Integrand size = 41, antiderivative size = 253 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {7 a^4 (7 A+8 B+10 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^4 (72 A+83 B+100 C) \tan (c+d x)}{15 d}+\frac {a^4 (417 A+488 B+550 C) \sec (c+d x) \tan (c+d x)}{240 d}+\frac {(43 A+52 B+50 C) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac {(37 A+48 B+30 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {a (2 A+3 B) (a+a \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d} \]
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Time = 0.92 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {3122, 3054, 3047, 3100, 2827, 3852, 8, 3855} \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {7 a^4 (7 A+8 B+10 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^4 (72 A+83 B+100 C) \tan (c+d x)}{15 d}+\frac {a^4 (417 A+488 B+550 C) \tan (c+d x) \sec (c+d x)}{240 d}+\frac {(43 A+52 B+50 C) \tan (c+d x) \sec ^2(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{60 d}+\frac {(37 A+48 B+30 C) \tan (c+d x) \sec ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{120 d}+\frac {a (2 A+3 B) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{15 d}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d} \]
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Rule 8
Rule 2827
Rule 3047
Rule 3054
Rule 3100
Rule 3122
Rule 3852
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {A (a+a \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {\int (a+a \cos (c+d x))^4 (2 a (2 A+3 B)+a (A+6 C) \cos (c+d x)) \sec ^6(c+d x) \, dx}{6 a} \\ & = \frac {a (2 A+3 B) (a+a \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {\int (a+a \cos (c+d x))^3 \left (a^2 (37 A+48 B+30 C)+3 a^2 (3 A+2 B+10 C) \cos (c+d x)\right ) \sec ^5(c+d x) \, dx}{30 a} \\ & = \frac {(37 A+48 B+30 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {a (2 A+3 B) (a+a \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {\int (a+a \cos (c+d x))^2 \left (6 a^3 (43 A+52 B+50 C)+a^3 (73 A+72 B+150 C) \cos (c+d x)\right ) \sec ^4(c+d x) \, dx}{120 a} \\ & = \frac {(43 A+52 B+50 C) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac {(37 A+48 B+30 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {a (2 A+3 B) (a+a \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {\int (a+a \cos (c+d x)) \left (3 a^4 (417 A+488 B+550 C)+3 a^4 (159 A+176 B+250 C) \cos (c+d x)\right ) \sec ^3(c+d x) \, dx}{360 a} \\ & = \frac {(43 A+52 B+50 C) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac {(37 A+48 B+30 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {a (2 A+3 B) (a+a \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {\int \left (3 a^5 (417 A+488 B+550 C)+\left (3 a^5 (159 A+176 B+250 C)+3 a^5 (417 A+488 B+550 C)\right ) \cos (c+d x)+3 a^5 (159 A+176 B+250 C) \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx}{360 a} \\ & = \frac {a^4 (417 A+488 B+550 C) \sec (c+d x) \tan (c+d x)}{240 d}+\frac {(43 A+52 B+50 C) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac {(37 A+48 B+30 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {a (2 A+3 B) (a+a \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {\int \left (48 a^5 (72 A+83 B+100 C)+315 a^5 (7 A+8 B+10 C) \cos (c+d x)\right ) \sec ^2(c+d x) \, dx}{720 a} \\ & = \frac {a^4 (417 A+488 B+550 C) \sec (c+d x) \tan (c+d x)}{240 d}+\frac {(43 A+52 B+50 C) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac {(37 A+48 B+30 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {a (2 A+3 B) (a+a \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {1}{16} \left (7 a^4 (7 A+8 B+10 C)\right ) \int \sec (c+d x) \, dx+\frac {1}{15} \left (a^4 (72 A+83 B+100 C)\right ) \int \sec ^2(c+d x) \, dx \\ & = \frac {7 a^4 (7 A+8 B+10 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^4 (417 A+488 B+550 C) \sec (c+d x) \tan (c+d x)}{240 d}+\frac {(43 A+52 B+50 C) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac {(37 A+48 B+30 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {a (2 A+3 B) (a+a \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {\left (a^4 (72 A+83 B+100 C)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{15 d} \\ & = \frac {7 a^4 (7 A+8 B+10 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^4 (72 A+83 B+100 C) \tan (c+d x)}{15 d}+\frac {a^4 (417 A+488 B+550 C) \sec (c+d x) \tan (c+d x)}{240 d}+\frac {(43 A+52 B+50 C) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac {(37 A+48 B+30 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {a (2 A+3 B) (a+a \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d} \\ \end{align*}
Time = 6.49 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.51 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {a^4 \left (105 (7 A+8 B+10 C) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (15 (49 A+56 B+54 C) \sec (c+d x)+10 (41 A+6 (4 B+C)) \sec ^3(c+d x)+40 A \sec ^5(c+d x)+16 \left (120 (A+B+C)+20 (3 A+2 B+C) \tan ^2(c+d x)+3 (4 A+B) \tan ^4(c+d x)\right )\right )\right )}{240 d} \]
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Time = 13.07 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.07
method | result | size |
