Integrand size = 33, antiderivative size = 228 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4 (14 A+11 C) \text {arctanh}(\sin (c+d x))}{4 d}+\frac {16 a^4 (14 A+11 C) \tan (c+d x)}{35 d}+\frac {27 a^4 (14 A+11 C) \sec (c+d x) \tan (c+d x)}{140 d}+\frac {a^4 (14 A+11 C) \sec ^3(c+d x) \tan (c+d x)}{70 d}+\frac {(21 A+4 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{105 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {2 C (a+a \sec (c+d x))^5 \tan (c+d x)}{21 a d}+\frac {8 a^4 (14 A+11 C) \tan ^3(c+d x)}{105 d} \]
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Time = 0.61 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {4174, 4095, 4086, 3876, 3855, 3852, 8, 3853} \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4 (14 A+11 C) \text {arctanh}(\sin (c+d x))}{4 d}+\frac {8 a^4 (14 A+11 C) \tan ^3(c+d x)}{105 d}+\frac {16 a^4 (14 A+11 C) \tan (c+d x)}{35 d}+\frac {a^4 (14 A+11 C) \tan (c+d x) \sec ^3(c+d x)}{70 d}+\frac {27 a^4 (14 A+11 C) \tan (c+d x) \sec (c+d x)}{140 d}+\frac {(21 A+4 C) \tan (c+d x) (a \sec (c+d x)+a)^4}{105 d}+\frac {C \tan (c+d x) \sec ^2(c+d x) (a \sec (c+d x)+a)^4}{7 d}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^5}{21 a d} \]
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Rule 8
Rule 3852
Rule 3853
Rule 3855
Rule 3876
Rule 4086
Rule 4095
Rule 4174
Rubi steps \begin{align*} \text {integral}& = \frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {\int \sec ^2(c+d x) (a+a \sec (c+d x))^4 (a (7 A+2 C)+4 a C \sec (c+d x)) \, dx}{7 a} \\ & = \frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {2 C (a+a \sec (c+d x))^5 \tan (c+d x)}{21 a d}+\frac {\int \sec (c+d x) (a+a \sec (c+d x))^4 \left (20 a^2 C+2 a^2 (21 A+4 C) \sec (c+d x)\right ) \, dx}{42 a^2} \\ & = \frac {(21 A+4 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{105 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {2 C (a+a \sec (c+d x))^5 \tan (c+d x)}{21 a d}+\frac {1}{35} (2 (14 A+11 C)) \int \sec (c+d x) (a+a \sec (c+d x))^4 \, dx \\ & = \frac {(21 A+4 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{105 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {2 C (a+a \sec (c+d x))^5 \tan (c+d x)}{21 a d}+\frac {1}{35} (2 (14 A+11 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 {(21 A+4 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{105 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {2 C (a+a \sec (c+d x))^5 \tan (c+d x)}{21 a d}+\frac {1}{35} \left (2 a^4 (14 A+11 C)\right ) \int \sec (c+d x) \, dx+\frac {1}{35} \left (2 a^4 (14 A+11 C)\right ) \int \sec ^5(c+d x) \, dx+\frac {1}{35} \left (8 a^4 (14 A+11 C)\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{35} \left (8 a^4 (14 A+11 C)\right ) \int \sec ^4(c+d x) \, dx+\frac {1}{35} \left (12 a^4 (14 A+11 C)\right ) \int \sec ^3(c+d x) \, dx \\ & = \frac {2 a^4 (14 A+11 C) \text {arctanh}(\sin (c+d x))}{35 d}+\frac {6 a^4 (14 A+11 C) \sec (c+d x) \tan (c+d x)}{35 d}+\frac {a^4 (14 A+11 C) \sec ^3(c+d x) \tan (c+d x)}{70 d}+\frac {(21 A+4 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{105 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {2 C (a+a \sec (c+d x))^5 \tan (c+d x)}{21 a d}+\frac {1}{70} \left (3 a^4 (14 A+11 C)\right ) \int \sec ^3(c+d x) \, dx+\frac {1}{35} \left (6 a^4 (14 A+11 C)\right ) \int \sec (c+d x) \, dx-\frac {\left (8 a^4 (14 A+11 C)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{35 d}-\frac {\left (8 a^4 (14 A+11 C)\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{35 d} \\ & = \frac {8 a^4 (14 A+11 C) \text {arctanh}(\sin (c+d x))}{35 d}+\frac {16 a^4 (14 A+11 C) \tan (c+d x)}{35 d}+\frac {27 a^4 (14 A+11 C) \sec (c+d x) \tan (c+d x)}{140 d}+\frac {a^4 (14 A+11 C) \sec ^3(c+d x) \tan (c+d x)}{70 d}+\frac {(21 A+4 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{105 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {2 C (a+a \sec (c+d x))^5 \tan (c+d x)}{21 a d}+\frac {8 a^4 (14 A+11 C) \tan ^3(c+d x)}{105 d}+\frac {1}{140} \left (3 a^4 (14 A+11 C)\right ) \int \sec (c+d x) \, dx \\ & = \frac {a^4 (14 A+11 C) \text {arctanh}(\sin (c+d x))}{4 d}+\frac {16 a^4 (14 A+11 C) \tan (c+d x)}{35 d}+\frac {27 a^4 (14 A+11 C) \sec (c+d x) \tan (c+d x)}{140 d}+\frac {a^4 (14 A+11 C) \sec ^3(c+d x) \tan (c+d x)}{70 d}+\frac {(21 A+4 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{105 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {2 C (a+a \sec (c+d x))^5 \tan (c+d x)}{21 a d}+\frac {8 a^4 (14 A+11 C) \tan ^3(c+d x)}{105 d} \\ \end{align*}
Time = 5.85 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.57 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4 \left (105 (14 A+11 C) \text {arctanh}(\sin (c+d x))+\left (4 (581 A+454 C)+105 (14 A+11 C) \sec (c+d x)+(952 A+908 C) \sec ^2(c+d x)+70 (6 A+11 C) \sec ^3(c+d x)+12 (7 A+48 C) \sec ^4(c+d x)+280 C \sec ^5(c+d x)+60 C \sec ^6(c+d x)\right ) \tan (c+d x)\right )}{420 d} \]
