \tan \left (f x +e \right )}{f}+\frac {\left (3 a^{3} c^{2} d +9 a^{3} c \,d^{2}+3 a^{3} d^{3}\right ) \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )}{f}+\frac {\left (3 c^{3} a^{3}+9 a^{3} c^{2} d +3 a^{3} c \,d^{2}\right ) \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )}{f}-\frac {\left (c^{3} a^{3}+9 a^{3} c^{2} d +9 a^{3} c \,d^{2}+a^{3} d^{3}\right ) \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )}{f}+\frac {a^{3} d^{3} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )}{f}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right ) a^{3} c^{3}}{f}\) | \(362\) |
derivativedivides | \(\frac {-c^{3} a^{3} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+3 a^{3} c^{2} 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 )-3 a^{3} c \,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 )+a^{3} d^{3} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )+3 c^{3} 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 )-9 a^{3} c^{2} d \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+9 a^{3} c \,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 a^{3} d^{3} \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^{3} a^{3} \tan \left (f x +e \right )+9 a^{3} c^{2} 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 )-9 a^{3} c \,d^{2} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+3 a^{3} d^{3} \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^{3} a^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+3 a^{3} c^{2} d \tan \left (f x +e \right )+3 a^{3} c \,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 )-a^{3} d^{3} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )}{f}\) | \(573\) |
default | \(\frac {-c^{3} a^{3} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+3 a^{3} c^{2} 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 )-3 a^{3} c \,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 )+a^{3} d^{3} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )+3 c^{3} 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 )-9 a^{3} c^{2} d \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+9 a^{3} c \,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 a^{3} d^{3} \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^{3} a^{3} \tan \left (f x +e \right )+9 a^{3} c^{2} 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 )-9 a^{3} c \,d^{2} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+3 a^{3} d^{3} \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^{3} a^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+3 a^{3} c^{2} d \tan \left (f x +e \right )+3 a^{3} c \,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 )-a^{3} d^{3} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )}{f}\) | \(573\) |
risch | \(-\frac {i a^{3} \left (-880 c^{3}-1824 c \,d^{2}-544 d^{3}-2160 c^{2} d -1170 c \,d^{2} {\mathrm e}^{i \left (f x +e \right )}-10944 c \,d^{2} {\mathrm e}^{2 i \left (f x +e \right )}-5670 c \,d^{2} {\mathrm e}^{3 i \left (f x +e \right )}-18240 c \,d^{2} {\mathrm e}^{6 i \left (f x +e \right )}-1350 c^{2} d \,{\mathrm e}^{i \left (f x +e \right )}+4500 c \,d^{2} {\mathrm e}^{7 i \left (f x +e \right )}-12240 c^{2} d \,{\mathrm e}^{2 i \left (f x +e \right )}+3420 c^{2} d \,{\mathrm e}^{7 i \left (f x +e \right )}-7920 c^{2} d \,{\mathrm e}^{8 i \left (f x +e \right )}-23040 c \,d^{2} {\mathrm e}^{4 i \left (f x +e \right )}-4770 c^{2} d \,{\mathrm e}^{3 i \left (f x +e \right )}+5670 c \,d^{2} {\mathrm e}^{9 i \left (f x +e \right )}-4500 c \,d^{2} {\mathrm e}^{5 i \left (f x +e \right )}+1170 c \,d^{2} {\mathrm e}^{11 i \left (f x +e \right )}-720 c^{2} d \,{\mathrm e}^{10 i \left (f x +e \right )}-3420 c^{2} d \,{\mathrm e}^{5 i \left (f x +e \right )}-4320 c \,d^{2} {\mathrm e}^{8 i \left (f x +e \right )}-21600 c^{2} d \,{\mathrm e}^{6 i \left (f x +e \right )}-24480 c^{2} d \,{\mathrm e}^{4 i \left (f x +e \right )}+1350 c^{2} d \,{\mathrm e}^{11 i \left (f x +e \right )}+4770 c^{2} d \,{\mathrm e}^{9 i \left (f x +e \right )}-345 d^{3} {\mathrm e}^{i \left (f x +e \right )}+1955 d^{3} {\mathrm e}^{9 i \left (f x +e \right )}-2250 d^{3} {\mathrm e}^{5 i \left (f x +e \right )}-8800 c^{3} {\mathrm e}^{6 i \left (f x +e \right )}-1955 d^{3} {\mathrm e}^{3 i \left (f x +e \right )}-5440 d^{3} {\mathrm e}^{6 i \left (f x +e \right )}+1080 c^{3} {\mathrm e}^{9 i \left (f x +e \right )}+720 c^{3} {\mathrm e}^{7 i \left (f x +e \right )}-480 d^{3} {\mathrm e}^{8 i \left (f x +e \right )}-720 c^{3} {\mathrm e}^{10 i \left (f x +e \right )}+345 d^{3} {\mathrm e}^{11 i \left (f x +e \right )}+360 c^{3} {\mathrm e}^{11 i \left (f x +e \right )}-720 c^{3} {\mathrm e}^{5 i \left (f x +e \right )}-3264 d^{3} {\mathrm e}^{2 i \left (f x +e \right )}-360 c^{3} {\mathrm e}^{i \left (f x +e \right )}+2250 d^{3} {\mathrm e}^{7 i \left (f x +e \right )}-7680 d^{3} {\mathrm e}^{4 i \left (f x +e \right )}-4560 c^{3} {\mathrm e}^{2 i \left (f x +e \right )}-1080 c^{3} {\mathrm e}^{3 i \left (f x +e \right )}-4080 c^{3} {\mathrm e}^{8 i \left (f x +e \right )}-9120 c^{3} {\mathrm e}^{4 i \left (f x +e \right )}\right )}{120 f \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{6}}+\frac {5 c^{3} a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{2 f}+\frac {45 a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) c^{2} d}{8 f}+\frac {39 a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) c \,d^{2}}{8 f}+\frac {23 a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) d^{3}}{16 f}-\frac {5 c^{3} a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{2 f}-\frac {45 a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) c^{2} d}{8 f}-\frac {39 a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) c \,d^{2}}{8 f}-\frac {23 a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) d^{3}}{16 f}\) | \(852\) |
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Time = 0.28 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.17 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3 \, dx=\frac {15 \, {\left (40 \, a^{3} c^{3} + 90 \, a^{3} c^{2} d + 78 \, a^{3} c d^{2} + 23 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )^{6} \log \left (\sin \left (f x + e\right ) + 1\right ) - 15 \, {\left (40 \, a^{3} c^{3} + 90 \, a^{3} c^{2} d + 78 \, a^{3} c d^{2} + 23 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )^{6} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (40 \, a^{3} d^{3} + 16 \, {\left (55 \, a^{3} c^{3} + 135 \, a^{3} c^{2} d + 114 \, a^{3} c d^{2} + 34 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )^{5} + 15 \, {\left (24 \, a^{3} c^{3} + 90 \, a^{3} c^{2} d + 78 \, a^{3} c d^{2} + 23 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )^{4} + 16 \, {\left (5 \, a^{3} c^{3} + 45 \, a^{3} c^{2} d + 57 \, a^{3} c d^{2} + 17 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )^{3} + 10 \, {\left (18 \, a^{3} c^{2} d + 54 \, a^{3} c d^{2} + 23 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )^{2} + 144 \, {\left (a^{3} c d^{2} + a^{3} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{480 \, f \cos \left (f x + e\right )^{6}} \]
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\[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3 \, dx=a^{3} \left (\int c^{3} \sec {\left (e + f x \right )}\, dx + \int 3 c^{3} \sec ^{2}{\left (e + f x \right )}\, dx + \int 3 c^{3} \sec ^{3}{\left (e + f x \right )}\, dx + \int c^{3} \sec ^{4}{\left (e + f x \right )}\, dx + \int d^{3} \sec ^{4}{\left (e + f x \right )}\, dx + \int 3 d^{3} \sec ^{5}{\left (e + f x \right )}\, dx + \int 3 d^{3} \sec ^{6}{\left (e + f x \right )}\, dx + \int d^{3} \sec ^{7}{\left (e + f x \right )}\, dx + \int 3 c d^{2} \sec ^{3}{\left (e + f x \right )}\, dx + \int 9 c d^{2} \sec ^{4}{\left (e + f x \right )}\, dx + \int 9 c d^{2} \sec ^{5}{\left (e + f x \right )}\, dx + \int 3 c d^{2} \sec ^{6}{\left (e + f x \right )}\, dx + \int 3 c^{2} d \sec ^{2}{\left (e + f x \right )}\, dx + \int 9 c^{2} d \sec ^{3}{\left (e + f x \right )}\, dx + \int 9 c^{2} d \sec ^{4}{\left (e + f x \right )}\, dx + \int 3 c^{2} d \sec ^{5}{\left (e + f x \right )}\, dx\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 701 vs. \(2 (273) = 546\).
