Integrand size = 41, antiderivative size = 274 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {1}{2} b \left (12 a^2 b B+b^3 B+8 a^3 C+4 a b^2 (2 A+C)\right ) x+\frac {a^2 \left (12 A b^2+8 a b B+a^2 (A+2 C)\right ) \text {arctanh}(\sin (c+d x))}{2 d}-\frac {b \left (12 a^3 B-24 a b^2 B+a^2 b (39 A-34 C)-2 b^3 (3 A+2 C)\right ) \sin (c+d x)}{6 d}-\frac {b^2 \left (6 a^2 B-3 b^2 B+2 a b (9 A-4 C)\right ) \cos (c+d x) \sin (c+d x)}{6 d}-\frac {b (15 A b+6 a B-2 b C) (a+b \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(2 A b+a B) (a+b \cos (c+d x))^3 \tan (c+d x)}{d}+\frac {A (a+b \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d} \] Output:
1/2*b*(12*B*a^2*b+B*b^3+8*a^3*C+4*a*b^2*(2*A+C))*x+1/2*a^2*(12*A*b^2+8*B*a *b+a^2*(A+2*C))*arctanh(sin(d*x+c))/d-1/6*b*(12*B*a^3-24*B*a*b^2+a^2*b*(39 *A-34*C)-2*b^3*(3*A+2*C))*sin(d*x+c)/d-1/6*b^2*(6*B*a^2-3*B*b^2+2*a*b*(9*A -4*C))*cos(d*x+c)*sin(d*x+c)/d-1/6*b*(15*A*b+6*B*a-2*C*b)*(a+b*cos(d*x+c)) ^2*sin(d*x+c)/d+(2*A*b+B*a)*(a+b*cos(d*x+c))^3*tan(d*x+c)/d+1/2*A*(a+b*cos (d*x+c))^4*sec(d*x+c)*tan(d*x+c)/d
Time = 9.07 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.34 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {6 b \left (12 a^2 b B+b^3 B+8 a^3 C+4 a b^2 (2 A+C)\right ) (c+d x)-6 a^2 \left (12 A b^2+8 a b B+a^2 (A+2 C)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+6 a^2 \left (12 A b^2+8 a b B+a^2 (A+2 C)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {3 a^4 A}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {12 a^3 (4 A b+a B) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}-\frac {3 a^4 A}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {12 a^3 (4 A b+a B) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}+3 b^2 \left (4 A b^2+16 a b B+24 a^2 C+3 b^2 C\right ) \sin (c+d x)+3 b^3 (b B+4 a C) \sin (2 (c+d x))+b^4 C \sin (3 (c+d x))}{12 d} \] 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]^3,x]
Output:
(6*b*(12*a^2*b*B + b^3*B + 8*a^3*C + 4*a*b^2*(2*A + C))*(c + d*x) - 6*a^2* (12*A*b^2 + 8*a*b*B + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/ 2]] + 6*a^2*(12*A*b^2 + 8*a*b*B + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] + Si n[(c + d*x)/2]] + (3*a^4*A)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (12* a^3*(4*A*b + a*B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (3*a^4*A)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (12*a^3*(4*A*b + a*B )*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 3*b^2*(4*A*b^2 + 16*a*b*B + 24*a^2*C + 3*b^2*C)*Sin[c + d*x] + 3*b^3*(b*B + 4*a*C)*Sin[2 *(c + d*x)] + b^4*C*Sin[3*(c + d*x)])/(12*d)
Time = 2.13 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.01, 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, 3528, 3042, 3512, 27, 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.
