Integrand size = 39, antiderivative size = 345 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+b \cos (c+d x))^4} \, dx=-\frac {\left (7 a^2 A b^5-2 A b^7-2 a^7 B-3 a^5 b^2 B-a^4 b^3 (8 A-C)+4 a^6 b (2 A+C)\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 (a-b)^{7/2} (a+b)^{7/2} d}+\frac {A \text {arctanh}(\sin (c+d x))}{a^4 d}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac {\left (3 A b^4+5 a^3 b B-2 a^4 C-a^2 b^2 (8 A+3 C)\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac {\left (17 a^2 A b^4-6 A b^6+11 a^5 b B+4 a^3 b^3 B-2 a^6 C-13 a^4 b^2 (2 A+C)\right ) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))} \] Output:
-(7*a^2*A*b^5-2*A*b^7-2*a^7*B-3*a^5*b^2*B-a^4*b^3*(8*A-C)+4*a^6*b*(2*A+C)) *arctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^4/(a-b)^(7/2)/(a+b)^ (7/2)/d+A*arctanh(sin(d*x+c))/a^4/d+1/3*(A*b^2-a*(B*b-C*a))*sin(d*x+c)/a/( a^2-b^2)/d/(a+b*cos(d*x+c))^3-1/6*(3*A*b^4+5*B*a^3*b-2*a^4*C-a^2*b^2*(8*A+ 3*C))*sin(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c))^2-1/6*(17*a^2*A*b^4-6* A*b^6+11*B*a^5*b+4*B*a^3*b^3-2*a^6*C-13*a^4*b^2*(2*A+C))*sin(d*x+c)/a^3/(a ^2-b^2)^3/d/(a+b*cos(d*x+c))
Result contains complex when optimal does not.
Time = 8.72 (sec) , antiderivative size = 587, normalized size of antiderivative = 1.70 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+b \cos (c+d x))^4} \, dx=\frac {\cos (c+d x) (B+C \cos (c+d x)+A \sec (c+d x)) \left (-\frac {6 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{a^4}+\frac {6 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{a^4}-\frac {6 i \left (-7 a^2 A b^5+2 A b^7+2 a^7 B+3 a^5 b^2 B+a^4 b^3 (8 A-C)-4 a^6 b (2 A+C)\right ) \arctan \left (\frac {(i \cos (c)+\sin (c)) \left (b \sin (c)+(-a+b \cos (c)) \tan \left (\frac {d x}{2}\right )\right )}{\sqrt {-\left (\left (a^2-b^2\right ) (\cos (c)-i \sin (c))^2\right )}}\right ) (\cos (c)-i \sin (c))}{a^4 \left (a^2-b^2\right )^3 \sqrt {-\left (\left (a^2-b^2\right ) (\cos (c)-i \sin (c))^2\right )}}+\frac {2 \left (A b^2+a (-b B+a C)\right ) \sec (c) (-a \sin (c)+b \sin (d x))}{a b \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}+\frac {\left (-17 a^2 A b^4+6 A b^6-11 a^5 b B-4 a^3 b^3 B+2 a^6 C+13 a^4 b^2 (2 A+C)\right ) \sec (c) \sin (d x)+3 a \left (-A b^5+2 a^5 B+3 a^3 b^2 B+a^2 b^3 (2 A-C)-2 a^4 b (3 A+2 C)\right ) \tan (c)}{\left (a^3-a b^2\right )^3 (a+b \cos (c+d x))}+\frac {\left (-3 A b^4-5 a^3 b B+2 a^4 C+a^2 b^2 (8 A+3 C)\right ) \sec (c) \sin (d x)+a \left (A b^3+3 a^3 B+2 a b^2 B-a^2 b (6 A+5 C)\right ) \tan (c)}{a^2 \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}\right )}{3 d (2 A+C+2 B \cos (c+d x)+C \cos (2 (c+d x)))} \] Input:
Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Co s[c + d*x])^4,x]
Output:
(Cos[c + d*x]*(B + C*Cos[c + d*x] + A*Sec[c + d*x])*((-6*A*Log[Cos[(c + d* x)/2] - Sin[(c + d*x)/2]])/a^4 + (6*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x) /2]])/a^4 - ((6*I)*(-7*a^2*A*b^5 + 2*A*b^7 + 2*a^7*B + 3*a^5*b^2*B + a^4*b ^3*(8*A - C) - 4*a^6*b*(2*A + C))*ArcTan[((I*Cos[c] + Sin[c])*(b*Sin[c] + (-a + b*Cos[c])*Tan[(d*x)/2]))/Sqrt[-((a^2 - b^2)*(Cos[c] - I*Sin[c])^2)]] *(Cos[c] - I*Sin[c]))/(a^4*(a^2 - b^2)^3*Sqrt[-((a^2 - b^2)*(Cos[c] - I*Si n[c])^2)]) + (2*(A*b^2 + a*(-(b*B) + a*C))*Sec[c]*(-(a*Sin[c]) + b*Sin[d*x ]))/(a*b*(a^2 - b^2)*(a + b*Cos[c + d*x])^3) + ((-17*a^2*A*b^4 + 6*A*b^6 - 11*a^5*b*B - 4*a^3*b^3*B + 2*a^6*C + 13*a^4*b^2*(2*A + C))*Sec[c]*Sin[d*x ] + 3*a*(-(A*b^5) + 2*a^5*B + 3*a^3*b^2*B + a^2*b^3*(2*A - C) - 2*a^4*b*(3 *A + 2*C))*Tan[c])/((a^3 - a*b^2)^3*(a + b*Cos[c + d*x])) + ((-3*A*b^4 - 5 *a^3*b*B + 2*a^4*C + a^2*b^2*(8*A + 3*C))*Sec[c]*Sin[d*x] + a*(A*b^3 + 3*a ^3*B + 2*a*b^2*B - a^2*b*(6*A + 5*C))*Tan[c])/(a^2*(a^2 - b^2)^2*(a + b*Co s[c + d*x])^2)))/(3*d*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*(c + d*x)]))
