Integrand size = 39, antiderivative size = 290 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {1}{8} \left (8 a^4 B+24 a^2 b^2 B+3 b^4 B+16 a^3 b (2 A+C)+4 a b^3 (4 A+3 C)\right ) x+\frac {a^4 A \text {arctanh}(\sin (c+d x))}{d}+\frac {\left (95 a^3 b B+80 a b^3 B+12 a^4 C+4 b^4 (5 A+4 C)+2 a^2 b^2 (85 A+56 C)\right ) \sin (c+d x)}{30 d}+\frac {b \left (130 a^2 b B+45 b^3 B+24 a^3 C+4 a b^2 (40 A+29 C)\right ) \cos (c+d x) \sin (c+d x)}{120 d}+\frac {\left (20 A b^2+35 a b B+12 a^2 C+16 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{60 d}+\frac {(5 b B+4 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac {C (a+b \cos (c+d x))^4 \sin (c+d x)}{5 d} \] Output:
1/8*(8*B*a^4+24*B*a^2*b^2+3*B*b^4+16*a^3*b*(2*A+C)+4*a*b^3*(4*A+3*C))*x+a^ 4*A*arctanh(sin(d*x+c))/d+1/30*(95*B*a^3*b+80*B*a*b^3+12*a^4*C+4*b^4*(5*A+ 4*C)+2*a^2*b^2*(85*A+56*C))*sin(d*x+c)/d+1/120*b*(130*B*a^2*b+45*B*b^3+24* a^3*C+4*a*b^2*(40*A+29*C))*cos(d*x+c)*sin(d*x+c)/d+1/60*(20*A*b^2+35*B*a*b +12*C*a^2+16*C*b^2)*(a+b*cos(d*x+c))^2*sin(d*x+c)/d+1/20*(5*B*b+4*C*a)*(a+ b*cos(d*x+c))^3*sin(d*x+c)/d+1/5*C*(a+b*cos(d*x+c))^4*sin(d*x+c)/d
Time = 3.91 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.32 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {1920 a^3 A b c+960 a A b^3 c+480 a^4 B c+1440 a^2 b^2 B c+180 b^4 B c+960 a^3 b c C+720 a b^3 c C+1920 a^3 A b d x+960 a A b^3 d x+480 a^4 B d x+1440 a^2 b^2 B d x+180 b^4 B d x+960 a^3 b C d x+720 a b^3 C d x-480 a^4 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+480 a^4 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+60 \left (32 a^3 b B+24 a b^3 B+8 a^4 C+12 a^2 b^2 (4 A+3 C)+b^4 (6 A+5 C)\right ) \sin (c+d x)+120 b \left (6 a^2 b B+b^3 B+4 a^3 C+4 a b^2 (A+C)\right ) \sin (2 (c+d x))+40 A b^4 \sin (3 (c+d x))+160 a b^3 B \sin (3 (c+d x))+240 a^2 b^2 C \sin (3 (c+d x))+50 b^4 C \sin (3 (c+d x))+15 b^4 B \sin (4 (c+d x))+60 a b^3 C \sin (4 (c+d x))+6 b^4 C \sin (5 (c+d x))}{480 d} \] 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],x]
Output:
(1920*a^3*A*b*c + 960*a*A*b^3*c + 480*a^4*B*c + 1440*a^2*b^2*B*c + 180*b^4 *B*c + 960*a^3*b*c*C + 720*a*b^3*c*C + 1920*a^3*A*b*d*x + 960*a*A*b^3*d*x + 480*a^4*B*d*x + 1440*a^2*b^2*B*d*x + 180*b^4*B*d*x + 960*a^3*b*C*d*x + 7 20*a*b^3*C*d*x - 480*a^4*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 480* a^4*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 60*(32*a^3*b*B + 24*a*b^3 *B + 8*a^4*C + 12*a^2*b^2*(4*A + 3*C) + b^4*(6*A + 5*C))*Sin[c + d*x] + 12 0*b*(6*a^2*b*B + b^3*B + 4*a^3*C + 4*a*b^2*(A + C))*Sin[2*(c + d*x)] + 40* A*b^4*Sin[3*(c + d*x)] + 160*a*b^3*B*Sin[3*(c + d*x)] + 240*a^2*b^2*C*Sin[ 3*(c + d*x)] + 50*b^4*C*Sin[3*(c + d*x)] + 15*b^4*B*Sin[4*(c + d*x)] + 60* a*b^3*C*Sin[4*(c + d*x)] + 6*b^4*C*Sin[5*(c + d*x)])/(480*d)
Time = 2.03 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.07, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 3528, 3042, 3528, 3042, 3528, 3042, 3512, 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.
