Integrand size = 31, antiderivative size = 194 \[ \int \frac {\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {(A-4 B) \text {arctanh}(\sin (c+d x))}{a^4 d}-\frac {(55 A-244 B) \tan (c+d x)}{105 a^4 d}+\frac {(25 A-88 B) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-4 B) \tan (c+d x)}{a^4 d (1+\sec (c+d x))}+\frac {(A-B) \sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(5 A-12 B) \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3} \]
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Time = 0.68 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4104, 4093, 3872, 3855, 3852, 8} \[ \int \frac {\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {(A-4 B) \text {arctanh}(\sin (c+d x))}{a^4 d}-\frac {(55 A-244 B) \tan (c+d x)}{105 a^4 d}+\frac {(25 A-88 B) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac {(A-4 B) \tan (c+d x)}{a^4 d (\sec (c+d x)+1)}+\frac {(A-B) \tan (c+d x) \sec ^4(c+d x)}{7 d (a \sec (c+d x)+a)^4}+\frac {(5 A-12 B) \tan (c+d x) \sec ^3(c+d x)}{35 a d (a \sec (c+d x)+a)^3} \]
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Rule 8
Rule 3852
Rule 3855
Rule 3872
Rule 4093
Rule 4104
Rubi steps \begin{align*} \text {integral}& = \frac {(A-B) \sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {\int \frac {\sec ^4(c+d x) (4 a (A-B)-a (A-8 B) \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx}{7 a^2} \\ & = \frac {(A-B) \sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(5 A-12 B) \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {\int \frac {\sec ^3(c+d x) \left (3 a^2 (5 A-12 B)-2 a^2 (5 A-26 B) \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{35 a^4} \\ & = \frac {(25 A-88 B) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}+\frac {(A-B) \sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(5 A-12 B) \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {\int \frac {\sec ^2(c+d x) \left (2 a^3 (25 A-88 B)-a^3 (55 A-244 B) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{105 a^6} \\ & = \frac {(25 A-88 B) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}+\frac {(A-B) \sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(5 A-12 B) \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {(A-4 B) \tan (c+d x)}{d \left (a^4+a^4 \sec (c+d x)\right )}-\frac {\int \sec (c+d x) \left (-105 a^4 (A-4 B)+a^4 (55 A-244 B) \sec (c+d x)\right ) \, dx}{105 a^8} \\ & = \frac {(25 A-88 B) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}+\frac {(A-B) \sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(5 A-12 B) \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {(A-4 B) \tan (c+d x)}{d \left (a^4+a^4 \sec (c+d x)\right )}-\frac {(55 A-244 B) \int \sec ^2(c+d x) \, dx}{105 a^4}+\frac {(A-4 B) \int \sec (c+d x) \, dx}{a^4} \\ & = \frac {(A-4 B) \text {arctanh}(\sin (c+d x))}{a^4 d}+\frac {(25 A-88 B) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}+\frac {(A-B) \sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(5 A-12 B) \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {(A-4 B) \tan (c+d x)}{d \left (a^4+a^4 \sec (c+d x)\right )}+\frac {(55 A-244 B) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{105 a^4 d} \\ & = \frac {(A-4 B) \text {arctanh}(\sin (c+d x))}{a^4 d}-\frac {(55 A-244 B) \tan (c+d x)}{105 a^4 d}+\frac {(25 A-88 B) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}+\frac {(A-B) \sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(5 A-12 B) \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {(A-4 B) \tan (c+d x)}{d \left (a^4+a^4 \sec (c+d x)\right )} \\ \end{align*}
Time = 3.35 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.75 \[ \int \frac {\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {\sec ^4(c+d x) \left (6720 (A-4 B) \text {arctanh}(\sin (c+d x)) \cos ^8\left (\frac {1}{2} (c+d x)\right )-(1480 A-6688 B+(2645 A-11444 B) \cos (c+d x)+120 (13 A-55 B) \cos (2 (c+d x))+535 A \cos (3 (c+d x))-2236 B \cos (3 (c+d x))+80 A \cos (4 (c+d x))-332 B \cos (4 (c+d x))) \tan (c+d x)\right )}{420 a^4 d (1+\sec (c+d x))^4} \]
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Time = 0.91 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.82
method | result | size |
parallelrisch | \(\frac {-3360 \cos \left (d x +c \right ) \left (A -4 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+3360 \cos \left (d x +c \right ) \left (A -4 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-80 \left (\left (\frac {39 A}{2}-\frac {165 B}{2}\right ) \cos \left (2 d x +2 c \right )+\left (\frac {107 A}{16}-\frac {559 B}{20}\right ) \cos \left (3 d x +3 c \right )+\left (A -\frac {83 B}{20}\right ) \cos \left (4 d x +4 c \right )+\left (\frac {529 A}{16}-\frac {2861 B}{20}\right ) \cos \left (d x +c \right )+\frac {37 A}{2}-\frac {418 B}{5}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3360 a^{4} d \cos \left (d x +c \right )}\) | \(160\) |
derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} B}{7}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A +\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}+\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}-15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\left (-32 B +8 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {8 B}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\left (32 B -8 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {8 B}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}}{8 d \,a^{4}}\) | \(190\) |