Time = 2.13 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.19, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3534, 3042, 3534, 3042, 3534, 27, 3042, 3480, 3042, 3138, 218, 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.
| \(\displaystyle \int \frac {\sec (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4}dx\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\int \frac {\left (2 \left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x)-3 a (A b+C b-a B) \cos (c+d x)+3 A \left (a^2-b^2\right )\right ) \sec (c+d x)}{(a+b \cos (c+d x))^3}dx}{3 a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {2 \left (A b^2-a (b B-a C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-3 a (A b+C b-a B) \sin \left (c+d x+\frac {\pi }{2}\right )+3 A \left (a^2-b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx}{3 a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\int \frac {\left (6 A \left (a^2-b^2\right )^2-\left (-2 C a^4+5 b B a^3-b^2 (8 A+3 C) a^2+3 A b^4\right ) \cos ^2(c+d x)+2 a \left (3 B a^3-b (6 A+5 C) a^2+2 b^2 B a+A b^3\right ) \cos (c+d x)\right ) \sec (c+d x)}{(a+b \cos (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (-2 a^4 C+5 a^3 b B-a^2 b^2 (8 A+3 C)+3 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {6 A \left (a^2-b^2\right )^2+\left (2 C a^4-5 b B a^3+b^2 (8 A+3 C) a^2-3 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 a \left (3 B a^3-b (6 A+5 C) a^2+2 b^2 B a+A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (-2 a^4 C+5 a^3 b B-a^2 b^2 (8 A+3 C)+3 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {3 \left (2 A \left (a^2-b^2\right )^3-a \left (-2 B a^5+(6 A b+4 C b) a^4-3 b^2 B a^3-b^3 (2 A-C) a^2+A b^5\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)}dx}{a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (-2 a^6 C+11 a^5 b B-13 a^4 b^2 (2 A+C)+4 a^3 b^3 B+17 a^2 A b^4-6 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (-2 a^4 C+5 a^3 b B-a^2 b^2 (8 A+3 C)+3 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {\left (2 A \left (a^2-b^2\right )^3-a \left (-2 B a^5+(6 A b+4 C b) a^4-3 b^2 B a^3-b^3 (2 A-C) a^2+A b^5\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)}dx}{a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (-2 a^6 C+11 a^5 b B-13 a^4 b^2 (2 A+C)+4 a^3 b^3 B+17 a^2 A b^4-6 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (-2 a^4 C+5 a^3 b B-a^2 b^2 (8 A+3 C)+3 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {2 A \left (a^2-b^2\right )^3-a \left (-2 B a^5+(6 A b+4 C b) a^4-3 b^2 B a^3-b^3 (2 A-C) a^2+A b^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (-2 a^6 C+11 a^5 b B-13 a^4 b^2 (2 A+C)+4 a^3 b^3 B+17 a^2 A b^4-6 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (-2 a^4 C+5 a^3 b B-a^2 b^2 (8 A+3 C)+3 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 A \left (a^2-b^2\right )^3 \int \sec (c+d x)dx}{a}-\frac {\left (-2 a^7 B+4 a^6 b (2 A+C)-3 a^5 b^2 B-a^4 b^3 (8 A-C)+7 a^2 A b^5-2 A b^7\right ) \int \frac {1}{a+b \cos (c+d x)}dx}{a}\right )}{a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (-2 a^6 