Integrand size = 33, antiderivative size = 339 \[ \int \cos ^7(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{4} a b \left (2 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) x+\frac {\left (12 a^4 (6 A+7 C)+b^4 (74 A+105 C)+3 a^2 b^2 (162 A+203 C)\right ) \sin (c+d x)}{105 d}+\frac {a b \left (2 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{4 d}+\frac {a b \left (6 A b^2+a^2 (103 A+126 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{210 d}+\frac {\left (2 A b^2+a^2 (6 A+7 C)\right ) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{35 d}+\frac {2 A b \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{21 d}+\frac {A \cos ^6(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{7 d}-\frac {\left (4 A b^4+4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)\right ) \sin ^3(c+d x)}{105 d} \] Output:
1/4*a*b*(2*b^2*(3*A+4*C)+a^2*(5*A+6*C))*x+1/105*(12*a^4*(6*A+7*C)+b^4*(74* A+105*C)+3*a^2*b^2*(162*A+203*C))*sin(d*x+c)/d+1/4*a*b*(2*b^2*(3*A+4*C)+a^ 2*(5*A+6*C))*cos(d*x+c)*sin(d*x+c)/d+1/210*a*b*(6*A*b^2+a^2*(103*A+126*C)) *cos(d*x+c)^3*sin(d*x+c)/d+1/35*(2*A*b^2+a^2*(6*A+7*C))*cos(d*x+c)^4*(a+b* sec(d*x+c))^2*sin(d*x+c)/d+2/21*A*b*cos(d*x+c)^5*(a+b*sec(d*x+c))^3*sin(d* x+c)/d+1/7*A*cos(d*x+c)^6*(a+b*sec(d*x+c))^4*sin(d*x+c)/d-1/105*(4*A*b^4+4 *a^4*(6*A+7*C)+3*a^2*b^2*(50*A+63*C))*sin(d*x+c)^3/d
Time = 1.76 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.04 \[ \int \cos ^7(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {8400 a^3 A b c+10080 a A b^3 c+10080 a^3 b c C+13440 a b^3 c C+8400 a^3 A b d x+10080 a A b^3 d x+10080 a^3 b C d x+13440 a b^3 C d x+105 \left (16 b^4 (3 A+4 C)+48 a^2 b^2 (5 A+6 C)+5 a^4 (7 A+8 C)\right ) \sin (c+d x)+420 a b \left (16 b^2 (A+C)+a^2 (15 A+16 C)\right ) \sin (2 (c+d x))+735 a^4 A \sin (3 (c+d x))+4200 a^2 A b^2 \sin (3 (c+d x))+560 A b^4 \sin (3 (c+d x))+700 a^4 C \sin (3 (c+d x))+3360 a^2 b^2 C \sin (3 (c+d x))+1260 a^3 A b \sin (4 (c+d x))+840 a A b^3 \sin (4 (c+d x))+840 a^3 b C \sin (4 (c+d x))+147 a^4 A \sin (5 (c+d x))+504 a^2 A b^2 \sin (5 (c+d x))+84 a^4 C \sin (5 (c+d x))+140 a^3 A b \sin (6 (c+d x))+15 a^4 A \sin (7 (c+d x))}{6720 d} \] Input:
Integrate[Cos[c + d*x]^7*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]
Output:
(8400*a^3*A*b*c + 10080*a*A*b^3*c + 10080*a^3*b*c*C + 13440*a*b^3*c*C + 84 00*a^3*A*b*d*x + 10080*a*A*b^3*d*x + 10080*a^3*b*C*d*x + 13440*a*b^3*C*d*x + 105*(16*b^4*(3*A + 4*C) + 48*a^2*b^2*(5*A + 6*C) + 5*a^4*(7*A + 8*C))*S in[c + d*x] + 420*a*b*(16*b^2*(A + C) + a^2*(15*A + 16*C))*Sin[2*(c + d*x) ] + 735*a^4*A*Sin[3*(c + d*x)] + 4200*a^2*A*b^2*Sin[3*(c + d*x)] + 560*A*b ^4*Sin[3*(c + d*x)] + 700*a^4*C*Sin[3*(c + d*x)] + 3360*a^2*b^2*C*Sin[3*(c + d*x)] + 1260*a^3*A*b*Sin[4*(c + d*x)] + 840*a*A*b^3*Sin[4*(c + d*x)] + 840*a^3*b*C*Sin[4*(c + d*x)] + 147*a^4*A*Sin[5*(c + d*x)] + 504*a^2*A*b^2* Sin[5*(c + d*x)] + 84*a^4*C*Sin[5*(c + d*x)] + 140*a^3*A*b*Sin[6*(c + d*x) ] + 15*a^4*A*Sin[7*(c + d*x)])/(6720*d)
Time = 2.38 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.98, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.576, Rules used = {3042, 4583, 3042, 4582, 27, 3042, 4582, 3042, 4562, 27, 3042, 4535, 3042, 3115, 24, 4532, 3042, 3492, 2009}
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.
