Integrand size = 21, antiderivative size = 224 \[ \int \frac {\sec ^3(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {31 \text {arctanh}(\sin (c+d x))}{2 a^5 d}-\frac {7664 \tan (c+d x)}{315 a^5 d}+\frac {31 \sec (c+d x) \tan (c+d x)}{2 a^5 d}-\frac {\sec (c+d x) \tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {17 \sec (c+d x) \tan (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {28 \sec (c+d x) \tan (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac {577 \sec (c+d x) \tan (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac {3832 \sec (c+d x) \tan (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )} \]
31/2*arctanh(sin(d*x+c))/a^5/d-7664/315*tan(d*x+c)/a^5/d+31/2*sec(d*x+c)*t an(d*x+c)/a^5/d-1/9*sec(d*x+c)*tan(d*x+c)/d/(a+a*cos(d*x+c))^5-17/63*sec(d *x+c)*tan(d*x+c)/a/d/(a+a*cos(d*x+c))^4-28/45*sec(d*x+c)*tan(d*x+c)/a^2/d/ (a+a*cos(d*x+c))^3-577/315*sec(d*x+c)*tan(d*x+c)/a^3/d/(a+a*cos(d*x+c))^2- 3832/315*sec(d*x+c)*tan(d*x+c)/d/(a^5+a^5*cos(d*x+c))
Leaf count is larger than twice the leaf count of optimal. \(507\) vs. \(2(224)=448\).
Time = 7.09 (sec) , antiderivative size = 507, normalized size of antiderivative = 2.26 \[ \int \frac {\sec ^3(c+d x)}{(a+a \cos (c+d x))^5} \, dx=-\frac {496 \cos ^{10}\left (\frac {c}{2}+\frac {d x}{2}\right ) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d (a+a \cos (c+d x))^5}+\frac {496 \cos ^{10}\left (\frac {c}{2}+\frac {d x}{2}\right ) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d (a+a \cos (c+d x))^5}+\frac {\cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec (c) \sec ^2(c+d x) \left (1472562 \sin \left (\frac {d x}{2}\right )-2822886 \sin \left (\frac {3 d x}{2}\right )+3057654 \sin \left (c-\frac {d x}{2}\right )-1885854 \sin \left (c+\frac {d x}{2}\right )+2644362 \sin \left (2 c+\frac {d x}{2}\right )+867048 \sin \left (c+\frac {3 d x}{2}\right )-1868436 \sin \left (2 c+\frac {3 d x}{2}\right )+1821498 \sin \left (3 c+\frac {3 d x}{2}\right )-2083537 \sin \left (c+\frac {5 d x}{2}\right )+339885 \sin \left (2 c+\frac {5 d x}{2}\right )-1456687 \sin \left (3 c+\frac {5 d x}{2}\right )+966735 \sin \left (4 c+\frac {5 d x}{2}\right )-1195641 \sin \left (2 c+\frac {7 d x}{2}\right )+46515 \sin \left (3 c+\frac {7 d x}{2}\right )-874341 \sin \left (4 c+\frac {7 d x}{2}\right )+367815 \sin \left (5 c+\frac {7 d x}{2}\right )-494579 \sin \left (3 c+\frac {9 d x}{2}\right )-31815 \sin \left (4 c+\frac {9 d x}{2}\right )-374879 \sin \left (5 c+\frac {9 d x}{2}\right )+87885 \sin \left (6 c+\frac {9 d x}{2}\right )-128187 \sin \left (4 c+\frac {11 d x}{2}\right )-18585 \sin \left (5 c+\frac {11 d x}{2}\right )-99837 \sin \left (6 c+\frac {11 d x}{2}\right )+9765 \sin \left (7 c+\frac {11 d x}{2}\right )-15328 \sin \left (5 c+\frac {13 d x}{2}\right )-3150 \sin \left (6 c+\frac {13 d x}{2}\right )-12178 \sin \left (7 c+\frac {13 d x}{2}\right )\right )}{40320 d (a+a \cos (c+d x))^5} \]
(-496*Cos[c/2 + (d*x)/2]^10*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]])/ (d*(a + a*Cos[c + d*x])^5) + (496*Cos[c/2 + (d*x)/2]^10*Log[Cos[c/2 + (d*x )/2] + Sin[c/2 + (d*x)/2]])/(d*(a + a*Cos[c + d*x])^5) + (Cos[c/2 + (d*x)/ 2]*Sec[c/2]*Sec[c]*Sec[c + d*x]^2*(1472562*Sin[(d*x)/2] - 2822886*Sin[(3*d *x)/2] + 3057654*Sin[c - (d*x)/2] - 1885854*Sin[c + (d*x)/2] + 2644362*Sin [2*c + (d*x)/2] + 867048*Sin[c + (3*d*x)/2] - 1868436*Sin[2*c + (3*d*x)/2] + 1821498*Sin[3*c + (3*d*x)/2] - 2083537*Sin[c + (5*d*x)/2] + 339885*Sin[ 2*c + (5*d*x)/2] - 1456687*Sin[3*c + (5*d*x)/2] + 966735*Sin[4*c + (5*d*x) /2] - 1195641*Sin[2*c + (7*d*x)/2] + 46515*Sin[3*c + (7*d*x)/2] - 874341*S in[4*c + (7*d*x)/2] + 367815*Sin[5*c + (7*d*x)/2] - 494579*Sin[3*c + (9*d* x)/2] - 31815*Sin[4*c + (9*d*x)/2] - 374879*Sin[5*c + (9*d*x)/2] + 87885*S in[6*c + (9*d*x)/2] - 128187*Sin[4*c + (11*d*x)/2] - 18585*Sin[5*c + (11*d *x)/2] - 99837*Sin[6*c + (11*d*x)/2] + 9765*Sin[7*c + (11*d*x)/2] - 15328* Sin[5*c + (13*d*x)/2] - 3150*Sin[6*c + (13*d*x)/2] - 12178*Sin[7*c + (13*d *x)/2]))/(40320*d*(a + a*Cos[c + d*x])^5)
