Integrand size = 29, antiderivative size = 180 \[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x))^3 \, dx=\frac {\left (8 a c^3+12 b c^2 d+12 a c d^2+3 b d^3\right ) \text {arctanh}(\sin (e+f x))}{8 f}+\frac {\left (4 a d \left (4 c^2+d^2\right )+3 b \left (c^3+4 c d^2\right )\right ) \tan (e+f x)}{6 f}+\frac {d \left (6 b c^2+20 a c d+9 b d^2\right ) \sec (e+f x) \tan (e+f x)}{24 f}+\frac {(3 b c+4 a d) (c+d \sec (e+f x))^2 \tan (e+f x)}{12 f}+\frac {b (c+d \sec (e+f x))^3 \tan (e+f x)}{4 f} \]
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Time = 0.40 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {4087, 4082, 3872, 3855, 3852, 8} \[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x))^3 \, dx=\frac {\left (8 a c^3+12 a c d^2+12 b c^2 d+3 b d^3\right ) \text {arctanh}(\sin (e+f x))}{8 f}+\frac {d \left (20 a c d+6 b c^2+9 b d^2\right ) \tan (e+f x) \sec (e+f x)}{24 f}+\frac {\left (4 a d \left (4 c^2+d^2\right )+3 b \left (c^3+4 c d^2\right )\right ) \tan (e+f x)}{6 f}+\frac {(4 a d+3 b c) \tan (e+f x) (c+d \sec (e+f x))^2}{12 f}+\frac {b \tan (e+f x) (c+d \sec (e+f x))^3}{4 f} \]
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Rule 8
Rule 3852
Rule 3855
Rule 3872
Rule 4082
Rule 4087
Rubi steps \begin{align*} \text {integral}& = \frac {b (c+d \sec (e+f x))^3 \tan (e+f x)}{4 f}+\frac {1}{4} \int \sec (e+f x) (c+d \sec (e+f x))^2 (4 a c+3 b d+(3 b c+4 a d) \sec (e+f x)) \, dx \\ & = \frac {(3 b c+4 a d) (c+d \sec (e+f x))^2 \tan (e+f x)}{12 f}+\frac {b (c+d \sec (e+f x))^3 \tan (e+f x)}{4 f}+\frac {1}{12} \int \sec (e+f x) (c+d \sec (e+f x)) \left (12 a c^2+15 b c d+8 a d^2+\left (6 b c^2+20 a c d+9 b d^2\right ) \sec (e+f x)\right ) \, dx \\ & = \frac {d \left (6 b c^2+20 a c d+9 b d^2\right ) \sec (e+f x) \tan (e+f x)}{24 f}+\frac {(3 b c+4 a d) (c+d \sec (e+f x))^2 \tan (e+f x)}{12 f}+\frac {b (c+d \sec (e+f x))^3 \tan (e+f x)}{4 f}+\frac {1}{24} \int \sec (e+f x) \left (3 \left (3 b d \left (4 c^2+d^2\right )+4 a \left (2 c^3+3 c d^2\right )\right )+4 \left (4 a d \left (4 c^2+d^2\right )+3 b \left (c^3+4 c d^2\right )\right ) \sec (e+f x)\right ) \, dx \\ & = \frac {d \left (6 b c^2+20 a c d+9 b d^2\right ) \sec (e+f x) \tan (e+f x)}{24 f}+\frac {(3 b c+4 a d) (c+d \sec (e+f x))^2 \tan (e+f x)}{12 f}+\frac {b (c+d \sec (e+f x))^3 \tan (e+f x)}{4 f}+\frac {1}{8} \left (8 a c^3+12 b c^2 d+12 a c d^2+3 b d^3\right ) \int \sec (e+f x) \, dx+\frac {1}{6} \left (4 a d \left (4 c^2+d^2\right )+3 b \left (c^3+4 c d^2\right )\right ) \int \sec ^2(e+f x) \, dx \\ & = \frac {\left (8 a c^3+12 b c^2 d+12 a c d^2+3 b d^3\right ) \text {arctanh}(\sin (e+f x))}{8 f}+\frac {d \left (6 b c^2+20 a c d+9 b d^2\right ) \sec (e+f x) \tan (e+f x)}{24 f}+\frac {(3 b c+4 a d) (c+d \sec (e+f x))^2 \tan (e+f x)}{12 f}+\frac {b (c+d \sec (e+f x))^3 \tan (e+f x)}{4 f}-\frac {\left (4 a d \left (4 c^2+d^2\right )+3 b \left (c^3+4 c d^2\right )\right ) \text {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{6 f} \\ & = \frac {\left (8 a c^3+12 b c^2 d+12 a c d^2+3 b d^3\right ) \text {arctanh}(\sin (e+f x))}{8 f}+\frac {\left (4 a d \left (4 c^2+d^2\right )+3 b \left (c^3+4 c d^2\right )\right ) \tan (e+f x)}{6 f}+\frac {d \left (6 b c^2+20 a c d+9 b d^2\right ) \sec (e+f x) \tan (e+f x)}{24 f}+\frac {(3 b c+4 a d) (c+d \sec (e+f x))^2 \tan (e+f x)}{12 f}+\frac {b (c+d \sec (e+f x))^3 \tan (e+f x)}{4 f} \\ \end{align*}
Time = 0.78 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.79 \[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x))^3 \, dx=\frac {3 \left (3 b d \left (4 c^2+d^2\right )+4 a \left (2 c^3+3 c d^2\right )\right ) \text {arctanh}(\sin (e+f x))+\tan (e+f x) \left (9 d \left (4 a c d+b \left (4 c^2+d^2\right )\right ) \sec (e+f x)+6 b d^3 \sec ^3(e+f x)+8 \left (3 a d \left (3 c^2+d^2\right )+3 b \left (c^3+3 c d^2\right )+d^2 (3 b c+a d) \tan ^2(e+f x)\right )\right )}{24 f} \]