C+11 a^5 b B-13 a^4 b^2 (2 A+C)+4 a^3 b^3 B+17 a^2 A b^4-6 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (-2 a^4 C+5 a^3 b B-a^2 b^2 (8 A+3 C)+3 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 A \left (a^2-b^2\right )^3 \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {\left (-2 a^7 B+4 a^6 b (2 A+C)-3 a^5 b^2 B-a^4 b^3 (8 A-C)+7 a^2 A b^5-2 A b^7\right ) \int \frac {1}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a}\right )}{a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (-2 a^6 C+11 a^5 b B-13 a^4 b^2 (2 A+C)+4 a^3 b^3 B+17 a^2 A b^4-6 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (-2 a^4 C+5 a^3 b B-a^2 b^2 (8 A+3 C)+3 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 A \left (a^2-b^2\right )^3 \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {2 \left (-2 a^7 B+4 a^6 b (2 A+C)-3 a^5 b^2 B-a^4 b^3 (8 A-C)+7 a^2 A b^5-2 A b^7\right ) \int \frac {1}{(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}\right )}{a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (-2 a^6 C+11 a^5 b B-13 a^4 b^2 (2 A+C)+4 a^3 b^3 B+17 a^2 A b^4-6 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (-2 a^4 C+5 a^3 b B-a^2 b^2 (8 A+3 C)+3 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 A \left (a^2-b^2\right )^3 \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {2 \left (-2 a^7 B+4 a^6 b (2 A+C)-3 a^5 b^2 B-a^4 b^3 (8 A-C)+7 a^2 A b^5-2 A b^7\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}\right )}{a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (-2 a^6 C+11 a^5 b B-13 a^4 b^2 (2 A+C)+4 a^3 b^3 B+17 a^2 A b^4-6 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (-2 a^4 C+5 a^3 b B-a^2 b^2 (8 A+3 C)+3 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}+\frac {\frac {\frac {3 \left (\frac {2 A \left (a^2-b^2\right )^3 \text {arctanh}(\sin (c+d x))}{a d}-\frac {2 \left (-2 a^7 B+4 a^6 b (2 A+C)-3 a^5 b^2 B-a^4 b^3 (8 A-C)+7 a^2 A b^5-2 A b^7\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}\right )}{a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (-2 a^6 C+11 a^5 b B-13 a^4 b^2 (2 A+C)+4 a^3 b^3 B+17 a^2 A b^4-6 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (-2 a^4 C+5 a^3 b B-a^2 b^2 (8 A+3 C)+3 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}\) |
Input:
Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^4,x]
Output:
((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d *x])^3) + (-1/2*((3*A*b^4 + 5*a^3*b*B - 2*a^4*C - a^2*b^2*(8*A + 3*C))*Sin [c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((3*((-2*(7*a^2*A*b^ 5 - 2*A*b^7 - 2*a^7*B - 3*a^5*b^2*B - a^4*b^3*(8*A - C) + 4*a^6*b*(2*A + C ))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt [a + b]*d) + (2*A*(a^2 - b^2)^3*ArcTanh[Sin[c + d*x]])/(a*d)))/(a*(a^2 - b ^2)) - ((17*a^2*A*b^4 - 6*A*b^6 + 11*a^5*b*B + 4*a^3*b^3*B - 2*a^6*C - 13* a^4*b^2*(2*A + C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])))/( 2*a*(a^2 - b^2)))/(3*a*(a^2 - b^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_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(A*b - a*B)/(b*c - a*d) Int[1/(a + b*Sin[e + f*x]), x], x] + Simp[(B*c - A*d)/ (b*c - a*d) Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
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[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*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] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 2.35 (sec) , antiderivative size = 580, normalized size of antiderivative = 1.68