| \(\displaystyle \int \sec (c+d x) (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {1}{5} \int (a+b \cos (c+d x))^3 \left ((5 b B+4 a C) \cos ^2(c+d x)+(5 A b+4 C b+5 a B) \cos (c+d x)+5 a A\right ) \sec (c+d x)dx+\frac {C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left ((5 b B+4 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(5 A b+4 C b+5 a B) \sin \left (c+d x+\frac {\pi }{2}\right )+5 a A\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \int (a+b \cos (c+d x))^2 \left (20 A a^2+\left (12 C a^2+35 b B a+20 A b^2+16 b^2 C\right ) \cos ^2(c+d x)+\left (20 B a^2+40 A b a+28 b C a+15 b^2 B\right ) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {(4 a C+5 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (20 A a^2+\left (12 C a^2+35 b B a+20 A b^2+16 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (20 B a^2+40 A b a+28 b C a+15 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {(4 a C+5 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \int (a+b \cos (c+d x)) \left (60 A a^3+\left (24 C a^3+130 b B a^2+4 b^2 (40 A+29 C) a+45 b^3 B\right ) \cos ^2(c+d x)+\left (60 B a^3+36 b (5 A+3 C) a^2+115 b^2 B a+8 b^3 (5 A+4 C)\right ) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {\sin (c+d x) \left (12 a^2 C+35 a b B+20 A b^2+16 b^2 C\right ) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {(4 a C+5 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (60 A a^3+\left (24 C a^3+130 b B a^2+4 b^2 (40 A+29 C) a+45 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (60 B a^3+36 b (5 A+3 C) a^2+115 b^2 B a+8 b^3 (5 A+4 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {\sin (c+d x) \left (12 a^2 C+35 a b B+20 A b^2+16 b^2 C\right ) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {(4 a C+5 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \left (120 A a^4+4 \left (12 C a^4+95 b B a^3+2 b^2 (85 A+56 C) a^2+80 b^3 B a+4 b^4 (5 A+4 C)\right ) \cos ^2(c+d x)+15 \left (8 B a^4+16 b (2 A+C) a^3+24 b^2 B a^2+4 b^3 (4 A+3 C) a+3 b^4 B\right ) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {b \sin (c+d x) \cos (c+d x) \left (24 a^3 C+130 a^2 b B+4 a b^2 (40 A+29 C)+45 b^3 B\right )}{2 d}\right )+\frac {\sin (c+d x) \left (12 a^2 C+35 a b B+20 A b^2+16 b^2 C\right ) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {(4 a C+5 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {120 A a^4+4 \left (12 C a^4+95 b B a^3+2 b^2 (85 A+56 C) a^2+80 b^3 B a+4 b^4 (5 A+4 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+15 \left (8 B a^4+16 b (2 A+C) a^3+24 b^2 B a^2+4 b^3 (4 A+3 C) a+3 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {b \sin (c+d x) \cos (c+d x) \left (24 a^3 C+130 a^2 b B+4 a b^2 (40 A+29 C)+45 b^3 B\right )}{2 d}\right )+\frac {\sin (c+d x) \left (12 a^2 C+35 a b B+20 A b^2+16 b^2 C\right ) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {(4 a C+5 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int 15 \left (8 A a^4+\left (8 B a^4+16 b (2 A+C) a^3+24 b^2 B a^2+4 b^3 (4 A+3 C) a+3 b^4 B\right ) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {4 \sin (c+d x) \left (12 a^4 C+95 a^3 b B+2 a^2 b^2 (85 A+56 C)+80 a b^3 B+4 b^4 (5 A+4 C)\right )}{d}\right )+\frac {b \sin (c+d x) \cos (c+d x) \left (24 a^3 C+130 a^2 b B+4 a b^2 (40 A+29 C)+45 b^3 B\right )}{2 d}\right )+\frac {\sin (c+d x) \left (12 a^2 C+35 a b B+20 A