Time = 1.68 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.11, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.905, Rules used = {3042, 3245, 3042, 3457, 3042, 3457, 3042, 3457, 27, 3042, 3457, 3042, 3227, 3042, 4254, 24, 4255, 3042, 4257}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sec ^3(c+d x)}{(a \cos (c+d x)+a)^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^5}dx\) |
| \(\Big \downarrow \) 3245 |
| \(\displaystyle \frac {\int \frac {(11 a-6 a \cos (c+d x)) \sec ^3(c+d x)}{(\cos (c+d x) a+a)^4}dx}{9 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {11 a-6 a \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4}dx}{9 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {\int \frac {\left (111 a^2-85 a^2 \cos (c+d x)\right ) \sec ^3(c+d x)}{(\cos (c+d x) a+a)^3}dx}{7 a^2}-\frac {17 a \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {111 a^2-85 a^2 \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}-\frac {17 a \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\left (947 a^3-784 a^3 \cos (c+d x)\right ) \sec ^3(c+d x)}{(\cos (c+d x) a+a)^2}dx}{5 a^2}-\frac {196 a^2 \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {17 a \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {947 a^3-784 a^3 \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}-\frac {196 a^2 \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {17 a \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {3 \left (2101 a^4-1731 a^4 \cos (c+d x)\right ) \sec ^3(c+d x)}{\cos (c+d x) a+a}dx}{3 a^2}-\frac {577 a^3 \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {196 a^2 \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {17 a \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {\left (2101 a^4-1731 a^4 \cos (c+d x)\right ) \sec ^3(c+d x)}{\cos (c+d x) a+a}dx}{a^2}-\frac {577 a^3 \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {196 a^2 \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {17 a \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {2101 a^4-1731 a^4 \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )}dx}{a^2}-\frac {577 a^3 \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {196 a^2 \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {17 a \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\int \left (9765 a^5-7664 a^5 \cos (c+d x)\right ) \sec ^3(c+d x)dx}{a^2}-\frac {3832 a^4 \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}}{a^2}-\frac {577 a^3 \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {196 a^2 \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {17 a \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\int \frac {9765 a^5-7664 a^5 \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx}{a^2}-\frac {3832 a^4 \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}}{a^2}-\frac {577 a^3 \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {196 a^2 \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {17 a \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {9765 a^5 \int \sec ^3(c+d x)dx-7664 a^5 \int \sec ^2(c+d x)dx}{a^2}-\frac {3832 a^4 \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}}{a^2}-\frac {577 a^3 \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {196 a^2 \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {17 a \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {9765 a^5 \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-7664 a^5 \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx}{a^2}-\frac {3832 a^4 \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}}{a^2}-\frac {577 a^3 \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {196 a^2 \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {17 a \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {7664 a^5 \int 1d(-\tan (c+d x))}{d}+9765 a^5 \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{a^2}-\frac {3832 a^4 \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}}{a^2}-\frac {577 a^3 \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {196 a^2 \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {17 a \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {9765 a^5 \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {7664 a^5 \tan (c+d x)}{d}}{a^2}-\frac {3832 a^4 \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}}{a^2}-\frac {577 a^3 \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {196 a^2 \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {17 a \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {9765 a^5 \left (\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {7664 a^5 \tan (c+d x)}{d}}{a^2}-\frac {3832 a^4 \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}}{a^2}-\frac {577 a^3 \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {196 a^2 \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {17 a \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {9765 a^5 \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {7664 a^5 \tan (c+d x)}{d}}{a^2}-\frac {3832 a^4 \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}}{a^2}-\frac {577 a^3 \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {196 a^2 \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {17 a \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {9765 a^5 \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {7664 a^5 \tan (c+d x)}{d}}{a^2}-\frac {3832 a^4 \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}}{a^2}-\frac {577 a^3 \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {196 a^2 \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {17 a \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}}{9 a^2}-\frac {\tan (c+d x) \sec (c+d x)}{9 d (a \cos (c+d x)+a)^5}\) |
-1/9*(Sec[c + d*x]*Tan[c + d*x])/(d*(a + a*Cos[c + d*x])^5) + ((-17*a*Sec[ c + d*x]*Tan[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((-196*a^2*Sec[c + d *x]*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((-577*a^3*Sec[c + d*x]*T an[c + d*x])/(d*(a + a*Cos[c + d*x])^2) + ((-3832*a^4*Sec[c + d*x]*Tan[c + d*x])/(d*(a + a*Cos[c + d*x])) + ((-7664*a^5*Tan[c + d*x])/d + 9765*a^5*( ArcTanh[Sin[c + d*x]]/(2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*d)))/a^2)/a^2 )/(5*a^2))/(7*a^2))/(9*a^2)
3.1.92.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^ m*((c + d*Sin[e + f*x])^(n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/( a*(2*m + 1)*(b*c - a*d)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[b*c*(m + 1) - a*d*(2*m + n + 2) + b*d*(m + n + 2)*Sin[e + f*x] , x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ [a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && !GtQ[n, 0] && (Intege rsQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^( n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/(a*(2*m + 1)*(b*c - a*d)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b *d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ [b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 1.22 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.71
| method | result | size |
| parallelrisch | \(\frac {\left (-1249920 \cos \left (2 d x +2 c \right )-1249920\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (1249920 \cos \left (2 d x +2 c \right )+1249920\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-7664 \left (\cos \left (6 d x +6 c \right )+\frac {871615 \cos \left (d x +c \right )}{3832}+\frac {155317 \cos \left (2 d x +2 c \right )}{958}+\frac {684135 \cos \left (3 d x +3 c \right )}{7664}+\frac {135111 \cos \left (4 d x +4 c \right )}{3832}+\frac {66875 \cos \left (5 d x +5 c \right )}{7664}+\frac {487469}{3832}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sec ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80640 a^{5} d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(160\) |
| derivativedivides | \(\frac {-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {10 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {48 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-50 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-351 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {8}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {88}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+248 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\frac {8}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {88}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}-248 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16 d \,a^{5}}\) | \(161\) |
