Integrand size = 38, antiderivative size = 108 \[ \int \frac {\cos (c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {B x}{a^3}-\frac {(B-C) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(7 B-2 C) \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {2 (11 B-C) \tan (c+d x)}{15 d \left (a^3+a^3 \sec (c+d x)\right )} \]
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Time = 0.30 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {4157, 4007, 4004, 3879} \[ \int \frac {\cos (c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=-\frac {2 (11 B-C) \tan (c+d x)}{15 d \left (a^3 \sec (c+d x)+a^3\right )}+\frac {B x}{a^3}-\frac {(7 B-2 C) \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac {(B-C) \tan (c+d x)}{5 d (a \sec (c+d x)+a)^3} \]
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Rule 3879
Rule 4004
Rule 4007
Rule 4157
Rubi steps \begin{align*} \text {integral}& = \int \frac {B+C \sec (c+d x)}{(a+a \sec (c+d x))^3} \, dx \\ & = -\frac {(B-C) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {\int \frac {-5 a B+2 a (B-C) \sec (c+d x)}{(a+a \sec (c+d x))^2} \, dx}{5 a^2} \\ & = -\frac {(B-C) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(7 B-2 C) \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}+\frac {\int \frac {15 a^2 B-a^2 (7 B-2 C) \sec (c+d x)}{a+a \sec (c+d x)} \, dx}{15 a^4} \\ & = \frac {B x}{a^3}-\frac {(B-C) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(7 B-2 C) \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(2 (11 B-C)) \int \frac {\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{15 a^2} \\ & = \frac {B x}{a^3}-\frac {(B-C) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(7 B-2 C) \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {2 (11 B-C) \tan (c+d x)}{15 d \left (a^3+a^3 \sec (c+d x)\right )} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(241\) vs. \(2(108)=216\).
Time = 1.05 (sec) , antiderivative size = 241, normalized size of antiderivative = 2.23 \[ \int \frac {\cos (c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {\sec \left (\frac {c}{2}\right ) \sec ^5\left (\frac {1}{2} (c+d x)\right ) \left (150 B d x \cos \left (\frac {d x}{2}\right )+150 B d x \cos \left (c+\frac {d x}{2}\right )+75 B d x \cos \left (c+\frac {3 d x}{2}\right )+75 B d x \cos \left (2 c+\frac {3 d x}{2}\right )+15 B d x \cos \left (2 c+\frac {5 d x}{2}\right )+15 B d x \cos \left (3 c+\frac {5 d x}{2}\right )-370 B \sin \left (\frac {d x}{2}\right )+80 C \sin \left (\frac {d x}{2}\right )+270 B \sin \left (c+\frac {d x}{2}\right )-60 C \sin \left (c+\frac {d x}{2}\right )-230 B \sin \left (c+\frac {3 d x}{2}\right )+40 C \sin \left (c+\frac {3 d x}{2}\right )+90 B \sin \left (2 c+\frac {3 d x}{2}\right )-30 C \sin \left (2 c+\frac {3 d x}{2}\right )-64 B \sin \left (2 c+\frac {5 d x}{2}\right )+14 C \sin \left (2 c+\frac {5 d x}{2}\right )\right )}{480 a^3 d} \]
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Time = 0.16 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.64
method | result | size |
parallelrisch | \(\frac {3 \left (-B +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+10 \left (2 B -C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+15 \left (-7 B +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+60 B x d}{60 a^{3} d}\) | \(69\) |
derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}+\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}-7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +8 B \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(102\) |
default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}+\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}-7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +8 B \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(102\) |
risch | \(\frac {B x}{a^{3}}-\frac {2 i \left (45 B \,{\mathrm e}^{4 i \left (d x +c \right )}-15 C \,{\mathrm e}^{4 i \left (d x +c \right )}+135 B \,{\mathrm e}^{3 i \left (d x +c \right )}-30 C \,{\mathrm e}^{3 i \left (d x +c \right )}+185 B \,{\mathrm e}^{2 i \left (d x +c \right )}-40 C \,{\mathrm e}^{2 i \left (d x +c \right )}+115 B \,{\mathrm e}^{i \left (d x +c \right )}-20 C \,{\mathrm e}^{i \left (d x +c \right )}+32 B -7 C \right )}{15 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{5}}\) | \(133\) |
norman | \(\frac {\frac {B x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{a}-\frac {B x}{a}-\frac {\left (B -C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{20 a d}-\frac {\left (2 B -C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 a d}+\frac {\left (2 B -C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{6 a d}+\frac {\left (7 B -C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}-\frac {\left (17 B -2 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{10 a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) a^{2}}\) | \(186\) |
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Time = 0.25 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.28 \[ \int \frac {\cos (c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {15 \, B d x \cos \left (d x + c\right )^{3} + 45 \, B d x \cos \left (d x + c\right )^{2} + 45 \, B d x \cos \left (d x + c\right ) + 15 \, B d x - {\left ({\left (32 \, B - 7 \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (17 \, B - 2 \, C\right )} \cos \left (d x + c\right ) + 22 \, B - 2 \, C\right )} \sin \left (d x + c\right )}{15 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
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\[ \int \frac {\cos (c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {\int \frac {B \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
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Time = 0.34 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.48 \[ \int \frac {\cos (c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=-\frac {B {\left (\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {20 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} - \frac {C {\left (\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{3}}}{60 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.12 \[ \int \frac {\cos (c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {\frac {60 \, {\left (d x + c\right )} B}{a^{3}} - \frac {3 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 20 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 10 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 105 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{60 \, d} \]
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Time = 15.92 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.23 \[ \int \frac {\cos (c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {B\,x}{a^3}+\frac {{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {B\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}-\frac {C\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{6}\right )-{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {7\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}-\frac {C\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}\right )-\frac {B\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{20}+\frac {C\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{20}}{a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \]
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