Integrand size = 41, antiderivative size = 252 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4 (56 A+49 B+44 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {4 a^4 (56 A+49 B+44 C) \tan (c+d x)}{35 d}+\frac {27 a^4 (56 A+49 B+44 C) \sec (c+d x) \tan (c+d x)}{560 d}+\frac {a^4 (56 A+49 B+44 C) \sec ^3(c+d x) \tan (c+d x)}{280 d}+\frac {(42 A-7 B+8 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{210 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \sec (c+d x))^5 \tan (c+d x)}{42 a d}+\frac {2 a^4 (56 A+49 B+44 C) \tan ^3(c+d x)}{105 d} \]
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Time = 0.59 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {4173, 4095, 4086, 3876, 3855, 3852, 8, 3853} \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4 (56 A+49 B+44 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {2 a^4 (56 A+49 B+44 C) \tan ^3(c+d x)}{105 d}+\frac {4 a^4 (56 A+49 B+44 C) \tan (c+d x)}{35 d}+\frac {a^4 (56 A+49 B+44 C) \tan (c+d x) \sec ^3(c+d x)}{280 d}+\frac {27 a^4 (56 A+49 B+44 C) \tan (c+d x) \sec (c+d x)}{560 d}+\frac {(42 A-7 B+8 C) \tan (c+d x) (a \sec (c+d x)+a)^4}{210 d}+\frac {(7 B+4 C) \tan (c+d x) (a \sec (c+d x)+a)^5}{42 a d}+\frac {C \tan (c+d x) \sec ^2(c+d x) (a \sec (c+d x)+a)^4}{7 d} \]
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Rule 8
Rule 3852
Rule 3853
Rule 3855
Rule 3876
Rule 4086
Rule 4095
Rule 4173
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)+a (7 B+4 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 {(7 B+4 C) (a+a \sec (c+d x))^5 \tan (c+d x)}{42 a d}+\frac {\int \sec (c+d x) (a+a \sec (c+d x))^4 \left (5 a^2 (7 B+4 C)+a^2 (42 A-7 B+8 C) \sec (c+d x)\right ) \, dx}{42 a^2} \\ & = \frac {(42 A-7 B+8 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{210 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \sec (c+d x))^5 \tan (c+d x)}{42 a d}+\frac {1}{70} (56 A+49 B+44 C) \int \sec (c+d x) (a+a \sec (c+d x))^4 \, dx \\ & = \frac {(42 A-7 B+8 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{210 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \sec (c+d x))^5 \tan (c+d x)}{42 a d}+\frac {1}{70} (56 A+49 B+44 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 {(42 A-7 B+8 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{210 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \sec (c+d x))^5 \tan (c+d x)}{42 a d}+\frac {1}{70} \left (a^4 (56 A+49 B+44 C)\right ) \int \sec (c+d x) \, dx+\frac {1}{70} \left (a^4 (56 A+49 B+44 C)\right ) \int \sec ^5(c+d x) \, dx+\frac {1}{35} \left (2 a^4 (56 A+49 B+44 C)\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{35} \left (2 a^4 (56 A+49 B+44 C)\right ) \int \sec ^4(c+d x) \, dx+\frac {1}{35} \left (3 a^4 (56 A+49 B+44 C)\right ) \int \sec ^3(c+d x) \, dx \\ & = \frac {a^4 (56 A+49 B+44 C) \text {arctanh}(\sin (c+d x))}{70 d}+\frac {3 a^4 (56 A+49 B+44 C) \sec (c+d x) \tan (c+d x)}{70 d}+\frac {a^4 (56 A+49 B+44 C) \sec ^3(c+d x) \tan (c+d x)}{280 d}+\frac {(42 A-7 B+8 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{210 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \sec (c+d x))^5 \tan (c+d x)}{42 a d}+\frac {1}{280} \left (3 a^4 (56 A+49 B+44 C)\right ) \int \sec ^3(c+d x) \, dx+\frac {1}{70} \left (3 a^4 (56 A+49 B+44 C)\right ) \int \sec (c+d x) \, dx-\frac {\left (2 a^4 (56 A+49 B+44 C)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{35 d}-\frac {\left (2 a^4 (56 A+49 B+44 C)\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{35 d} \\ & = \frac {2 a^4 (56 A+49 B+44 C) \text {arctanh}(\sin (c+d x))}{35 d}+\frac {4 a^4 (56 A+49 B+44 C) \tan (c+d x)}{35 d}+\frac {27 a^4 (56 A+49 B+44 C) \sec (c+d x) \tan (c+d x)}{560 d}+\frac {a^4 (56 A+49 B+44 C) \sec ^3(c+d x) \tan (c+d x)}{280 d}+\frac {(42 A-7 B+8 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{210 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \sec (c+d x))^5 \tan (c+d x)}{42 a d}+\frac {2 a^4 (56 A+49 B+44 C) \tan ^3(c+d x)}{105 d}+\frac {1}{560} \left (3 a^4 (56 A+49 B+44 C)\right ) \int \sec (c+d x) \, dx \\ & = \frac {a^4 (56 A+49 B+44 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {4 a^4 (56 A+49 B+44 C) \tan (c+d x)}{35 d}+\frac {27 a^4 (56 A+49 B+44 C) \sec (c+d x) \tan (c+d x)}{560 d}+\frac {a^4 (56 A+49 B+44 C) \sec ^3(c+d x) \tan (c+d x)}{280 d}+\frac {(42 A-7 B+8 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{210 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \sec (c+d x))^5 \tan (c+d x)}{42 a d}+\frac {2 a^4 (56 A+49 B+44 C) \tan ^3(c+d x)}{105 d} \\ \end{align*}
