Integrand size = 33, antiderivative size = 248 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=-\frac {2 (11 A+2 C) x}{a^4}+\frac {4 (454 A+83 C) \sin (c+d x)}{35 a^4 d}-\frac {2 (11 A+2 C) \cos (c+d x) \sin (c+d x)}{a^4 d}-\frac {(178 A+31 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {4 (11 A+2 C) \cos ^2(c+d x) \sin (c+d x)}{3 a^4 d (1+\sec (c+d x))}-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {2 (8 A+C) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {4 (454 A+83 C) \sin ^3(c+d x)}{105 a^4 d} \]
-2*(11*A+2*C)*x/a^4+4/35*(454*A+83*C)*sin(d*x+c)/a^4/d-2*(11*A+2*C)*cos(d* x+c)*sin(d*x+c)/a^4/d-1/105*(178*A+31*C)*cos(d*x+c)^2*sin(d*x+c)/a^4/d/(1+ sec(d*x+c))^2-4/3*(11*A+2*C)*cos(d*x+c)^2*sin(d*x+c)/a^4/d/(1+sec(d*x+c))- 1/7*(A+C)*cos(d*x+c)^2*sin(d*x+c)/d/(a+a*sec(d*x+c))^4-2/35*(8*A+C)*cos(d* x+c)^2*sin(d*x+c)/a/d/(a+a*sec(d*x+c))^3-4/105*(454*A+83*C)*sin(d*x+c)^3/a ^4/d
Leaf count is larger than twice the leaf count of optimal. \(575\) vs. \(2(248)=496\).
Time = 8.87 (sec) , antiderivative size = 575, normalized size of antiderivative = 2.32 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=-\frac {\sec \left (\frac {c}{2}\right ) \sec ^7\left (\frac {1}{2} (c+d x)\right ) \left (58800 (11 A+2 C) d x \cos \left (\frac {d x}{2}\right )+58800 (11 A+2 C) d x \cos \left (c+\frac {d x}{2}\right )+388080 A d x \cos \left (c+\frac {3 d x}{2}\right )+70560 C d x \cos \left (c+\frac {3 d x}{2}\right )+388080 A d x \cos \left (2 c+\frac {3 d x}{2}\right )+70560 C d x \cos \left (2 c+\frac {3 d x}{2}\right )+129360 A d x \cos \left (2 c+\frac {5 d x}{2}\right )+23520 C d x \cos \left (2 c+\frac {5 d x}{2}\right )+129360 A d x \cos \left (3 c+\frac {5 d x}{2}\right )+23520 C d x \cos \left (3 c+\frac {5 d x}{2}\right )+18480 A d x \cos \left (3 c+\frac {7 d x}{2}\right )+3360 C d x \cos \left (3 c+\frac {7 d x}{2}\right )+18480 A d x \cos \left (4 c+\frac {7 d x}{2}\right )+3360 C d x \cos \left (4 c+\frac {7 d x}{2}\right )-1010660 A \sin \left (\frac {d x}{2}\right )-243320 C \sin \left (\frac {d x}{2}\right )+687260 A \sin \left (c+\frac {d x}{2}\right )+184520 C \sin \left (c+\frac {d x}{2}\right )-814107 A \sin \left (c+\frac {3 d x}{2}\right )-184464 C \sin \left (c+\frac {3 d x}{2}\right )+204645 A \sin \left (2 c+\frac {3 d x}{2}\right )+72240 C \sin \left (2 c+\frac {3 d x}{2}\right )-357609 A \sin \left (2 c+\frac {5 d x}{2}\right )-77168 C \sin \left (2 c+\frac {5 d x}{2}\right )-18025 A \sin \left (3 c+\frac {5 d x}{2}\right )+8400 C \sin \left (3 c+\frac {5 d x}{2}\right )-72522 A \sin \left (3 c+\frac {7 d x}{2}\right )-15164 C \sin \left (3 c+\frac {7 d x}{2}\right )-24010 A \sin \left (4 c+\frac {7 d x}{2}\right )-2940 C \sin \left (4 c+\frac {7 d x}{2}\right )-2310 A \sin \left (4 c+\frac {9 d x}{2}\right )-420 C \sin \left (4 c+\frac {9 d x}{2}\right )-2310 A \sin \left (5 c+\frac {9 d x}{2}\right )-420 C \sin \left (5 c+\frac {9 d x}{2}\right )+175 A \sin \left (5 c+\frac {11 d x}{2}\right )+175 A \sin \left (6 c+\frac {11 d x}{2}\right )-35 A \sin \left (6 c+\frac {13 d x}{2}\right )-35 A \sin \left (7 c+\frac {13 d x}{2}\right )\right )}{107520 a^4 d} \]
-1/107520*(Sec[c/2]*Sec[(c + d*x)/2]^7*(58800*(11*A + 2*C)*d*x*Cos[(d*x)/2 ] + 58800*(11*A + 2*C)*d*x*Cos[c + (d*x)/2] + 388080*A*d*x*Cos[c + (3*d*x) /2] + 70560*C*d*x*Cos[c + (3*d*x)/2] + 388080*A*d*x*Cos[2*c + (3*d*x)/2] + 70560*C*d*x*Cos[2*c + (3*d*x)/2] + 129360*A*d*x*Cos[2*c + (5*d*x)/2] + 23 520*C*d*x*Cos[2*c + (5*d*x)/2] + 129360*A*d*x*Cos[3*c + (5*d*x)/2] + 23520 *C*d*x*Cos[3*c + (5*d*x)/2] + 18480*A*d*x*Cos[3*c + (7*d*x)/2] + 3360*C*d* x*Cos[3*c + (7*d*x)/2] + 18480*A*d*x*Cos[4*c + (7*d*x)/2] + 3360*C*d*x*Cos [4*c + (7*d*x)/2] - 1010660*A*Sin[(d*x)/2] - 243320*C*Sin[(d*x)/2] + 68726 0*A*Sin[c + (d*x)/2] + 184520*C*Sin[c + (d*x)/2] - 814107*A*Sin[c + (3*d*x )/2] - 184464*C*Sin[c + (3*d*x)/2] + 204645*A*Sin[2*c + (3*d*x)/2] + 72240 *C*Sin[2*c + (3*d*x)/2] - 357609*A*Sin[2*c + (5*d*x)/2] - 77168*C*Sin[2*c + (5*d*x)/2] - 18025*A*Sin[3*c + (5*d*x)/2] + 8400*C*Sin[3*c + (5*d*x)/2] - 72522*A*Sin[3*c + (7*d*x)/2] - 15164*C*Sin[3*c + (7*d*x)/2] - 24010*A*Si n[4*c + (7*d*x)/2] - 2940*C*Sin[4*c + (7*d*x)/2] - 2310*A*Sin[4*c + (9*d*x )/2] - 420*C*Sin[4*c + (9*d*x)/2] - 2310*A*Sin[5*c + (9*d*x)/2] - 420*C*Si n[5*c + (9*d*x)/2] + 175*A*Sin[5*c + (11*d*x)/2] + 175*A*Sin[6*c + (11*d*x )/2] - 35*A*Sin[6*c + (13*d*x)/2] - 35*A*Sin[7*c + (13*d*x)/2]))/(a^4*d)
Time = 1.57 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.04, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {3042, 4573, 25, 3042, 4508, 3042, 4508, 27, 3042, 4508, 27, 3042, 4274, 3042, 3113, 2009, 3115, 24}
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.