| \(\displaystyle \int \sec ^3(c+d x) (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{2} \int (a+b \cos (c+d x))^3 \left (-b (3 A-2 C) \cos ^2(c+d x)+(2 b B+a (A+2 C)) \cos (c+d x)+2 (2 A b+a B)\right ) \sec ^2(c+d x)dx+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^4}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (-b (3 A-2 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(2 b B+a (A+2 C)) \sin \left (c+d x+\frac {\pi }{2}\right )+2 (2 A b+a B)\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^4}{2 d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{2} \left (\int (a+b \cos (c+d x))^2 \left ((A+2 C) a^2+8 b B a+12 A b^2-b (15 A b-2 C b+6 a B) \cos ^2(c+d x)+2 b (b B-a (A-2 C)) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {2 (a B+2 A b) \tan (c+d x) (a+b \cos (c+d x))^3}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^4}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (\int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left ((A+2 C) a^2+8 b B a+12 A b^2-b (15 A b-2 C b+6 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 b (b B-a (A-2 C)) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 (a B+2 A b) \tan (c+d x) (a+b \cos (c+d x))^3}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^4}{2 d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \int (a+b \cos (c+d x)) \left (-2 b \left (6 B a^2+2 b (9 A-4 C) a-3 b^2 B\right ) \cos ^2(c+d x)+b \left (-3 (A-6 C) a^2+18 b B a+2 b^2 (3 A+2 C)\right ) \cos (c+d x)+3 a \left ((A+2 C) a^2+8 b B a+12 A b^2\right )\right ) \sec (c+d x)dx-\frac {b \sin (c+d x) (6 a B+15 A b-2 b C) (a+b \cos (c+d x))^2}{3 d}+\frac {2 (a B+2 A b) \tan (c+d x) (a+b \cos (c+d x))^3}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^4}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (-2 b \left (6 B a^2+2 b (9 A-4 C) a-3 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b \left (-3 (A-6 C) a^2+18 b B a+2 b^2 (3 A+2 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 a \left ((A+2 C) a^2+8 b B a+12 A b^2\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {b \sin (c+d x) (6 a B+15 A b-2 b C) (a+b \cos (c+d x))^2}{3 d}+\frac {2 (a B+2 A b) \tan (c+d x) (a+b \cos (c+d x))^3}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^4}{2 d}\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{2} \int 2 \left (3 \left ((A+2 C) a^2+8 b B a+12 A b^2\right ) a^2-b \left (12 B a^3+b (39 A-34 C) a^2-24 b^2 B a-2 b^3 (3 A+2 C)\right ) \cos ^2(c+d x)+3 b \left (8 C a^3+12 b B a^2+4 b^2 (2 A+C) a+b^3 B\right ) \cos (c+d x)\right ) \sec (c+d x)dx-\frac {b^2 \sin (c+d x) \cos (c+d x) \left (6 a^2 B+2 a b (9 A-4 C)-3 b^2 B\right )}{d}\right )-\frac {b \sin (c+d x) (6 a B+15 A b-2 b C) (a+b \cos (c+d x))^2}{3 d}+\frac {2 (a B+2 A b) \tan (c+d x) (a+b \cos (c+d x))^3}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^4}{2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\int \left (3 \left ((A+2 C) a^2+8 b B a+12 A b^2\right ) a^2-b \left (12 B a^3+b (39 A-34 C) a^2-24 b^2 B a-2 b^3 (3 A+2 C)\right ) \cos ^2(c+d x)+3 b \left (8 C a^3+12 b B a^2+4 b^2 (2 A+C) a+b^3 B\right ) \cos (c+d x)\right ) \sec (c+d x)dx-\frac {b^2 \sin (c+d x) \cos (c+d x) \left (6 a^2 B+2 a b (9 A-4 C)-3 b^2 B\right )}{d}\right )-\frac {b \sin (c+d x) (6 a B+15 A b-2 b C) (a+b \cos (c+d x))^2}{3 d}+\frac {2 (a B+2 A b) \tan (c+d x) (a+b \cos (c+d x))^3}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^4}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\int \frac {3 \left ((A+2 C) a^2+8 b B a+12 A b^2\right ) a^2-b \left (12 B a^3+b (39 A-34 C) a^2-24 b^2 B a-2 b^3 (3 A+2 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+3 b \left (8 C a^3+12 b B a^2+4 b^2 (2 A+C) a+b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {b^2 \sin (c+d x) \cos (c+d x) \left (6 a^2 B+2 a b (9 A-4 C)-3 b^2 B\right )}{d}\right )-\frac {b \sin (c+d x) (6 a B+15 A b-2 b C) (a+b \cos (c+d x))^2}{3 d}+\frac {2 (a B+2 A b) \tan (c+d x) (a+b \cos (c+d x))^3}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^4}{2 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\int 3 \left (\left ((A+2 C) a^2+8 b B a+12 A b^2\right ) a^2+b \left (8 C a^3+12 b B a^2+4 b^2 (2 A+C) a+b^3 B\right ) \cos (c+d x)\right ) \sec (c+d x)dx-\frac {b^2 \sin (c+d x) \cos (c+d x) \left (6 a^2 B+2 a b (9 A-4 C)-3 b^2 B\right )}{d}-\frac {b \sin (c+d x) \left (12 a^3 B+a^2 b (39 A-34 C)-24 a b^2 B-2 b^3 (3 A+2 C)\right )}{d}\right )-\frac {b \sin (c+d x) (6 a B+15 A b-2 b C) (a+b \cos (c+d x))^2}{3 d}+\frac {2 (a B+2 A b) \tan (c+d x) (a+b \cos (c+d x))^3}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^4}{2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (3 \int \left (\left ((A+2 C) a^2+8 b B a+12 A b^2\right ) a^2+b \left (8 C a^3+12 b B a^2+4 b^2 (2 A+C) a+b^3 B\right ) \cos (c+d x)\right ) \sec (c+d x)dx-\frac {b^2 \sin (c+d x) \cos (c+d x) \left (6 a^2 B+2 a b (9 A-4 C)-3 b^2 B\right )}{d}-\frac {b \sin (c+d x) \left (12 a^3 B+a^2 b (39 A-34 C)-24 a b^2 B-2 b^3 (3 A+2 C)\right )}{d}\right )-\frac {b \sin (c+d x) (6 a B+15 A b-2 b C) (a+b \cos (c+d x))^2}{3 d}+\frac {2 (a B+2 A b) \tan (c+d x) (a+b \cos (c+d x))^3}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^4}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (3 \int \frac {\left ((A+2 C) a^2+8 b B a+12 A b^2\right ) a^2+b \left (8 C a^3+12 b B a^2+4 b^2 (2 A+C) a+b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {b^2 \sin (c+d x) \cos (c+d x) \left (6 a^2 B+2 a b (9 A-4 C)-3 b^2 B\right )}{d}-\frac {b \sin (c+d x) \left (12 a^3 B+a^2 b (39 A-34 C)-24 a b^2 B-2 b^3 (3 A+2 C)\right )}{d}\right )-\frac {b \sin (c+d x) (6 a B+15 A b-2 b C) (a+b \cos (c+d x))^2}{3 d}+\frac {2 (a B+2 A b) \tan (c+d x) (a+b \cos (c+d x))^3}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^4}{2 d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (3 \left (a^2 \left (a^2 (A+2 C)+8 a b B+12 A b^2\right ) \int \sec (c+d x)dx+b x \left (8 a^3 C+12 a^2 b B+4 a b^2 (2 A+C)+b^3 B\right )\right )-\frac {b^2 \sin (c+d x) \cos (c+d x) \left (6 a^2 B+2 a b (9 A-4 C)-3 b^2 B\right )}{d}-\frac {b \sin (c+d x) \left (12 a^3 B+a^2 b (39 A-34 C)-24 a b^2 B-2 b^3 (3 A+2 C)\right )}{d}\right )-\frac {b \sin (c+d x) (6 a B+15 A b-2 b C) (a+b \cos (c+d x))^2}{3 d}+\frac {2 (a B+2 A b) \tan (c+d x) (a+b \cos (c+d x))^3}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^4}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (3 \left (a^2 \left (a^2 (A+2 C)+8 a b B+12 A b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+b x \left (8 a^3 C+12 a^2 b B+4 a b^2 (2 A+C)+b^3 B\right )\right )-\frac {b^2 \sin (c+d x) \cos (c+d x) \left (6 a^2 B+2 a b (9 A-4 C)-3 b^2 B\right )}{d}-\frac {b \sin (c+d x) \left (12 a^3 B+a^2 b (39 A-34 C)-24 a b^2 B-2 b^3 (3 A+2 C)\right )}{d}\right )-\frac {b \sin (c+d x) (6 a B+15 A b-2 b C) (a+b \cos (c+d x))^2}{3 d}+\frac {2 (a B+2 A b) \tan (c+d x) (a+b \cos (c+d