Integrand size = 41, antiderivative size = 293 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=b^3 (b B+4 a C) x+\frac {\left (8 A b^4+16 a^3 b B+32 a b^3 B+24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) \text {arctanh}(\sin (c+d x))}{8 d}-\frac {b^2 \left (32 a b B+2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \sin (c+d x)}{24 d}+\frac {a \left (12 A b^3+8 a^3 B+36 a b^2 B+a^2 b (23 A+36 C)\right ) \tan (c+d x)}{12 d}+\frac {\left (4 A b^2+8 a b B+a^2 (3 A+4 C)\right ) (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {(A b+a B) (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d} \] Output:
b^3*(B*b+4*C*a)*x+1/8*(8*A*b^4+16*B*a^3*b+32*B*a*b^3+24*a^2*b^2*(A+2*C)+a^ 4*(3*A+4*C))*arctanh(sin(d*x+c))/d-1/24*b^2*(32*B*a*b+2*b^2*(13*A-12*C)+3* a^2*(3*A+4*C))*sin(d*x+c)/d+1/12*a*(12*A*b^3+8*B*a^3+36*B*a*b^2+a^2*b*(23* A+36*C))*tan(d*x+c)/d+1/8*(4*A*b^2+8*B*a*b+a^2*(3*A+4*C))*(a+b*cos(d*x+c)) ^2*sec(d*x+c)*tan(d*x+c)/d+1/3*(A*b+B*a)*(a+b*cos(d*x+c))^3*sec(d*x+c)^2*t an(d*x+c)/d+1/4*A*(a+b*cos(d*x+c))^4*sec(d*x+c)^3*tan(d*x+c)/d
Time = 6.52 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.58 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {-12 \left (8 A b^4+16 a^3 b B+32 a b^3 B+24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\sec ^4(c+d x) \left (36 b^4 B c+144 a b^3 c C+36 b^4 B d x+144 a b^3 C d x+48 b^3 (b B+4 a C) (c+d x) \cos (2 (c+d x))+12 b^3 (b B+4 a C) (c+d x) \cos (4 (c+d x))+9 a^4 A \sin (3 (c+d x))+72 a^2 A b^2 \sin (3 (c+d x))+48 a^3 b B \sin (3 (c+d x))+12 a^4 C \sin (3 (c+d x))+18 b^4 C \sin (3 (c+d x))+32 a^3 A b \sin (4 (c+d x))+48 a A b^3 \sin (4 (c+d x))+8 a^4 B \sin (4 (c+d x))+72 a^2 b^2 B \sin (4 (c+d x))+48 a^3 b C \sin (4 (c+d x))+6 b^4 C \sin (5 (c+d x))\right )+32 a \left (6 A b^3+2 a^3 B+9 a b^2 B+a^2 (8 A b+6 b C)\right ) \sec ^2(c+d x) \tan (c+d x)+3 \left (24 a^2 A b^2+16 a^3 b B+4 b^4 C+a^4 (11 A+4 C)\right ) \sec ^3(c+d x) \tan (c+d x)}{96 d} \] Input:
Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*S ec[c + d*x]^5,x]
Output:
(-12*(8*A*b^4 + 16*a^3*b*B + 32*a*b^3*B + 24*a^2*b^2*(A + 2*C) + a^4*(3*A + 4*C))*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c + d*x]^4*(36*b^4*B*c + 144*a*b^3*c*C + 36*b^4* B*d*x + 144*a*b^3*C*d*x + 48*b^3*(b*B + 4*a*C)*(c + d*x)*Cos[2*(c + d*x)] + 12*b^3*(b*B + 4*a*C)*(c + d*x)*Cos[4*(c + d*x)] + 9*a^4*A*Sin[3*(c + d*x )] + 72*a^2*A*b^2*Sin[3*(c + d*x)] + 48*a^3*b*B*Sin[3*(c + d*x)] + 12*a^4* C*Sin[3*(c + d*x)] + 18*b^4*C*Sin[3*(c + d*x)] + 32*a^3*A*b*Sin[4*(c + d*x )] + 48*a*A*b^3*Sin[4*(c + d*x)] + 8*a^4*B*Sin[4*(c + d*x)] + 72*a^2*b^2*B *Sin[4*(c + d*x)] + 48*a^3*b*C*Sin[4*(c + d*x)] + 6*b^4*C*Sin[5*(c + d*x)] ) + 32*a*(6*A*b^3 + 2*a^3*B + 9*a*b^2*B + a^2*(8*A*b + 6*b*C))*Sec[c + d*x ]^2*Tan[c + d*x] + 3*(24*a^2*A*b^2 + 16*a^3*b*B + 4*b^4*C + a^4*(11*A + 4* C))*Sec[c + d*x]^3*Tan[c + d*x])/(96*d)
Time = 2.28 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.04, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.390, Rules used = {3042, 3526, 3042, 3526, 3042, 3526, 3042, 3510, 25, 3042, 3502, 27, 3042, 3214, 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.