| \(\displaystyle \int \cos ^7(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^7}dx\) |
| \(\Big \downarrow \) 4583 |
| \(\displaystyle \frac {1}{7} \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (b (2 A+7 C) \sec ^2(c+d x)+a (6 A+7 C) \sec (c+d x)+4 A b\right )dx+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (b (2 A+7 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+a (6 A+7 C) \csc \left (c+d x+\frac {\pi }{2}\right )+4 A b\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^6}dx+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{6} \int 2 \cos ^5(c+d x) (a+b \sec (c+d x))^2 \left (b^2 (10 A+21 C) \sec ^2(c+d x)+2 a b (17 A+21 C) \sec (c+d x)+3 \left ((6 A+7 C) a^2+2 A b^2\right )\right )dx+\frac {2 A b \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{3 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \int \cos ^5(c+d x) (a+b \sec (c+d x))^2 \left (b^2 (10 A+21 C) \sec ^2(c+d x)+2 a b (17 A+21 C) \sec (c+d x)+3 \left ((6 A+7 C) a^2+2 A b^2\right )\right )dx+\frac {2 A b \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{3 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (b^2 (10 A+21 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 a b (17 A+21 C) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left ((6 A+7 C) a^2+2 A b^2\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^5}dx+\frac {2 A b \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{3 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {1}{5} \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (b \left (6 (6 A+7 C) a^2+b^2 (62 A+105 C)\right ) \sec ^2(c+d x)+a \left (12 (6 A+7 C) a^2+b^2 (244 A+315 C)\right ) \sec (c+d x)+2 b \left ((103 A+126 C) a^2+6 A b^2\right )\right )dx+\frac {3 \left (a^2 (6 A+7 C)+2 A b^2\right ) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 A b \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{3 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {1}{5} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (b \left (6 (6 A+7 C) a^2+b^2 (62 A+105 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+a \left (12 (6 A+7 C) a^2+b^2 (244 A+315 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+2 b \left ((103 A+126 C) a^2+6 A b^2\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {3 \left (a^2 (6 A+7 C)+2 A b^2\right ) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 A b \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{3 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 4562 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {a b \left (a^2 (103 A+126 C)+6 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{2 d}-\frac {1}{4} \int -2 \cos ^3(c+d x) \left (2 b^2 \left (6 (6 A+7 C) a^2+b^2 (62 A+105 C)\right ) \sec ^2(c+d x)+105 a b \left ((5 A+6 C) a^2+2 b^2 (3 A+4 C)\right ) \sec (c+d x)+6 \left (4 (6 A+7 C) a^4+3 b^2 (50 A+63 C) a^2+4 A b^4\right )\right )dx\right )+\frac {3 \left (a^2 (6 A+7 C)+2 A b^2\right ) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 A b \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{3 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{2} \int \cos ^3(c+d x) \left (2 b^2 \left (6 (6 A+7 C) a^2+b^2 (62 A+105 C)\right ) \sec ^2(c+d x)+105 a b \left ((5 A+6 C) a^2+2 b^2 (3 A+4 C)\right ) \sec (c+d x)+6 \left (4 (6 A+7 C) a^4+3 b^2 (50 A+63 C) a^2+4 A b^4\right )\right )dx+\frac {a b \left (a^2 (103 A+126 C)+6 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{2 d}\right )+\frac {3 \left (a^2 (6 A+7 C)+2 A b^2\right ) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 A b \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{3 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{2} \int \frac {2 b^2 \left (6 (6 A+7 C) a^2+b^2 (62 A+105 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+105 a b \left ((5 A+6 C) a^2+2 b^2 (3 A+4 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+6 \left (4 (6 A+7 C) a^4+3 b^2 (50 A+63 C) a^2+4 A b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {a b \left (a^2 (103 A+126 C)+6 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{2 d}\right )+\frac {3 \left (a^2 (6 A+7 C)+2 A b^2\right ) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 A b \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{3 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{2} \left (105 a b \left (a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right ) \int \cos ^2(c+d x)dx+\int \cos ^3(c+d x) \left (2 b^2 \left (6 (6 A+7 C) a^2+b^2 (62 A+105 C)\right ) \sec ^2(c+d x)+6 \left (4 (6 A+7 C) a^4+3 b^2 (50 A+63 C) a^2+4 A b^4\right )\right )dx\right )+\frac {a b \left (a^2 (103 A+126 C)+6 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{2 d}\right )+\frac {3 \left (a^2 (6 A+7 C)+2 A b^2\right ) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 A b \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{3 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{2} \left (105 a b \left (a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\int \frac {2 b^2 \left (6 (6 A+7 C) a^2+b^2 (62 A+105 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+6 \left (4 (6 A+7 C) a^4+3 b^2 (50 A+63 C) a^2+4 A b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx\right )+\frac {a b \left (a^2 (103 A+126 C)+6 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{2 d}\right )+\frac {3 \left (a^2 (6 A+7 C)+2 A b^2\right ) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 A b \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{3 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{2} \left (105 a b \left (a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right ) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\int \frac {2 b^2 \left (6 (6 A+7 C) a^2+b^2 (62 A+105 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+6 \left (4 (6 A+7 C) a^4+3 b^2 (50 A+63 C) a^2+4 A b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx\right )+\frac {a b \left (a^2 (103 A+126 C)+6 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{2 d}\right )+\frac {3 \left (a^2 (6 A+7 C)+2 A b^2\right ) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 A b \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{3 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{2} \left (\int \frac {2 b^2 \left (6 (6 A+7 C) a^2+b^2 (62 A+105 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+6 \left (4 (6 A+7 C) a^4+3 b^2 (50 A+63 C) a^2+4 A b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+105 a b \left (a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right ) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )+\frac {a b \left (a^2 (103 A+126 C)+6 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{2 d}\right )+\frac {3 \left (a^2 (6 A+7 C)+2 A b^2\right ) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 A b \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{3 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 4532 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{2} \left (\int \cos (c+d x) \left (2 \left (6 (6 A+7 C) a^2+b^2 (62 A+105 C)\right ) b^2+6 \left (4 (6 A+7 C) a^4+3 b^2 (50 A+63 C) a^2+4 A b^4\right ) \cos ^2(c+d x)\right )dx+105 a b \left (a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right ) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )+\frac {a b \left (a^2 (103 A+126 C)+6 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{2 d}\right )+\frac {3 \left (a^2 (6 A+7 C)+2 A b^2\right ) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 A b \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{3 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{2} \left (\int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (2 \left (6 (6 A+7 C) a^2+b^2 (62 A+105 C)\right ) b^2+6 \left (4 (6 A+7 C) a^4+3 b^2 (50 A+63 C) a^2+4 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+105 a b \left (a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right ) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )+\frac {a b \left (a^2 (103 A+126 C)+6 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{2 d}\right )+\frac {3 \left (a^2 (6 A+7 C)+2 A b^2\right ) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 A b \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{3 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 3492 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{2} \left (105 a b \left (a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right ) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {\int \left (2 \left (12 (6 A+7 C) a^4+3 b^2 (162 A+203 C) a^2+b^4 (74 A+105 C)\right )-6 \left (4 (6 A+7 C) a^4+3 b^2 (50 A+63 C) a^2+4 A b^4\right ) \sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}\right )+\frac {a b \left (a^2 (103 A+126 C)+6 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{2 d}\right )+\frac {3 \left (a^2 (6 A+7 C)+2 A b^2\right ) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 A b \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{3 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {3 \left (a^2 (6 A+7 C)+2 A b^2\right ) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}+\frac {1}{5} \left (\frac {a b \left (a^2 (103 A+126 C)+6 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{2 d}+\frac {1}{2} \left (105 a b \left (a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right ) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {2 \left (4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)+4 A b^4\right ) \sin ^3(c+d x)-2 \left (12 a^4 (6 A+7 C)+3 a^2 b^2 (162 A+203 C)+b^4 (74 A+105 C)\right ) \sin (c+d x)}{d}\right )\right )\right )+\frac {2 A b \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{3 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
Input:
Int[Cos[c + d*x]^7*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]
Output:
(A*Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(7*d) + ((2*A*b*Cos [c + d*x]^5*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) + ((3*(2*A*b^2 + a^ 2*(6*A + 7*C))*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) + ((a*b*(6*A*b^2 + a^2*(103*A + 126*C))*Cos[c + d*x]^3*Sin[c + d*x])/(2*d) + (105*a*b*(2*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*(x/2 + (Cos[c + d*x]*Sin[ c + d*x])/(2*d)) - (-2*(12*a^4*(6*A + 7*C) + b^4*(74*A + 105*C) + 3*a^2*b^ 2*(162*A + 203*C))*Sin[c + d*x] + 2*(4*A*b^4 + 4*a^4*(6*A + 7*C) + 3*a^2*b ^2*(50*A + 63*C))*Sin[c + d*x]^3)/d)/2)/5)/3)/7
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[(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[sin[(e_.) + (f_.)*(x_)]^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[-f^(-1) Subst[Int[(1 - x^2)^((m - 1)/2)*(A + C - C*x^2 ), x], x, Cos[e + f*x]], x] /; FreeQ[{e, f, A, C}, x] && IGtQ[(m + 1)/2, 0]
Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Int[(C + A*Sin[e + f*x]^2)/Sin[e + f*x]^(m + 2), x] /; FreeQ[ {e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && ILtQ[(m + 1)/2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]* (B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.)), x_Symbol] :> Simp[B/b Int[(b*Cs c[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x]^2) , x] /; FreeQ[{b, e, f, A, B, C, m}, x]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _)), x_Symbol] :> Simp[A*a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n)), x] + Si mp[1/(d*n) Int[(d*Csc[e + f*x])^(n + 1)*Simp[n*(B*a + A*b) + (n*(a*C + B* b) + A*a*(n + 1))*Csc[e + f*x] + b*C*n*Csc[e + f*x]^2, x], x], x] /; FreeQ[ {a, b, d, e, f, A, B, C}, x] && LtQ[n, -1]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d* Csc[e + f*x])^(n + 1)*Simp[A*b*m - a*B*n - (b*B*n + a*(C*n + A*(n + 1)))*Cs c[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a , b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[n, -1]
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*Co t[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/( d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Simp[A*b*m - a*(C*n + A*(n + 1))*Csc[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^ 2, x], x], x] /; FreeQ[{a, b, d, e, f, A, C}, x] && NeQ[a^2 - b^2, 0] && Gt Q[m, 0] && LeQ[n, -1]