x))^3}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^4}{2 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (-\frac {b^2 \sin (c+d x) \cos (c+d x) \left (6 a^2 B+2 a b (9 A-4 C)-3 b^2 B\right )}{d}+3 \left (\frac {a^2 \left (a^2 (A+2 C)+8 a b B+12 A b^2\right ) \text {arctanh}(\sin (c+d x))}{d}+b x \left (8 a^3 C+12 a^2 b B+4 a b^2 (2 A+C)+b^3 B\right )\right )-\frac {b \sin (c+d x) \left (12 a^3 B+a^2 b (39 A-34 C)-24 a b^2 B-2 b^3 (3 A+2 C)\right )}{d}\right )-\frac {b \sin (c+d x) (6 a B+15 A b-2 b C) (a+b \cos (c+d x))^2}{3 d}+\frac {2 (a B+2 A b) \tan (c+d x) (a+b \cos (c+d x))^3}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^4}{2 d}\) |
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]^3,x]
Output:
(A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (-1/3*(b*(15* A*b + 6*a*B - 2*b*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/d + (3*(b*(12*a^ 2*b*B + b^3*B + 8*a^3*C + 4*a*b^2*(2*A + C))*x + (a^2*(12*A*b^2 + 8*a*b*B + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/d) - (b*(12*a^3*B - 24*a*b^2*B + a ^2*b*(39*A - 34*C) - 2*b^3*(3*A + 2*C))*Sin[c + d*x])/d - (b^2*(6*a^2*B - 3*b^2*B + 2*a*b*(9*A - 4*C))*Cos[c + d*x]*Sin[c + d*x])/d)/3 + (2*(2*A*b + a*B)*(a + b*Cos[c + d*x])^3*Tan[c + d*x])/d)/2
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^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[((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 = 12.97 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.85
| method | result | size |
| parts | \(\frac {A \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (B \,b^{4}+4 C a \,b^{3}\right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (A \,b^{4}+4 B a \,b^{3}+6 C \,a^{2} b^{2}\right ) \sin \left (d x +c \right )}{d}+\frac {\left (4 a A \,b^{3}+6 B \,a^{2} b^{2}+4 a^{3} b C \right ) \left (d x +c \right )}{d}+\frac {\left (6 A \,a^{2} b^{2}+4 B \,a^{3} b +a^{4} C \right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {C \,b^{4} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3 d}\) | \(233\) |
| derivativedivides | \(\frac {A \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+B \,a^{4} \tan \left (d x +c \right )+a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 A \,a^{3} b \tan \left (d x +c \right )+4 B \,a^{3} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a^{3} b C \left (d x +c \right )+6 A \,a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 B \,a^{2} b^{2} \left (d x +c \right )+6 C \sin \left (d x +c \right ) a^{2} b^{2}+4 a A \,b^{3} \left (d x +c \right )+4 B \sin \left (d x +c \right ) a \,b^{3}+4 C a \,b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+A \sin \left (d x +c \right ) b^{4}+B \,b^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {C \,b^{4} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(284\) |
| default | \(\frac {A \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+B \,a^{4} \tan \left (d x +c \right )+a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 A \,a^{3} b \tan \left (d x +c \right )+4 B \,a^{3} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a^{3} b C \left (d x +c \right )+6 A \,a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 B \,a^{2} b^{2} \left (d x +c \right )+6 C \sin \left (d x +c \right ) a^{2} b^{2}+4 a A \,b^{3} \left (d x +c \right )+4 B \sin \left (d x +c \right ) a \,b^{3}+4 C a \,b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+A \sin \left (d x +c \right ) b^{4}+B \,b^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {C \,b^{4} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(284\) |