| \(\displaystyle \int \sec ^5(c+d x) (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^5}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{4} \int (a+b \cos (c+d x))^3 \left (-b (A-4 C) \cos ^2(c+d x)+(3 a A+4 b B+4 a C) \cos (c+d x)+4 (A b+a B)\right ) \sec ^4(c+d x)dx+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (-b (A-4 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(3 a A+4 b B+4 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+4 (A b+a B)\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int (a+b \cos (c+d x))^2 \left (-b (7 A b-12 C b+4 a B) \cos ^2(c+d x)+2 \left (4 B a^2+b (7 A+12 C) a+6 b^2 B\right ) \cos (c+d x)+3 \left ((3 A+4 C) a^2+8 b B a+4 A b^2\right )\right ) \sec ^3(c+d x)dx+\frac {4 (a B+A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (-b (7 A b-12 C b+4 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (4 B a^2+b (7 A+12 C) a+6 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left ((3 A+4 C) a^2+8 b B a+4 A b^2\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {4 (a B+A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int (a+b \cos (c+d x)) \left (-b \left (3 (3 A+4 C) a^2+32 b B a+2 b^2 (13 A-12 C)\right ) \cos ^2(c+d x)+\left (3 (3 A+4 C) a^3+32 b B a^2+2 b^2 (13 A+36 C) a+24 b^3 B\right ) \cos (c+d x)+2 \left (8 B a^3+\frac {1}{2} (46 A b+72 C b) a^2+36 b^2 B a+12 A b^3\right )\right ) \sec ^2(c+d x)dx+\frac {3 \tan (c+d x) \sec (c+d x) \left (a^2 (3 A+4 C)+8 a b B+4 A b^2\right ) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {4 (a B+A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (-b \left (3 (3 A+4 C) a^2+32 b B a+2 b^2 (13 A-12 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (3 (3 A+4 C) a^3+32 b B a^2+2 b^2 (13 A+36 C) a+24 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 \left (8 B a^3+\frac {1}{2} (46 A b+72 C b) a^2+36 b^2 B a+12 A b^3\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {3 \tan (c+d x) \sec (c+d x) \left (a^2 (3 A+4 C)+8 a b B+4 A b^2\right ) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {4 (a B+A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\frac {2 a \tan (c+d x) \left (8 a^3 B+a^2 b (23 A+36 C)+36 a b^2 B+12 A b^3\right )}{d}-\int -\left (\left (24 (b B+4 a C) \cos (c+d x) b^3-\left (3 (3 A+4 C) a^2+32 b B a+2 b^2 (13 A-12 C)\right ) \cos ^2(c+d x) b^2+3 \left ((3 A+4 C) a^4+16 b B a^3+24 b^2 (A+2 C) a^2+32 b^3 B a+8 A b^4\right )\right ) \sec (c+d x)\right )dx\right )+\frac {3 \tan (c+d x) \sec (c+d x) \left (a^2 (3 A+4 C)+8 a b B+4 A b^2\right ) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {4 (a B+A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int \left (24 (b B+4 a C) \cos (c+d x) b^3-\left (3 (3 A+4 C) a^2+32 b B a+2 b^2 (13 A-12 C)\right ) \cos ^2(c+d x) b^2+3 \left ((3 A+4 C) a^4+16 b B a^3+24 b^2 (A+2 C) a^2+32 b^3 B a+8 A b^4\right )\right ) \sec (c+d x)dx+\frac {2 a \tan (c+d x) \left (8 a^3 B+a^2 b (23 A+36 C)+36 a b^2 B+12 A b^3\right )}{d}\right )+\frac {3 \tan (c+d x) \sec (c+d x) \left (a^2 (3 A+4 C)+8 a b B+4 A b^2\right ) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {4 (a B+A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int \frac {24 (b B+4 a C) \sin \left (c+d x+\frac {\pi }{2}\right ) b^3-\left (3 (3 A+4 C) a^2+32 b B a+2 b^2 (13 A-12 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 b^2+3 \left ((3 A+4 C) a^4+16 b B a^3+24 b^2 (A+2 C) a^2+32 b^3 B a+8 A b^4\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 a \tan (c+d x) \left (8 a^3 B+a^2 b (23 A+36 C)+36 a b^2 B+12 A b^3\right )}{d}\right )+\frac {3 \tan (c+d x) \sec (c+d x) \left (a^2 (3 A+4 C)+8 a b B+4 A b^2\right ) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {4 (a B+A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int 3 \left ((3 A+4 C) a^4+16 b B a^3+24 b^2 (A+2 C) a^2+32 b^3 B a+8 A b^4+8 b^3 (b B+4 a C) \cos (c+d x)\right ) \sec (c+d x)dx-\frac {b^2 \sin (c+d x) \left (3 a^2 (3 A+4 C)+32 a b B+2 b^2 (13 A-12 C)\right )}{d}+\frac {2 a \tan (c+d x) \left (8 a^3 B+a^2 b (23 A+36 C)+36 a b^2 B+12 A b^3\right )}{d}\right )+\frac {3 \tan (c+d x) \sec (c+d x) \left (a^2 (3 A+4 C)+8 a b B+4 A b^2\right ) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {4 (a B+A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 \int \left ((3 A+4 C) a^4+16 b B a^3+24 b^2 (A+2 C) a^2+32 b^3 B a+8 A b^4+8 b^3 (b B+4 a C) \cos (c+d x)\right ) \sec (c+d x)dx-\frac {b^2 \sin (c+d x) \left (3 a^2 (3 A+4 C)+32 a b B+2 b^2 (13 A-12 C)\right )}{d}+\frac {2 a \tan (c+d x) \left (8 a^3 B+a^2 b (23 A+36 C)+36 a b^2 B+12 A b^3\right )}{d}\right )+\frac {3 \tan (c+d x) \sec (c+d x) \left (a^2 (3 A+4 C)+8 a b B+4 A b^2\right ) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {4 (a B+A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 \int \frac {(3 A+4 C) a^4+16 b B a^3+24 b^2 (A+2 C) a^2+32 b^3 B a+8 A b^4+8 b^3 (b B+4 a C) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {b^2 \sin (c+d x) \left (3 a^2 (3 A+4 C)+32 a b B+2 b^2 (13 A-12 C)\right )}{d}+\frac {2 a \tan (c+d x) \left (8 a^3 B+a^2 b (23 A+36 C)+36 a b^2 B+12 A b^3\right )}{d}\right )+\frac {3 \tan (c+d x) \sec (c+d x) \left (a^2 (3 A+4 C)+8 a b B+4 A b^2\right ) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {4 (a B+A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 \left (\left (a^4 (3 A+4 C)+16 a^3 b B+24 a^2 b^2 (A+2 C)+32 a b^3 B+8 A b^4\right ) \int \sec (c+d x)dx+8 b^3 x (4 a C+b B)\right )-\frac {b^2 \sin (c+d x) \left (3 a^2 (3 A+4 C)+32 a b B+2 b^2 (13 A-12 C)\right )}{d}+\frac {2 a \tan (c+d x) \left (8 a^3 B+a^2 b (23 A+36 C)+36 a b^2 B+12 A b^3\right )}{d}\right )+\frac {3 \tan (c+d x) \sec (c+d x) \left (a^2 (3 A+4 C)+8 a b B+4 A b^2\right ) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {4 (a B+A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 \left (\left (a^4 (3 A+4 C)+16 a^3 b B+24 a^2 b^2 (A+2 C)+32 a b^3 B+8 A b^4\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+8 b^3 x (4 a C+b B)\right )-\frac {b^2 \sin (c+d x) \left (3 a^2 (3 A+4 C)+32 a b B+2 b^2 (13 A-12 C)\right )}{d}+\frac {2 a \tan (c+d x) \left (8 a^3 B+a^2 b (23 A+36 C)+36 a b^2 B+12 A b^3\right )}{d}\right )+\frac {3 \tan (c+d x) \sec (c+d x) \left (a^2 (3 A+4 C)+8 a b B+4 A b^2\right ) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {4 (a B+A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3 \tan (c+d x) \sec (c+d x) \left (a^2 (3 A+4 C)+8 a b B+4 A b^2\right ) (a+b \cos (c+d x))^2}{2 d}+\frac {1}{2} \left (-\frac {b^2 \sin (c+d x) \left (3 a^2 (3 A+4 C)+32 a b B+2 b^2 (13 A-12 C)\right )}{d}+\frac {2 a \tan (c+d x) \left (8 a^3 B+a^2 b (23 A+36 C)+36 a b^2 B+12 A b^3\right )}{d}+3 \left (\frac {\left (a^4 (3 A+4 C)+16 a^3 b B+24 a^2 b^2 (A+2 C)+32 a b^3 B+8 A b^4\right ) \text {arctanh}(\sin (c+d x))}{d}+8 b^3 x (4 a C+b B)\right )\right )\right )+\frac {4 (a B+A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
Input:
Int[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]
Output:
(A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + ((4*(A*b + a*B)*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + ((3*(4*A* b^2 + 8*a*b*B + a^2*(3*A + 4*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (3*(8*b^3*(b*B + 4*a*C)*x + ((8*A*b^4 + 16*a^3*b*B + 32*a *b^3*B + 24*a^2*b^2*(A + 2*C) + a^4*(3*A + 4*C))*ArcTanh[Sin[c + d*x]])/d) - (b^2*(32*a*b*B + 2*b^2*(13*A - 12*C) + 3*a^2*(3*A + 4*C))*Sin[c + d*x]) /d + (2*a*(12*A*b^3 + 8*a^3*B + 36*a*b^2*B + a^2*b*(23*A + 36*C))*Tan[c + d*x])/d)/2)/3)/4
