Integrand size = 41, antiderivative size = 461 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\frac {(b B-4 a C) x}{b^5}-\frac {\left (2 A b^8+2 a^7 b B-7 a^5 b^3 B+8 a^3 b^5 B-8 a b^7 B-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+a^2 b^6 (3 A+20 C)\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} b^5 (a+b)^{7/2} d}-\frac {\left (5 A b^4+3 a^3 b B-8 a b^3 B-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \sin (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {\left (3 A b^4+a^3 b B-6 a b^3 B-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {a \left (2 A b^6-a^5 b B+2 a^3 b^3 B-6 a b^5 B+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))} \] Output:
(B*b-4*C*a)*x/b^5-(2*A*b^8+2*a^7*b*B-7*a^5*b^3*B+8*a^3*b^5*B-8*a*b^7*B-8*a ^8*C+28*a^6*b^2*C-35*a^4*b^4*C+a^2*b^6*(3*A+20*C))*arctan((a-b)^(1/2)*tan( 1/2*d*x+1/2*c)/(a+b)^(1/2))/(a-b)^(7/2)/b^5/(a+b)^(7/2)/d-1/6*(5*A*b^4+3*B *a^3*b-8*B*a*b^3-12*C*a^4+23*C*a^2*b^2-6*C*b^4)*sin(d*x+c)/b^4/(a^2-b^2)^2 /d-1/3*(A*b^2-a*(B*b-C*a))*cos(d*x+c)^3*sin(d*x+c)/b/(a^2-b^2)/d/(a+b*cos( d*x+c))^3+1/6*(3*A*b^4+B*a^3*b-6*B*a*b^3-4*a^4*C+a^2*b^2*(2*A+9*C))*cos(d* x+c)^2*sin(d*x+c)/b^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c))^2+1/2*a*(2*A*b^6-B*a^ 5*b+2*B*a^3*b^3-6*B*a*b^5+4*a^6*C-11*a^4*b^2*C+3*a^2*b^4*(A+4*C))*sin(d*x+ c)/b^4/(a^2-b^2)^3/d/(a+b*cos(d*x+c))
Time = 10.20 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.15 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\frac {(b B-4 a C) (c+d x)}{b^5 d}-\frac {\left (-3 a^2 A b^6-2 A b^8-2 a^7 b B+7 a^5 b^3 B-8 a^3 b^5 B+8 a b^7 B+8 a^8 C-28 a^6 b^2 C+35 a^4 b^4 C-20 a^2 b^6 C\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{b^5 \left (a^2-b^2\right )^3 \sqrt {-a^2+b^2} d}+\frac {C \sin (c+d x)}{b^4 d}+\frac {-a^3 A b^2 \sin (c+d x)+a^4 b B \sin (c+d x)-a^5 C \sin (c+d x)}{3 b^4 \left (-a^2+b^2\right ) d (a+b \cos (c+d x))^3}+\frac {-4 a^4 A b^2 \sin (c+d x)+9 a^2 A b^4 \sin (c+d x)+7 a^5 b B \sin (c+d x)-12 a^3 b^3 B \sin (c+d x)-10 a^6 C \sin (c+d x)+15 a^4 b^2 C \sin (c+d x)}{6 b^4 \left (-a^2+b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {-2 a^5 A b^2 \sin (c+d x)+5 a^3 A b^4 \sin (c+d x)-18 a A b^6 \sin (c+d x)+11 a^6 b B \sin (c+d x)-32 a^4 b^3 B \sin (c+d x)+36 a^2 b^5 B \sin (c+d x)-26 a^7 C \sin (c+d x)+71 a^5 b^2 C \sin (c+d x)-60 a^3 b^4 C \sin (c+d x)}{6 b^4 \left (-a^2+b^2\right )^3 d (a+b \cos (c+d x))} \] Input:
Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b* Cos[c + d*x])^4,x]
Output:
((b*B - 4*a*C)*(c + d*x))/(b^5*d) - ((-3*a^2*A*b^6 - 2*A*b^8 - 2*a^7*b*B + 7*a^5*b^3*B - 8*a^3*b^5*B + 8*a*b^7*B + 8*a^8*C - 28*a^6*b^2*C + 35*a^4*b ^4*C - 20*a^2*b^6*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]]) /(b^5*(a^2 - b^2)^3*Sqrt[-a^2 + b^2]*d) + (C*Sin[c + d*x])/(b^4*d) + (-(a^ 3*A*b^2*Sin[c + d*x]) + a^4*b*B*Sin[c + d*x] - a^5*C*Sin[c + d*x])/(3*b^4* (-a^2 + b^2)*d*(a + b*Cos[c + d*x])^3) + (-4*a^4*A*b^2*Sin[c + d*x] + 9*a^ 2*A*b^4*Sin[c + d*x] + 7*a^5*b*B*Sin[c + d*x] - 12*a^3*b^3*B*Sin[c + d*x] - 10*a^6*C*Sin[c + d*x] + 15*a^4*b^2*C*Sin[c + d*x])/(6*b^4*(-a^2 + b^2)^2 *d*(a + b*Cos[c + d*x])^2) + (-2*a^5*A*b^2*Sin[c + d*x] + 5*a^3*A*b^4*Sin[ c + d*x] - 18*a*A*b^6*Sin[c + d*x] + 11*a^6*b*B*Sin[c + d*x] - 32*a^4*b^3* B*Sin[c + d*x] + 36*a^2*b^5*B*Sin[c + d*x] - 26*a^7*C*Sin[c + d*x] + 71*a^ 5*b^2*C*Sin[c + d*x] - 60*a^3*b^4*C*Sin[c + d*x])/(6*b^4*(-a^2 + b^2)^3*d* (a + b*Cos[c + d*x]))
Time = 2.66 (sec) , antiderivative size = 518, normalized size of antiderivative = 1.12, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.366, Rules used = {3042, 3526, 3042, 3526, 3042, 3510, 25, 3042, 3502, 27, 3042, 3214, 3042, 3138, 218}
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.