| method | result | size |
| derivativedivides | \(\frac {\frac {A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{4}}-\frac {2 \left (\frac {-\frac {\left (12 A \,a^{4} b^{2}+4 A \,a^{3} b^{3}-6 a^{2} A \,b^{4}-A a \,b^{5}+2 A \,b^{6}-6 B \,a^{5} b -3 B \,a^{4} b^{2}-2 B \,a^{3} b^{3}+2 a^{6} C +2 C \,a^{5} b +6 a^{4} b^{2} C +C \,a^{3} b^{3}\right ) a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {2 \left (18 A \,a^{4} b^{2}-11 a^{2} A \,b^{4}+3 A \,b^{6}-9 B \,a^{5} b -B \,a^{3} b^{3}+3 a^{6} C +7 a^{4} b^{2} C \right ) a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (12 A \,a^{4} b^{2}-4 A \,a^{3} b^{3}-6 a^{2} A \,b^{4}+A a \,b^{5}+2 A \,b^{6}-6 B \,a^{5} b +3 B \,a^{4} b^{2}-2 B \,a^{3} b^{3}+2 a^{6} C -2 C \,a^{5} b +6 a^{4} b^{2} C -C \,a^{3} b^{3}\right ) a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{3}}+\frac {\left (8 A \,a^{6} b -8 A \,a^{4} b^{3}+7 a^{2} A \,b^{5}-2 A \,b^{7}-2 a^{7} B -3 a^{5} b^{2} B +4 C \,a^{6} b +C \,a^{4} b^{3}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{4}}-\frac {A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{4}}}{d}\) | \(580\) |
| default | \(\frac {\frac {A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{4}}-\frac {2 \left (\frac {-\frac {\left (12 A \,a^{4} b^{2}+4 A \,a^{3} b^{3}-6 a^{2} A \,b^{4}-A a \,b^{5}+2 A \,b^{6}-6 B \,a^{5} b -3 B \,a^{4} b^{2}-2 B \,a^{3} b^{3}+2 a^{6} C +2 C \,a^{5} b +6 a^{4} b^{2} C +C \,a^{3} b^{3}\right ) a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {2 \left (18 A \,a^{4} b^{2}-11 a^{2} A \,b^{4}+3 A \,b^{6}-9 B \,a^{5} b -B \,a^{3} b^{3}+3 a^{6} C +7 a^{4} b^{2} C \right ) a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (12 A \,a^{4} b^{2}-4 A \,a^{3} b^{3}-6 a^{2} A \,b^{4}+A a \,b^{5}+2 A \,b^{6}-6 B \,a^{5} b +3 B \,a^{4} b^{2}-2 B \,a^{3} b^{3}+2 a^{6} C -2 C \,a^{5} b +6 a^{4} b^{2} C -C \,a^{3} b^{3}\right ) a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{3}}+\frac {\left (8 A \,a^{6} b -8 A \,a^{4} b^{3}+7 a^{2} A \,b^{5}-2 A \,b^{7}-2 a^{7} B -3 a^{5} b^{2} B +4 C \,a^{6} b +C \,a^{4} b^{3}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{4}}-\frac {A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{4}}}{d}\) | \(580\) |
| risch | \(\text {Expression too large to display}\) | \(2287\) |
Input:
int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)/(a+b*cos(d*x+c))^4,x,method =_RETURNVERBOSE)
Output:
1/d*(A/a^4*ln(tan(1/2*d*x+1/2*c)+1)-2/a^4*((-1/2*(12*A*a^4*b^2+4*A*a^3*b^3 -6*A*a^2*b^4-A*a*b^5+2*A*b^6-6*B*a^5*b-3*B*a^4*b^2-2*B*a^3*b^3+2*C*a^6+2*C *a^5*b+6*C*a^4*b^2+C*a^3*b^3)*a/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d* x+1/2*c)^5-2/3*(18*A*a^4*b^2-11*A*a^2*b^4+3*A*b^6-9*B*a^5*b-B*a^3*b^3+3*C* a^6+7*C*a^4*b^2)*a/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3-1/ 2*(12*A*a^4*b^2-4*A*a^3*b^3-6*A*a^2*b^4+A*a*b^5+2*A*b^6-6*B*a^5*b+3*B*a^4* b^2-2*B*a^3*b^3+2*C*a^6-2*C*a^5*b+6*C*a^4*b^2-C*a^3*b^3)*a/(a+b)/(a^3-3*a^ 2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c))/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1 /2*c)^2*b+a+b)^3+1/2*(8*A*a^6*b-8*A*a^4*b^3+7*A*a^2*b^5-2*A*b^7-2*B*a^7-3* B*a^5*b^2+4*C*a^6*b+C*a^4*b^3)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b)) ^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a-b)*(a+b))^(1/2)))-A/a^4*ln(tan( 1/2*d*x+1/2*c)-1))
Leaf count of result is larger than twice the leaf count of optimal. 1194 vs. \(2 (326) = 652\).
