Integrand size = 33, antiderivative size = 183 \[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=-\frac {4 C \text {arctanh}(\sin (c+d x))}{a^4 d}+\frac {2 (3 A+122 C) \tan (c+d x)}{105 a^4 d}+\frac {(3 A-88 C) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}+\frac {4 C \tan (c+d x)}{a^4 d (1+\sec (c+d x))}-\frac {(A+C) \sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {2 (A-6 C) \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3} \]
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Time = 0.80 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {4170, 4104, 4093, 3872, 3855, 3852, 8} \[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {2 (3 A+122 C) \tan (c+d x)}{105 a^4 d}+\frac {(3 A-88 C) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac {4 C \text {arctanh}(\sin (c+d x))}{a^4 d}+\frac {4 C \tan (c+d x)}{a^4 d (\sec (c+d x)+1)}-\frac {(A+C) \tan (c+d x) \sec ^4(c+d x)}{7 d (a \sec (c+d x)+a)^4}+\frac {2 (A-6 C) \tan (c+d x) \sec ^3(c+d x)}{35 a d (a \sec (c+d x)+a)^3} \]
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Rule 8
Rule 3852
Rule 3855
Rule 3872
Rule 4093
Rule 4104
Rule 4170
Rubi steps \begin{align*} \text {integral}& = -\frac {(A+C) \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) (-a (3 A-4 C)-a (A+8 C) \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx}{7 a^2} \\ & = -\frac {(A+C) \sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {2 (A-6 C) \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 (-6 a^2 (A-6 C)-a^2 (3 A+52 C) \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{35 a^4} \\ & = \frac {(3 A-88 C) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A+C) \sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {2 (A-6 C) \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 (3 A-88 C)-2 a^3 (3 A+122 C) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{105 a^6} \\ & = \frac {(3 A-88 C) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A+C) \sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {2 (A-6 C) \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {4 C \tan (c+d x)}{d \left (a^4+a^4 \sec (c+d x)\right )}+\frac {\int \sec (c+d x) \left (-420 a^4 C+2 a^4 (3 A+122 C) \sec (c+d x)\right ) \, dx}{105 a^8} \\ & = \frac {(3 A-88 C) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A+C) \sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {2 (A-6 C) \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {4 C \tan (c+d x)}{d \left (a^4+a^4 \sec (c+d x)\right )}-\frac {(4 C) \int \sec (c+d x) \, dx}{a^4}+\frac {(2 (3 A+122 C)) \int \sec ^2(c+d x) \, dx}{105 a^4} \\ & = -\frac {4 C \text {arctanh}(\sin (c+d x))}{a^4 d}+\frac {(3 A-88 C) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A+C) \sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {2 (A-6 C) \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {4 C \tan (c+d x)}{d \left (a^4+a^4 \sec (c+d x)\right )}-\frac {(2 (3 A+122 C)) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{105 a^4 d} \\ & = -\frac {4 C \text {arctanh}(\sin (c+d x))}{a^4 d}+\frac {2 (3 A+122 C) \tan (c+d x)}{105 a^4 d}+\frac {(3 A-88 C) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A+C) \sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {2 (A-6 C) \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {4 C \tan (c+d x)}{d \left (a^4+a^4 \sec (c+d x)\right )} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(544\) vs. \(2(183)=366\).
Time = 6.80 (sec) , antiderivative size = 544, normalized size of antiderivative = 2.97 \[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right ) \left (107520 C \cos ^7\left (\frac {1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\sec \left (\frac {c}{2}\right ) \sec (c) \sec (c+d x) \left (-70 (3 A+154 C) \sin \left (\frac {d x}{2}\right )+28 (9 A+671 C) \sin \left (\frac {3 d x}{2}\right )-126 A \sin \left (c-\frac {d x}{2}\right )-20524 C \sin \left (c-\frac {d x}{2}\right )+126 A \sin \left (c+\frac {d x}{2}\right )+14644 C \sin \left (c+\frac {d x}{2}\right )-210 A \sin \left (2 c+\frac {d x}{2}\right )-16660 C \sin \left (2 c+\frac {d x}{2}\right )-4690 C \sin \left (c+\frac {3 d x}{2}\right )+252 A \sin \left (2 c+\frac {3 d x}{2}\right )+14378 C \sin \left (2 c+\frac {3 d x}{2}\right )-9100 C \sin \left (3 c+\frac {3 d x}{2}\right )+132 A \sin \left (c+\frac {5 d x}{2}\right )+11668 C \sin \left (c+\frac {5 d x}{2}\right )-630 C \sin \left (2 c+\frac {5 d x}{2}\right )+132 A \sin \left (3 c+\frac {5 d x}{2}\right )+9358 C \sin \left (3 c+\frac {5 d x}{2}\right )-2940 C \sin \left (4 c+\frac {5 d x}{2}\right )+42 A \sin \left (2 c+\frac {7 d x}{2}\right )+4228 C \sin \left (2 c+\frac {7 d x}{2}\right )+315 C \sin \left (3 c+\frac {7 d x}{2}\right )+42 A \sin \left (4 c+\frac {7 d x}{2}\right )+3493 C \sin \left (4 c+\frac {7 d x}{2}\right )-420 C \sin \left (5 c+\frac {7 d x}{2}\right )+6 A \sin \left (3 c+\frac {9 d x}{2}\right )+664 C \sin \left (3 c+\frac {9 d x}{2}\right )+105 C \sin \left (4 c+\frac {9 d x}{2}\right )+6 A \sin \left (5 c+\frac {9 d x}{2}\right )+559 C \sin \left (5 c+\frac {9 d x}{2}\right )\right )\right )}{840 a^4 d (A+2 C+A \cos (2 (c+d x))) (1+\sec (c+d x))^4} \]
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Time = 0.40 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.83
method | result | size |