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Time = 3.81 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.03
method | result | size |
parts | \(-\frac {\left (a \,d^{3}+3 b c \,d^{2}\right ) \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )}{f}+\frac {\left (3 a c \,d^{2}+3 b \,c^{2} d \right ) \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )}{f}+\frac {\left (3 a \,c^{2} d +b \,c^{3}\right ) \tan \left (f x +e \right )}{f}+\frac {a \,c^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{f}+\frac {b \,d^{3} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )}{f}\) | \(185\) |
derivativedivides | \(\frac {a \,c^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+3 a \,c^{2} d \tan \left (f x +e \right )+3 a c \,d^{2} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-a \,d^{3} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+b \,c^{3} \tan \left (f x +e \right )+3 b \,c^{2} d \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-3 b c \,d^{2} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+b \,d^{3} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )}{f}\) | \(223\) |
default | \(\frac {a \,c^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+3 a \,c^{2} d \tan \left (f x +e \right )+3 a c \,d^{2} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-a \,d^{3} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+b \,c^{3} \tan \left (f x +e \right )+3 b \,c^{2} d \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-3 b c \,d^{2} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+b \,d^{3} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )}{f}\) | \(223\) |
parallelrisch | \(\frac {-96 \left (\frac {3}{4}+\frac {\cos \left (4 f x +4 e \right )}{4}+\cos \left (2 f x +2 e \right )\right ) \left (a \,c^{3}+\frac {3}{2} a c \,d^{2}+\frac {3}{2} b \,c^{2} d +\frac {3}{8} b \,d^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )+96 \left (\frac {3}{4}+\frac {\cos \left (4 f x +4 e \right )}{4}+\cos \left (2 f x +2 e \right )\right ) \left (a \,c^{3}+\frac {3}{2} a c \,d^{2}+\frac {3}{2} b \,c^{2} d +\frac {3}{8} b \,d^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )+\left (144 a \,c^{2} d +64 a \,d^{3}+48 b \,c^{3}+192 b c \,d^{2}\right ) \sin \left (2 f x +2 e \right )+\left (72 a \,c^{2} d +16 a \,d^{3}+24 b \,c^{3}+48 b c \,d^{2}\right ) \sin \left (4 f x +4 e \right )+72 d \left (\left (a c d +b \,c^{2}+\frac {1}{4} b \,d^{2}\right ) \sin \left (3 f x +3 e \right )+\sin \left (f x +e \right ) \left (a c d +b \,c^{2}+\frac {11}{12} b \,d^{2}\right )\right )}{24 f \left (3+\cos \left (4 f x +4 e \right )+4 \cos \left (2 f x +2 e \right )\right )}\) | \(282\) |
norman | \(\frac {-\frac {\left (24 a \,c^{2} d -12 a c \,d^{2}+8 a \,d^{3}+8 b \,c^{3}-12 b \,c^{2} d +24 b c \,d^{2}-5 b \,d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{4 f}+\frac {\left (24 a \,c^{2} d +12 a c \,d^{2}+8 a \,d^{3}+8 b \,c^{3}+12 b \,c^{2} d +24 b c \,d^{2}+5 b \,d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}+\frac {\left (216 a \,c^{2} d -36 a c \,d^{2}+40 a \,d^{3}+72 b \,c^{3}-36 b \,c^{2} d +120 b c \,d^{2}+9 b \,d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{12 f}-\frac {\left (216 a \,c^{2} d +36 a c \,d^{2}+40 a \,d^{3}+72 b \,c^{3}+36 b \,c^{2} d +120 b c \,d^{2}-9 b \,d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{12 f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{4}}-\frac {\left (8 a \,c^{3}+12 a c \,d^{2}+12 b \,c^{2} d +3 b \,d^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{8 f}+\frac {\left (8 a \,c^{3}+12 a c \,d^{2}+12 b \,c^{2} d +3 b \,d^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{8 f}\) | \(357\) |