| default | \(\frac {-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {10 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {48 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-50 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-351 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {8}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {88}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+248 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\frac {8}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {88}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}-248 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16 d \,a^{5}}\) | \(161\) |
| norman | \(\frac {-\frac {495 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d a}+\frac {207 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}-\frac {1303 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d a}-\frac {141 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{70 d a}-\frac {2159 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5040 d a}-\frac {19 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{252 d a}-\frac {\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )}{144 d a}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2} a^{4}}-\frac {31 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{5} d}+\frac {31 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{5} d}\) | \(193\) |
| risch | \(-\frac {i \left (9765 \,{\mathrm e}^{12 i \left (d x +c \right )}+87885 \,{\mathrm e}^{11 i \left (d x +c \right )}+367815 \,{\mathrm e}^{10 i \left (d x +c \right )}+966735 \,{\mathrm e}^{9 i \left (d x +c \right )}+1821498 \,{\mathrm e}^{8 i \left (d x +c \right )}+2644362 \,{\mathrm e}^{7 i \left (d x +c \right )}+3057654 \,{\mathrm e}^{6 i \left (d x +c \right )}+2822886 \,{\mathrm e}^{5 i \left (d x +c \right )}+2083537 \,{\mathrm e}^{4 i \left (d x +c \right )}+1195641 \,{\mathrm e}^{3 i \left (d x +c \right )}+494579 \,{\mathrm e}^{2 i \left (d x +c \right )}+128187 \,{\mathrm e}^{i \left (d x +c \right )}+15328\right )}{315 d \,a^{5} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{9}}-\frac {31 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 a^{5} d}+\frac {31 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 a^{5} d}\) | \(213\) |
1/80640*((-1249920*cos(2*d*x+2*c)-1249920)*ln(tan(1/2*d*x+1/2*c)-1)+(12499 20*cos(2*d*x+2*c)+1249920)*ln(tan(1/2*d*x+1/2*c)+1)-7664*(cos(6*d*x+6*c)+8 71615/3832*cos(d*x+c)+155317/958*cos(2*d*x+2*c)+684135/7664*cos(3*d*x+3*c) +135111/3832*cos(4*d*x+4*c)+66875/7664*cos(5*d*x+5*c)+487469/3832)*tan(1/2 *d*x+1/2*c)*sec(1/2*d*x+1/2*c)^8)/a^5/d/(1+cos(2*d*x+2*c))
Time = 0.27 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.31 \[ \int \frac {\sec ^3(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {9765 \, {\left (\cos \left (d x + c\right )^{7} + 5 \, \cos \left (d x + c\right )^{6} + 10 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 9765 \, {\left (\cos \left (d x + c\right )^{7} + 5 \, \cos \left (d x + c\right )^{6} + 10 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (15328 \, \cos \left (d x + c\right )^{6} + 66875 \, \cos \left (d x + c\right )^{5} + 112119 \, \cos \left (d x + c\right )^{4} + 87440 \, \cos \left (d x + c\right )^{3} + 28828 \, \cos \left (d x + c\right )^{2} + 1575 \, \cos \left (d x + c\right ) - 315\right )} \sin \left (d x + c\right )}{1260 \, {\left (a^{5} d \cos \left (d x + c\right )^{7} + 5 \, a^{5} d \cos \left (d x + c\right )^{6} + 10 \, a^{5} d \cos \left (d x + c\right )^{5} + 10 \, a^{5} d \cos \left (d x + c\right )^{4} + 5 \, a^{5} d \cos \left (d x + c\right )^{3} + a^{5} d \cos \left (d x + c\right )^{2}\right )}} \]
1/1260*(9765*(cos(d*x + c)^7 + 5*cos(d*x + c)^6 + 10*cos(d*x + c)^5 + 10*c os(d*x + c)^4 + 5*cos(d*x + c)^3 + cos(d*x + c)^2)*log(sin(d*x + c) + 1) - 9765*(cos(d*x + c)^7 + 5*cos(d*x + c)^6 + 10*cos(d*x + c)^5 + 10*cos(d*x + c)^4 + 5*cos(d*x + c)^3 + cos(d*x + c)^2)*log(-sin(d*x + c) + 1) - 2*(15 328*cos(d*x + c)^6 + 66875*cos(d*x + c)^5 + 112119*cos(d*x + c)^4 + 87440* cos(d*x + c)^3 + 28828*cos(d*x + c)^2 + 1575*cos(d*x + c) - 315)*sin(d*x + c))/(a^5*d*cos(d*x + c)^7 + 5*a^5*d*cos(d*x + c)^6 + 10*a^5*d*cos(d*x + c )^5 + 10*a^5*d*cos(d*x + c)^4 + 5*a^5*d*cos(d*x + c)^3 + a^5*d*cos(d*x + c )^2)