b^2+16 b^2 C\right ) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {(4 a C+5 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (15 \int \left (8 A a^4+\left (8 B a^4+16 b (2 A+C) a^3+24 b^2 B a^2+4 b^3 (4 A+3 C) a+3 b^4 B\right ) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {4 \sin (c+d x) \left (12 a^4 C+95 a^3 b B+2 a^2 b^2 (85 A+56 C)+80 a b^3 B+4 b^4 (5 A+4 C)\right )}{d}\right )+\frac {b \sin (c+d x) \cos (c+d x) \left (24 a^3 C+130 a^2 b B+4 a b^2 (40 A+29 C)+45 b^3 B\right )}{2 d}\right )+\frac {\sin (c+d x) \left (12 a^2 C+35 a b B+20 A b^2+16 b^2 C\right ) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {(4 a C+5 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (15 \int \frac {8 A a^4+\left (8 B a^4+16 b (2 A+C) a^3+24 b^2 B a^2+4 b^3 (4 A+3 C) a+3 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {4 \sin (c+d x) \left (12 a^4 C+95 a^3 b B+2 a^2 b^2 (85 A+56 C)+80 a b^3 B+4 b^4 (5 A+4 C)\right )}{d}\right )+\frac {b \sin (c+d x) \cos (c+d x) \left (24 a^3 C+130 a^2 b B+4 a b^2 (40 A+29 C)+45 b^3 B\right )}{2 d}\right )+\frac {\sin (c+d x) \left (12 a^2 C+35 a b B+20 A b^2+16 b^2 C\right ) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {(4 a C+5 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (15 \left (8 a^4 A \int \sec (c+d x)dx+x \left (8 a^4 B+16 a^3 b (2 A+C)+24 a^2 b^2 B+4 a b^3 (4 A+3 C)+3 b^4 B\right )\right )+\frac {4 \sin (c+d x) \left (12 a^4 C+95 a^3 b B+2 a^2 b^2 (85 A+56 C)+80 a b^3 B+4 b^4 (5 A+4 C)\right )}{d}\right )+\frac {b \sin (c+d x) \cos (c+d x) \left (24 a^3 C+130 a^2 b B+4 a b^2 (40 A+29 C)+45 b^3 B\right )}{2 d}\right )+\frac {\sin (c+d x) \left (12 a^2 C+35 a b B+20 A b^2+16 b^2 C\right ) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {(4 a C+5 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (15 \left (8 a^4 A \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+x \left (8 a^4 B+16 a^3 b (2 A+C)+24 a^2 b^2 B+4 a b^3 (4 A+3 C)+3 b^4 B\right )\right )+\frac {4 \sin (c+d x) \left (12 a^4 C+95 a^3 b B+2 a^2 b^2 (85 A+56 C)+80 a b^3 B+4 b^4 (5 A+4 C)\right )}{d}\right )+\frac {b \sin (c+d x) \cos (c+d x) \left (24 a^3 C+130 a^2 b B+4 a b^2 (40 A+29 C)+45 b^3 B\right )}{2 d}\right )+\frac {\sin (c+d x) \left (12 a^2 C+35 a b B+20 A b^2+16 b^2 C\right ) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {(4 a C+5 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (12 a^2 C+35 a b B+20 A b^2+16 b^2 C\right ) (a+b \cos (c+d x))^2}{3 d}+\frac {1}{3} \left (\frac {b \sin (c+d x) \cos (c+d x) \left (24 a^3 C+130 a^2 b B+4 a b^2 (40 A+29 C)+45 b^3 B\right )}{2 d}+\frac {1}{2} \left (15 \left (\frac {8 a^4 A \text {arctanh}(\sin (c+d x))}{d}+x \left (8 a^4 B+16 a^3 b (2 A+C)+24 a^2 b^2 B+4 a b^3 (4 A+3 C)+3 b^4 B\right )\right )+\frac {4 \sin (c+d x) \left (12 a^4 C+95 a^3 b B+2 a^2 b^2 (85 A+56 C)+80 a b^3 B+4 b^4 (5 A+4 C)\right )}{d}\right )\right )\right )+\frac {(4 a C+5 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
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],x]
Output:
(C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*d) + (((5*b*B + 4*a*C)*(a + b*C os[c + d*x])^3*Sin[c + d*x])/(4*d) + (((20*A*b^2 + 35*a*b*B + 12*a^2*C + 1 6*b^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d) + ((b*(130*a^2*b*B + 4 5*b^3*B + 24*a^3*C + 4*a*b^2*(40*A + 29*C))*Cos[c + d*x]*Sin[c + d*x])/(2* d) + (15*((8*a^4*B + 24*a^2*b^2*B + 3*b^4*B + 16*a^3*b*(2*A + C) + 4*a*b^3 *(4*A + 3*C))*x + (8*a^4*A*ArcTanh[Sin[c + d*x]])/d) + (4*(95*a^3*b*B + 80 *a*b^3*B + 12*a^4*C + 4*b^4*(5*A + 4*C) + 2*a^2*b^2*(85*A + 56*C))*Sin[c + d*x])/d)/2)/3)/4)/5