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-(b*c - a*d))*(A*b^2 - a*b*B + a^2*C)*Cos[ e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - S imp[1/(b^2*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*( m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1)) ))*Sin[e + f*x] - b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; F reeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d , 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 13.44 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.88
| method | result | size |
| parts | \(\frac {A \,a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (B \,b^{4}+4 C a \,b^{3}\right ) \left (d x +c \right )}{d}+\frac {\left (A \,b^{4}+4 B a \,b^{3}+6 C \,a^{2} b^{2}\right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (4 a A \,b^{3}+6 B \,a^{2} b^{2}+4 a^{3} b C \right ) \tan \left (d x +c \right )}{d}+\frac {\left (6 A \,a^{2} b^{2}+4 B \,a^{3} b +a^{4} C \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {C \,b^{4} \sin \left (d x +c \right )}{d}\) | \(257\) |
| derivativedivides | \(\frac {A \,a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-B \,a^{4} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 A \,a^{3} b \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 B \,a^{3} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 a^{3} b C \tan \left (d x +c \right )+6 A \,a^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 B \,a^{2} b^{2} \tan \left (d x +c \right )+6 C \,a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a A \,b^{3} \tan \left (d x +c \right )+4 B a \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 C a \,b^{3} \left (d x +c \right )+A \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,b^{4} \left (d x +c \right )+C \sin \left (d x +c \right ) b^{4}}{d}\) | \(355\) |
| default | \(\frac {A \,a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-B \,a^{4} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 A \,a^{3} b \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 B \,a^{3} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 a^{3} b C \tan \left (d x +c \right )+6 A \,a^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 B \,a^{2} b^{2} \tan \left (d x +c \right )+6 C \,a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a A \,b^{3} \tan \left (d x +c \right )+4 B a \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 C a \,b^{3} \left (d x +c \right )+A \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,b^{4} \left (d x +c \right )+C \sin \left (d x +c \right ) b^{4}}{d}\) | \(355\) |
| parallelrisch | \(\frac {-9 \left (\left (A +\frac {4 C}{3}\right ) a^{4}+\frac {16 B \,a^{3} b}{3}+8 a^{2} b^{2} \left (A +2 C \right )+\frac {32 B a \,b^{3}}{3}+\frac {8 A \,b^{4}}{3}\right ) \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+9 \left (\left (A +\frac {4 C}{3}\right ) a^{4}+\frac {16 B \,a^{3} b}{3}+8 a^{2} b^{2} \left (A +2 C \right )+\frac {32 B a \,b^{3}}{3}+\frac {8 A \,b^{4}}{3}\right ) \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+96 b^{3} d x \left (B b +4 C a \right ) \cos \left (2 d x +2 c \right )+24 b^{3} d x \left (B b +4 C a \right ) \cos \left (4 d x +4 c \right )+256 a \left (\frac {B \,a^{3}}{4}+a^{2} \left (A +\frac {3 C}{4}\right ) b +\frac {9 B a \,b^{2}}{8}+\frac {3 A \,b^{3}}{4}\right ) \sin \left (2 d x +2 c \right )+\left (\left (18 A +24 C \right ) a^{4}+96 B \,a^{3} b +144 A \,a^{2} b^{2}+36 C \,b^{4}\right ) \sin \left (3 d x +3 c \right )+64 a \left (\frac {B \,a^{3}}{4}+b \left (A +\frac {3 C}{2}\right ) a^{2}+\frac {9 B a \,b^{2}}{4}+\frac {3 A \,b^{3}}{2}\right ) \sin \left (4 d x +4 c \right )+12 C \sin \left (5 d x +5 c \right ) b^{4}+\left (\left (66 A +24 C \right ) a^{4}+96 B \,a^{3} b +144 A \,a^{2} b^{2}+24 C \,b^{4}\right ) \sin \left (d x +c \right )+72 b^{3} d x \left (B b +4 C a \right )}{24 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(436\) |