default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} B}{7}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A +\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}+\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}-15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\left (-32 B +8 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {8 B}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\left (32 B -8 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {8 B}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}}{8 d \,a^{4}}\) | \(190\) |
risch | \(-\frac {2 i \left (105 A \,{\mathrm e}^{8 i \left (d x +c \right )}-420 B \,{\mathrm e}^{8 i \left (d x +c \right )}+735 A \,{\mathrm e}^{7 i \left (d x +c \right )}-2940 B \,{\mathrm e}^{7 i \left (d x +c \right )}+2275 A \,{\mathrm e}^{6 i \left (d x +c \right )}-9100 B \,{\mathrm e}^{6 i \left (d x +c \right )}+4165 A \,{\mathrm e}^{5 i \left (d x +c \right )}-16660 B \,{\mathrm e}^{5 i \left (d x +c \right )}+4795 A \,{\mathrm e}^{4 i \left (d x +c \right )}-20524 B \,{\mathrm e}^{4 i \left (d x +c \right )}+4445 A \,{\mathrm e}^{3 i \left (d x +c \right )}-18788 B \,{\mathrm e}^{3 i \left (d x +c \right )}+2785 A \,{\mathrm e}^{2 i \left (d x +c \right )}-11668 B \,{\mathrm e}^{2 i \left (d x +c \right )}+1015 \,{\mathrm e}^{i \left (d x +c \right )} A -4228 B \,{\mathrm e}^{i \left (d x +c \right )}+160 A -664 B \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{a^{4} d}+\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{a^{4} d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{a^{4} d}-\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{a^{4} d}\) | \(323\) |
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Time = 0.28 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.63 \[ \int \frac {\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {105 \, {\left ({\left (A - 4 \, B\right )} \cos \left (d x + c\right )^{5} + 4 \, {\left (A - 4 \, B\right )} \cos \left (d x + c\right )^{4} + 6 \, {\left (A - 4 \, B\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (A - 4 \, B\right )} \cos \left (d x + c\right )^{2} + {\left (A - 4 \, B\right )} \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left ({\left (A - 4 \, B\right )} \cos \left (d x + c\right )^{5} + 4 \, {\left (A - 4 \, B\right )} \cos \left (d x + c\right )^{4} + 6 \, {\left (A - 4 \, B\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (A - 4 \, B\right )} \cos \left (d x + c\right )^{2} + {\left (A - 4 \, B\right )} \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (8 \, {\left (20 \, A - 83 \, B\right )} \cos \left (d x + c\right )^{4} + {\left (535 \, A - 2236 \, B\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (155 \, A - 659 \, B\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (65 \, A - 296 \, B\right )} \cos \left (d x + c\right ) - 105 \, B\right )} \sin \left (d x + c\right )}{210 \, {\left (a^{4} d \cos \left (d x + c\right )^{5} + 4 \, a^{4} d \cos \left (d x + c\right )^{4} + 6 \, a^{4} d \cos \left (d x + c\right )^{3} + 4 \, a^{4} d \cos \left (d x + c\right )^{2} + a^{4} d \cos \left (d x + c\right )\right )}} \]
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\[ \int \frac {\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {\int \frac {A \sec ^{5}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \sec ^{6}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
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Time = 0.22 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.68 \[ \int \frac {\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {B {\left (\frac {1680 \, \sin \left (d x + c\right )}{{\left (a^{4} - \frac {a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )} - 5 \, A {\left (\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {168 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {168 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )}}{840 \, d} \]
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Time = 0.34 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.13 \[ \int \frac {\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {\frac {840 \, {\left (A - 4 \, B\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {840 \, {\left (A - 4 \, B\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} - \frac {1680 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} a^{4}} - \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 105 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 147 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 385 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 805 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1575 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5145 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
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Time = 13.90 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.22 \[ \int \frac {\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (A-4\,B\right )}{a^4\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {A-B}{20\,a^4}+\frac {3\,A-5\,B}{40\,a^4}\right )}{d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {A-B}{2\,a^4}+\frac {3\,\left (3\,A-5\,B\right )}{8\,a^4}+\frac {2\,A-10\,B}{4\,a^4}-\frac {2\,A+10\,B}{8\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A-B\right )}{56\,a^4\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A-B}{8\,a^4}+\frac {3\,A-5\,B}{12\,a^4}+\frac {2\,A-10\,B}{24\,a^4}\right )}{d}-\frac {2\,B\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-a^4\right )} \]
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