Time = 9.26 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.70
| method | result | size |
| parallelrisch | \(\frac {\left (\left (735 A +700 C \right ) a^{4}+4200 b^{2} \left (A +\frac {4 C}{5}\right ) a^{2}+560 A \,b^{4}\right ) \sin \left (3 d x +3 c \right )+6300 \left (\left (A +\frac {16 C}{15}\right ) a^{2}+\frac {16 b^{2} \left (A +C \right )}{15}\right ) a b \sin \left (2 d x +2 c \right )+1260 a \left (a^{2} \left (A +\frac {2 C}{3}\right )+\frac {2 A \,b^{2}}{3}\right ) b \sin \left (4 d x +4 c \right )+\left (\left (147 A +84 C \right ) a^{4}+504 A \,a^{2} b^{2}\right ) \sin \left (5 d x +5 c \right )+140 A \,a^{3} b \sin \left (6 d x +6 c \right )+15 a^{4} A \sin \left (7 d x +7 c \right )+\left (\left (3675 A +4200 C \right ) a^{4}+25200 b^{2} \left (A +\frac {6 C}{5}\right ) a^{2}+5040 b^{4} \left (A +\frac {4 C}{3}\right )\right ) \sin \left (d x +c \right )+8400 a b x \left (a^{2} \left (A +\frac {6 C}{5}\right )+\frac {6 b^{2} \left (A +\frac {4 C}{3}\right )}{5}\right ) d}{6720 d}\) | \(238\) |
| derivativedivides | \(\frac {\frac {a^{4} A \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}+\frac {a^{4} C \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+4 A \,a^{3} b \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+4 a^{3} b C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {6 A \,a^{2} b^{2} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+2 C \,a^{2} b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+4 a A \,b^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 C a \,b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {A \,b^{4} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+C \,b^{4} \sin \left (d x +c \right )}{d}\) | \(332\) |
| default | \(\frac {\frac {a^{4} A \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}+\frac {a^{4} C \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+4 A \,a^{3} b \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+4 a^{3} b C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {6 A \,a^{2} b^{2} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+2 C \,a^{2} b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+4 a A \,b^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 C a \,b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {A \,b^{4} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+C \,b^{4} \sin \left (d x +c \right )}{d}\) | \(332\) |
| risch | \(\frac {5 a^{3} A b x}{4}+\frac {a^{4} A \sin \left (7 d x +7 c \right )}{448 d}+\frac {35 \sin \left (d x +c \right ) a^{4} A}{64 d}+\frac {5 \sin \left (d x +c \right ) a^{4} C}{8 d}+\frac {7 a^{4} A \sin \left (3 d x +3 c \right )}{64 d}+\frac {5 \sin \left (3 d x +3 c \right ) a^{4} C}{48 d}+\frac {7 a^{4} A \sin \left (5 d x +5 c \right )}{320 d}+\frac {\sin \left (5 d x +5 c \right ) a^{4} C}{80 d}+\frac {\sin \left (3 d x +3 c \right ) A \,b^{4}}{12 d}+\frac {3 A \,a^{3} b \sin \left (4 d x +4 c \right )}{16 d}+2 C a \,b^{3} x +\frac {3 \sin \left (d x +c \right ) A \,b^{4}}{4 d}+\frac {\sin \left (d x +c \right ) C \,b^{4}}{d}+\frac {3 A a \,b^{3} x}{2}+\frac {3 C \,a^{3} b x}{2}+\frac {\sin \left (2 d x +2 c \right ) C a \,b^{3}}{d}+\frac {\sin \left (3 d x +3 c \right ) C \,a^{2} b^{2}}{2 d}+\frac {15 \sin \left (d x +c \right ) A \,a^{2} b^{2}}{4 d}+\frac {9 \sin \left (d x +c \right ) C \,a^{2} b^{2}}{2 d}+\frac {3 \sin \left (5 d x +5 c \right ) A \,a^{2} b^{2}}{40 d}+\frac {\sin \left (4 d x +4 c \right ) a A \,b^{3}}{8 d}+\frac {\sin \left (4 d x +4 c \right ) a^{3} b C}{8 d}+\frac {A \,a^{3} b \sin \left (6 d x +6 c \right )}{48 d}+\frac {5 \sin \left (3 d x +3 c \right ) A \,a^{2} b^{2}}{8 d}+\frac {15 \sin \left (2 d x +2 c \right ) A \,a^{3} b}{16 d}+\frac {\sin \left (2 d x +2 c \right ) a A \,b^{3}}{d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} b C}{d}\) | \(449\) |