Time = 70.10 (sec) , antiderivative size = 2458, normalized size of antiderivative = 7.12 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)/(a+b*cos(d*x+c))^4,x, algorithm="fricas")
Output:
[1/12*(3*(2*B*a^10 - 4*(2*A + C)*a^9*b + 3*B*a^8*b^2 + (8*A - C)*a^7*b^3 - 7*A*a^5*b^5 + 2*A*a^3*b^7 + (2*B*a^7*b^3 - 4*(2*A + C)*a^6*b^4 + 3*B*a^5* b^5 + (8*A - C)*a^4*b^6 - 7*A*a^2*b^8 + 2*A*b^10)*cos(d*x + c)^3 + 3*(2*B* a^8*b^2 - 4*(2*A + C)*a^7*b^3 + 3*B*a^6*b^4 + (8*A - C)*a^5*b^5 - 7*A*a^3* b^7 + 2*A*a*b^9)*cos(d*x + c)^2 + 3*(2*B*a^9*b - 4*(2*A + C)*a^8*b^2 + 3*B *a^7*b^3 + (8*A - C)*a^6*b^4 - 7*A*a^4*b^6 + 2*A*a^2*b^8)*cos(d*x + c))*sq rt(-a^2 + b^2)*log((2*a*b*cos(d*x + c) + (2*a^2 - b^2)*cos(d*x + c)^2 - 2* sqrt(-a^2 + b^2)*(a*cos(d*x + c) + b)*sin(d*x + c) - a^2 + 2*b^2)/(b^2*cos (d*x + c)^2 + 2*a*b*cos(d*x + c) + a^2)) + 6*(A*a^11 - 4*A*a^9*b^2 + 6*A*a ^7*b^4 - 4*A*a^5*b^6 + A*a^3*b^8 + (A*a^8*b^3 - 4*A*a^6*b^5 + 6*A*a^4*b^7 - 4*A*a^2*b^9 + A*b^11)*cos(d*x + c)^3 + 3*(A*a^9*b^2 - 4*A*a^7*b^4 + 6*A* a^5*b^6 - 4*A*a^3*b^8 + A*a*b^10)*cos(d*x + c)^2 + 3*(A*a^10*b - 4*A*a^8*b ^3 + 6*A*a^6*b^5 - 4*A*a^4*b^7 + A*a^2*b^9)*cos(d*x + c))*log(sin(d*x + c) + 1) - 6*(A*a^11 - 4*A*a^9*b^2 + 6*A*a^7*b^4 - 4*A*a^5*b^6 + A*a^3*b^8 + (A*a^8*b^3 - 4*A*a^6*b^5 + 6*A*a^4*b^7 - 4*A*a^2*b^9 + A*b^11)*cos(d*x + c )^3 + 3*(A*a^9*b^2 - 4*A*a^7*b^4 + 6*A*a^5*b^6 - 4*A*a^3*b^8 + A*a*b^10)*c os(d*x + c)^2 + 3*(A*a^10*b - 4*A*a^8*b^3 + 6*A*a^6*b^5 - 4*A*a^4*b^7 + A* a^2*b^9)*cos(d*x + c))*log(-sin(d*x + c) + 1) + 2*(6*C*a^11 - 18*B*a^10*b + 4*(9*A + C)*a^9*b^2 + 23*B*a^8*b^3 - (68*A + 11*C)*a^7*b^4 - 7*B*a^6*b^5 + (43*A + C)*a^5*b^6 + 2*B*a^4*b^7 - 11*A*a^3*b^8 + (2*C*a^9*b^2 - 11*...