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[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si n[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3)) Int[(a + b*Si n[e + f*x])^m*Simp[a*C*d + A*b*c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e + f*x]^2 , x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, 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)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 12.78 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.88
| method | result | size |
| parallelrisch | \(\frac {-480 A \,a^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+480 A \,a^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+480 \left (\frac {B \,b^{3}}{4}+a \left (A +C \right ) b^{2}+\frac {3 B \,a^{2} b}{2}+a^{3} C \right ) b \sin \left (2 d x +2 c \right )+40 b^{2} \left (\left (A +\frac {5 C}{4}\right ) b^{2}+4 B a b +6 a^{2} C \right ) \sin \left (3 d x +3 c \right )+\left (15 B \,b^{4}+60 C a \,b^{3}\right ) \sin \left (4 d x +4 c \right )+6 C \sin \left (5 d x +5 c \right ) b^{4}+\left (\left (360 A +300 C \right ) b^{4}+1440 B a \,b^{3}+2880 \left (A +\frac {3 C}{4}\right ) a^{2} b^{2}+1920 B \,a^{3} b +480 a^{4} C \right ) \sin \left (d x +c \right )+1920 \left (\frac {3 B \,b^{4}}{32}+\frac {\left (A +\frac {3 C}{4}\right ) a \,b^{3}}{2}+\frac {3 B \,a^{2} b^{2}}{4}+a^{3} \left (A +\frac {C}{2}\right ) b +\frac {B \,a^{4}}{4}\right ) x d}{480 d}\) | \(255\) |
| parts | \(\frac {A \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) \left (d x +c \right )}{d}+\frac {\left (B \,b^{4}+4 C a \,b^{3}\right ) \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {\left (A \,b^{4}+4 B a \,b^{3}+6 C \,a^{2} b^{2}\right ) \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (4 a A \,b^{3}+6 B \,a^{2} b^{2}+4 a^{3} b C \right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (6 A \,a^{2} b^{2}+4 B \,a^{3} b +a^{4} C \right ) \sin \left (d x +c \right )}{d}+\frac {C \,b^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5 d}\) | \(256\) |
| derivativedivides | \(\frac {A \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{4} \left (d x +c \right )+C \sin \left (d x +c \right ) a^{4}+4 A \,a^{3} b \left (d x +c \right )+4 B \sin \left (d x +c \right ) a^{3} b +4 a^{3} b C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 A \sin \left (d x +c \right ) a^{2} b^{2}+6 B \,a^{2} b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 C \,a^{2} b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+4 a A \,b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 B a \,b^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 C a \,b^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {A \,b^{4} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B \,b^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {C \,b^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}}{d}\) | \(355\) |