Input:
int(cos(d*x+c)^7*(a+b*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x,method=_RETURNVER BOSE)
Output:
1/6720*(((735*A+700*C)*a^4+4200*b^2*(A+4/5*C)*a^2+560*A*b^4)*sin(3*d*x+3*c )+6300*((A+16/15*C)*a^2+16/15*b^2*(A+C))*a*b*sin(2*d*x+2*c)+1260*a*(a^2*(A +2/3*C)+2/3*A*b^2)*b*sin(4*d*x+4*c)+((147*A+84*C)*a^4+504*A*a^2*b^2)*sin(5 *d*x+5*c)+140*A*a^3*b*sin(6*d*x+6*c)+15*a^4*A*sin(7*d*x+7*c)+((3675*A+4200 *C)*a^4+25200*b^2*(A+6/5*C)*a^2+5040*b^4*(A+4/3*C))*sin(d*x+c)+8400*a*b*x* (a^2*(A+6/5*C)+6/5*b^2*(A+4/3*C))*d)/d
Time = 0.10 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.74 \[ \int \cos ^7(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} b + 2 \, {\left (3 \, A + 4 \, C\right )} a b^{3}\right )} d x + {\left (60 \, A a^{4} \cos \left (d x + c\right )^{6} + 280 \, A a^{3} b \cos \left (d x + c\right )^{5} + 32 \, {\left (6 \, A + 7 \, C\right )} a^{4} + 336 \, {\left (4 \, A + 5 \, C\right )} a^{2} b^{2} + 140 \, {\left (2 \, A + 3 \, C\right )} b^{4} + 12 \, {\left ({\left (6 \, A + 7 \, C\right )} a^{4} + 42 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{4} + 70 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} b + 6 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (4 \, {\left (6 \, A + 7 \, C\right )} a^{4} + 42 \, {\left (4 \, A + 5 \, C\right )} a^{2} b^{2} + 35 \, A b^{4}\right )} \cos \left (d x + c\right )^{2} + 105 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} b + 2 \, {\left (3 \, A + 4 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{420 \, d} \] Input:
integrate(cos(d*x+c)^7*(a+b*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x, algorithm= "fricas")
Output:
1/420*(105*((5*A + 6*C)*a^3*b + 2*(3*A + 4*C)*a*b^3)*d*x + (60*A*a^4*cos(d *x + c)^6 + 280*A*a^3*b*cos(d*x + c)^5 + 32*(6*A + 7*C)*a^4 + 336*(4*A + 5 *C)*a^2*b^2 + 140*(2*A + 3*C)*b^4 + 12*((6*A + 7*C)*a^4 + 42*A*a^2*b^2)*co s(d*x + c)^4 + 70*((5*A + 6*C)*a^3*b + 6*A*a*b^3)*cos(d*x + c)^3 + 4*(4*(6 *A + 7*C)*a^4 + 42*(4*A + 5*C)*a^2*b^2 + 35*A*b^4)*cos(d*x + c)^2 + 105*(( 5*A + 6*C)*a^3*b + 2*(3*A + 4*C)*a*b^3)*cos(d*x + c))*sin(d*x + c))/d
Timed out. \[ \int \cos ^7(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**7*(a+b*sec(d*x+c))**4*(A+C*sec(d*x+c)**2),x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 329, normalized size of antiderivative = 0.97 \[ \int \cos ^7(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {48 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} A a^{4} - 112 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{4} + 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} b - 210 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} b - 672 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{2} b^{2} + 3360 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} b^{2} - 210 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a b^{3} - 1680 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a b^{3} + 560 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A b^{4} - 1680 \, C b^{4} \sin \left (d x + c\right )}{1680 \, d} \] Input:
integrate(cos(d*x+c)^7*(a+b*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x, algorithm= "maxima")
Output:
-1/1680*(48*(5*sin(d*x + c)^7 - 21*sin(d*x + c)^5 + 35*sin(d*x + c)^3 - 35 *sin(d*x + c))*A*a^4 - 112*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin( d*x + c))*C*a^4 + 35*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 4*c) - 48*sin(2*d*x + 2*c))*A*a^3*b - 210*(12*d*x + 12*c + sin(4*d*x + 4* c) + 8*sin(2*d*x + 2*c))*C*a^3*b - 672*(3*sin(d*x + c)^5 - 10*sin(d*x + c) ^3 + 15*sin(d*x + c))*A*a^2*b^2 + 3360*(sin(d*x + c)^3 - 3*sin(d*x + c))*C *a^2*b^2 - 210*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a *b^3 - 1680*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a*b^3 + 560*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*b^4 - 1680*C*b^4*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 1228 vs. \(2 (323) = 646\).