\[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+b \cos (c+d x))^4} \, dx=\int \frac {\left (A + B \cos {\left (c + d x \right )} + C \cos ^{2}{\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}}{\left (a + b \cos {\left (c + d x \right )}\right )^{4}}\, dx \] Input:
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c)/(a+b*cos(d*x+c))**4, x)
Output:
Integral((A + B*cos(c + d*x) + C*cos(c + d*x)**2)*sec(c + d*x)/(a + b*cos( c + d*x))**4, x)
Exception generated. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)/(a+b*cos(d*x+c))^4,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 1127 vs. \(2 (326) = 652\).
Time = 0.25 (sec) , antiderivative size = 1127, normalized size of antiderivative = 3.27 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)/(a+b*cos(d*x+c))^4,x, algorithm="giac")
Output:
1/3*(3*(2*B*a^7 - 8*A*a^6*b - 4*C*a^6*b + 3*B*a^5*b^2 + 8*A*a^4*b^3 - C*a^ 4*b^3 - 7*A*a^2*b^5 + 2*A*b^7)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(2*a - 2*b) + arctan((a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(a^2 - b^2)))/((a^10 - 3*a^8*b^2 + 3*a^6*b^4 - a^4*b^6)*sqrt(a^2 - b^2)) + 3*A* log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^4 - 3*A*log(abs(tan(1/2*d*x + 1/2*c) - 1))/a^4 + (6*C*a^8*tan(1/2*d*x + 1/2*c)^5 - 18*B*a^7*b*tan(1/2*d*x + 1/2 *c)^5 - 6*C*a^7*b*tan(1/2*d*x + 1/2*c)^5 + 36*A*a^6*b^2*tan(1/2*d*x + 1/2* c)^5 + 27*B*a^6*b^2*tan(1/2*d*x + 1/2*c)^5 + 12*C*a^6*b^2*tan(1/2*d*x + 1/ 2*c)^5 - 60*A*a^5*b^3*tan(1/2*d*x + 1/2*c)^5 - 6*B*a^5*b^3*tan(1/2*d*x + 1 /2*c)^5 - 27*C*a^5*b^3*tan(1/2*d*x + 1/2*c)^5 - 6*A*a^4*b^4*tan(1/2*d*x + 1/2*c)^5 + 3*B*a^4*b^4*tan(1/2*d*x + 1/2*c)^5 + 12*C*a^4*b^4*tan(1/2*d*x + 1/2*c)^5 + 45*A*a^3*b^5*tan(1/2*d*x + 1/2*c)^5 - 6*B*a^3*b^5*tan(1/2*d*x + 1/2*c)^5 + 3*C*a^3*b^5*tan(1/2*d*x + 1/2*c)^5 - 6*A*a^2*b^6*tan(1/2*d*x + 1/2*c)^5 - 15*A*a*b^7*tan(1/2*d*x + 1/2*c)^5 + 6*A*b^8*tan(1/2*d*x + 1/2 *c)^5 + 12*C*a^8*tan(1/2*d*x + 1/2*c)^3 - 36*B*a^7*b*tan(1/2*d*x + 1/2*c)^ 3 + 72*A*a^6*b^2*tan(1/2*d*x + 1/2*c)^3 + 16*C*a^6*b^2*tan(1/2*d*x + 1/2*c )^3 + 32*B*a^5*b^3*tan(1/2*d*x + 1/2*c)^3 - 116*A*a^4*b^4*tan(1/2*d*x + 1/ 2*c)^3 - 28*C*a^4*b^4*tan(1/2*d*x + 1/2*c)^3 + 4*B*a^3*b^5*tan(1/2*d*x + 1 /2*c)^3 + 56*A*a^2*b^6*tan(1/2*d*x + 1/2*c)^3 - 12*A*b^8*tan(1/2*d*x + 1/2 *c)^3 + 6*C*a^8*tan(1/2*d*x + 1/2*c) - 18*B*a^7*b*tan(1/2*d*x + 1/2*c) ...