| \(\displaystyle \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a \sec (c+d x)+a)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^4}dx\) |
| \(\Big \downarrow \) 4573 |
| \(\displaystyle -\frac {\int -\frac {\cos ^3(c+d x) (a (10 A+3 C)-a (6 A-C) \sec (c+d x))}{(\sec (c+d x) a+a)^3}dx}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\cos ^3(c+d x) (a (10 A+3 C)-a (6 A-C) \sec (c+d x))}{(\sec (c+d x) a+a)^3}dx}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a (10 A+3 C)-a (6 A-C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4508 |
| \(\displaystyle \frac {\frac {\int \frac {\cos ^3(c+d x) \left (7 a^2 (14 A+3 C)-10 a^2 (8 A+C) \sec (c+d x)\right )}{(\sec (c+d x) a+a)^2}dx}{5 a^2}-\frac {2 a (8 A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {7 a^2 (14 A+3 C)-10 a^2 (8 A+C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}-\frac {2 a (8 A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4508 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {4 \cos ^3(c+d x) \left (3 a^3 (69 A+13 C)-a^3 (178 A+31 C) \sec (c+d x)\right )}{\sec (c+d x) a+a}dx}{3 a^2}-\frac {(178 A+31 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {2 a (8 A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {4 \int \frac {\cos ^3(c+d x) \left (3 a^3 (69 A+13 C)-a^3 (178 A+31 C) \sec (c+d x)\right )}{\sec (c+d x) a+a}dx}{3 a^2}-\frac {(178 A+31 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {2 a (8 A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {4 \int \frac {3 a^3 (69 A+13 C)-a^3 (178 A+31 C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )}dx}{3 a^2}-\frac {(178 A+31 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {2 a (8 A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4508 |
| \(\displaystyle \frac {\frac {\frac {4 \left (\frac {\int 3 \cos ^3(c+d x) \left (a^4 (454 A+83 C)-35 a^4 (11 A+2 C) \sec (c+d x)\right )dx}{a^2}-\frac {35 a^3 (11 A+2 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}\right )}{3 a^2}-\frac {(178 A+31 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {2 a (8 A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {4 \left (\frac {3 \int \cos ^3(c+d x) \left (a^4 (454 A+83 C)-35 a^4 (11 A+2 C) \sec (c+d x)\right )dx}{a^2}-\frac {35 a^3 (11 A+2 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}\right )}{3 a^2}-\frac {(178 A+31 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {2 a (8 A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {4 \left (\frac {3 \int \frac {a^4 (454 A+83 C)-35 a^4 (11 A+2 C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx}{a^2}-\frac {35 a^3 (11 A+2 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}\right )}{3 a^2}-\frac {(178 A+31 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {2 a (8 A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {\frac {\frac {4 \left (\frac {3 \left (a^4 (454 A+83 C) \int \cos ^3(c+d x)dx-35 a^4 (11 A+2 C) \int \cos ^2(c+d x)dx\right )}{a^2}-\frac {35 a^3 (11 A+2 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}\right )}{3 a^2}-\frac {(178 A+31 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {2 a (8 A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {4 \left (\frac {3 \left (a^4 (454 A+83 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^3dx-35 a^4 (11 A+2 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx\right )}{a^2}-\frac {35 a^3 (11 A+2 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}\right )}{3 a^2}-\frac {(178 A+31 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {2 a (8 A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {\frac {\frac {4 \left (\frac {3 \left (-\frac {a^4 (454 A+83 C) \int \left (1-\sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}-35 a^4 (11 A+2 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx\right )}{a^2}-\frac {35 a^3 (11 A+2 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}\right )}{3 a^2}-\frac {(178 A+31 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {2 a (8 A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {4 \left (\frac {3 \left (-35 a^4 (11 A+2 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {a^4 (454 A+83 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )}{a^2}-\frac {35 a^3 (11 A+2 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}\right )}{3 a^2}-\frac {(178 