Time = 6.28 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.61 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4 \left (105 (56 A+49 B+44 C) \text {arctanh}(\sin (c+d x))+\left (16 (581 A+504 B+454 C)+105 (56 A+49 B+44 C) \sec (c+d x)+16 (238 A+252 B+227 C) \sec ^2(c+d x)+70 (24 A+41 B+44 C) \sec ^3(c+d x)+48 (7 A+28 B+48 C) \sec ^4(c+d x)+280 (B+4 C) \sec ^5(c+d x)+240 C \sec ^6(c+d x)\right ) \tan (c+d x)\right )}{1680 d} \]
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Time = 1.08 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.11
method | result | size |
norman | \(\frac {\frac {128 a^{4} \left (56 A +49 B +44 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{35 d}-\frac {283 a^{4} \left (56 A +49 B +44 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{120 d}+\frac {5 a^{4} \left (56 A +49 B +44 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{6 d}-\frac {a^{4} \left (56 A +49 B +44 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{8 d}-\frac {a^{4} \left (200 A +207 B +212 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {a^{4} \left (616 A +523 B +420 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 d}-\frac {a^{4} \left (22808 A +19157 B +18012 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{120 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{7}}-\frac {a^{4} \left (56 A +49 B +44 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16 d}+\frac {a^{4} \left (56 A +49 B +44 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16 d}\) | \(280\) |
parallelrisch | \(-\frac {7 a^{4} \left (\left (\frac {\cos \left (7 d x +7 c \right )}{7}+3 \cos \left (3 d x +3 c \right )+5 \cos \left (d x +c \right )+\cos \left (5 d x +5 c \right )\right ) \left (A +\frac {7 B}{8}+\frac {11 C}{14}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\left (\frac {\cos \left (7 d x +7 c \right )}{7}+3 \cos \left (3 d x +3 c \right )+5 \cos \left (d x +c \right )+\cos \left (5 d x +5 c \right )\right ) \left (A +\frac {7 B}{8}+\frac {11 C}{14}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (-\frac {67 C}{21}-\frac {102 A}{49}-\frac {31 B}{12}\right ) \sin \left (2 d x +2 c \right )+\left (-\frac {848 B}{245}-\frac {124 C}{35}-\frac {802 A}{245}\right ) \sin \left (3 d x +3 c \right )+\left (-\frac {220 C}{147}-\frac {72 A}{49}-\frac {229 B}{147}\right ) \sin \left (4 d x +4 c \right )+\left (-\frac {48 B}{35}-\frac {1102 A}{735}-\frac {908 C}{735}\right ) \sin \left (5 d x +5 c \right )+\left (-\frac {B}{4}-\frac {2 A}{7}-\frac {11 C}{49}\right ) \sin \left (6 d x +6 c \right )+\left (-\frac {908 C}{5145}-\frac {166 A}{735}-\frac {48 B}{245}\right ) \sin \left (7 d x +7 c \right )-2 \sin \left (d x +c \right ) \left (A +\frac {8 B}{7}+\frac {10 C}{7}\right )\right )}{2 d \left (\frac {\cos \left (7 d x +7 c \right )}{7}+3 \cos \left (3 d x +3 c \right )+5 \cos \left (d x +c \right )+\cos \left (5 d x +5 c \right )\right )}\) | \(311\) |
parts | \(\frac {\left (4 a^{4} A +B \,a^{4}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (B \,a^{4}+4 a^{4} C \right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}-\frac {\left (a^{4} A +4 B \,a^{4}+6 a^{4} C \right ) \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (4 a^{4} A +6 B \,a^{4}+4 a^{4} C \right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (6 a^{4} A +4 B \,a^{4}+a^{4} C \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {a^{4} A \tan \left (d x +c \right )}{d}-\frac {a^{4} C \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )}{d}\) | \(339\) |
risch | \(-\frac {i a^{4} \left (-7264 C -9296 A -8064 B -121520 A \,{\mathrm e}^{8 i \left (d x +c \right )}-42840 A \,{\mathrm e}^{5 i \left (d x +c \right )}-30240 A \,{\mathrm e}^{3 i \left (d x +c \right )}-5880 A \,{\mathrm e}^{i \left (d x +c \right )}-155904 B \,{\mathrm e}^{4 i \left (d x +c \right )}-203840 A \,{\mathrm e}^{6 i \left (d x +c \right )}-185920 C \,{\mathrm e}^{6 i \left (d x +c \right )}-164976 A \,{\mathrm e}^{4 i \left (d x +c \right )}-149184 C \,{\mathrm e}^{4 i \left (d x +c \right )}-63392 A \,{\mathrm e}^{2 i \left (d x +c \right )}-50848 C \,{\mathrm e}^{2 i \left (d x +c \right )}-65660 C \,{\mathrm e}^{5 i \left (d x +c \right )}-30800 C \,{\mathrm e}^{3 i \left (d x +c \right )}-56448 B \,{\mathrm e}^{2 i \left (d x +c \right )}-4620 C \,{\mathrm e}^{i \left (d x +c \right )}-1680 A \,{\mathrm e}^{12 i \left (d x +c \right )}+4620 C \,{\mathrm e}^{13 