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Time = 0.80 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.11
method | result | size |
norman | \(\frac {\frac {512 a^{4} \left (14 A +11 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{35 d}-\frac {283 a^{4} \left (14 A +11 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{30 d}+\frac {10 a^{4} \left (14 A +11 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{3 d}-\frac {a^{4} \left (14 A +11 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{2 d}+\frac {14 a^{4} \left (22 A +15 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}-\frac {a^{4} \left (50 A +53 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}-\frac {a^{4} \left (5702 A +4503 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{30 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{7}}-\frac {a^{4} \left (14 A +11 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{4 d}+\frac {a^{4} \left (14 A +11 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{4 d}\) | \(253\) |
parallelrisch | \(-\frac {7 a^{4} \left (\left (\cos \left (7 d x +7 c \right )+7 \cos \left (5 d x +5 c \right )+21 \cos \left (3 d x +3 c \right )+35 \cos \left (d x +c \right )\right ) \left (A +\frac {11 C}{14}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\left (\cos \left (7 d x +7 c \right )+7 \cos \left (5 d x +5 c \right )+21 \cos \left (3 d x +3 c \right )+35 \cos \left (d x +c \right )\right ) \left (A +\frac {11 C}{14}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (-\frac {102 A}{7}-\frac {67 C}{3}\right ) \sin \left (2 d x +2 c \right )+\left (-\frac {802 A}{35}-\frac {124 C}{5}\right ) \sin \left (3 d x +3 c \right )+\left (-\frac {72 A}{7}-\frac {220 C}{21}\right ) \sin \left (4 d x +4 c \right )+\left (-\frac {1102 A}{105}-\frac {908 C}{105}\right ) \sin \left (5 d x +5 c \right )+\left (-2 A -\frac {11 C}{7}\right ) \sin \left (6 d x +6 c \right )+\left (-\frac {908 C}{735}-\frac {166 A}{105}\right ) \sin \left (7 d x +7 c \right )-14 \left (A +\frac {10 C}{7}\right ) \sin \left (d x +c \right )\right )}{2 d \left (\cos \left (7 d x +7 c \right )+7 \cos \left (5 d x +5 c \right )+21 \cos \left (3 d x +3 c \right )+35 \cos \left (d x +c \right )\right )}\) | \(284\) |
parts | \(-\frac {\left (a^{4} A +6 a^{4} C \right ) \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (4 a^{4} A +4 a^{4} C \right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (6 a^{4} A +a^{4} C \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {a^{4} A \tan \left (d x +c \right )}{d}-\frac {a^{4} C \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )}{d}+\frac {2 A \sec \left (d x +c \right ) \tan \left (d x +c \right ) a^{4}}{d}+\frac {2 A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right ) a^{4}}{d}+\frac {4 a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\) | \(311\) |
derivativedivides | \(\frac {a^{4} A \tan \left (d x +c \right )-a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-6 a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-6 a^{4} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+4 a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+4 a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-a^{4} A \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )-a^{4} C \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )}{d}\) | \(374\) |
default | \(\frac {a^{4} A \tan \left (d x +c \right )-a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-6 a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-6 a^{4} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+4 a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+4 a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-a^{4} A \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )-a^{4} C \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )}{d}\) | \(374\) |
risch | \(-\frac {i a^{4} \left (-1816 C -2324 A -30380 A \,{\mathrm e}^{8 i \left (d x +c \right )}-10710 A \,{\mathrm e}^{5 i \left (d x +c \right )}-7560 A \,{\mathrm e}^{3 i \left (d x +c \right )}-1470 A \,{\mathrm e}^{i \left (d x +c \right )}-50960 A \,{\mathrm e}^{6 i \left (d x +c \right )}-46480 C \,{\mathrm e}^{6 i \left (d x +c \right )}-41244 A \,{\mathrm e}^{4 i \left (d x +c \right )}-37296 C \,{\mathrm e}^{4 i \left (d x +c \right )}-15848 A \,{\mathrm e}^{2 i \left (d x +c \right )}-12712 C \,{\mathrm e}^{2 i \left (d x +c \right )}-16415 C \,{\mathrm e}^{5 i \left (d x +c \right )}-7700 C \,{\mathrm e}^{3 i \left (d x +c \right )}-1155 C \,{\mathrm e}^{i \left (d x +c \right )}-420 A \,{\mathrm e}^{12 i \left (d x +c \right )}+1155 C \,{\mathrm e}^{13 i \left (d x +c \right )}+1470 A \,{\mathrm e}^{13 i \left (d x +c \right )}-840 C \,{\mathrm e}^{10 i \left (d x +c \right )}+7560 A \,{\mathrm e}^{11 i \left (d x +c \right )}+7700 C \,{\mathrm e}^{11 i \left (d x +c \right )}-7560 A \,{\mathrm e}^{10 i \left (d x +c \right )}+10710 A \,{\mathrm e}^{9 i \left (d x +c \right )}+16415 C \,{\mathrm e}^{9 i \left (d x +c \right )}-17080 C \,{\mathrm e}^{8 i \left (d x +c \right )}\right )}{210 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{7}}+\frac {7 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{2 d}+\frac {11 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{4 d}-\frac {7 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{2 d}-\frac {11 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{4 d}\) | \(395\) |