A+31 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {2 a (8 A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {\frac {4 \left (\frac {3 \left (-35 a^4 (11 A+2 C) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {a^4 (454 A+83 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )}{a^2}-\frac {35 a^3 (11 A+2 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}\right )}{3 a^2}-\frac {(178 A+31 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {2 a (8 A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {\frac {4 \left (\frac {3 \left (-\frac {a^4 (454 A+83 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}-35 a^4 (11 A+2 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{a^2}-\frac {35 a^3 (11 A+2 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}\right )}{3 a^2}-\frac {(178 A+31 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {2 a (8 A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
-1/7*((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])^4) + (( -2*a*(8*A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + (-1/3*((178*A + 31*C)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(1 + Sec[c + d*x])^ 2) + (4*((-35*a^3*(11*A + 2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Sec[ c + d*x])) + (3*(-35*a^4*(11*A + 2*C)*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/( 2*d)) - (a^4*(454*A + 83*C)*(-Sin[c + d*x] + Sin[c + d*x]^3/3))/d))/a^2))/ (3*a^2))/(5*a^2))/(7*a^2)
3.2.54.3.1 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[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-(A*b - a*B))*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(b*f*(2*m + 1))), x] - Simp[1/(a^2*(2*m + 1)) Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Cs c[e + f*x])^n*Simp[b*B*n - a*A*(2*m + n + 1) + (A*b - a*B)*(m + n + 1)*Csc[ e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B , 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)] && !GtQ[n, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_. ))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-a) *(A + C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(a*f*(2*m + 1))), x] + Simp[1/(a*b*(2*m + 1)) Int[(a + b*Csc[e + f*x])^(m + 1)*(d*C sc[e + f*x])^n*Simp[b*C*n + A*b*(2*m + n + 1) - (a*(A*(m + n + 1) - C*(m - n)))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, C, n}, x] && EqQ[ a^2 - b^2, 0] && LtQ[m, -2^(-1)]
Time = 0.48 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.56
| method | result | size |
| parallelrisch | \(\frac {31 \left (\frac {\left (\frac {238253 A}{2}+21928 C \right ) \cos \left (2 d x +2 c \right )}{1085}+\frac {2 \left (12609 A +2368 C \right ) \cos \left (3 d x +3 c \right )}{1085}+\left (A +\frac {6 C}{31}\right ) \cos \left (4 d x +4 c \right )-\frac {2 A \cos \left (5 d x +5 c \right )}{31}+\frac {A \cos \left (6 d x +6 c \right )}{62}+\frac {4 \left (67873 A +12496 C \right ) \cos \left (d x +c \right )}{1085}+\frac {176171 A}{1085}+\frac {32342 C}{1085}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-8448 x d \left (A +\frac {2 C}{11}\right )}{384 a^{4} d}\) | \(138\) |
| derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} C}{7}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}-\frac {59 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}-\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}+209 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -\frac {16 \left (\left (-13 A -C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {62 A}{3}-2 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-9 A -C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}-32 \left (11 A +2 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(210\) |
| default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} C}{7}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}-\frac {59 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}-\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}+209 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -\frac {16 \left (\left (-13 A -C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {62 A}{3}-2 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-9 A -C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}-32 \left (11 A +2 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(210\) |