parallelrisch | \(-\frac {49 a^{4} \left (\left (A +\frac {8 B}{7}+\frac {10 C}{7}\right ) \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\left (A +\frac {8 B}{7}+\frac {10 C}{7}\right ) \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (-\frac {704 C}{49}-\frac {128 A}{7}-16 B \right ) \sin \left (2 d x +2 c \right )+\left (-\frac {1538 A}{147}-\frac {464 B}{49}-\frac {356 C}{49}\right ) \sin \left (3 d x +3 c \right )+\left (-\frac {2304 A}{245}-\frac {2496 B}{245}-\frac {512 C}{49}\right ) \sin \left (4 d x +4 c \right )+\left (-2 A -\frac {16 B}{7}-\frac {108 C}{49}\right ) \sin \left (5 d x +5 c \right )+\left (-\frac {384 A}{245}-\frac {320 C}{147}-\frac {1328 B}{735}\right ) \sin \left (6 d x +6 c \right )-\frac {500 \left (\frac {88 B}{125}+\frac {62 C}{125}+A \right ) \sin \left (d x +c \right )}{49}\right )}{16 d \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right )}\) | \(270\) |
parts | \(\frac {a^{4} A \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}-\frac {\left (4 a^{4} A +B \,a^{4}\right ) \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (B \,a^{4}+4 C \,a^{4}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (a^{4} A +4 B \,a^{4}+6 C \,a^{4}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {\left (4 a^{4} A +6 B \,a^{4}+4 C \,a^{4}\right ) \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (6 a^{4} A +4 B \,a^{4}+C \,a^{4}\right ) \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right ) a^{4}}{d}\) | \(314\) |
derivativedivides | \(\frac {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 )+B \,a^{4} \tan \left (d x +c \right )+C \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )-4 a^{4} A \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 B \,a^{4} \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 C \,a^{4} \tan \left (d x +c \right )+6 a^{4} A \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{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 )-6 B \,a^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+6 C \,a^{4} \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 \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+4 B \,a^{4} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{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 C \,a^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+a^{4} A \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{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 )-B \,a^{4} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+C \,a^{4} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{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}\) | \(506\) |
default | \(\frac {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 )+B \,a^{4} \tan \left (d x +c \right )+C \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )-4 a^{4} A \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 B \,a^{4} \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 C \,a^{4} \tan \left (d x +c \right )+6 a^{4} A \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{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 )-6 B \,a^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+6 C \,a^{4} \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 \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+4 B \,a^{4} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{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 C \,a^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+a^{4} A \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{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 )-B \,a^{4} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+C \,a^{4} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{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}\) | \(506\) |
risch | \(-\frac {i a^{4} \left (-1920 A \,{\mathrm e}^{8 i \left (d x +c \right )}-1600 C -1152 A -1328 B -15360 A \,{\mathrm e}^{4 i \left (d x +c \right )}-15840 B \,{\mathrm e}^{4 i \left (d x +c \right )}-7728 B \,{\mathrm e}^{2 i \left (d x +c \right )}-17280 C \,{\mathrm e}^{4 i \left (d x +c \right )}-8640 C \,{\mathrm e}^{2 i \left (d x +c \right )}-240 B \,{\mathrm e}^{10 i \left (d x +c \right )}+2640 B \,{\mathrm e}^{7 i \left (d x +c \right )}-3845 A \,{\mathrm e}^{3 i \left (d x +c \right )}-3750 A \,{\mathrm e}^{5 i \left (d x +c \right )}-2640 B \,{\mathrm e}^{5 i \left (d x +c \right )}-6912 A \,{\mathrm e}^{2 i \left (d x +c \right )}-735 A \,{\mathrm e}^{i \left (d x +c \right )}-840 B \,{\mathrm e}^{i \left (d x +c \right )}-13280 B \,{\mathrm e}^{6 i \left (d x +c \right )}-3480 B \,{\mathrm e}^{3 i \left (d x +c \right )}-11520 A \,{\mathrm e}^{6 i \left (d x +c \right )}-16000 C \,{\mathrm e}^{6 i \left (d x +c \right )}+3750 A \,{\mathrm e}^{7 i \left (d x +c \right )}-810 C \,{\mathrm e}^{i \left (d x +c \right )}-4080 B \,{\mathrm e}^{8 i \left (d x +c \right )}+1860 C \,{\mathrm e}^{7 i \left (d x +c \right )}-1860 C \,{\mathrm e}^{5 i \left (d x +c \right )}-2670 C \,{\mathrm e}^{3 i \left (d x +c \right )}+840 B \,{\mathrm e}^{11 i \left (d x +c \right )}+735 A \,{\mathrm e}^{11 i \left (d x +c \right )}+810 C \,{\mathrm e}^{11 i \left (d x +c \right )}-960 C \,{\mathrm e}^{10 i \left (d x +c \right )}-6720 C \,{\mathrm e}^{8 i \left (d x +c \right )}+2670 C \,{\mathrm e}^{9 i \left (d x +c \right )}+3845 A \,{\mathrm e}^{9 i \left (d x +c \right )}+3480 B \,{\mathrm e}^{9 i \left (d x +c \right )}\right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}+\frac {49 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{16 d}+\frac {7 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{2 d}+\frac {35 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{8 d}-\frac {49 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{16 d}-\frac {7 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{2 d}-\frac {35 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{8 d}\) | \(550\) |