Time = 0.24 (sec) , antiderivative size = 701, normalized size of antiderivative = 2.43 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3 \, dx=\frac {160 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} c^{3} + 1440 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} c^{2} d + 96 \, {\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 d^{2} + 1440 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} c d^{2} + 96 \, {\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^{3} + 160 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} d^{3} - 5 \, a^{3} d^{3} {\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 )} - 90 \, a^{3} c^{2} 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 )} - 270 \, a^{3} c 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 )} - 90 \, a^{3} d^{3} {\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 )} - 360 \, a^{3} c^{3} {\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 )} - 1080 \, a^{3} c^{2} 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 )} - 360 \, a^{3} c 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 )} + 480 \, a^{3} c^{3} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) + 1440 \, a^{3} c^{3} \tan \left (f x + e\right ) + 1440 \, a^{3} c^{2} d \tan \left (f x + e\right )}{480 \, f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 584 vs. \(2 (273) = 546\).
Time = 0.44 (sec) , antiderivative size = 584, normalized size of antiderivative = 2.03 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3 \, dx=\frac {15 \, {\left (40 \, a^{3} c^{3} + 90 \, a^{3} c^{2} d + 78 \, a^{3} c d^{2} + 23 \, a^{3} d^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) - 15 \, {\left (40 \, a^{3} c^{3} + 90 \, a^{3} c^{2} d + 78 \, a^{3} c d^{2} + 23 \, a^{3} d^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) - \frac {2 \, {\left (600 \, a^{3} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} + 1350 \, a^{3} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} + 1170 \, a^{3} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} + 345 \, a^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} - 3400 \, a^{3} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} - 7650 \, a^{3} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} - 6630 \, a^{3} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} - 1955 \, a^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 7920 \, a^{3} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 17820 \, a^{3} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 15444 \, a^{3} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 4554 \, a^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 9360 \, a^{3} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 22500 \, a^{3} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 17964 \, a^{3} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 5814 \, a^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 5560 \, a^{3} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15390 \, a^{3} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 12570 \, a^{3} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 3165 \, a^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 1320 \, a^{3} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 4410 \, a^{3} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 4590 \, a^{3} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1575 \, a^{3} d^{3} \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 )}^{6}}}{240 \, f} \]
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Time = 17.05 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.43 \[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3 \, dx=\frac {a^3\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (40\,c^3+90\,c^2\,d+78\,c\,d^2+23\,d^3\right )}{4\,\left (10\,c^3+\frac {45\,c^2\,d}{2}+\frac {39\,c\,d^2}{2}+\frac {23\,d^3}{4}\right )}\right )\,\left (40\,c^3+90\,c^2\,d+78\,c\,d^2+23\,d^3\right )}{8\,f}-\frac {\left (5\,a^3\,c^3+\frac {45\,a^3\,c^2\,d}{4}+\frac {39\,a^3\,c\,d^2}{4}+\frac {23\,a^3\,d^3}{8}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}+\left (-\frac {85\,a^3\,c^3}{3}-\frac {255\,a^3\,c^2\,d}{4}-\frac {221\,a^3\,c\,d^2}{4}-\frac {391\,a^3\,d^3}{24}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9+\left (66\,a^3\,c^3+\frac {297\,a^3\,c^2\,d}{2}+\frac {1287\,a^3\,c\,d^2}{10}+\frac {759\,a^3\,d^3}{20}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+\left (-78\,a^3\,c^3-\frac {375\,a^3\,c^2\,d}{2}-\frac {1497\,a^3\,c\,d^2}{10}-\frac {969\,a^3\,d^3}{20}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+\left (\frac {139\,a^3\,c^3}{3}+\frac {513\,a^3\,c^2\,d}{4}+\frac {419\,a^3\,c\,d^2}{4}+\frac {211\,a^3\,d^3}{8}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+\left (-11\,a^3\,c^3-\frac {147\,a^3\,c^2\,d}{4}-\frac {153\,a^3\,c\,d^2}{4}-\frac {105\,a^3\,d^3}{8}\right )\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \]
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