\[ \int \frac {\sec ^3(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {\int \frac {\sec ^{3}{\left (c + d x \right )}}{\cos ^{5}{\left (c + d x \right )} + 5 \cos ^{4}{\left (c + d x \right )} + 10 \cos ^{3}{\left (c + d x \right )} + 10 \cos ^{2}{\left (c + d x \right )} + 5 \cos {\left (c + d x \right )} + 1}\, dx}{a^{5}} \]
Integral(sec(c + d*x)**3/(cos(c + d*x)**5 + 5*cos(c + d*x)**4 + 10*cos(c + d*x)**3 + 10*cos(c + d*x)**2 + 5*cos(c + d*x) + 1), x)/a**5
Time = 0.26 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.12 \[ \int \frac {\sec ^3(c+d x)}{(a+a \cos (c+d x))^5} \, dx=-\frac {\frac {5040 \, {\left (\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {11 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{5} - \frac {2 \, a^{5} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{5} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {110565 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {15750 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3024 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {450 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{5}} - \frac {78120 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{5}} + \frac {78120 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{5}}}{5040 \, d} \]
-1/5040*(5040*(9*sin(d*x + c)/(cos(d*x + c) + 1) - 11*sin(d*x + c)^3/(cos( d*x + c) + 1)^3)/(a^5 - 2*a^5*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a^5*si n(d*x + c)^4/(cos(d*x + c) + 1)^4) + (110565*sin(d*x + c)/(cos(d*x + c) + 1) + 15750*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3024*sin(d*x + c)^5/(cos( d*x + c) + 1)^5 + 450*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 35*sin(d*x + c )^9/(cos(d*x + c) + 1)^9)/a^5 - 78120*log(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/a^5 + 78120*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/a^5)/d
Time = 0.42 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.76 \[ \int \frac {\sec ^3(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {\frac {78120 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{5}} - \frac {78120 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{5}} + \frac {5040 \, {\left (11 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{5}} - \frac {35 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 450 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 3024 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15750 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 110565 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{45}}}{5040 \, d} \]
1/5040*(78120*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^5 - 78120*log(abs(tan(1 /2*d*x + 1/2*c) - 1))/a^5 + 5040*(11*tan(1/2*d*x + 1/2*c)^3 - 9*tan(1/2*d* x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 - 1)^2*a^5) - (35*a^40*tan(1/2*d*x + 1/2*c)^9 + 450*a^40*tan(1/2*d*x + 1/2*c)^7 + 3024*a^40*tan(1/2*d*x + 1/2*c )^5 + 15750*a^40*tan(1/2*d*x + 1/2*c)^3 + 110565*a^40*tan(1/2*d*x + 1/2*c) )/a^45)/d
Time = 14.23 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.80 \[ \int \frac {\sec ^3(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {31\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^5\,d}-\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{5\,a^5\,d}-\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{56\,a^5\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{144\,a^5\,d}-\frac {25\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8\,a^5\,d}-\frac {9\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{d\,\left (a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^5\right )}-\frac {351\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,a^5\,d} \]
(31*atanh(tan(c/2 + (d*x)/2)))/(a^5*d) - (3*tan(c/2 + (d*x)/2)^5)/(5*a^5*d ) - (5*tan(c/2 + (d*x)/2)^7)/(56*a^5*d) - tan(c/2 + (d*x)/2)^9/(144*a^5*d) - (25*tan(c/2 + (d*x)/2)^3)/(8*a^5*d) - (9*tan(c/2 + (d*x)/2) - 11*tan(c/ 2 + (d*x)/2)^3)/(d*(a^5*tan(c/2 + (d*x)/2)^4 - 2*a^5*tan(c/2 + (d*x)/2)^2 + a^5)) - (351*tan(c/2 + (d*x)/2))/(16*a^5*d)