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Time = 0.28 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.88 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (14 \, A + 11 \, C\right )} a^{4} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (14 \, A + 11 \, C\right )} a^{4} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (4 \, {\left (581 \, A + 454 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} + 105 \, {\left (14 \, A + 11 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 4 \, {\left (238 \, A + 227 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \, {\left (6 \, A + 11 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 12 \, {\left (7 \, A + 48 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 280 \, C a^{4} \cos \left (d x + c\right ) + 60 \, C a^{4}\right )} \sin \left (d x + c\right )}{840 \, d \cos \left (d x + c\right )^{7}} \]
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\[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=a^{4} \left (\int A \sec ^{2}{\left (c + d x \right )}\, dx + \int 4 A \sec ^{3}{\left (c + d x \right )}\, dx + \int 6 A \sec ^{4}{\left (c + d x \right )}\, dx + \int 4 A \sec ^{5}{\left (c + d x \right )}\, dx + \int A \sec ^{6}{\left (c + d x \right )}\, dx + \int C \sec ^{4}{\left (c + d x \right )}\, dx + \int 4 C \sec ^{5}{\left (c + d x \right )}\, dx + \int 6 C \sec ^{6}{\left (c + d x \right )}\, dx + \int 4 C \sec ^{7}{\left (c + d x \right )}\, dx + \int C \sec ^{8}{\left (c + d x \right )}\, dx\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 462 vs. \(2 (212) = 424\).
Time = 0.22 (sec) , antiderivative size = 462, normalized size of antiderivative = 2.03 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {56 \, {\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} + 1680 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 24 \, {\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} C a^{4} + 336 \, {\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} + 280 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{4} - 35 \, 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 )} - 210 \, 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 )} - 210 \, 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 )} - 840 \, 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 )} + 840 \, A a^{4} \tan \left (d x + c\right )}{840 \, d} \]
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Time = 0.37 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.38 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (14 \, A a^{4} + 11 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 105 \, {\left (14 \, A a^{4} + 11 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (1470 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 1155 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 9800 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 7700 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 27734 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 21791 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 43008 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 33792 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 39914 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 31521 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 21560 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 14700 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5250 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5565 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{7}}}{420 \, d} \]
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Time = 18.20 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.32 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (14\,A+11\,C\right )}{2\,d}-\frac {\left (7\,A\,a^4+\frac {11\,C\,a^4}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (-\frac {140\,A\,a^4}{3}-\frac {110\,C\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {1981\,A\,a^4}{15}+\frac {3113\,C\,a^4}{30}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-\frac {1024\,A\,a^4}{5}-\frac {5632\,C\,a^4}{35}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {2851\,A\,a^4}{15}+\frac {1501\,C\,a^4}{10}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {308\,A\,a^4}{3}-70\,C\,a^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (25\,A\,a^4+\frac {53\,C\,a^4}{2}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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