Integrand size = 31, antiderivative size = 170 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx=-\frac {(10 A-7 B) x}{2 a^2}+\frac {4 (3 A-2 B) \sin (c+d x)}{a^2 d}-\frac {(10 A-7 B) \cos (c+d x) \sin (c+d x)}{2 a^2 d}-\frac {(10 A-7 B) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d (1+\sec (c+d x))}-\frac {(A-B) \cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac {4 (3 A-2 B) \sin ^3(c+d x)}{3 a^2 d} \]
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Time = 0.52 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {4105, 3872, 2713, 2715, 8} \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx=-\frac {4 (3 A-2 B) \sin ^3(c+d x)}{3 a^2 d}+\frac {4 (3 A-2 B) \sin (c+d x)}{a^2 d}-\frac {(10 A-7 B) \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac {(10 A-7 B) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d (\sec (c+d x)+1)}-\frac {x (10 A-7 B)}{2 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2} \]
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Rule 8
Rule 2713
Rule 2715
Rule 3872
Rule 4105
Rubi steps \begin{align*} \text {integral}& = -\frac {(A-B) \cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac {\int \frac {\cos ^3(c+d x) (3 a (2 A-B)-4 a (A-B) \sec (c+d x))}{a+a \sec (c+d x)} \, dx}{3 a^2} \\ & = -\frac {(10 A-7 B) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d (1+\sec (c+d x))}-\frac {(A-B) \cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac {\int \cos ^3(c+d x) \left (12 a^2 (3 A-2 B)-3 a^2 (10 A-7 B) \sec (c+d x)\right ) \, dx}{3 a^4} \\ & = -\frac {(10 A-7 B) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d (1+\sec (c+d x))}-\frac {(A-B) \cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac {(10 A-7 B) \int \cos ^2(c+d x) \, dx}{a^2}+\frac {(4 (3 A-2 B)) \int \cos ^3(c+d x) \, dx}{a^2} \\ & = -\frac {(10 A-7 B) \cos (c+d x) \sin (c+d x)}{2 a^2 d}-\frac {(10 A-7 B) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d (1+\sec (c+d x))}-\frac {(A-B) \cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac {(10 A-7 B) \int 1 \, dx}{2 a^2}-\frac {(4 (3 A-2 B)) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{a^2 d} \\ & = -\frac {(10 A-7 B) x}{2 a^2}+\frac {4 (3 A-2 B) \sin (c+d x)}{a^2 d}-\frac {(10 A-7 B) \cos (c+d x) \sin (c+d x)}{2 a^2 d}-\frac {(10 A-7 B) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d (1+\sec (c+d x))}-\frac {(A-B) \cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac {4 (3 A-2 B) \sin ^3(c+d x)}{3 a^2 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(369\) vs. \(2(170)=340\).
Time = 2.03 (sec) , antiderivative size = 369, normalized size of antiderivative = 2.17 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \left (-36 (10 A-7 B) d x \cos \left (\frac {d x}{2}\right )-36 (10 A-7 B) d x \cos \left (c+\frac {d x}{2}\right )-120 A d x \cos \left (c+\frac {3 d x}{2}\right )+84 B d x \cos \left (c+\frac {3 d x}{2}\right )-120 A d x \cos \left (2 c+\frac {3 d x}{2}\right )+84 B d x \cos \left (2 c+\frac {3 d x}{2}\right )+516 A \sin \left (\frac {d x}{2}\right )-381 B \sin \left (\frac {d x}{2}\right )-156 A \sin \left (c+\frac {d x}{2}\right )+147 B \sin \left (c+\frac {d x}{2}\right )+342 A \sin \left (c+\frac {3 d x}{2}\right )-239 B \sin \left (c+\frac {3 d x}{2}\right )+118 A \sin \left (2 c+\frac {3 d x}{2}\right )-63 B \sin \left (2 c+\frac {3 d x}{2}\right )+30 A \sin \left (2 c+\frac {5 d x}{2}\right )-15 B \sin \left (2 c+\frac {5 d x}{2}\right )+30 A \sin \left (3 c+\frac {5 d x}{2}\right )-15 B \sin \left (3 c+\frac {5 d x}{2}\right )-3 A \sin \left (3 c+\frac {7 d x}{2}\right )+3 B \sin \left (3 c+\frac {7 d x}{2}\right )-3 A \sin \left (4 c+\frac {7 d x}{2}\right )+3 B \sin \left (4 c+\frac {7 d x}{2}\right )+A \sin \left (4 c+\frac {9 d x}{2}\right )+A \sin \left (5 c+\frac {9 d x}{2}\right )\right )}{48 a^2 d (1+\cos (c+d x))^2} \]
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Time = 1.01 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.62
method | result | size |
parallelrisch | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\left (28 A -12 B \right ) \cos \left (2 d x +2 c \right )+\left (-2 A +3 B \right ) \cos \left (3 d x +3 c \right )+A \cos \left (4 d x +4 c \right )+\left (258 A -163 B \right ) \cos \left (d x +c \right )+219 A -140 B \right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-240 d x \left (A -\frac {7 B}{10}\right )}{48 a^{2} d}\) | \(106\) |
derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}+9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B -\frac {8 \left (\left (-\frac {5 A}{2}+\frac {5 B}{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {10 A}{3}+2 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-\frac {3 A}{2}+\frac {3 B}{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}-2 \left (10 A -7 B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{2}}\) | \(154\) |