| parallelrisch | \(\frac {-12 a^{2} \left (1+\cos \left (2 d x +2 c \right )\right ) \left (12 A \,b^{2}+8 B a b +a^{2} \left (A +2 C \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+12 a^{2} \left (1+\cos \left (2 d x +2 c \right )\right ) \left (12 A \,b^{2}+8 B a b +a^{2} \left (A +2 C \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+96 b x \left (a^{3} C +\frac {3 B \,a^{2} b}{2}+a \left (A +\frac {C}{2}\right ) b^{2}+\frac {B \,b^{3}}{8}\right ) d \cos \left (2 d x +2 c \right )+6 \left (16 A \,a^{3} b +4 B \,a^{4}+B \,b^{4}+4 C a \,b^{3}\right ) \sin \left (2 d x +2 c \right )+12 \left (6 a^{2} C +4 B a b +b^{2} \left (A +\frac {11 C}{12}\right )\right ) b^{2} \sin \left (3 d x +3 c \right )+3 \left (B \,b^{4}+4 C a \,b^{3}\right ) \sin \left (4 d x +4 c \right )+C \sin \left (5 d x +5 c \right ) b^{4}+12 \left (2 A \,a^{4}+6 C \,a^{2} b^{2}+4 B a \,b^{3}+b^{4} \left (A +\frac {5 C}{6}\right )\right ) \sin \left (d x +c \right )+96 b x \left (a^{3} C +\frac {3 B \,a^{2} b}{2}+a \left (A +\frac {C}{2}\right ) b^{2}+\frac {B \,b^{3}}{8}\right ) d}{24 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(345\) |
| risch | \(-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} C a \,b^{3}}{2 d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} C a \,b^{3}}{2 d}-\frac {i C \,b^{4} {\mathrm e}^{3 i \left (d x +c \right )}}{24 d}-\frac {i a^{3} \left (A a \,{\mathrm e}^{3 i \left (d x +c \right )}-8 A b \,{\mathrm e}^{2 i \left (d x +c \right )}-2 B a \,{\mathrm e}^{2 i \left (d x +c \right )}-a A \,{\mathrm e}^{i \left (d x +c \right )}-8 A b -2 B a \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} B \,b^{4}}{8 d}+\frac {i C \,b^{4} {\mathrm e}^{-3 i \left (d x +c \right )}}{24 d}-\frac {6 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,b^{2}}{d}-\frac {4 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B b}{d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} B \,b^{4}}{8 d}+\frac {6 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,b^{2}}{d}+\frac {4 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B b}{d}+2 a \,b^{3} C x +4 x a A \,b^{3}+6 x B \,a^{2} b^{2}+4 x \,a^{3} b C +\frac {A \,a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}-\frac {A \,a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}+\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}-\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}+\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} B a \,b^{3}}{d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} C \,a^{2} b^{2}}{d}-\frac {2 i {\mathrm e}^{i \left (d x +c \right )} B a \,b^{3}}{d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} C \,a^{2} b^{2}}{d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} A \,b^{4}}{2 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} C \,b^{4}}{8 d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} A \,b^{4}}{2 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} C \,b^{4}}{8 d}+\frac {x B \,b^{4}}{2}\) | \(584\) |
| norman | \(\text {Expression too large to display}\) | \(1168\) |
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)^3,x,meth od=_RETURNVERBOSE)
Output:
A*a^4/d*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))+(4*A*a^3 *b+B*a^4)/d*tan(d*x+c)+(B*b^4+4*C*a*b^3)/d*(1/2*cos(d*x+c)*sin(d*x+c)+1/2* d*x+1/2*c)+(A*b^4+4*B*a*b^3+6*C*a^2*b^2)/d*sin(d*x+c)+(4*A*a*b^3+6*B*a^2*b ^2+4*C*a^3*b)/d*(d*x+c)+(6*A*a^2*b^2+4*B*a^3*b+C*a^4)/d*ln(sec(d*x+c)+tan( d*x+c))+1/3*C*b^4/d*(2+cos(d*x+c)^2)*sin(d*x+c)
