Integrand size = 31, antiderivative size = 188 \[ \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {7 a^4 (10 A+7 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {4 a^4 (10 A+7 C) \tan (c+d x)}{5 d}+\frac {27 a^4 (10 A+7 C) \sec (c+d x) \tan (c+d x)}{80 d}+\frac {a^4 (10 A+7 C) \sec ^3(c+d x) \tan (c+d x)}{40 d}-\frac {C (a+a \sec (c+d x))^4 \tan (c+d x)}{30 d}+\frac {C (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac {2 a^4 (10 A+7 C) \tan ^3(c+d x)}{15 d} \]
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Time = 0.39 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4168, 4086, 3876, 3855, 3852, 8, 3853} \[ \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {7 a^4 (10 A+7 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {2 a^4 (10 A+7 C) \tan ^3(c+d x)}{15 d}+\frac {4 a^4 (10 A+7 C) \tan (c+d x)}{5 d}+\frac {a^4 (10 A+7 C) \tan (c+d x) \sec ^3(c+d x)}{40 d}+\frac {27 a^4 (10 A+7 C) \tan (c+d x) \sec (c+d x)}{80 d}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^5}{6 a d}-\frac {C \tan (c+d x) (a \sec (c+d x)+a)^4}{30 d} \]
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Rule 8
Rule 3852
Rule 3853
Rule 3855
Rule 3876
Rule 4086
Rule 4168
Rubi steps \begin{align*} \text {integral}& = \frac {C (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac {\int \sec (c+d x) (a+a \sec (c+d x))^4 (a (6 A+5 C)-a C \sec (c+d x)) \, dx}{6 a} \\ & = -\frac {C (a+a \sec (c+d x))^4 \tan (c+d x)}{30 d}+\frac {C (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac {1}{10} (10 A+7 C) \int \sec (c+d x) (a+a \sec (c+d x))^4 \, dx \\ & = -\frac {C (a+a \sec (c+d x))^4 \tan (c+d x)}{30 d}+\frac {C (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac {1}{10} (10 A+7 C) \int \left (a^4 \sec (c+d x)+4 a^4 \sec ^2(c+d x)+6 a^4 \sec ^3(c+d x)+4 a^4 \sec ^4(c+d x)+a^4 \sec ^5(c+d x)\right ) \, dx \\ & = -\frac {C (a+a \sec (c+d x))^4 \tan (c+d x)}{30 d}+\frac {C (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac {1}{10} \left (a^4 (10 A+7 C)\right ) \int \sec (c+d x) \, dx+\frac {1}{10} \left (a^4 (10 A+7 C)\right ) \int \sec ^5(c+d x) \, dx+\frac {1}{5} \left (2 a^4 (10 A+7 C)\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{5} \left (2 a^4 (10 A+7 C)\right ) \int \sec ^4(c+d x) \, dx+\frac {1}{5} \left (3 a^4 (10 A+7 C)\right ) \int \sec ^3(c+d x) \, dx \\ & = \frac {a^4 (10 A+7 C) \text {arctanh}(\sin (c+d x))}{10 d}+\frac {3 a^4 (10 A+7 C) \sec (c+d x) \tan (c+d x)}{10 d}+\frac {a^4 (10 A+7 C) \sec ^3(c+d x) \tan (c+d x)}{40 d}-\frac {C (a+a \sec (c+d x))^4 \tan (c+d x)}{30 d}+\frac {C (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac {1}{40} \left (3 a^4 (10 A+7 C)\right ) \int \sec ^3(c+d x) \, dx+\frac {1}{10} \left (3 a^4 (10 A+7 C)\right ) \int \sec (c+d x) \, dx-\frac {\left (2 a^4 (10 A+7 C)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{5 d}-\frac {\left (2 a^4 (10 A+7 C)\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{5 d} \\ & = \frac {2 a^4 (10 A+7 C) \text {arctanh}(\sin (c+d x))}{5 d}+\frac {4 a^4 (10 A+7 C) \tan (c+d x)}{5 d}+\frac {27 a^4 (10 A+7 C) \sec (c+d x) \tan (c+d x)}{80 d}+\frac {a^4 (10 A+7 C) \sec ^3(c+d x) \tan (c+d x)}{40 d}-\frac {C (a+a \sec (c+d x))^4 \tan (c+d x)}{30 d}+\frac {C (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac {2 a^4 (10 A+7 C) \tan ^3(c+d x)}{15 d}+\frac {1}{80} \left (3 a^4 (10 A+7 C)\right ) \int \sec (c+d x) \, dx \\ & = \frac {7 a^4 (10 A+7 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {4 a^4 (10 A+7 C) \tan (c+d x)}{5 d}+\frac {27 a^4 (10 A+7 C) \sec (c+d x) \tan (c+d x)}{80 d}+\frac {a^4 (10 A+7 C) \sec ^3(c+d x) \tan (c+d x)}{40 d}-\frac {C (a+a \sec (c+d x))^4 \tan (c+d x)}{30 d}+\frac {C (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac {2 a^4 (10 A+7 C) \tan ^3(c+d x)}{15 d} \\ \end{align*}
Time = 5.13 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.29 \[ \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {35 a^4 A \text {arctanh}(\sin (c+d x))}{8 d}+\frac {49 a^4 C \text {arctanh}(\sin (c+d x))}{16 d}+\frac {8 a^4 A \tan (c+d x)}{d}+\frac {8 a^4 C \tan (c+d x)}{d}+\frac {27 a^4 A \sec (c+d x) \tan (c+d x)}{8 d}+\frac {49 a^4 C \sec (c+d x) \tan (c+d x)}{16 d}+\frac {a^4 A \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {41 a^4 C \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {a^4 C \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {4 a^4 A \tan ^3(c+d x)}{3 d}+\frac {4 a^4 C \tan ^3(c+d x)}{d}+\frac {4 a^4 C \tan ^5(c+d x)}{5 d} \]
