Integrand size = 23, antiderivative size = 108 \[ \int \frac {A+B \sec (c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {A x}{a^3}-\frac {(A-B) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(7 A-2 B) \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {2 (11 A-B) \tan (c+d x)}{15 d \left (a^3+a^3 \sec (c+d x)\right )} \]
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Time = 0.22 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4007, 4004, 3879} \[ \int \frac {A+B \sec (c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {2 (11 A-B) \tan (c+d x)}{15 d \left (a^3 \sec (c+d x)+a^3\right )}+\frac {A x}{a^3}-\frac {(7 A-2 B) \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac {(A-B) \tan (c+d x)}{5 d (a \sec (c+d x)+a)^3} \]
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Rule 3879
Rule 4004
Rule 4007
Rubi steps \begin{align*} \text {integral}& = -\frac {(A-B) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {\int \frac {-5 a A+2 a (A-B) \sec (c+d x)}{(a+a \sec (c+d x))^2} \, dx}{5 a^2} \\ & = -\frac {(A-B) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(7 A-2 B) \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}+\frac {\int \frac {15 a^2 A-a^2 (7 A-2 B) \sec (c+d x)}{a+a \sec (c+d x)} \, dx}{15 a^4} \\ & = \frac {A x}{a^3}-\frac {(A-B) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(7 A-2 B) \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(2 (11 A-B)) \int \frac {\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{15 a^2} \\ & = \frac {A x}{a^3}-\frac {(A-B) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(7 A-2 B) \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {2 (11 A-B) \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 = 0.94 (sec) , antiderivative size = 241, normalized size of antiderivative = 2.23 \[ \int \frac {A+B \sec (c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\sec \left (\frac {c}{2}\right ) \sec ^5\left (\frac {1}{2} (c+d x)\right ) \left (150 A d x \cos \left (\frac {d x}{2}\right )+150 A d x \cos \left (c+\frac {d x}{2}\right )+75 A d x \cos \left (c+\frac {3 d x}{2}\right )+75 A d x \cos \left (2 c+\frac {3 d x}{2}\right )+15 A d x \cos \left (2 c+\frac {5 d x}{2}\right )+15 A d x \cos \left (3 c+\frac {5 d x}{2}\right )-370 A \sin \left (\frac {d x}{2}\right )+80 B \sin \left (\frac {d x}{2}\right )+270 A \sin \left (c+\frac {d x}{2}\right )-60 B \sin \left (c+\frac {d x}{2}\right )-230 A \sin \left (c+\frac {3 d x}{2}\right )+40 B \sin \left (c+\frac {3 d x}{2}\right )+90 A \sin \left (2 c+\frac {3 d x}{2}\right )-30 B \sin \left (2 c+\frac {3 d x}{2}\right )-64 A \sin \left (2 c+\frac {5 d x}{2}\right )+14 B \sin \left (2 c+\frac {5 d x}{2}\right )\right )}{480 a^3 d} \]
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Time = 0.66 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.64
method | result | size |
parallelrisch | \(\frac {3 \left (-A +B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+10 \left (2 A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+15 \left (-7 A +B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+60 A x d}{60 a^{3} d}\) | \(69\) |
norman | \(\frac {\frac {A x}{a}-\frac {\left (A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 a d}+\frac {\left (2 A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 a d}-\frac {\left (7 A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}}{a^{2}}\) | \(86\) |
derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}+\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}-7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +8 A \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(102\) |
default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}+\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}-7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +8 A \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(102\) |
risch | \(\frac {A x}{a^{3}}-\frac {2 i \left (45 A \,{\mathrm e}^{4 i \left (d x +c \right )}-15 B \,{\mathrm e}^{4 i \left (d x +c \right )}+135 A \,{\mathrm e}^{3 i \left (d x +c \right )}-30 B \,{\mathrm e}^{3 i \left (d x +c \right )}+185 A \,{\mathrm e}^{2 i \left (d x +c \right )}-40 B \,{\mathrm e}^{2 i \left (d x +c \right )}+115 \,{\mathrm e}^{i \left (d x +c \right )} A -20 B \,{\mathrm e}^{i \left (d x +c \right )}+32 A -7 B \right )}{15 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{5}}\) | \(133\) |
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Time = 0.28 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.28 \[ \int \frac {A+B \sec (c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {15 \, A d x \cos \left (d x + c\right )^{3} + 45 \, A d x \cos \left (d x + c\right )^{2} + 45 \, A d x \cos \left (d x + c\right ) + 15 \, A d x - {\left ({\left (32 \, A - 7 \, B\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (17 \, A - 2 \, B\right )} \cos \left (d x + c\right ) + 22 \, A - 2 \, B\right )} \sin \left (d x + c\right )}{15 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
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\[ \int \frac {A+B \sec (c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\int \frac {A}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \sec {\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
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Time = 0.34 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.48 \[ \int \frac {A+B \sec (c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {A {\left (\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {20 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} - \frac {B {\left (\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{3}}}{60 \, d} \]
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Time = 0.32 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.12 \[ \int \frac {A+B \sec (c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\frac {60 \, {\left (d x + c\right )} A}{a^{3}} - \frac {3 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 20 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 10 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 105 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{60 \, d} \]
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Time = 13.98 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.23 \[ \int \frac {A+B \sec (c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {A\,x}{a^3}+\frac {{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {A\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}-\frac {B\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{6}\right )-{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {7\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}-\frac {B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}\right )-\frac {A\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{20}+\frac {B\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{20}}{a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \]
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