Time = 0.11 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.96 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {6 \, {\left (8 \, C a^{3} b + 12 \, B a^{2} b^{2} + 4 \, {\left (2 \, A + C\right )} a b^{3} + B b^{4}\right )} d x \cos \left (d x + c\right )^{2} + 3 \, {\left ({\left (A + 2 \, C\right )} a^{4} + 8 \, B a^{3} b + 12 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left ({\left (A + 2 \, C\right )} a^{4} + 8 \, B a^{3} b + 12 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, C b^{4} \cos \left (d x + c\right )^{4} + 3 \, A a^{4} + 3 \, {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (18 \, C a^{2} b^{2} + 12 \, B a b^{3} + {\left (3 \, A + 2 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 6 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{2}} \] 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)^3, x, algorithm="fricas")
Output:
1/12*(6*(8*C*a^3*b + 12*B*a^2*b^2 + 4*(2*A + C)*a*b^3 + B*b^4)*d*x*cos(d*x + c)^2 + 3*((A + 2*C)*a^4 + 8*B*a^3*b + 12*A*a^2*b^2)*cos(d*x + c)^2*log( sin(d*x + c) + 1) - 3*((A + 2*C)*a^4 + 8*B*a^3*b + 12*A*a^2*b^2)*cos(d*x + c)^2*log(-sin(d*x + c) + 1) + 2*(2*C*b^4*cos(d*x + c)^4 + 3*A*a^4 + 3*(4* C*a*b^3 + B*b^4)*cos(d*x + c)^3 + 2*(18*C*a^2*b^2 + 12*B*a*b^3 + (3*A + 2* C)*b^4)*cos(d*x + c)^2 + 6*(B*a^4 + 4*A*a^3*b)*cos(d*x + c))*sin(d*x + c)) /(d*cos(d*x + c)^2)
Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\text {Timed out} \] 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)* *3,x)
Output:
Timed out
Time = 0.05 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.14 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {48 \, {\left (d x + c\right )} C a^{3} b + 72 \, {\left (d x + c\right )} B a^{2} b^{2} + 48 \, {\left (d x + c\right )} A a b^{3} + 12 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a b^{3} + 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B b^{4} - 4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C b^{4} - 3 \, A a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, C a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, B a^{3} b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 36 \, A a^{2} b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 72 \, C a^{2} b^{2} \sin \left (d x + c\right ) + 48 \, B a b^{3} \sin \left (d x + c\right ) + 12 \, A b^{4} \sin \left (d x + c\right ) + 12 \, B a^{4} \tan \left (d x + c\right ) + 48 \, A a^{3} b \tan \left (d x + c\right )}{12 \, d} \] 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)^3, x, algorithm="maxima")
Output:
1/12*(48*(d*x + c)*C*a^3*b + 72*(d*x + c)*B*a^2*b^2 + 48*(d*x + c)*A*a*b^3 + 12*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a*b^3 + 3*(2*d*x + 2*c + sin(2*d* x + 2*c))*B*b^4 - 4*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*b^4 - 3*A*a^4*(2*s in(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c ) - 1)) + 6*C*a^4*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 24*B*a ^3*b*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 36*A*a^2*b^2*(log(s in(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 72*C*a^2*b^2*sin(d*x + c) + 48 *B*a*b^3*sin(d*x + c) + 12*A*b^4*sin(d*x + c) + 12*B*a^4*tan(d*x + c) + 48 *A*a^3*b*tan(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 541 vs. \(2 (262) = 524\).
