Integrand size = 31, antiderivative size = 193 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx=\frac {d^2 \left (12 c^2-16 c d+7 d^2\right ) \text {arctanh}(\sin (e+f x))}{2 a^2 f}+\frac {(c-d) (c+8 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )}+\frac {(c-d) (c+d \sec (e+f x))^3 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}-\frac {d \left (4 \left (c^3+8 c^2 d-20 c d^2+8 d^3\right )+d \left (2 c^2+16 c d-21 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{6 a^2 f} \]
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Time = 0.37 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.29, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4072, 100, 155, 152, 65, 223, 209} \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx=-\frac {d \tan (e+f x) \left (d \left (2 c^2+16 c d-21 d^2\right ) \sec (e+f x)+4 \left (c^3+8 c^2 d-20 c d^2+8 d^3\right )\right )}{6 a^2 f}+\frac {(c-d) (c+8 d) \tan (e+f x) (c+d \sec (e+f x))^2}{3 f \left (a^2 \sec (e+f x)+a^2\right )}+\frac {d^2 \left (12 c^2-16 c d+7 d^2\right ) \tan (e+f x) \arctan \left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a (\sec (e+f x)+1)}}\right )}{a f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {(c-d) \tan (e+f x) (c+d \sec (e+f x))^3}{3 f (a \sec (e+f x)+a)^2} \]
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Rule 65
Rule 100
Rule 152
Rule 155
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 {(c+d x)^4}{\sqrt {a-a x} (a+a x)^{5/2}} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {(c-d) (c+d \sec (e+f x))^3 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}+\frac {\tan (e+f x) \text {Subst}\left (\int \frac {(c+d x)^2 \left (-a^2 \left (c^2+5 c d-3 d^2\right )+a^2 (2 c-5 d) d x\right )}{\sqrt {a-a x} (a+a x)^{3/2}} \, dx,x,\sec (e+f x)\right )}{3 a f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {(c-d) (c+8 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )}+\frac {(c-d) (c+d \sec (e+f x))^3 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}+\frac {\tan (e+f x) \text {Subst}\left (\int \frac {(c+d x) \left (-a^4 (19 c-16 d) d^2+a^4 d \left (2 c^2+16 c d-21 d^2\right ) x\right )}{\sqrt {a-a x} \sqrt {a+a x}} \, dx,x,\sec (e+f x)\right )}{3 a^4 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {(c-d) (c+8 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )}+\frac {(c-d) (c+d \sec (e+f x))^3 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}-\frac {d \left (4 \left (c^3+8 c^2 d-20 c d^2+8 d^3\right )+d \left (2 c^2+16 c d-21 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{6 a^2 f}-\frac {\left (d^2 \left (12 c^2-16 c d+7 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 )}{2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {(c-d) (c+8 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )}+\frac {(c-d) (c+d \sec (e+f x))^3 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}-\frac {d \left (4 \left (c^3+8 c^2 d-20 c d^2+8 d^3\right )+d \left (2 c^2+16 c d-21 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{6 a^2 f}+\frac {\left (d^2 \left (12 c^2-16 c d+7 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 )}{a f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {(c-d) (c+8 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )}+\frac {(c-d) (c+d \sec (e+f x))^3 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}-\frac {d \left (4 \left (c^3+8 c^2 d-20 c d^2+8 d^3\right )+d \left (2 c^2+16 c d-21 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{6 a^2 f}+\frac {\left (d^2 \left (12 c^2-16 c d+7 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 )}{a f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {d^2 \left (12 c^2-16 c d+7 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)}{a f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {(c-d) (c+8 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )}+\frac {(c-d) (c+d \sec (e+f x))^3 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}-\frac {d \left (4 \left (c^3+8 c^2 d-20 c d^2+8 d^3\right )+d \left (2 c^2+16 c d-21 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{6 a^2 f} \\ \end{align*}