risch | \(-\frac {i \left (-48 b c \,d^{2}-72 a \,c^{2} d -16 a \,d^{3}-24 b \,c^{3}-33 b \,d^{3} {\mathrm e}^{3 i \left (f x +e \right )}-9 d^{3} b \,{\mathrm e}^{i \left (f x +e \right )}-72 b \,c^{3} {\mathrm e}^{4 i \left (f x +e \right )}+33 b \,d^{3} {\mathrm e}^{5 i \left (f x +e \right )}-48 a \,d^{3} {\mathrm e}^{4 i \left (f x +e \right )}-72 b \,c^{3} {\mathrm e}^{2 i \left (f x +e \right )}-64 a \,d^{3} {\mathrm e}^{2 i \left (f x +e \right )}+9 b \,d^{3} {\mathrm e}^{7 i \left (f x +e \right )}-24 b \,c^{3} {\mathrm e}^{6 i \left (f x +e \right )}-36 a c \,d^{2} {\mathrm e}^{i \left (f x +e \right )}-192 b c \,d^{2} {\mathrm e}^{2 i \left (f x +e \right )}-144 b c \,d^{2} {\mathrm e}^{4 i \left (f x +e \right )}+36 a c \,d^{2} {\mathrm e}^{7 i \left (f x +e \right )}-36 a c \,d^{2} {\mathrm e}^{3 i \left (f x +e \right )}-36 b \,c^{2} d \,{\mathrm e}^{3 i \left (f x +e \right )}-216 a \,c^{2} d \,{\mathrm e}^{2 i \left (f x +e \right )}-216 a \,c^{2} d \,{\mathrm e}^{4 i \left (f x +e \right )}+36 b \,c^{2} d \,{\mathrm e}^{5 i \left (f x +e \right )}-36 b \,c^{2} d \,{\mathrm e}^{i \left (f x +e \right )}+36 b \,c^{2} d \,{\mathrm e}^{7 i \left (f x +e \right )}-72 a \,c^{2} d \,{\mathrm e}^{6 i \left (f x +e \right )}+36 a c \,d^{2} {\mathrm e}^{5 i \left (f x +e \right )}\right )}{12 f \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{4}}-\frac {a \,c^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{f}-\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) a c \,d^{2}}{2 f}-\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) b \,c^{2} d}{2 f}-\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) b \,d^{3}}{8 f}+\frac {a \,c^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{f}+\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) a c \,d^{2}}{2 f}+\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) b \,c^{2} d}{2 f}+\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) b \,d^{3}}{8 f}\) | \(570\) |
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Time = 0.28 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.17 \[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x))^3 \, dx=\frac {3 \, {\left (8 \, a c^{3} + 12 \, b c^{2} d + 12 \, a c d^{2} + 3 \, b d^{3}\right )} \cos \left (f x + e\right )^{4} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \, {\left (8 \, a c^{3} + 12 \, b c^{2} d + 12 \, a c d^{2} + 3 \, b d^{3}\right )} \cos \left (f x + e\right )^{4} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (6 \, b d^{3} + 8 \, {\left (3 \, b c^{3} + 9 \, a c^{2} d + 6 \, b c d^{2} + 2 \, a d^{3}\right )} \cos \left (f x + e\right )^{3} + 9 \, {\left (4 \, b c^{2} d + 4 \, a c d^{2} + b d^{3}\right )} \cos \left (f x + e\right )^{2} + 8 \, {\left (3 \, b c d^{2} + a d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{48 \, f \cos \left (f x + e\right )^{4}} \]
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\[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x))^3 \, dx=\int \left (a + b \sec {\left (e + f x \right )}\right ) \left (c + d \sec {\left (e + f x \right )}\right )^{3} \sec {\left (e + f x \right )}\, dx \]
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Time = 0.25 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.48 \[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x))^3 \, dx=\frac {48 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} b c d^{2} + 16 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a d^{3} - 3 \, b d^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (f x + e\right )^{3} - 5 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{4} - 2 \, \sin \left (f x + e\right )^{2} + 1} - 3 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 3 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 36 \, b c^{2} d {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 36 \, a c d^{2} {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 48 \, a c^{3} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) + 48 \, b c^{3} \tan \left (f x + e\right ) + 144 \, a c^{2} d \tan \left (f x + e\right )}{48 \, f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 586 vs. \(2 (170) = 340\).