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Time = 0.27 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.80 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {105 \, {\left (7 \, A + 8 \, B + 10 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (7 \, A + 8 \, B + 10 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (72 \, A + 83 \, B + 100 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 15 \, {\left (49 \, A + 56 \, B + 54 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 32 \, {\left (18 \, A + 17 \, B + 10 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 10 \, {\left (41 \, A + 24 \, B + 6 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 48 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right ) + 40 \, A a^{4}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \]
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Timed out. \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 645 vs. \(2 (239) = 478\).
Time = 0.23 (sec) , antiderivative size = 645, normalized size of antiderivative = 2.55 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {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 )} A a^{4} + 640 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 32 \, {\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 )} B a^{4} + 960 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{4} + 640 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{4} - 5 \, A 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 )} - 180 \, 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 )} - 120 \, B 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 )} - 30 \, 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 )} - 120 \, 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 )} - 480 \, B 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 )} - 720 \, 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 )} + 240 \, C a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 480 \, B a^{4} \tan \left (d x + c\right ) + 1920 \, C a^{4} \tan \left (d x + c\right )}{480 \, d} \]
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Time = 0.40 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.55 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {105 \, {\left (7 \, A a^{4} + 8 \, B a^{4} + 10 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 105 \, {\left (7 \, A a^{4} + 8 \, B a^{4} + 10 \, 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 (735 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 840 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1050 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 4165 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 4760 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 5950 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 9702 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 11088 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 13860 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 11802 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 13488 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 16860 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 7355 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9320 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 10690 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3105 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3000 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2790 \, 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 = 5.08 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.34 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {7\,a^4\,\mathrm {atanh}\left (\frac {7\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (7\,A+8\,B+10\,C\right )}{4\,\left (\frac {49\,A\,a^4}{4}+14\,B\,a^4+\frac {35\,C\,a^4}{2}\right )}\right )\,\left (7\,A+8\,B+10\,C\right )}{8\,d}-\frac {\left (\frac {49\,A\,a^4}{8}+7\,B\,a^4+\frac {35\,C\,a^4}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (-\frac {833\,A\,a^4}{24}-\frac {119\,B\,a^4}{3}-\frac {595\,C\,a^4}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {1617\,A\,a^4}{20}+\frac {462\,B\,a^4}{5}+\frac {231\,C\,a^4}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (-\frac {1967\,A\,a^4}{20}-\frac {562\,B\,a^4}{5}-\frac {281\,C\,a^4}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {1471\,A\,a^4}{24}+\frac {233\,B\,a^4}{3}+\frac {1069\,C\,a^4}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (-\frac {207\,A\,a^4}{8}-25\,B\,a^4-\frac {93\,C\,a^4}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
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