Time = 0.21 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.97 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx =\text {Too large to display} \] 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)^3, x, algorithm="giac")
Output:
1/6*(3*(8*C*a^3*b + 12*B*a^2*b^2 + 8*A*a*b^3 + 4*C*a*b^3 + B*b^4)*(d*x + c ) + 3*(A*a^4 + 2*C*a^4 + 8*B*a^3*b + 12*A*a^2*b^2)*log(abs(tan(1/2*d*x + 1 /2*c) + 1)) - 3*(A*a^4 + 2*C*a^4 + 8*B*a^3*b + 12*A*a^2*b^2)*log(abs(tan(1 /2*d*x + 1/2*c) - 1)) + 6*(A*a^4*tan(1/2*d*x + 1/2*c)^3 - 2*B*a^4*tan(1/2* d*x + 1/2*c)^3 - 8*A*a^3*b*tan(1/2*d*x + 1/2*c)^3 + A*a^4*tan(1/2*d*x + 1/ 2*c) + 2*B*a^4*tan(1/2*d*x + 1/2*c) + 8*A*a^3*b*tan(1/2*d*x + 1/2*c))/(tan (1/2*d*x + 1/2*c)^2 - 1)^2 + 2*(36*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 24*B *a*b^3*tan(1/2*d*x + 1/2*c)^5 - 12*C*a*b^3*tan(1/2*d*x + 1/2*c)^5 + 6*A*b^ 4*tan(1/2*d*x + 1/2*c)^5 - 3*B*b^4*tan(1/2*d*x + 1/2*c)^5 + 6*C*b^4*tan(1/ 2*d*x + 1/2*c)^5 + 72*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 + 48*B*a*b^3*tan(1/ 2*d*x + 1/2*c)^3 + 12*A*b^4*tan(1/2*d*x + 1/2*c)^3 + 4*C*b^4*tan(1/2*d*x + 1/2*c)^3 + 36*C*a^2*b^2*tan(1/2*d*x + 1/2*c) + 24*B*a*b^3*tan(1/2*d*x + 1 /2*c) + 12*C*a*b^3*tan(1/2*d*x + 1/2*c) + 6*A*b^4*tan(1/2*d*x + 1/2*c) + 3 *B*b^4*tan(1/2*d*x + 1/2*c) + 6*C*b^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^3)/d
Time = 2.95 (sec) , antiderivative size = 4837, normalized size of antiderivative = 17.65 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\text {Too large to display} \] 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)^3,x)
Output:
(tan(c/2 + (d*x)/2)^3*(4*A*a^4 + 4*B*a^4 - 2*B*b^4 - (8*C*b^4)/3 + 16*A*a^ 3*b - 8*C*a*b^3) + tan(c/2 + (d*x)/2)^7*(4*A*a^4 - 4*B*a^4 + 2*B*b^4 - (8* C*b^4)/3 - 16*A*a^3*b + 8*C*a*b^3) + tan(c/2 + (d*x)/2)^9*(A*a^4 + 2*A*b^4 - 2*B*a^4 - B*b^4 + 2*C*b^4 + 12*C*a^2*b^2 - 8*A*a^3*b + 8*B*a*b^3 - 4*C* a*b^3) + tan(c/2 + (d*x)/2)*(A*a^4 + 2*A*b^4 + 2*B*a^4 + B*b^4 + 2*C*b^4 + 12*C*a^2*b^2 + 8*A*a^3*b + 8*B*a*b^3 + 4*C*a*b^3) - tan(c/2 + (d*x)/2)^5* (4*A*b^4 - 6*A*a^4 - (4*C*b^4)/3 + 24*C*a^2*b^2 + 16*B*a*b^3))/(d*(tan(c/2 + (d*x)/2)^2 - 2*tan(c/2 + (d*x)/2)^4 - 2*tan(c/2 + (d*x)/2)^6 + tan(c/2 + (d*x)/2)^8 + tan(c/2 + (d*x)/2)^10 + 1)) - (atan(((((A*a^4)/2 + C*a^4 + 6*A*a^2*b^2 + 4*B*a^3*b)*(16*A*a^4 + 16*B*b^4 + 32*C*a^4 + 192*A*a^2*b^2 + 192*B*a^2*b^2 + 128*A*a*b^3 + 128*B*a^3*b + 64*C*a*b^3 + 128*C*a^3*b) + t an(c/2 + (d*x)/2)*(8*A^2*a^8 + 8*B^2*b^8 + 32*C^2*a^8 + 512*A^2*a^2*b^6 + 1152*A^2*a^4*b^4 + 192*A^2*a^6*b^2 + 192*B^2*a^2*b^6 + 1152*B^2*a^4*b^4 + 512*B^2*a^6*b^2 + 128*C^2*a^2*b^6 + 512*C^2*a^4*b^4 + 512*C^2*a^6*b^2 + 32 *A*C*a^8 + 128*A*B*a*b^7 + 128*A*B*a^7*b + 64*B*C*a*b^7 + 256*B*C*a^7*b + 1536*A*B*a^3*b^5 + 1536*A*B*a^5*b^3 + 512*A*C*a^2*b^6 + 1024*A*C*a^4*b^4 + 384*A*C*a^6*b^2 + 896*B*C*a^3*b^5 + 1536*B*C*a^5*b^3))*((A*a^4)/2 + C*a^4 + 6*A*a^2*b^2 + 4*B*a^3*b)*1i - (((A*a^4)/2 + C*a^4 + 6*A*a^2*b^2 + 4*B*a ^3*b)*(16*A*a^4 + 16*B*b^4 + 32*C*a^4 + 192*A*a^2*b^2 + 192*B*a^2*b^2 + 12 8*A*a*b^3 + 128*B*a^3*b + 64*C*a*b^3 + 128*C*a^3*b) - tan(c/2 + (d*x)/2...
Time = 0.18 (sec) , antiderivative size = 783, normalized size of antiderivative = 2.86 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx =\text {Too large to display} \] 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)^3,x)
Output:
(12*cos(c + d*x)**2*sin(c + d*x)**3*a*b**3*c + 3*cos(c + d*x)**2*sin(c + d *x)**3*b**5 - 12*cos(c + d*x)**2*sin(c + d*x)*a*b**3*c - 3*cos(c + d*x)**2 *sin(c + d*x)*b**5 - 3*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x) **2*a**5 - 6*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a**4*c - 60*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a**3*b**2 + 3 *cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a**5 + 6*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a**4*c + 60*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a**3*b**2 + 3*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*a**5 + 6*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*a**4*c + 60*cos(c + d*x) *log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*a**3*b**2 - 3*cos(c + d*x)*log( tan((c + d*x)/2) + 1)*a**5 - 6*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*a**4 *c - 60*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*a**3*b**2 - 2*cos(c + d*x)* sin(c + d*x)**5*b**4*c + 36*cos(c + d*x)*sin(c + d*x)**3*a**2*b**2*c + 30* cos(c + d*x)*sin(c + d*x)**3*a*b**4 + 8*cos(c + d*x)*sin(c + d*x)**3*b**4* c + 24*cos(c + d*x)*sin(c + d*x)**2*a**3*b*c*d*x + 60*cos(c + d*x)*sin(c + d*x)**2*a**2*b**3*d*x + 12*cos(c + d*x)*sin(c + d*x)**2*a*b**3*c*d*x + 3* cos(c + d*x)*sin(c + d*x)**2*b**5*d*x - 3*cos(c + d*x)*sin(c + d*x)*a**5 - 36*cos(c + d*x)*sin(c + d*x)*a**2*b**2*c - 30*cos(c + d*x)*sin(c + d*x)*a *b**4 - 6*cos(c + d*x)*sin(c + d*x)*b**4*c - 24*cos(c + d*x)*a**3*b*c*d*x - 60*cos(c + d*x)*a**2*b**3*d*x - 12*cos(c + d*x)*a*b**3*c*d*x - 3*cos(...