parallelrisch | \(\frac {13440 C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (d x +c \right )-13440 C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (d x +c \right )+3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\left (30 A +2200 C \right ) \cos \left (2 d x +2 c \right )+\left (8 A +\frac {2236 C}{3}\right ) \cos \left (3 d x +3 c \right )+\left (A +\frac {332 C}{3}\right ) \cos \left (4 d x +4 c \right )+\left (72 A +\frac {11444 C}{3}\right ) \cos \left (d x +c \right )+29 A +\frac {6688 C}{3}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3360 d \,a^{4} \cos \left (d x +c \right )}\) | \(152\) |
derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} C}{7}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A +\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -\frac {8 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+32 C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {8 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-32 C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d \,a^{4}}\) | \(178\) |
default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} C}{7}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A +\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -\frac {8 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+32 C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {8 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-32 C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d \,a^{4}}\) | \(178\) |
risch | \(\frac {4 i \left (210 C \,{\mathrm e}^{8 i \left (d x +c \right )}+1470 C \,{\mathrm e}^{7 i \left (d x +c \right )}+4550 C \,{\mathrm e}^{6 i \left (d x +c \right )}+105 A \,{\mathrm e}^{5 i \left (d x +c \right )}+8330 C \,{\mathrm e}^{5 i \left (d x +c \right )}+63 A \,{\mathrm e}^{4 i \left (d x +c \right )}+10262 C \,{\mathrm e}^{4 i \left (d x +c \right )}+126 A \,{\mathrm e}^{3 i \left (d x +c \right )}+9394 C \,{\mathrm e}^{3 i \left (d x +c \right )}+66 A \,{\mathrm e}^{2 i \left (d x +c \right )}+5834 C \,{\mathrm e}^{2 i \left (d x +c \right )}+21 A \,{\mathrm e}^{i \left (d x +c \right )}+2114 C \,{\mathrm e}^{i \left (d x +c \right )}+3 A +332 C \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{a^{4} d}-\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{a^{4} d}\) | \(244\) |
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Time = 0.27 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.51 \[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=-\frac {210 \, {\left (C \cos \left (d x + c\right )^{5} + 4 \, C \cos \left (d x + c\right )^{4} + 6 \, C \cos \left (d x + c\right )^{3} + 4 \, C \cos \left (d x + c\right )^{2} + C \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 210 \, {\left (C \cos \left (d x + c\right )^{5} + 4 \, C \cos \left (d x + c\right )^{4} + 6 \, C \cos \left (d x + c\right )^{3} + 4 \, C \cos \left (d x + c\right )^{2} + C \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - {\left (2 \, {\left (3 \, A + 332 \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (6 \, A + 559 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (39 \, A + 2636 \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (9 \, A + 296 \, C\right )} \cos \left (d x + c\right ) + 105 \, C\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{5} + 4 \, a^{4} d \cos \left (d x + c\right )^{4} + 6 \, a^{4} d \cos \left (d x + c\right )^{3} + 4 \, a^{4} d \cos \left (d x + c\right )^{2} + a^{4} d \cos \left (d x + c\right )\right )}} \]
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\[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {\int \frac {A \sec ^{4}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \sec ^{6}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
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Time = 0.22 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.50 \[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {C {\left (\frac {1680 \, \sin \left (d x + c\right )}{{\left (a^{4} - \frac {a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )} + \frac {3 \, A {\left (\frac {35 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}}}{840 \, d} \]
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Time = 0.34 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.16 \[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=-\frac {\frac {3360 \, C \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {3360 \, C \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} + \frac {1680 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} a^{4}} - \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 63 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 147 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 105 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 805 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 105 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5145 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
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Time = 15.06 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.11 \[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {A+C}{20\,a^4}+\frac {A+5\,C}{40\,a^4}\right )}{d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {A+C}{2\,a^4}+\frac {3\,\left (A+5\,C\right )}{8\,a^4}-\frac {3\,\left (2\,A-10\,C\right )}{8\,a^4}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A+C}{8\,a^4}+\frac {A+5\,C}{12\,a^4}-\frac {2\,A-10\,C}{24\,a^4}\right )}{d}-\frac {8\,C\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^4\,d}-\frac {2\,C\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-a^4\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A+C\right )}{56\,a^4\,d} \]
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