| risch | \(x B \,b^{4}+4 a \,b^{3} C x +\frac {i {\mathrm e}^{-i \left (d x +c \right )} C \,b^{4}}{2 d}+\frac {i a \left (96 a^{2} b C +144 B a \,b^{2}+96 A \,b^{3}+16 B \,a^{3}+288 A \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+192 A \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+256 A \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+432 B a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+288 C \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+72 A a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+48 B \,a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}+72 A a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}+432 B a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+288 C \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+48 B \,a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}-72 A a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-48 B \,a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}+144 B a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+96 C \,a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}-72 A a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-48 B \,a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}+64 A \,a^{2} b -12 C \,a^{3} {\mathrm e}^{5 i \left (d x +c \right )}+48 B \,a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+64 B \,a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+9 A \,a^{3} {\mathrm e}^{i \left (d x +c \right )}+33 A \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}+12 C \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}+96 A \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-33 A \,a^{3} {\mathrm e}^{5 i \left (d x +c \right )}-12 C \,a^{3} {\mathrm e}^{7 i \left (d x +c \right )}+12 C \,a^{3} {\mathrm e}^{i \left (d x +c \right )}+288 A \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-9 A \,a^{3} {\mathrm e}^{7 i \left (d x +c \right )}\right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {i {\mathrm e}^{i \left (d x +c \right )} C \,b^{4}}{2 d}+\frac {3 A \,a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}+\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,b^{2}}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,b^{4}}{d}+\frac {2 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B b}{d}+\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B a \,b^{3}}{d}+\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{2 d}+\frac {6 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C \,a^{2} b^{2}}{d}-\frac {3 A \,a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,b^{2}}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,b^{4}}{d}-\frac {2 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B b}{d}-\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B a \,b^{3}}{d}-\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{2 d}-\frac {6 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C \,a^{2} b^{2}}{d}\) | \(882\) |
Input:
int((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5,x,meth od=_RETURNVERBOSE)
Output:
A*a^4/d*(-(-1/4*sec(d*x+c)^3-3/8*sec(d*x+c))*tan(d*x+c)+3/8*ln(sec(d*x+c)+ tan(d*x+c)))-(4*A*a^3*b+B*a^4)/d*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+(B*b^4 +4*C*a*b^3)/d*(d*x+c)+(A*b^4+4*B*a*b^3+6*C*a^2*b^2)/d*ln(sec(d*x+c)+tan(d* x+c))+(4*A*a*b^3+6*B*a^2*b^2+4*C*a^3*b)/d*tan(d*x+c)+(6*A*a^2*b^2+4*B*a^3* b+C*a^4)/d*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))+C*b^4 /d*sin(d*x+c)