Time = 0.36 (sec) , antiderivative size = 586, normalized size of antiderivative = 3.26 \[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x))^3 \, dx=\frac {3 \, {\left (8 \, a c^{3} + 12 \, b c^{2} d + 12 \, a c d^{2} + 3 \, b d^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) - 3 \, {\left (8 \, a c^{3} + 12 \, b c^{2} d + 12 \, a c d^{2} + 3 \, b d^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) - \frac {2 \, {\left (24 \, b c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 72 \, a c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 36 \, b c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 36 \, a c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 72 \, b c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 24 \, a d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 15 \, b d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 72 \, b c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 216 \, a c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 36 \, b c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 36 \, a c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 120 \, b c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 40 \, a d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 9 \, b d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 72 \, b c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 216 \, a c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 36 \, b c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 36 \, a c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 120 \, b c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 40 \, a d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 9 \, b d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 24 \, b c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 72 \, a c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 36 \, b c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 36 \, a c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 72 \, b c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 24 \, a d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 15 \, b 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 )}^{4}}}{24 \, f} \]
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Time = 17.20 (sec) , antiderivative size = 395, normalized size of antiderivative = 2.19 \[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x))^3 \, dx=\frac {\mathrm {atanh}\left (\frac {4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (a\,c^3+\frac {3\,b\,c^2\,d}{2}+\frac {3\,a\,c\,d^2}{2}+\frac {3\,b\,d^3}{8}\right )}{4\,a\,c^3+6\,b\,c^2\,d+6\,a\,c\,d^2+\frac {3\,b\,d^3}{2}}\right )\,\left (2\,a\,c^3+3\,b\,c^2\,d+3\,a\,c\,d^2+\frac {3\,b\,d^3}{4}\right )}{f}-\frac {\left (2\,a\,d^3+2\,b\,c^3-\frac {5\,b\,d^3}{4}-3\,a\,c\,d^2+6\,a\,c^2\,d+6\,b\,c\,d^2-3\,b\,c^2\,d\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+\left (3\,a\,c\,d^2-6\,b\,c^3-\frac {3\,b\,d^3}{4}-\frac {10\,a\,d^3}{3}-18\,a\,c^2\,d-10\,b\,c\,d^2+3\,b\,c^2\,d\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+\left (\frac {10\,a\,d^3}{3}+6\,b\,c^3-\frac {3\,b\,d^3}{4}+3\,a\,c\,d^2+18\,a\,c^2\,d+10\,b\,c\,d^2+3\,b\,c^2\,d\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+\left (-2\,a\,d^3-2\,b\,c^3-\frac {5\,b\,d^3}{4}-3\,a\,c\,d^2-6\,a\,c^2\,d-6\,b\,c\,d^2-3\,b\,c^2\,d\right )\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \]
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