i \left (d x +c \right )}+5880 A \,{\mathrm e}^{13 i \left (d x +c \right )}-3360 C \,{\mathrm e}^{10 i \left (d x +c \right )}+30240 A \,{\mathrm e}^{11 i \left (d x +c \right )}+30800 C \,{\mathrm e}^{11 i \left (d x +c \right )}-30240 A \,{\mathrm e}^{10 i \left (d x +c \right )}+42840 A \,{\mathrm e}^{9 i \left (d x +c \right )}+65660 C \,{\mathrm e}^{9 i \left (d x +c \right )}-68320 C \,{\mathrm e}^{8 i \left (d x +c \right )}-5145 B \,{\mathrm e}^{i \left (d x +c \right )}-53165 B \,{\mathrm e}^{5 i \left (d x +c \right )}-188160 B \,{\mathrm e}^{6 i \left (d x +c \right )}+32060 B \,{\mathrm e}^{11 i \left (d x +c \right )}+5145 B \,{\mathrm e}^{13 i \left (d x +c \right )}-32060 B \,{\mathrm e}^{3 i \left (d x +c \right )}-13440 B \,{\mathrm e}^{10 i \left (d x +c \right )}+53165 B \,{\mathrm e}^{9 i \left (d x +c \right )}-94080 B \,{\mathrm e}^{8 i \left (d x +c \right )}\right )}{840 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 {49 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{16 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 {49 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{16 d}-\frac {11 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{4 d}\) | \(574\) |
derivativedivides | \(\frac {-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 )+B \,a^{4} \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} 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 )+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 B \,a^{4} \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} 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 )-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 B \,a^{4} \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} 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 (\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 B \,a^{4} \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 (-\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} A \tan \left (d x +c \right )+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 )-a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(577\) |
default | \(\frac {-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 )+B \,a^{4} \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} 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 )+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 B \,a^{4} \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} 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 )-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 B \,a^{4} \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} 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 (\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 B \,a^{4} \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 (-\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} A \tan \left (d x +c \right )+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 )-a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(577\) |
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Time = 0.29 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.90 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (56 \, A + 49 \, B + 44 \, C\right )} a^{4} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (56 \, A + 49 \, B + 44 \, C\right )} a^{4} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (581 \, A + 504 \, B + 454 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} + 105 \, {\left (56 \, A + 49 \, B + 44 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 16 \, {\left (238 \, A + 252 \, B + 227 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \, {\left (24 \, A + 41 \, B + 44 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 48 \, {\left (7 \, A + 28 \, B + 48 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 280 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right ) + 240 \, C a^{4}\right )} \sin \left (d x + c\right )}{3360 \, d \cos \left (d x + c\right )^{7}} \]
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\[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=a^{4} \left (\int A \sec ^{2}{\left (c + d x \right )}\, dx + \int 4 A \sec ^{3}{\left (c + d x \right )}\, dx + \int 6 A \sec ^{4}{\left (c + d x \right )}\, dx + \int 4 A \sec ^{5}{\left (c + d x \right )}\, dx + \int A \sec ^{6}{\left (c + d x \right )}\, dx + \int B \sec ^{3}{\left (c + d x \right )}\, dx + \int 4 B \sec ^{4}{\left (c + d x \right )}\, dx + \int 6 B \sec ^{5}{\left (c + d x \right )}\, dx + \int 4 B \sec ^{6}{\left (c + d x \right )}\, dx + \int B \sec ^{7}{\left (c + d x \right )}\, dx + \int C \sec ^{4}{\left (c + d x \right )}\, dx + \int 4 C \sec ^{5}{\left (c + d x \right )}\, dx + \int 6 C \sec ^{6}{\left (c + d x \right )}\, dx + \int 4 C \sec ^{7}{\left (c + d x \right )}\, dx + \int C \sec ^{8}{\left (c + d x \right )}\, dx\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 731 vs. \(2 (236) = 472\).