Time = 0.12 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.03 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {48 \, {\left (4 \, C a b^{3} + B b^{4}\right )} d x \cos \left (d x + c\right )^{4} + 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{4} + 16 \, B a^{3} b + 24 \, {\left (A + 2 \, C\right )} a^{2} b^{2} + 32 \, B a b^{3} + 8 \, A b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{4} + 16 \, B a^{3} b + 24 \, {\left (A + 2 \, C\right )} a^{2} b^{2} + 32 \, B a b^{3} + 8 \, A b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (24 \, C b^{4} \cos \left (d x + c\right )^{4} + 6 \, A a^{4} + 16 \, {\left (B a^{4} + 2 \, {\left (2 \, A + 3 \, C\right )} a^{3} b + 9 \, B a^{2} b^{2} + 6 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{4} + 16 \, B a^{3} b + 24 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \] Input:
integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5, x, algorithm="fricas")
Output:
1/48*(48*(4*C*a*b^3 + B*b^4)*d*x*cos(d*x + c)^4 + 3*((3*A + 4*C)*a^4 + 16* B*a^3*b + 24*(A + 2*C)*a^2*b^2 + 32*B*a*b^3 + 8*A*b^4)*cos(d*x + c)^4*log( sin(d*x + c) + 1) - 3*((3*A + 4*C)*a^4 + 16*B*a^3*b + 24*(A + 2*C)*a^2*b^2 + 32*B*a*b^3 + 8*A*b^4)*cos(d*x + c)^4*log(-sin(d*x + c) + 1) + 2*(24*C*b ^4*cos(d*x + c)^4 + 6*A*a^4 + 16*(B*a^4 + 2*(2*A + 3*C)*a^3*b + 9*B*a^2*b^ 2 + 6*A*a*b^3)*cos(d*x + c)^3 + 3*((3*A + 4*C)*a^4 + 16*B*a^3*b + 24*A*a^2 *b^2)*cos(d*x + c)^2 + 8*(B*a^4 + 4*A*a^3*b)*cos(d*x + c))*sin(d*x + c))/( d*cos(d*x + c)^4)
Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**4*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c)* *5,x)
Output:
Timed out
Time = 0.05 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.47 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{4} + 64 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{3} b + 192 \, {\left (d x + c\right )} C a b^{3} + 48 \, {\left (d x + c\right )} B b^{4} - 3 \, A a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, C a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 48 \, B a^{3} b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 72 \, A a^{2} b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 144 \, C a^{2} b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 96 \, B a b^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, A b^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, C b^{4} \sin \left (d x + c\right ) + 192 \, C a^{3} b \tan \left (d x + c\right ) + 288 \, B a^{2} b^{2} \tan \left (d x + c\right ) + 192 \, A a b^{3} \tan \left (d x + c\right )}{48 \, d} \] Input:
integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5, x, algorithm="maxima")
Output:
1/48*(16*(tan(d*x + c)^3 + 3*tan(d*x + c))*B*a^4 + 64*(tan(d*x + c)^3 + 3* tan(d*x + c))*A*a^3*b + 192*(d*x + c)*C*a*b^3 + 48*(d*x + c)*B*b^4 - 3*A*a ^4*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c) ^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 12*C*a^4*(2 *sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 48*B*a^3*b*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 72*A*a^2*b^2*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 144*C*a^2* b^2*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 96*B*a*b^3*(log(sin( d*x + c) + 1) - log(sin(d*x + c) - 1)) + 24*A*b^4*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 48*C*b^4*sin(d*x + c) + 192*C*a^3*b*tan(d*x + c) + 288*B*a^2*b^2*tan(d*x + c) + 192*A*a*b^3*tan(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 840 vs. \(2 (281) = 562\).