Time = 5.10 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.61 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx=\frac {-24 d^2 \left (12 c^2-16 c d+7 d^2\right ) \cos ^4\left (\frac {1}{2} (e+f x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )\right )+2 \cos \left (\frac {1}{2} (e+f x)\right ) \left (2 c^4+16 c^3 d-60 c^2 d^2+112 c d^3-37 d^4+6 \left (c^4+2 c^3 d-12 c^2 d^2+28 c d^3-10 d^4\right ) \cos (e+f x)+\left (2 c^4+16 c^3 d-60 c^2 d^2+112 c d^3-43 d^4\right ) \cos (2 (e+f x))+2 c^4 \cos (3 (e+f x))+4 c^3 d \cos (3 (e+f x))-24 c^2 d^2 \cos (3 (e+f x))+40 c d^3 \cos (3 (e+f x))-16 d^4 \cos (3 (e+f x))\right ) \sec ^2(e+f x) \sin \left (\frac {1}{2} (e+f x)\right )}{12 a^2 f (1+\cos (e+f x))^2} \]
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Time = 1.01 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.42
method | result | size |
parallelrisch | \(\frac {-12 \left (1+\cos \left (2 f x +2 e \right )\right ) \left (c^{2}-\frac {4}{3} c d +\frac {7}{12} d^{2}\right ) d^{2} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )+12 \left (1+\cos \left (2 f x +2 e \right )\right ) \left (c^{2}-\frac {4}{3} c d +\frac {7}{12} d^{2}\right ) d^{2} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )+\sec \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) \left (\left (\frac {1}{3} c^{4}+\frac {8}{3} c^{3} d -10 c^{2} d^{2}+\frac {56}{3} c \,d^{3}-\frac {43}{6} d^{4}\right ) \cos \left (2 f x +2 e \right )+\left (\frac {1}{3} c^{4}+\frac {2}{3} c^{3} d -4 c^{2} d^{2}+\frac {20}{3} c \,d^{3}-\frac {8}{3} d^{4}\right ) \cos \left (3 f x +3 e \right )+\left (c^{4}+2 c^{3} d -12 c^{2} d^{2}+28 c \,d^{3}-10 d^{4}\right ) \cos \left (f x +e \right )+\frac {c^{4}}{3}+\frac {8 c^{3} d}{3}-10 c^{2} d^{2}+\frac {56 c \,d^{3}}{3}-\frac {37 d^{4}}{6}\right )}{2 f \,a^{2} \left (1+\cos \left (2 f x +2 e \right )\right )}\) | \(275\) |
derivativedivides | \(\frac {-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} c^{4}}{3}+\frac {4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} c^{3} d}{3}-2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} c^{2} d^{2}+\frac {4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} c \,d^{3}}{3}-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} d^{4}}{3}+\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c^{4}+4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c^{3} d -18 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c^{2} d^{2}+20 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c \,d^{3}-7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d^{4}-\frac {d^{3} \left (8 c -5 d \right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}+\frac {d^{4}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-d^{2} \left (12 c^{2}-16 c d +7 d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )+d^{2} \left (12 c^{2}-16 c d +7 d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )-\frac {d^{3} \left (8 c -5 d \right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {d^{4}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}}{2 f \,a^{2}}\) | \(317\) |
default | \(\frac {-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} c^{4}}{3}+\frac {4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} c^{3} d}{3}-2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} c^{2} d^{2}+\frac {4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} c \,d^{3}}{3}-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} d^{4}}{3}+\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c^{4}+4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c^{3} d -18 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c^{2} d^{2}+20 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c \,d^{3}-7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d^{4}-\frac {d^{3} \left (8 c -5 d \right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}+\frac {d^{4}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-d^{2} \left (12 c^{2}-16 c d +7 d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )+d^{2} \left (12 c^{2}-16 c d +7 d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )-\frac {d^{3} \left (8 c -5 d \right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {d^{4}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}}{2 f \,a^{2}}\) | \(317\) |