Time = 8.68 (sec) , antiderivative size = 11939, normalized size of antiderivative = 34.61 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:
int((A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)*(a + b*cos(c + d *x))^4),x)
Output:
- ((tan(c/2 + (d*x)/2)*(2*A*b^6 + 2*C*a^6 - 6*A*a^2*b^4 - 4*A*a^3*b^3 + 12 *A*a^4*b^2 - 2*B*a^3*b^3 + 3*B*a^4*b^2 - C*a^3*b^3 + 6*C*a^4*b^2 + A*a*b^5 - 6*B*a^5*b - 2*C*a^5*b))/((a + b)*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)) - (4*tan(c/2 + (d*x)/2)^3*(3*A*b^6 + 3*C*a^6 - 11*A*a^2*b^4 + 18*A*a^4*b^ 2 - B*a^3*b^3 + 7*C*a^4*b^2 - 9*B*a^5*b))/(3*(a + b)^2*(a^5 - 2*a^4*b + a^ 3*b^2)) + (tan(c/2 + (d*x)/2)^5*(2*A*b^6 + 2*C*a^6 - 6*A*a^2*b^4 + 4*A*a^3 *b^3 + 12*A*a^4*b^2 - 2*B*a^3*b^3 - 3*B*a^4*b^2 + C*a^3*b^3 + 6*C*a^4*b^2 - A*a*b^5 - 6*B*a^5*b + 2*C*a^5*b))/((a^3*b - a^4)*(a + b)^3))/(d*(3*a*b^2 - tan(c/2 + (d*x)/2)^4*(3*a*b^2 + 3*a^2*b - 3*a^3 - 3*b^3) - tan(c/2 + (d *x)/2)^2*(3*a*b^2 - 3*a^2*b - 3*a^3 + 3*b^3) + 3*a^2*b + a^3 + b^3 + tan(c /2 + (d*x)/2)^6*(3*a*b^2 - 3*a^2*b + a^3 - b^3))) - (A*atan(((A*((8*tan(c/ 2 + (d*x)/2)*(4*A^2*a^14 + 8*A^2*b^14 + 4*B^2*a^14 - 8*A^2*a*b^13 - 8*A^2* a^13*b - 48*A^2*a^2*b^12 + 48*A^2*a^3*b^11 + 117*A^2*a^4*b^10 - 120*A^2*a^ 5*b^9 - 164*A^2*a^6*b^8 + 160*A^2*a^7*b^7 + 156*A^2*a^8*b^6 - 120*A^2*a^9* b^5 - 92*A^2*a^10*b^4 + 48*A^2*a^11*b^3 + 44*A^2*a^12*b^2 + 9*B^2*a^10*b^4 + 12*B^2*a^12*b^2 + C^2*a^8*b^6 + 8*C^2*a^10*b^4 + 16*C^2*a^12*b^2 - 32*A *B*a^13*b - 16*B*C*a^13*b + 12*A*B*a^5*b^9 - 34*A*B*a^7*b^7 + 20*A*B*a^9*b ^5 - 16*A*B*a^11*b^3 - 4*A*C*a^4*b^10 - 2*A*C*a^6*b^8 + 40*A*C*a^8*b^6 - 4 8*A*C*a^10*b^4 + 64*A*C*a^12*b^2 - 6*B*C*a^9*b^5 - 28*B*C*a^11*b^3))/(a^16 *b + a^17 - a^6*b^11 - a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - 10*a^10*b^7 -...
Time = 0.22 (sec) , antiderivative size = 3572, normalized size of antiderivative = 10.35 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+b \cos (c+d x))^4} \, dx =\text {Too large to display} \] Input:
int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)/(a+b*cos(d*x+c))^4,x)
Output:
( - 36*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sq rt(a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**2*a**6*b**4 - 24*sqrt(a**2 - b **2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos (c + d*x)*sin(c + d*x)**2*a**5*b**4*c + 66*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**2*a**4*b**6 - 6*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**2*a**3*b**6*c - 42*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt( a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**2*a**2*b**8 + 12*sqrt(a**2 - b**2 )*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**2*b**10 + 108*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2 )*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*a**8*b**2 + 72*s qrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*a**7*b**2*c - 162*sqrt(a**2 - b**2)*atan((tan((c + d *x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*a**6*b**4 + 42*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt( a**2 - b**2))*cos(c + d*x)*a**5*b**4*c + 60*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*a**4*b* *6 + 6*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sq rt(a**2 - b**2))*cos(c + d*x)*a**3*b**6*c + 6*sqrt(a**2 - b**2)*atan((t...