default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}+9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B -\frac {8 \left (\left (-\frac {5 A}{2}+\frac {5 B}{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {10 A}{3}+2 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-\frac {3 A}{2}+\frac {3 B}{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}-2 \left (10 A -7 B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{2}}\) | \(154\) |
norman | \(\frac {\frac {\left (4 A -3 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{a d}+\frac {\left (23 A -15 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{a d}-\frac {\left (10 A -7 B \right ) x}{2 a}-\frac {\left (A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{6 a d}-\frac {3 \left (10 A -7 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 a}-\frac {3 \left (10 A -7 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2 a}-\frac {\left (10 A -7 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2 a}+\frac {5 \left (16 A -11 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 a d}+\frac {\left (21 A -13 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} a}\) | \(230\) |
risch | \(-\frac {5 A x}{a^{2}}+\frac {7 x B}{2 a^{2}}+\frac {i A \,{\mathrm e}^{2 i \left (d x +c \right )}}{4 a^{2} d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} B}{8 a^{2} d}-\frac {15 i A \,{\mathrm e}^{i \left (d x +c \right )}}{8 a^{2} d}+\frac {i {\mathrm e}^{i \left (d x +c \right )} B}{a^{2} d}+\frac {15 i A \,{\mathrm e}^{-i \left (d x +c \right )}}{8 a^{2} d}-\frac {i {\mathrm e}^{-i \left (d x +c \right )} B}{a^{2} d}-\frac {i A \,{\mathrm e}^{-2 i \left (d x +c \right )}}{4 a^{2} d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} B}{8 a^{2} d}+\frac {2 i \left (15 A \,{\mathrm e}^{2 i \left (d x +c \right )}-12 B \,{\mathrm e}^{2 i \left (d x +c \right )}+27 \,{\mathrm e}^{i \left (d x +c \right )} A -21 B \,{\mathrm e}^{i \left (d x +c \right )}+14 A -11 B \right )}{3 d \,a^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{3}}+\frac {A \sin \left (3 d x +3 c \right )}{12 a^{2} d}\) | \(263\) |
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Time = 0.28 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.92 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx=-\frac {3 \, {\left (10 \, A - 7 \, B\right )} d x \cos \left (d x + c\right )^{2} + 6 \, {\left (10 \, A - 7 \, B\right )} d x \cos \left (d x + c\right ) + 3 \, {\left (10 \, A - 7 \, B\right )} d x - {\left (2 \, A \cos \left (d x + c\right )^{4} - {\left (2 \, A - 3 \, B\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (2 \, A - B\right )} \cos \left (d x + c\right )^{2} + {\left (66 \, A - 43 \, B\right )} \cos \left (d x + c\right ) + 48 \, A - 32 \, B\right )} \sin \left (d x + c\right )}{6 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
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\[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx=\frac {\int \frac {A \cos ^{3}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 372 vs. \(2 (160) = 320\).
Time = 0.30 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.19 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx=\frac {A {\left (\frac {4 \, {\left (\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {20 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{2} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {\frac {27 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {60 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )} - B {\left (\frac {6 \, {\left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {5 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{2} + \frac {2 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {42 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )}}{6 \, d} \]
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Time = 0.29 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.13 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx=-\frac {\frac {3 \, {\left (d x + c\right )} {\left (10 \, A - 7 \, B\right )}}{a^{2}} - \frac {2 \, {\left (30 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 9 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{2}} + \frac {A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 27 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 21 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{6 \, d} \]
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Time = 13.86 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.10 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx=\frac {\left (10\,A-5\,B\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {40\,A}{3}-8\,B\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (6\,A-3\,B\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^2\right )}-\frac {x\,\left (10\,A-7\,B\right )}{2\,a^2}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {2\,\left (A-B\right )}{a^2}+\frac {5\,A-3\,B}{2\,a^2}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (A-B\right )}{6\,a^2\,d} \]
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