| default | \(\frac {A \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{4} \left (d x +c \right )+C \sin \left (d x +c \right ) a^{4}+4 A \,a^{3} b \left (d x +c \right )+4 B \sin \left (d x +c \right ) a^{3} b +4 a^{3} b C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 A \sin \left (d x +c \right ) a^{2} b^{2}+6 B \,a^{2} b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 C \,a^{2} b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+4 a A \,b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 B a \,b^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 C a \,b^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {A \,b^{4} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B \,b^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {C \,b^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}}{d}\) | \(355\) |
| risch | \(\frac {3 a \,b^{3} C x}{2}+\frac {\sin \left (4 d x +4 c \right ) B \,b^{4}}{32 d}+\frac {\sin \left (3 d x +3 c \right ) A \,b^{4}}{12 d}+\frac {5 \sin \left (3 d x +3 c \right ) C \,b^{4}}{48 d}+\frac {\sin \left (2 d x +2 c \right ) B \,b^{4}}{4 d}+4 x A \,a^{3} b +2 x a A \,b^{3}+3 x B \,a^{2} b^{2}+2 x \,a^{3} b C +\frac {C \,b^{4} \sin \left (5 d x +5 c \right )}{80 d}+\frac {A \,a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {A \,a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {\sin \left (2 d x +2 c \right ) a A \,b^{3}}{d}+\frac {3 \sin \left (2 d x +2 c \right ) B \,a^{2} b^{2}}{2 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} b C}{d}+\frac {\sin \left (2 d x +2 c \right ) C a \,b^{3}}{d}+\frac {\sin \left (4 d x +4 c \right ) C a \,b^{3}}{8 d}+\frac {\sin \left (3 d x +3 c \right ) B a \,b^{3}}{3 d}+\frac {\sin \left (3 d x +3 c \right ) C \,a^{2} b^{2}}{2 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} A \,a^{2} b^{2}}{d}+\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} B \,a^{3} b}{d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} B a \,b^{3}}{2 d}+\frac {9 i {\mathrm e}^{-i \left (d x +c \right )} C \,a^{2} b^{2}}{4 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} A \,a^{2} b^{2}}{d}-\frac {2 i {\mathrm e}^{i \left (d x +c \right )} B \,a^{3} b}{d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} B a \,b^{3}}{2 d}-\frac {9 i {\mathrm e}^{i \left (d x +c \right )} C \,a^{2} b^{2}}{4 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} A \,b^{4}}{8 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} a^{4} C}{2 d}+\frac {5 i {\mathrm e}^{-i \left (d x +c \right )} C \,b^{4}}{16 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} A \,b^{4}}{8 d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} a^{4} C}{2 d}-\frac {5 i {\mathrm e}^{i \left (d x +c \right )} C \,b^{4}}{16 d}+x B \,a^{4}+\frac {3 x B \,b^{4}}{8}\) | \(606\) |
| norman | \(\text {Expression too large to display}\) | \(1104\) |
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),x,method =_RETURNVERBOSE)
Output:
1/480*(-480*A*a^4*ln(tan(1/2*d*x+1/2*c)-1)+480*A*a^4*ln(tan(1/2*d*x+1/2*c) +1)+480*(1/4*B*b^3+a*(A+C)*b^2+3/2*B*a^2*b+a^3*C)*b*sin(2*d*x+2*c)+40*b^2* ((A+5/4*C)*b^2+4*B*a*b+6*a^2*C)*sin(3*d*x+3*c)+(15*B*b^4+60*C*a*b^3)*sin(4 *d*x+4*c)+6*C*sin(5*d*x+5*c)*b^4+((360*A+300*C)*b^4+1440*B*a*b^3+2880*(A+3 /4*C)*a^2*b^2+1920*B*a^3*b+480*a^4*C)*sin(d*x+c)+1920*(3/32*B*b^4+1/2*(A+3 /4*C)*a*b^3+3/4*B*a^2*b^2+a^3*(A+1/2*C)*b+1/4*B*a^4)*x*d)/d