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Time = 0.59 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.21
method | result | size |
norman | \(\frac {\frac {281 a^{4} \left (10 A +7 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 d}-\frac {231 a^{4} \left (10 A +7 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{20 d}+\frac {119 a^{4} \left (10 A +7 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}-\frac {7 a^{4} \left (10 A +7 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}+\frac {3 a^{4} \left (62 A +69 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {a^{4} \left (2138 A +1471 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{6}}-\frac {7 a^{4} \left (10 A +7 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16 d}+\frac {7 a^{4} \left (10 A +7 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16 d}\) | \(227\) |
parallelrisch | \(\frac {31 a^{4} \left (-\frac {525 \left (A +\frac {7 C}{10}\right ) \left (\frac {\cos \left (6 d x +6 c \right )}{15}+\frac {2 \cos \left (4 d x +4 c \right )}{5}+\cos \left (2 d x +2 c \right )+\frac {2}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{124}+\frac {525 \left (A +\frac {7 C}{10}\right ) \left (\frac {\cos \left (6 d x +6 c \right )}{15}+\frac {2 \cos \left (4 d x +4 c \right )}{5}+\cos \left (2 d x +2 c \right )+\frac {2}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{124}+\left (\frac {88 A}{31}+\frac {112 C}{31}\right ) \sin \left (2 d x +2 c \right )+\left (\frac {89 A}{62}+\frac {769 C}{372}\right ) \sin \left (3 d x +3 c \right )+\left (\frac {64 A}{31}+\frac {288 C}{155}\right ) \sin \left (4 d x +4 c \right )+\left (\frac {27 A}{62}+\frac {49 C}{124}\right ) \sin \left (5 d x +5 c \right )+\left (\frac {40 A}{93}+\frac {48 C}{155}\right ) \sin \left (6 d x +6 c \right )+\sin \left (d x +c \right ) \left (\frac {125 C}{62}+A \right )\right )}{2 d \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right )}\) | \(246\) |
parts | \(\frac {\left (a^{4} A +6 a^{4} C \right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (4 a^{4} A +4 a^{4} C \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (6 a^{4} A +a^{4} C \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right ) a^{4}}{d}+\frac {a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {4 a^{4} A \tan \left (d x +c \right )}{d}-\frac {4 a^{4} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(281\) |
derivativedivides | \(\frac {a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 a^{4} A \tan \left (d x +c \right )-4 a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+6 a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-4 a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-4 a^{4} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\) | \(350\) |
default | \(\frac {a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 a^{4} A \tan \left (d x +c \right )-4 a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+6 a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-4 a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-4 a^{4} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\) | \(350\) |
risch | \(-\frac {i a^{4} \left (810 A \,{\mathrm e}^{11 i \left (d x +c \right )}+735 C \,{\mathrm e}^{11 i \left (d x +c \right )}-960 A \,{\mathrm e}^{10 i \left (d x +c \right )}+2670 A \,{\mathrm e}^{9 i \left (d x +c \right )}+3845 C \,{\mathrm e}^{9 i \left (d x +c \right )}-6720 A \,{\mathrm e}^{8 i \left (d x +c \right )}-1920 C \,{\mathrm e}^{8 i \left (d x +c \right )}+1860 A \,{\mathrm e}^{7 i \left (d x +c \right )}+3750 C \,{\mathrm e}^{7 i \left (d x +c \right )}-16000 A \,{\mathrm e}^{6 i \left (d x +c \right )}-11520 C \,{\mathrm e}^{6 i \left (d x +c \right )}-1860 A \,{\mathrm e}^{5 i \left (d x +c \right )}-3750 C \,{\mathrm e}^{5 i \left (d x +c \right )}-17280 A \,{\mathrm e}^{4 i \left (d x +c \right )}-15360 C \,{\mathrm e}^{4 i \left (d x +c \right )}-2670 A \,{\mathrm e}^{3 i \left (d x +c \right )}-3845 C \,{\mathrm e}^{3 i \left (d x +c \right )}-8640 A \,{\mathrm e}^{2 i \left (d x +c \right )}-6912 C \,{\mathrm e}^{2 i \left (d x +c \right )}-810 A \,{\mathrm e}^{i \left (d x +c \right )}-735 C \,{\mathrm e}^{i \left (d x +c \right )}-1600 A -1152 C \right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}-\frac {35 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{8 d}-\frac {49 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{16 d}+\frac {35 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{8 d}+\frac {49 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{16 d}\) | \(371\) |