Time = 0.24 (sec) , antiderivative size = 731, normalized size of antiderivative = 2.90 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {224 \, {\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} + 6720 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 896 \, {\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} + 4480 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{4} + 96 \, {\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} + 1344 \, {\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} + 1120 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{4} - 35 \, B 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 )} - 140 \, 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 )} - 840 \, 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 )} - 1260 \, 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 )} - 840 \, 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 )} - 3360 \, 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 \, 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 )} + 3360 \, A a^{4} \tan \left (d x + c\right )}{3360 \, d} \]
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Time = 0.41 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.76 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (56 \, A a^{4} + 49 \, B a^{4} + 44 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 105 \, {\left (56 \, A a^{4} + 49 \, B a^{4} + 44 \, 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 (5880 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 5145 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 4620 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 39200 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 34300 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 30800 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 110936 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 97069 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 87164 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 172032 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 150528 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 135168 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 159656 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 134099 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 126084 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 86240 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 73220 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 58800 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21000 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 21735 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 22260 \, 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}}}{1680 \, d} \]
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Time = 20.05 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.51 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4\,\mathrm {atanh}\left (\frac {a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (56\,A+49\,B+44\,C\right )}{4\,\left (14\,A\,a^4+\frac {49\,B\,a^4}{4}+11\,C\,a^4\right )}\right )\,\left (56\,A+49\,B+44\,C\right )}{8\,d}-\frac {\left (7\,A\,a^4+\frac {49\,B\,a^4}{8}+\frac {11\,C\,a^4}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (-\frac {140\,A\,a^4}{3}-\frac {245\,B\,a^4}{6}-\frac {110\,C\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {1981\,A\,a^4}{15}+\frac {13867\,B\,a^4}{120}+\frac {3113\,C\,a^4}{30}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-\frac {1024\,A\,a^4}{5}-\frac {896\,B\,a^4}{5}-\frac {5632\,C\,a^4}{35}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {2851\,A\,a^4}{15}+\frac {19157\,B\,a^4}{120}+\frac {1501\,C\,a^4}{10}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {308\,A\,a^4}{3}-\frac {523\,B\,a^4}{6}-70\,C\,a^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (25\,A\,a^4+\frac {207\,B\,a^4}{8}+\frac {53\,C\,a^4}{2}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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