Time = 0.10 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.90 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {60 \, A a^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 60 \, A a^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 15 \, {\left (8 \, B a^{4} + 16 \, {\left (2 \, A + C\right )} a^{3} b + 24 \, B a^{2} b^{2} + 4 \, {\left (4 \, A + 3 \, C\right )} a b^{3} + 3 \, B b^{4}\right )} d x + {\left (24 \, C b^{4} \cos \left (d x + c\right )^{4} + 120 \, C a^{4} + 480 \, B a^{3} b + 240 \, {\left (3 \, A + 2 \, C\right )} a^{2} b^{2} + 320 \, B a b^{3} + 16 \, {\left (5 \, A + 4 \, C\right )} b^{4} + 30 \, {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left (30 \, C a^{2} b^{2} + 20 \, B a b^{3} + {\left (5 \, A + 4 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (16 \, C a^{3} b + 24 \, B a^{2} b^{2} + 4 \, {\left (4 \, A + 3 \, C\right )} a b^{3} + 3 \, B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, d} \] 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),x, algorithm="fricas")
Output:
1/120*(60*A*a^4*log(sin(d*x + c) + 1) - 60*A*a^4*log(-sin(d*x + c) + 1) + 15*(8*B*a^4 + 16*(2*A + C)*a^3*b + 24*B*a^2*b^2 + 4*(4*A + 3*C)*a*b^3 + 3* B*b^4)*d*x + (24*C*b^4*cos(d*x + c)^4 + 120*C*a^4 + 480*B*a^3*b + 240*(3*A + 2*C)*a^2*b^2 + 320*B*a*b^3 + 16*(5*A + 4*C)*b^4 + 30*(4*C*a*b^3 + B*b^4 )*cos(d*x + c)^3 + 8*(30*C*a^2*b^2 + 20*B*a*b^3 + (5*A + 4*C)*b^4)*cos(d*x + c)^2 + 15*(16*C*a^3*b + 24*B*a^2*b^2 + 4*(4*A + 3*C)*a*b^3 + 3*B*b^4)*c os(d*x + c))*sin(d*x + c))/d
Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\text {Timed out} \] 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), x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.17 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {480 \, {\left (d x + c\right )} B a^{4} + 1920 \, {\left (d x + c\right )} A a^{3} b + 480 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} b + 720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} b^{2} - 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} b^{2} + 480 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a b^{3} - 640 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a b^{3} + 60 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a b^{3} - 160 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A b^{4} + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B b^{4} + 32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C b^{4} + 480 \, A a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 480 \, C a^{4} \sin \left (d x + c\right ) + 1920 \, B a^{3} b \sin \left (d x + c\right ) + 2880 \, A a^{2} b^{2} \sin \left (d x + c\right )}{480 \, d} \] 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),x, algorithm="maxima")
Output:
1/480*(480*(d*x + c)*B*a^4 + 1920*(d*x + c)*A*a^3*b + 480*(2*d*x + 2*c + s in(2*d*x + 2*c))*C*a^3*b + 720*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a^2*b^2 - 960*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^2*b^2 + 480*(2*d*x + 2*c + sin (2*d*x + 2*c))*A*a*b^3 - 640*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a*b^3 + 6 0*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*a*b^3 - 160*(s in(d*x + c)^3 - 3*sin(d*x + c))*A*b^4 + 15*(12*d*x + 12*c + sin(4*d*x + 4* c) + 8*sin(2*d*x + 2*c))*B*b^4 + 32*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*C*b^4 + 480*A*a^4*log(sec(d*x + c) + tan(d*x + c)) + 48 0*C*a^4*sin(d*x + c) + 1920*B*a^3*b*sin(d*x + c) + 2880*A*a^2*b^2*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 1094 vs. \(2 (278) = 556\).