Time = 0.32 (sec) , antiderivative size = 1228, normalized size of antiderivative = 3.62 \[ \int \cos ^7(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^7*(a+b*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x, algorithm= "giac")
Output:
1/420*(105*(5*A*a^3*b + 6*C*a^3*b + 6*A*a*b^3 + 8*C*a*b^3)*(d*x + c) + 2*( 420*A*a^4*tan(1/2*d*x + 1/2*c)^13 + 420*C*a^4*tan(1/2*d*x + 1/2*c)^13 - 11 55*A*a^3*b*tan(1/2*d*x + 1/2*c)^13 - 1050*C*a^3*b*tan(1/2*d*x + 1/2*c)^13 + 2520*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^13 + 2520*C*a^2*b^2*tan(1/2*d*x + 1/ 2*c)^13 - 1050*A*a*b^3*tan(1/2*d*x + 1/2*c)^13 - 840*C*a*b^3*tan(1/2*d*x + 1/2*c)^13 + 420*A*b^4*tan(1/2*d*x + 1/2*c)^13 + 420*C*b^4*tan(1/2*d*x + 1 /2*c)^13 + 840*A*a^4*tan(1/2*d*x + 1/2*c)^11 + 1400*C*a^4*tan(1/2*d*x + 1/ 2*c)^11 - 980*A*a^3*b*tan(1/2*d*x + 1/2*c)^11 - 2520*C*a^3*b*tan(1/2*d*x + 1/2*c)^11 + 8400*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^11 + 11760*C*a^2*b^2*tan( 1/2*d*x + 1/2*c)^11 - 2520*A*a*b^3*tan(1/2*d*x + 1/2*c)^11 - 3360*C*a*b^3* tan(1/2*d*x + 1/2*c)^11 + 1960*A*b^4*tan(1/2*d*x + 1/2*c)^11 + 2520*C*b^4* tan(1/2*d*x + 1/2*c)^11 + 3612*A*a^4*tan(1/2*d*x + 1/2*c)^9 + 3164*C*a^4*t an(1/2*d*x + 1/2*c)^9 - 2975*A*a^3*b*tan(1/2*d*x + 1/2*c)^9 - 1890*C*a^3*b *tan(1/2*d*x + 1/2*c)^9 + 18984*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 + 24360*C *a^2*b^2*tan(1/2*d*x + 1/2*c)^9 - 1890*A*a*b^3*tan(1/2*d*x + 1/2*c)^9 - 42 00*C*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 4060*A*b^4*tan(1/2*d*x + 1/2*c)^9 + 63 00*C*b^4*tan(1/2*d*x + 1/2*c)^9 + 2544*A*a^4*tan(1/2*d*x + 1/2*c)^7 + 4368 *C*a^4*tan(1/2*d*x + 1/2*c)^7 + 26208*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 + 3 0240*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 + 5040*A*b^4*tan(1/2*d*x + 1/2*c)^7 + 8400*C*b^4*tan(1/2*d*x + 1/2*c)^7 + 3612*A*a^4*tan(1/2*d*x + 1/2*c)^5...
Time = 15.18 (sec) , antiderivative size = 751, normalized size of antiderivative = 2.22 \[ \int \cos ^7(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:
int(cos(c + d*x)^7*(A + C/cos(c + d*x)^2)*(a + b/cos(c + d*x))^4,x)
Output:
(tan(c/2 + (d*x)/2)*(2*A*a^4 + 2*A*b^4 + 2*C*a^4 + 2*C*b^4 + 12*A*a^2*b^2 + 12*C*a^2*b^2 + 5*A*a*b^3 + (11*A*a^3*b)/2 + 4*C*a*b^3 + 5*C*a^3*b) + tan (c/2 + (d*x)/2)^7*((424*A*a^4)/35 + 24*A*b^4 + (104*C*a^4)/5 + 40*C*b^4 + (624*A*a^2*b^2)/5 + 144*C*a^2*b^2) + tan(c/2 + (d*x)/2)^13*(2*A*a^4 + 2*A* b^4 + 2*C*a^4 + 2*C*b^4 + 12*A*a^2*b^2 + 12*C*a^2*b^2 - 5*A*a*b^3 - (11*A* a^3*b)/2 - 4*C*a*b^3 - 5*C*a^3*b) + tan(c/2 + (d*x)/2)^3*(4*A*a^4 + (28*A* b^4)/3 + (20*C*a^4)/3 + 12*C*b^4 + 40*A*a^2*b^2 + 56*C*a^2*b^2 + 12*A*a*b^ 3 + (14*A*a^3*b)/3 + 16*C*a*b^3 + 12*C*a^3*b) + tan(c/2 + (d*x)/2)^11*(4*A *a^4 + (28*A*b^4)/3 + (20*C*a^4)/3 + 12*C*b^4 + 40*A*a^2*b^2 + 56*C*a^2*b^ 2 - 12*A*a*b^3 - (14*A*a^3*b)/3 - 16*C*a*b^3 - 12*C*a^3*b) + tan(c/2 + (d* x)/2)^5*((86*A*a^4)/5 + (58*A*b^4)/3 + (226*C*a^4)/15 + 30*C*b^4 + (452*A* a^2*b^2)/5 + 116*C*a^2*b^2 + 