| norman | \(\frac {\frac {2 \left (11 A +2 C \right ) x}{a}-\frac {\left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{56 a d}+\frac {4 \left (11 A +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a}-\frac {4 \left (11 A +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{a}-\frac {2 \left (11 A +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{a}+\frac {\left (67 A +39 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{280 a d}-\frac {\left (229 A +73 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{120 a d}-\frac {\left (353 A +65 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}-\frac {\left (1755 A +319 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 a d}+\frac {\left (2397 A +449 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{120 a d}+\frac {\left (3569 A +713 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{168 a d}+\frac {\left (21761 A +3897 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{280 a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) a^{3}}\) | \(323\) |
| risch | \(-\frac {22 A x}{a^{4}}-\frac {4 x C}{a^{4}}-\frac {i A \,{\mathrm e}^{3 i \left (d x +c \right )}}{24 a^{4} d}+\frac {i A \,{\mathrm e}^{2 i \left (d x +c \right )}}{2 a^{4} d}-\frac {43 i A \,{\mathrm e}^{i \left (d x +c \right )}}{8 a^{4} d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} C}{2 a^{4} d}+\frac {43 i A \,{\mathrm e}^{-i \left (d x +c \right )}}{8 a^{4} d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} C}{2 a^{4} d}-\frac {i A \,{\mathrm e}^{-2 i \left (d x +c \right )}}{2 a^{4} d}+\frac {i A \,{\mathrm e}^{-3 i \left (d x +c \right )}}{24 a^{4} d}+\frac {2 i \left (3675 A \,{\mathrm e}^{6 i \left (d x +c \right )}+1050 C \,{\mathrm e}^{6 i \left (d x +c \right )}+19845 A \,{\mathrm e}^{5 i \left (d x +c \right )}+5250 C \,{\mathrm e}^{5 i \left (d x +c \right )}+46550 A \,{\mathrm e}^{4 i \left (d x +c \right )}+11900 C \,{\mathrm e}^{4 i \left (d x +c \right )}+59570 A \,{\mathrm e}^{3 i \left (d x +c \right )}+14840 C \,{\mathrm e}^{3 i \left (d x +c \right )}+43827 A \,{\mathrm e}^{2 i \left (d x +c \right )}+10794 C \,{\mathrm e}^{2 i \left (d x +c \right )}+17549 A \,{\mathrm e}^{i \left (d x +c \right )}+4298 C \,{\mathrm e}^{i \left (d x +c \right )}+3032 A +764 C \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\) | \(341\) |
1/384*(31*(1/1085*(238253/2*A+21928*C)*cos(2*d*x+2*c)+2/1085*(12609*A+2368 *C)*cos(3*d*x+3*c)+(A+6/31*C)*cos(4*d*x+4*c)-2/31*A*cos(5*d*x+5*c)+1/62*A* cos(6*d*x+6*c)+4/1085*(67873*A+12496*C)*cos(d*x+c)+176171/1085*A+32342/108 5*C)*tan(1/2*d*x+1/2*c)*sec(1/2*d*x+1/2*c)^6-8448*x*d*(A+2/11*C))/a^4/d
Time = 0.29 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.02 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=-\frac {210 \, {\left (11 \, A + 2 \, C\right )} d x \cos \left (d x + c\right )^{4} + 840 \, {\left (11 \, A + 2 \, C\right )} d x \cos \left (d x + c\right )^{3} + 1260 \, {\left (11 \, A + 2 \, C\right )} d x \cos \left (d x + c\right )^{2} + 840 \, {\left (11 \, A + 2 \, C\right )} d x \cos \left (d x + c\right ) + 210 \, {\left (11 \, A + 2 \, C\right )} d x - {\left (35 \, A \cos \left (d x + c\right )^{6} - 70 \, A \cos \left (d x + c\right )^{5} + 35 \, {\left (14 \, A + 3 \, C\right )} \cos \left (d x + c\right )^{4} + 8 \, {\left (799 \, A + 148 \, C\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (3592 \, A + 659 \, C\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (6109 \, A + 1118 \, C\right )} \cos \left (d x + c\right ) + 3632 \, A + 664 \, C\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
-1/105*(210*(11*A + 2*C)*d*x*cos(d*x + c)^4 + 840*(11*A + 2*C)*d*x*cos(d*x + c)^3 + 1260*(11*A + 2*C)*d*x*cos(d*x + c)^2 + 840*(11*A + 2*C)*d*x*cos( d*x + c) + 210*(11*A + 2*C)*d*x - (35*A*cos(d*x + c)^6 - 70*A*cos(d*x + c) ^5 + 35*(14*A + 3*C)*cos(d*x + c)^4 + 8*(799*A + 148*C)*cos(d*x + c)^3 + 4 *(3592*A + 659*C)*cos(d*x + c)^2 + 2*(6109*A + 1118*C)*cos(d*x + c) + 3632 *A + 664*C)*sin(d*x + c))/(a^4*d*cos(d*x + c)^4 + 4*a^4*d*cos(d*x + c)^3 + 6*a^4*d*cos(d*x + c)^2 + 4*a^4*d*cos(d*x + c) + a^4*d)