Time = 0.19 (sec) , antiderivative size = 1094, normalized size of antiderivative = 3.77 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\text {Too large to display} \] 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),x, algorithm="giac")
Output:
1/120*(120*A*a^4*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 120*A*a^4*log(abs(ta n(1/2*d*x + 1/2*c) - 1)) + 15*(8*B*a^4 + 32*A*a^3*b + 16*C*a^3*b + 24*B*a^ 2*b^2 + 16*A*a*b^3 + 12*C*a*b^3 + 3*B*b^4)*(d*x + c) + 2*(120*C*a^4*tan(1/ 2*d*x + 1/2*c)^9 + 480*B*a^3*b*tan(1/2*d*x + 1/2*c)^9 - 240*C*a^3*b*tan(1/ 2*d*x + 1/2*c)^9 + 720*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 - 360*B*a^2*b^2*ta n(1/2*d*x + 1/2*c)^9 + 720*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 - 240*A*a*b^3* tan(1/2*d*x + 1/2*c)^9 + 480*B*a*b^3*tan(1/2*d*x + 1/2*c)^9 - 300*C*a*b^3* tan(1/2*d*x + 1/2*c)^9 + 120*A*b^4*tan(1/2*d*x + 1/2*c)^9 - 75*B*b^4*tan(1 /2*d*x + 1/2*c)^9 + 120*C*b^4*tan(1/2*d*x + 1/2*c)^9 + 480*C*a^4*tan(1/2*d *x + 1/2*c)^7 + 1920*B*a^3*b*tan(1/2*d*x + 1/2*c)^7 - 480*C*a^3*b*tan(1/2* d*x + 1/2*c)^7 + 2880*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 - 720*B*a^2*b^2*tan (1/2*d*x + 1/2*c)^7 + 1920*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 - 480*A*a*b^3* tan(1/2*d*x + 1/2*c)^7 + 1280*B*a*b^3*tan(1/2*d*x + 1/2*c)^7 - 120*C*a*b^3 *tan(1/2*d*x + 1/2*c)^7 + 320*A*b^4*tan(1/2*d*x + 1/2*c)^7 - 30*B*b^4*tan( 1/2*d*x + 1/2*c)^7 + 160*C*b^4*tan(1/2*d*x + 1/2*c)^7 + 720*C*a^4*tan(1/2* d*x + 1/2*c)^5 + 2880*B*a^3*b*tan(1/2*d*x + 1/2*c)^5 + 4320*A*a^2*b^2*tan( 1/2*d*x + 1/2*c)^5 + 2400*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 1600*B*a*b^3* tan(1/2*d*x + 1/2*c)^5 + 400*A*b^4*tan(1/2*d*x + 1/2*c)^5 + 464*C*b^4*tan( 1/2*d*x + 1/2*c)^5 + 480*C*a^4*tan(1/2*d*x + 1/2*c)^3 + 1920*B*a^3*b*tan(1 /2*d*x + 1/2*c)^3 + 480*C*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 2880*A*a^2*b^2...
Time = 2.75 (sec) , antiderivative size = 4118, normalized size of antiderivative = 14.20 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\text {Too large to display} \] 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),x)
Output:
(atan(((tan(c/2 + (d*x)/2)*(32*A^2*a^8 + 32*B^2*a^8 + (9*B^2*b^8)/2 + 128* A^2*a^2*b^6 + 512*A^2*a^4*b^4 + 512*A^2*a^6*b^2 + 72*B^2*a^2*b^6 + 312*B^2 *a^4*b^4 + 192*B^2*a^6*b^2 + 72*C^2*a^2*b^6 + 192*C^2*a^4*b^4 + 128*C^2*a^ 6*b^2 + 48*A*B*a*b^7 + 256*A*B*a^7*b + 36*B*C*a*b^7 + 128*B*C*a^7*b + 480* A*B*a^3*b^5 + 896*A*B*a^5*b^3 + 192*A*C*a^2*b^6 + 640*A*C*a^4*b^4 + 512*A* C*a^6*b^2 + 336*B*C*a^3*b^5 + 480*B*C*a^5*b^3) + (B*a^4*1i + (B*b^4*3i)/8 + B*a^2*b^2*3i + A*a*b^3*2i + A*a^3*b*4i + (C*a*b^3*3i)/2 + C*a^3*b*2i)*(3 2*A*a^4 + 32*B*a^4 + 12*B*b^4 + 96*B*a^2*b^2 + 64*A*a*b^3 + 128*A*a^3*b + 48*C*a*b^3 + 64*C*a^3*b))*(B*a^4*1i + (B*b^4*3i)/8 + B*a^2*b^2*3i + A*a*b^ 3*2i + A*a^3*b*4i + (C*a*b^3*3i)/2 + C*a^3*b*2i)*1i + (tan(c/2 + (d*x)/2)* (32*A^2*a^8 + 32*B^2*a^8 + (9*B^2*b^8)/2 + 128*A^2*a^2*b^6 + 512*A^2*a^4*b ^4 + 512*A^2*a^6*b^2 + 72*B^2*a^2*b^6 + 312*B^2*a^4*b^4 + 192*B^2*a^6*b^2 + 72*C^2*a^2*b^6 + 192*C^2*a^4*b^4 + 128*C^2*a^6*b^2 + 48*A*B*a*b^7 + 256* A*B*a^7*b + 36*B*C*a*b^7 + 128*B*C*a^7*b + 480*A*B*a^3*b^5 + 896*A*B*a^5*b ^3 + 192*A*C*a^2*b^6 + 640*A*C*a^4*b^4 + 512*A*C*a^6*b^2 + 336*B*C*a^3*b^5 + 480*B*C*a^5*b^3) - (B*a^4*1i + (B*b^4*3i)/8 + B*a^2*b^2*3i + A*a*b^3*2i + A*a^3*b*4i + (C*a*b^3*3i)/2 + C*a^3*b*2i)*(32*A*a^4 + 32*B*a^4 + 12*B*b ^4 + 96*B*a^2*b^2 + 64*A*a*b^3 + 128*A*a^3*b + 48*C*a*b^3 + 64*C*a^3*b))*( B*a^4*1i + (B*b^4*3i)/8 + B*a^2*b^2*3i + A*a*b^3*2i + A*a^3*b*4i + (C*a*b^ 3*3i)/2 + C*a^3*b*2i)*1i)/((tan(c/2 + (d*x)/2)*(32*A^2*a^8 + 32*B^2*a^8...
Time = 0.19 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.11 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {-120 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a \,b^{3} c -30 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b^{5}+240 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{3} b c +600 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2} b^{3}+300 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a \,b^{3} c +75 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b^{5}-120 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a^{5}+120 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a^{5}+24 \sin \left (d x +c \right )^{5} b^{4} c -240 \sin \left (d x +c \right )^{3} a^{2} b^{2} c -200 \sin \left (d x +c \right )^{3} a \,b^{4}-80 \sin \left (d x +c \right )^{3} b^{4} c +120 \sin \left (d x +c \right ) a^{4} c +1200 \sin \left (d x +c \right ) a^{3} b^{2}+720 \sin \left (d x +c \right ) a^{2} b^{2} c +600 \sin \left (d x +c \right ) a \,b^{4}+120 \sin \left (d x +c \right ) b^{4} c +600 a^{4} b d x +240 a^{3} b c d x +600 a^{2} b^{3} d x +180 a \,b^{3} c d x +45 b^{5} d x}{120 d} \] 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),x)
Output:
( - 120*cos(c + d*x)*sin(c + d*x)**3*a*b**3*c - 30*cos(c + d*x)*sin(c + d* x)**3*b**5 + 240*cos(c + d*x)*sin(c + d*x)*a**3*b*c + 600*cos(c + d*x)*sin (c + d*x)*a**2*b**3 + 300*cos(c + d*x)*sin(c + d*x)*a*b**3*c + 75*cos(c + d*x)*sin(c + d*x)*b**5 - 120*log(tan((c + d*x)/2) - 1)*a**5 + 120*log(tan( (c + d*x)/2) + 1)*a**5 + 24*sin(c + d*x)**5*b**4*c - 240*sin(c + d*x)**3*a **2*b**2*c - 200*sin(c + d*x)**3*a*b**4 - 80*sin(c + d*x)**3*b**4*c + 120* sin(c + d*x)*a**4*c + 1200*sin(c + d*x)*a**3*b**2 + 720*sin(c + d*x)*a**2* b**2*c + 600*sin(c + d*x)*a*b**4 + 120*sin(c + d*x)*b**4*c + 600*a**4*b*d* x + 240*a**3*b*c*d*x + 600*a**2*b**3*d*x + 180*a*b**3*c*d*x + 45*b**5*d*x) /(120*d)