norman | \(\frac {-\frac {\left (c^{4}-4 c^{3} d +6 c^{2} d^{2}-4 c \,d^{3}+d^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{6 a f}+\frac {\left (c^{4}+4 c^{3} d -18 c^{2} d^{2}+36 c \,d^{3}-13 d^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 a f}-\frac {\left (3 c^{4}+4 c^{3} d -30 c^{2} d^{2}+44 c \,d^{3}-18 d^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{a f}+\frac {\left (7 c^{4}-4 c^{3} d -30 c^{2} d^{2}+44 c \,d^{3}-17 d^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{6 a f}+\frac {\left (11 c^{4}+28 c^{3} d -150 c^{2} d^{2}+244 c \,d^{3}-100 d^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{3 a f}-\frac {\left (13 c^{4}+44 c^{3} d -210 c^{2} d^{2}+380 c \,d^{3}-149 d^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{6 a f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{4} a}-\frac {d^{2} \left (12 c^{2}-16 c d +7 d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{2 a^{2} f}+\frac {d^{2} \left (12 c^{2}-16 c d +7 d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{2 a^{2} f}\) | \(390\) |
risch | \(\frac {i \left (-36 c^{2} d^{2} {\mathrm e}^{6 i \left (f x +e \right )}-120 c^{2} d^{2} {\mathrm e}^{4 i \left (f x +e \right )}+256 c \,d^{3} {\mathrm e}^{2 i \left (f x +e \right )}+16 c^{3} d \,{\mathrm e}^{2 i \left (f x +e \right )}+144 c \,d^{3} {\mathrm e}^{5 i \left (f x +e \right )}+224 c \,d^{3} {\mathrm e}^{4 i \left (f x +e \right )}-108 c^{2} d^{2} {\mathrm e}^{i \left (f x +e \right )}+192 c \,d^{3} {\mathrm e}^{i \left (f x +e \right )}+48 c \,d^{3} {\mathrm e}^{6 i \left (f x +e \right )}+8 c^{3} d \,{\mathrm e}^{4 i \left (f x +e \right )}-108 c^{2} d^{2} {\mathrm e}^{5 i \left (f x +e \right )}-216 c^{2} d^{2} {\mathrm e}^{3 i \left (f x +e \right )}+336 c \,d^{3} {\mathrm e}^{3 i \left (f x +e \right )}-132 c^{2} d^{2} {\mathrm e}^{2 i \left (f x +e \right )}-32 d^{4}+4 c^{4}+48 c^{3} d \,{\mathrm e}^{3 i \left (f x +e \right )}+24 c^{3} d \,{\mathrm e}^{i \left (f x +e \right )}+24 c^{3} d \,{\mathrm e}^{5 i \left (f x +e \right )}-63 d^{4} {\mathrm e}^{5 i \left (f x +e \right )}-75 d^{4} {\mathrm e}^{i \left (f x +e \right )}-97 d^{4} {\mathrm e}^{2 i \left (f x +e \right )}+6 c^{4} {\mathrm e}^{6 i \left (f x +e \right )}-126 d^{4} {\mathrm e}^{3 i \left (f x +e \right )}-98 d^{4} {\mathrm e}^{4 i \left (f x +e \right )}+6 c^{4} {\mathrm e}^{i \left (f x +e \right )}-21 d^{4} {\mathrm e}^{6 i \left (f x +e \right )}+14 c^{4} {\mathrm e}^{2 i \left (f x +e \right )}+16 c^{4} {\mathrm e}^{4 i \left (f x +e \right )}+8 c^{3} d -48 c^{2} d^{2}+80 c \,d^{3}+12 c^{4} {\mathrm e}^{3 i \left (f x +e \right )}+6 c^{4} {\mathrm e}^{5 i \left (f x +e \right )}\right )}{3 f \,a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3} \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{2}}+\frac {6 d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) c^{2}}{a^{2} f}-\frac {8 d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) c}{a^{2} f}+\frac {7 d^{4} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{2 a^{2} f}-\frac {6 d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) c^{2}}{a^{2} f}+\frac {8 d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) c}{a^{2} f}-\frac {7 d^{4} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{2 a^{2} f}\) | \(655\) |
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Time = 0.31 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.87 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx=\frac {3 \, {\left ({\left (12 \, c^{2} d^{2} - 16 \, c d^{3} + 7 \, d^{4}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (12 \, c^{2} d^{2} - 16 \, c d^{3} + 7 \, d^{4}\right )} \cos \left (f x + e\right )^{3} + {\left (12 \, c^{2} d^{2} - 16 \, c d^{3} + 7 \, d^{4}\right )} \cos \left (f x + e\right )^{2}\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \, {\left ({\left (12 \, c^{2} d^{2} - 16 \, c d^{3} + 7 \, d^{4}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (12 \, c^{2} d^{2} - 16 \, c d^{3} + 7 \, d^{4}\right )} \cos \left (f x + e\right )^{3} + {\left (12 \, c^{2} d^{2} - 16 \, c d^{3} + 7 \, d^{4}\right )} \cos \left (f x + e\right )^{2}\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (3 \, d^{4} + 4 \, {\left (c^{4} + 2 \, c^{3} d - 12 \, c^{2} d^{2} + 20 \, c d^{3} - 8 \, d^{4}\right )} \cos \left (f x + e\right )^{3} + {\left (2 \, c^{4} + 16 \, c^{3} d - 60 \, c^{2} d^{2} + 112 \, c d^{3} - 43 \, d^{4}\right )} \cos \left (f x + e\right )^{2} + 6 \, {\left (4 \, c d^{3} - d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{12 \, {\left (a^{2} f \cos \left (f x + e\right )^{4} + 2 \, a^{2} f \cos \left (f x + e\right )^{3} + a^{2} f \cos \left (f x + e\right )^{2}\right )}} \]