Timed out. \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.63 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {A {\left (\frac {560 \, {\left (\frac {27 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {62 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {39 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{4} + \frac {3 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{4} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {\frac {21945 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2065 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {231 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {36960 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} + C {\left (\frac {1680 \, \sin \left (d x + c\right )}{{\left (a^{4} + \frac {a^{4} \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 )}} + \frac {\frac {5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {6720 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )}}{840 \, d} \]
1/840*(A*(560*(27*sin(d*x + c)/(cos(d*x + c) + 1) + 62*sin(d*x + c)^3/(cos (d*x + c) + 1)^3 + 39*sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/(a^4 + 3*a^4*si n(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*a^4*sin(d*x + c)^4/(cos(d*x + c) + 1 )^4 + a^4*sin(d*x + c)^6/(cos(d*x + c) + 1)^6) + (21945*sin(d*x + c)/(cos( d*x + c) + 1) - 2065*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 231*sin(d*x + c )^5/(cos(d*x + c) + 1)^5 - 15*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4 - 3 6960*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^4) + C*(1680*sin(d*x + c)/( (a^4 + a^4*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)*(cos(d*x + c) + 1)) + (514 5*sin(d*x + c)/(cos(d*x + c) + 1) - 805*sin(d*x + c)^3/(cos(d*x + c) + 1)^ 3 + 147*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 15*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4 - 6720*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^4))/d
Time = 0.35 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.05 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=-\frac {\frac {1680 \, {\left (d x + c\right )} {\left (11 \, A + 2 \, C\right )}}{a^{4}} - \frac {560 \, {\left (39 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 62 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 27 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{4}} + \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 231 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 147 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2065 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 805 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 21945 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5145 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
-1/840*(1680*(d*x + c)*(11*A + 2*C)/a^4 - 560*(39*A*tan(1/2*d*x + 1/2*c)^5 + 3*C*tan(1/2*d*x + 1/2*c)^5 + 62*A*tan(1/2*d*x + 1/2*c)^3 + 6*C*tan(1/2* d*x + 1/2*c)^3 + 27*A*tan(1/2*d*x + 1/2*c) + 3*C*tan(1/2*d*x + 1/2*c))/((t an(1/2*d*x + 1/2*c)^2 + 1)^3*a^4) + (15*A*a^24*tan(1/2*d*x + 1/2*c)^7 + 15 *C*a^24*tan(1/2*d*x + 1/2*c)^7 - 231*A*a^24*tan(1/2*d*x + 1/2*c)^5 - 147*C *a^24*tan(1/2*d*x + 1/2*c)^5 + 2065*A*a^24*tan(1/2*d*x + 1/2*c)^3 + 805*C* a^24*tan(1/2*d*x + 1/2*c)^3 - 21945*A*a^24*tan(1/2*d*x + 1/2*c) - 5145*C*a ^24*tan(1/2*d*x + 1/2*c))/a^28)/d
Time = 15.84 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.80 \[ \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {209\,A}{8}+\frac {49\,C}{8}\right )}{a^4\,d}-\frac {22\,A\,d\,x+4\,C\,d\,x}{a^4\,d}+\frac {\left (26\,A+2\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {124\,A}{3}+4\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (18\,A+2\,C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^4\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^3}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {11\,A}{40}+\frac {7\,C}{40}\right )}{a^4\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {A}{56}+\frac {C}{56}\right )}{a^4\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {59\,A}{24}+\frac {23\,C}{24}\right )}{a^4\,d} \]
(tan(c/2 + (d*x)/2)*((209*A)/8 + (49*C)/8))/(a^4*d) - (22*A*d*x + 4*C*d*x) /(a^4*d) + (tan(c/2 + (d*x)/2)^5*(26*A + 2*C) + tan(c/2 + (d*x)/2)^3*((124 *A)/3 + 4*C) + tan(c/2 + (d*x)/2)*(18*A + 2*C))/(a^4*d*(tan(c/2 + (d*x)/2) ^2 + 1)^3) + (tan(c/2 + (d*x)/2)^5*((11*A)/40 + (7*C)/40))/(a^4*d) - (tan( c/2 + (d*x)/2)^7*(A/56 + C/56))/(a^4*d) - (tan(c/2 + (d*x)/2)^3*((59*A)/24 + (23*C)/24))/(a^4*d)