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Time = 0.28 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.96 \[ \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (10 \, A + 7 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (10 \, A + 7 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (64 \, {\left (25 \, A + 18 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 15 \, {\left (54 \, A + 49 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 64 \, {\left (5 \, A + 9 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 10 \, {\left (6 \, A + 41 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 192 \, C a^{4} \cos \left (d x + c\right ) + 40 \, C a^{4}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \]
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\[ \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=a^{4} \left (\int A \sec {\left (c + d x \right )}\, dx + \int 4 A \sec ^{2}{\left (c + d x \right )}\, dx + \int 6 A \sec ^{3}{\left (c + d x \right )}\, dx + \int 4 A \sec ^{4}{\left (c + d x \right )}\, dx + \int A \sec ^{5}{\left (c + d x \right )}\, dx + \int C \sec ^{3}{\left (c + d x \right )}\, dx + \int 4 C \sec ^{4}{\left (c + d x \right )}\, dx + \int 6 C \sec ^{5}{\left (c + d x \right )}\, dx + \int 4 C \sec ^{6}{\left (c + d x \right )}\, dx + \int C \sec ^{7}{\left (c + d x \right )}\, dx\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 449 vs. \(2 (174) = 348\).
Time = 0.21 (sec) , antiderivative size = 449, normalized size of antiderivative = 2.39 \[ \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {640 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 128 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} C a^{4} + 640 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{4} - 5 \, C a^{4} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 30 \, A a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 180 \, C a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 720 \, A a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 120 \, C a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 480 \, A a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 1920 \, A a^{4} \tan \left (d x + c\right )}{480 \, d} \]
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Time = 0.37 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.49 \[ \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (10 \, A a^{4} + 7 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 105 \, {\left (10 \, A a^{4} + 7 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (1050 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 735 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 5950 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 4165 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 13860 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 9702 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 16860 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 11802 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 10690 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 7355 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2790 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3105 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{6}}}{240 \, d} \]
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Time = 18.20 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.39 \[ \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (-\frac {35\,A\,a^4}{4}-\frac {49\,C\,a^4}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {595\,A\,a^4}{12}+\frac {833\,C\,a^4}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-\frac {231\,A\,a^4}{2}-\frac {1617\,C\,a^4}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {281\,A\,a^4}{2}+\frac {1967\,C\,a^4}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {1069\,A\,a^4}{12}-\frac {1471\,C\,a^4}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {93\,A\,a^4}{4}+\frac {207\,C\,a^4}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {7\,a^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (10\,A+7\,C\right )}{8\,d} \]
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