Integrand size = 29, antiderivative size = 98 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx=-\frac {(2 A-B) x}{a^2}+\frac {2 (5 A-2 B) \sin (c+d x)}{3 a^2 d}-\frac {(2 A-B) \sin (c+d x)}{a^2 d (1+\sec (c+d x))}-\frac {(A-B) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2} \]
[Out]
Time = 0.36 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {4105, 3872, 2717, 8} \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx=\frac {2 (5 A-2 B) \sin (c+d x)}{3 a^2 d}-\frac {(2 A-B) \sin (c+d x)}{a^2 d (\sec (c+d x)+1)}-\frac {x (2 A-B)}{a^2}-\frac {(A-B) \sin (c+d x)}{3 d (a \sec (c+d x)+a)^2} \]
[In]
[Out]
Rule 8
Rule 2717
Rule 3872
Rule 4105
Rubi steps \begin{align*} \text {integral}& = -\frac {(A-B) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac {\int \frac {\cos (c+d x) (a (4 A-B)-2 a (A-B) \sec (c+d x))}{a+a \sec (c+d x)} \, dx}{3 a^2} \\ & = -\frac {(2 A-B) \sin (c+d x)}{a^2 d (1+\sec (c+d x))}-\frac {(A-B) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac {\int \cos (c+d x) \left (2 a^2 (5 A-2 B)-3 a^2 (2 A-B) \sec (c+d x)\right ) \, dx}{3 a^4} \\ & = -\frac {(2 A-B) \sin (c+d x)}{a^2 d (1+\sec (c+d x))}-\frac {(A-B) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac {(2 (5 A-2 B)) \int \cos (c+d x) \, dx}{3 a^2}-\frac {(2 A-B) \int 1 \, dx}{a^2} \\ & = -\frac {(2 A-B) x}{a^2}+\frac {2 (5 A-2 B) \sin (c+d x)}{3 a^2 d}-\frac {(2 A-B) \sin (c+d x)}{a^2 d (1+\sec (c+d x))}-\frac {(A-B) \sin (c+d x)}{3 d (a+a \sec (c+d x))^2} \\ \end{align*}
Time = 1.16 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.05 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx=\frac {\sin (c+d x) \left (4 A-B+\frac {-A+B}{(1+\sec (c+d x))^2}+\frac {3 (2 A-B) \left (\arcsin (\cos (c+d x)) (1+\cos (c+d x))+\sqrt {\sin ^2(c+d x)}\right )}{\sqrt {1-\cos (c+d x)} (1+\cos (c+d x))^{3/2}}\right )}{3 a^2 d} \]
[In]
[Out]
Time = 0.97 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.74
method | result | size |
parallelrisch | \(\frac {\left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (3 A \cos \left (2 d x +2 c \right )+28 A \cos \left (d x +c \right )+23 A +2 B \right )-20 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-24 d \left (A -\frac {B}{2}\right ) x}{12 a^{2} d}\) | \(73\) |
derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}+5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\frac {4 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-4 \left (2 A -B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{2}}\) | \(108\) |
default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}+5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\frac {4 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-4 \left (2 A -B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{2}}\) | \(108\) |
risch | \(-\frac {2 A x}{a^{2}}+\frac {x B}{a^{2}}-\frac {i A \,{\mathrm e}^{i \left (d x +c \right )}}{2 a^{2} d}+\frac {i A \,{\mathrm e}^{-i \left (d x +c \right )}}{2 a^{2} d}+\frac {2 i \left (9 A \,{\mathrm e}^{2 i \left (d x +c \right )}-6 B \,{\mathrm e}^{2 i \left (d x +c \right )}+15 \,{\mathrm e}^{i \left (d x +c \right )} A -9 B \,{\mathrm e}^{i \left (d x +c \right )}+8 A -5 B \right )}{3 d \,a^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{3}}\) | \(130\) |
norman | \(\frac {-\frac {\left (2 A -B \right ) x}{a}-\frac {\left (A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{6 a d}-\frac {\left (2 A -B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a}+\frac {3 \left (3 A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a d}+\frac {\left (7 A -4 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) a}\) | \(132\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.26 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx=-\frac {3 \, {\left (2 \, A - B\right )} d x \cos \left (d x + c\right )^{2} + 6 \, {\left (2 \, A - B\right )} d x \cos \left (d x + c\right ) + 3 \, {\left (2 \, A - B\right )} d x - {\left (3 \, A \cos \left (d x + c\right )^{2} + {\left (14 \, A - 5 \, B\right )} \cos \left (d x + c\right ) + 10 \, A - 4 \, B\right )} \sin \left (d x + c\right )}{3 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
[In]
[Out]
\[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx=\frac {\int \frac {A \cos {\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (94) = 188\).
Time = 0.31 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.95 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx=\frac {A {\left (\frac {\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {24 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} + \frac {12 \, \sin \left (d x + c\right )}{{\left (a^{2} + \frac {a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} - B {\left (\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {12 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )}}{6 \, d} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.23 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx=-\frac {\frac {6 \, {\left (d x + c\right )} {\left (2 \, A - B\right )}}{a^{2}} - \frac {12 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{2}} + \frac {A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{6 \, d} \]
[In]
[Out]
Time = 13.58 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.11 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {A-B}{a^2}+\frac {3\,A-B}{2\,a^2}\right )}{d}-\frac {x\,\left (2\,A-B\right )}{a^2}+\frac {2\,A\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^2\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (A-B\right )}{6\,a^2\,d} \]
[In]
[Out]