9*A*a*b^3 + (85*A*a^3*b)/6 + 20*C*a*b^3 + 9*C *a^3*b) + tan(c/2 + (d*x)/2)^9*((86*A*a^4)/5 + (58*A*b^4)/3 + (226*C*a^4)/ 15 + 30*C*b^4 + (452*A*a^2*b^2)/5 + 116*C*a^2*b^2 - 9*A*a*b^3 - (85*A*a^3* b)/6 - 20*C*a*b^3 - 9*C*a^3*b))/(d*(7*tan(c/2 + (d*x)/2)^2 + 21*tan(c/2 + (d*x)/2)^4 + 35*tan(c/2 + (d*x)/2)^6 + 35*tan(c/2 + (d*x)/2)^8 + 21*tan(c/ 2 + (d*x)/2)^10 + 7*tan(c/2 + (d*x)/2)^12 + tan(c/2 + (d*x)/2)^14 + 1)) + (a*b*atan((a*b*tan(c/2 + (d*x)/2)*(5*A*a^2 + 6*A*b^2 + 6*C*a^2 + 8*C*b^2)) /(2*(3*A*a*b^3 + (5*A*a^3*b)/2 + 4*C*a*b^3 + 3*C*a^3*b)))*(5*A*a^2 + 6*A*b ^2 + 6*C*a^2 + 8*C*b^2))/(2*d)
Time = 0.16 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.20 \[ \int \cos ^7(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {84 \sin \left (d x +c \right )^{5} a^{4} c +504 \sin \left (d x +c \right )^{5} a^{3} b^{2}-280 \sin \left (d x +c \right )^{3} a^{4} c -1680 \sin \left (d x +c \right )^{3} a^{3} b^{2}-140 \sin \left (d x +c \right )^{3} a \,b^{4}+420 \sin \left (d x +c \right ) a^{4} c +2520 \sin \left (d x +c \right ) a^{3} b^{2}+420 \sin \left (d x +c \right ) a \,b^{4}+420 \sin \left (d x +c \right ) b^{4} c +840 a \,b^{3} c d x +280 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a^{4} b -910 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{4} b -420 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{2} b^{3}+1155 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{4} b +1050 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2} b^{3}-840 \sin \left (d x +c \right )^{3} a^{2} b^{2} c +2520 \sin \left (d x +c \right ) a^{2} b^{2} c +525 a^{4} b d x -60 \sin \left (d x +c \right )^{7} a^{5}+252 \sin \left (d x +c \right )^{5} a^{5}-420 \sin \left (d x +c \right )^{3} a^{5}+420 \sin \left (d x +c \right ) a^{5}+630 a^{3} b c d x +630 a^{2} b^{3} d x -420 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{3} b c +1050 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{3} b c +840 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a \,b^{3} c}{420 d} \] Input:
int(cos(d*x+c)^7*(a+b*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x)
Output:
(280*cos(c + d*x)*sin(c + d*x)**5*a**4*b - 910*cos(c + d*x)*sin(c + d*x)** 3*a**4*b - 420*cos(c + d*x)*sin(c + d*x)**3*a**3*b*c - 420*cos(c + d*x)*si n(c + d*x)**3*a**2*b**3 + 1155*cos(c + d*x)*sin(c + d*x)*a**4*b + 1050*cos (c + d*x)*sin(c + d*x)*a**3*b*c + 1050*cos(c + d*x)*sin(c + d*x)*a**2*b**3 + 840*cos(c + d*x)*sin(c + d*x)*a*b**3*c - 60*sin(c + d*x)**7*a**5 + 252* sin(c + d*x)**5*a**5 + 84*sin(c + d*x)**5*a**4*c + 504*sin(c + d*x)**5*a** 3*b**2 - 420*sin(c + d*x)**3*a**5 - 280*sin(c + d*x)**3*a**4*c - 1680*sin( c + d*x)**3*a**3*b**2 - 840*sin(c + d*x)**3*a**2*b**2*c - 140*sin(c + d*x) **3*a*b**4 + 420*sin(c + d*x)*a**5 + 420*sin(c + d*x)*a**4*c + 2520*sin(c + d*x)*a**3*b**2 + 2520*sin(c + d*x)*a**2*b**2*c + 420*sin(c + d*x)*a*b**4 + 420*sin(c + d*x)*b**4*c + 525*a**4*b*d*x + 630*a**3*b*c*d*x + 630*a**2* b**3*d*x + 840*a*b**3*c*d*x)/(420*d)