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\[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx=\frac {\int \frac {c^{4} \sec {\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{4} \sec ^{5}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {4 c d^{3} \sec ^{4}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {6 c^{2} d^{2} \sec ^{3}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {4 c^{3} d \sec ^{2}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx}{a^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 536 vs. \(2 (184) = 368\).
Time = 0.23 (sec) , antiderivative size = 536, normalized size of antiderivative = 2.78 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx=-\frac {d^{4} {\left (\frac {6 \, {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {5 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2} - \frac {2 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}} + \frac {\frac {21 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a^{2}} - \frac {21 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{2}} + \frac {21 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{2}}\right )} - 4 \, c d^{3} {\left (\frac {\frac {15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a^{2}} - \frac {12 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{2}} + \frac {12 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{2}} + \frac {12 \, \sin \left (f x + e\right )}{{\left (a^{2} - \frac {a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (f x + e\right ) + 1\right )}}\right )} + 6 \, c^{2} d^{2} {\left (\frac {\frac {9 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a^{2}} - \frac {6 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{2}} + \frac {6 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{2}}\right )} - \frac {4 \, c^{3} d {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2}} - \frac {c^{4} {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2}}}{6 \, f} \]
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Time = 0.36 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.86 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx=\frac {\frac {3 \, {\left (12 \, c^{2} d^{2} - 16 \, c d^{3} + 7 \, d^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a^{2}} - \frac {3 \, {\left (12 \, c^{2} d^{2} - 16 \, c d^{3} + 7 \, d^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a^{2}} - \frac {6 \, {\left (8 \, c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 5 \, d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 8 \, c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{2} a^{2}} - \frac {a^{4} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, a^{4} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 \, a^{4} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, a^{4} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + a^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, a^{4} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 12 \, a^{4} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 54 \, a^{4} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 60 \, a^{4} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 21 \, a^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a^{6}}}{6 \, f} \]
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Time = 13.70 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx=\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (8\,c\,d^3-3\,d^4\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (8\,c\,d^3-5\,d^4\right )}{f\,\left (a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-2\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+a^2\right )}-\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {3\,{\left (c-d\right )}^4}{2\,a^2}-\frac {2\,\left (c+d\right )\,{\left (c-d\right )}^3}{a^2}\right )}{f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,{\left (c-d\right )}^4}{6\,a^2\,f}+\frac {d^2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )\,\left (12\,c^2-16\,c\,d+7\,d^2\right )}{a^2\,f} \]
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