Time = 0.21 (sec) , antiderivative size = 840, normalized size of antiderivative = 2.87 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Too large to display} \] Input:
integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5, x, algorithm="giac")
Output:
1/24*(48*C*b^4*tan(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1/2*c)^2 + 1) + 24*(4*C *a*b^3 + B*b^4)*(d*x + c) + 3*(3*A*a^4 + 4*C*a^4 + 16*B*a^3*b + 24*A*a^2*b ^2 + 48*C*a^2*b^2 + 32*B*a*b^3 + 8*A*b^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1 )) - 3*(3*A*a^4 + 4*C*a^4 + 16*B*a^3*b + 24*A*a^2*b^2 + 48*C*a^2*b^2 + 32* B*a*b^3 + 8*A*b^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 2*(15*A*a^4*tan(1/ 2*d*x + 1/2*c)^7 - 24*B*a^4*tan(1/2*d*x + 1/2*c)^7 + 12*C*a^4*tan(1/2*d*x + 1/2*c)^7 - 96*A*a^3*b*tan(1/2*d*x + 1/2*c)^7 + 48*B*a^3*b*tan(1/2*d*x + 1/2*c)^7 - 96*C*a^3*b*tan(1/2*d*x + 1/2*c)^7 + 72*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 - 144*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 - 96*A*a*b^3*tan(1/2*d*x + 1/2*c)^7 + 9*A*a^4*tan(1/2*d*x + 1/2*c)^5 + 40*B*a^4*tan(1/2*d*x + 1/2*c) ^5 - 12*C*a^4*tan(1/2*d*x + 1/2*c)^5 + 160*A*a^3*b*tan(1/2*d*x + 1/2*c)^5 - 48*B*a^3*b*tan(1/2*d*x + 1/2*c)^5 + 288*C*a^3*b*tan(1/2*d*x + 1/2*c)^5 - 72*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 432*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^ 5 + 288*A*a*b^3*tan(1/2*d*x + 1/2*c)^5 + 9*A*a^4*tan(1/2*d*x + 1/2*c)^3 - 40*B*a^4*tan(1/2*d*x + 1/2*c)^3 - 12*C*a^4*tan(1/2*d*x + 1/2*c)^3 - 160*A* a^3*b*tan(1/2*d*x + 1/2*c)^3 - 48*B*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 288*C*a ^3*b*tan(1/2*d*x + 1/2*c)^3 - 72*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 - 432*B* a^2*b^2*tan(1/2*d*x + 1/2*c)^3 - 288*A*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 15*A *a^4*tan(1/2*d*x + 1/2*c) + 24*B*a^4*tan(1/2*d*x + 1/2*c) + 12*C*a^4*tan(1 /2*d*x + 1/2*c) + 96*A*a^3*b*tan(1/2*d*x + 1/2*c) + 48*B*a^3*b*tan(1/2*...
Time = 3.21 (sec) , antiderivative size = 4710, normalized size of antiderivative = 16.08 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Too large to display} \] Input:
int(((a + b*cos(c + d*x))^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^5,x)
Output:
(atan(((((3*A*a^4)/8 + A*b^4 + (C*a^4)/2 + 3*A*a^2*b^2 + 6*C*a^2*b^2 + 4*B *a*b^3 + 2*B*a^3*b)*(12*A*a^4 + 32*A*b^4 + 32*B*b^4 + 16*C*a^4 + 96*A*a^2* b^2 + 192*C*a^2*b^2 + 128*B*a*b^3 + 64*B*a^3*b + 128*C*a*b^3) + tan(c/2 + (d*x)/2)*((9*A^2*a^8)/2 + 32*A^2*b^8 + 32*B^2*b^8 + 8*C^2*a^8 + 192*A^2*a^ 2*b^6 + 312*A^2*a^4*b^4 + 72*A^2*a^6*b^2 + 512*B^2*a^2*b^6 + 512*B^2*a^4*b ^4 + 128*B^2*a^6*b^2 + 512*C^2*a^2*b^6 + 1152*C^2*a^4*b^4 + 192*C^2*a^6*b^ 2 + 12*A*C*a^8 + 256*A*B*a*b^7 + 48*A*B*a^7*b + 256*B*C*a*b^7 + 64*B*C*a^7 *b + 896*A*B*a^3*b^5 + 480*A*B*a^5*b^3 + 384*A*C*a^2*b^6 + 1184*A*C*a^4*b^ 4 + 240*A*C*a^6*b^2 + 1536*B*C*a^3*b^5 + 896*B*C*a^5*b^3))*((3*A*a^4)/8 + A*b^4 + (C*a^4)/2 + 3*A*a^2*b^2 + 6*C*a^2*b^2 + 4*B*a*b^3 + 2*B*a^3*b)*1i - (((3*A*a^4)/8 + A*b^4 + (C*a^4)/2 + 3*A*a^2*b^2 + 6*C*a^2*b^2 + 4*B*a*b^ 3 + 2*B*a^3*b)*(12*A*a^4 + 32*A*b^4 + 32*B*b^4 + 16*C*a^4 + 96*A*a^2*b^2 + 192*C*a^2*b^2 + 128*B*a*b^3 + 64*B*a^3*b + 128*C*a*b^3) - tan(c/2 + (d*x) /2)*((9*A^2*a^8)/2 + 32*A^2*b^8 + 32*B^2*b^8 + 8*C^2*a^8 + 192*A^2*a^2*b^6 + 312*A^2*a^4*b^4 + 72*A^2*a^6*b^2 + 512*B^2*a^2*b^6 + 512*B^2*a^4*b^4 + 128*B^2*a^6*b^2 + 512*C^2*a^2*b^6 + 1152*C^2*a^4*b^4 + 192*C^2*a^6*b^2 + 1 2*A*C*a^8 + 256*A*B*a*b^7 + 48*A*B*a^7*b + 256*B*C*a*b^7 + 64*B*C*a^7*b + 896*A*B*a^3*b^5 + 480*A*B*a^5*b^3 + 384*A*C*a^2*b^6 + 1184*A*C*a^4*b^4 + 2 40*A*C*a^6*b^2 + 1536*B*C*a^3*b^5 + 896*B*C*a^5*b^3))*((3*A*a^4)/8 + A*b^4 + (C*a^4)/2 + 3*A*a^2*b^2 + 6*C*a^2*b^2 + 4*B*a*b^3 + 2*B*a^3*b)*1i)/(...
Time = 0.25 (sec) , antiderivative size = 1359, normalized size of antiderivative = 4.64 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx =\text {Too large to display} \] Input:
int((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5,x)
Output:
( - 9*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a**5 - 12*cos (c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a**4*c - 120*cos(c + d *x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a**3*b**2 - 144*cos(c + d*x) *log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a**2*b**2*c - 120*cos(c + d*x)* log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a*b**4 + 18*cos(c + d*x)*log(tan ((c + d*x)/2) - 1)*sin(c + d*x)**2*a**5 + 24*cos(c + d*x)*log(tan((c + d*x )/2) - 1)*sin(c + d*x)**2*a**4*c + 240*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a**3*b**2 + 288*cos(c + d*x)*log(tan((c + d*x)/2) - 1) *sin(c + d*x)**2*a**2*b**2*c + 240*cos(c + d*x)*log(tan((c + d*x)/2) - 1)* sin(c + d*x)**2*a*b**4 - 9*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a**5 - 1 2*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a**4*c - 120*cos(c + d*x)*log(tan ((c + d*x)/2) - 1)*a**3*b**2 - 144*cos(c + d*x)*log(tan((c + d*x)/2) - 1)* a**2*b**2*c - 120*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a*b**4 + 9*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4*a**5 + 12*cos(c + d*x)*lo g(tan((c + d*x)/2) + 1)*sin(c + d*x)**4*a**4*c + 120*cos(c + d*x)*log(tan( (c + d*x)/2) + 1)*sin(c + d*x)**4*a**3*b**2 + 144*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4*a**2*b**2*c + 120*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4*a*b**4 - 18*cos(c + d*x)*log(tan((c + d*x)/2 ) + 1)*sin(c + d*x)**2*a**5 - 24*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*si n(c + d*x)**2*a**4*c - 240*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c...