| \(\displaystyle \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle -\frac {\int \frac {\cos ^2(c+d x) \left (-\left (\left (4 C a^2-b B a+A b^2-3 b^2 C\right ) \cos ^2(c+d x)\right )+3 b (b B-a (A+C)) \cos (c+d x)+3 \left (A b^2-a (b B-a C)\right )\right )}{(a+b \cos (c+d x))^3}dx}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (\left (-4 C a^2+b B a-A b^2+3 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+3 b (b B-a (A+C)) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (A b^2-a (b B-a C)\right )\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle -\frac {-\frac {\int \frac {\cos (c+d x) \left (-\left (\left (-12 C a^4+3 b B a^3+23 b^2 C a^2-8 b^3 B a+5 A b^4-6 b^4 C\right ) \cos ^2(c+d x)\right )+2 b \left (C a^3+2 b B a^2-b^2 (5 A+6 C) a+3 b^3 B\right ) \cos (c+d x)+2 \left (-4 C a^4+b B a^3+b^2 (2 A+9 C) a^2-6 b^3 B a+3 A b^4\right )\right )}{(a+b \cos (c+d x))^2}dx}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (\left (12 C a^4-3 b B a^3-23 b^2 C a^2+8 b^3 B a-5 A b^4+6 b^4 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 b \left (C a^3+2 b B a^2-b^2 (5 A+6 C) a+3 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 \left (-4 C a^4+b B a^3+b^2 (2 A+9 C) a^2-6 b^3 B a+3 A b^4\right )\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle -\frac {-\frac {\frac {\int -\frac {b \left (a^2-b^2\right ) \left (-12 C a^4+3 b B a^3+23 b^2 C a^2-8 b^3 B a+5 A b^4-6 b^4 C\right ) \cos ^2(c+d x)-\left (a^2-b^2\right ) \left (-12 C a^5+3 b B a^4+25 b^2 C a^3-4 b^3 B a^2-b^4 (5 A+18 C) a+6 b^5 B\right ) \cos (c+d x)+3 b \left (4 C a^6-b B a^5-11 b^2 C a^4+2 b^3 B a^3+3 b^4 (A+4 C) a^2-6 b^5 B a+2 A b^6\right )}{a+b \cos (c+d x)}dx}{b^2 \left (a^2-b^2\right )}+\frac {3 a \sin (c+d x) \left (4 a^6 C-a^5 b B-11 a^4 b^2 C+2 a^3 b^3 B+3 a^2 b^4 (A+4 C)-6 a b^5 B+2 A b^6\right )}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\frac {3 a \sin (c+d x) \left (4 a^6 C-a^5 b B-11 a^4 b^2 C+2 a^3 b^3 B+3 a^2 b^4 (A+4 C)-6 a b^5 B+2 A b^6\right )}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\int \frac {b \left (a^2-b^2\right ) \left (-12 C a^4+3 b B a^3+23 b^2 C a^2-8 b^3 B a+5 A b^4-6 b^4 C\right ) \cos ^2(c+d x)-\left (a^2-b^2\right ) \left (-12 C a^5+3 b B a^4+25 b^2 C a^3-4 b^3 B a^2-b^4 (5 A+18 C) a+6 b^5 B\right ) \cos (c+d x)+3 b \left (4 C a^6-b B a^5-11 b^2 C a^4+2 b^3 B a^3+3 b^4 (A+4 C) a^2-6 b^5 B a+2 A b^6\right )}{a+b \cos (c+d x)}dx}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\frac {3 a \sin (c+d x) \left (4 a^6 C-a^5 b B-11 a^4 b^2 C+2 a^3 b^3 B+3 a^2 b^4 (A+4 C)-6 a b^5 B+2 A b^6\right )}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\int \frac {b \left (a^2-b^2\right ) \left (-12 C a^4+3 b B a^3+23 b^2 C a^2-8 b^3 B a+5 A b^4-6 b^4 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-\left (a^2-b^2\right ) \left (-12 C a^5+3 b B a^4+25 b^2 C a^3-4 b^3 B a^2-b^4 (5 A+18 C) a+6 b^5 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 b \left (4 C a^6-b B a^5-11 b^2 C a^4+2 b^3 B a^3+3 b^4 (A+4 C) a^2-6 b^5 B a+2 A b^6\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle -\frac {-\frac {\frac {3 a \sin (c+d x) \left (4 a^6 C-a^5 b B-11 a^4 b^2 C+2 a^3 b^3 B+3 a^2 b^4 (A+4 C)-6 a b^5 B+2 A b^6\right )}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\int \frac {3 \left (b^2 \left (4 C a^6-b B a^5-11 b^2 C a^4+2 b^3 B a^3+3 b^4 (A+4 C) a^2-6 b^5 B a+2 A b^6\right )-2 b \left (a^2-b^2\right )^3 (b B-4 a C) \cos (c+d x)\right )}{a+b \cos (c+d x)}dx}{b}+\frac {\left (a^2-b^2\right ) \sin (c+d x) \left (-12 a^4 C+3 a^3 b B+23 a^2 b^2 C-8 a b^3 B+5 A b^4-6 b^4 C\right )}{d}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {3 a \sin (c+d x) \left (4 a^6 C-a^5 b B-11 a^4 b^2 C+2 a^3 b^3 B+3 a^2 b^4 (A+4 C)-6 a b^5 B+2 A b^6\right )}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \int \frac {b^2 \left (4 C a^6-b B a^5-11 b^2 C a^4+2 b^3 B a^3+3 b^4 (A+4 C) a^2-6 b^5 B a+2 A b^6\right )-2 b \left (a^2-b^2\right )^3 (b B-4 a C) \cos (c+d x)}{a+b \cos (c+d x)}dx}{b}+\frac {\left (a^2-b^2\right ) \sin (c+d x) \left (-12 a^4 C+3 a^3 b B+23 a^2 b^2 C-8 a b^3 B+5 A b^4-6 b^4 C\right )}{d}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\frac {3 a \sin (c+d x) \left (4 a^6 C-a^5 b B-11 a^4 b^2 C+2 a^3 b^3 B+3 a^2 b^4 (A+4 C)-6 a b^5 B+2 A b^6\right )}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \int \frac {b^2 \left (4 C a^6-b B a^5-11 b^2 C a^4+2 b^3 B a^3+3 b^4 (A+4 C) a^2-6 b^5 B a+2 A b^6\right )-2 b \left (a^2-b^2\right )^3 (b B-4 a C) \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}+\frac {\left (a^2-b^2\right ) \sin (c+d x) \left (-12 a^4 C+3 a^3 b B+23 a^2 b^2 C-8 a b^3 B+5 A b^4-6 b^4 C\right )}{d}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle -\frac {-\frac {\frac {3 a \sin (c+d x) \left (4 a^6 C-a^5 b B-11 a^4 b^2 C+2 a^3 b^3 B+3 a^2 b^4 (A+4 C)-6 a b^5 B+2 A b^6\right )}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \left (\left (-8 a^8 C+2 a^7 b B+28 a^6 b^2 C-7 a^5 b^3 B-35 a^4 b^4 C+8 a^3 b^5 B+a^2 b^6 (3 A+20 C)-8 a b^7 B+2 A b^8\right ) \int \frac {1}{a+b \cos (c+d x)}dx-2 x \left (a^2-b^2\right )^3 (b B-4 a C)\right )}{b}+\frac {\left (a^2-b^2\right ) \sin (c+d x) \left (-12 a^4 C+3 a^3 b B+23 a^2 b^2 C-8 a b^3 B+5 A b^4-6 b^4 C\right )}{d}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\frac {3 a \sin (c+d x) \left (4 a^6 C-a^5 b B-11 a^4 b^2 C+2 a^3 b^3 B+3 a^2 b^4 (A+4 C)-6 a b^5 B+2 A b^6\right )}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \left (\left (-8 a^8 C+2 a^7 b B+28 a^6 b^2 C-7 a^5 b^3 B-35 a^4 b^4 C+8 a^3 b^5 B+a^2 b^6 (3 A+20 C)-8 a b^7 B+2 A b^8\right ) \int \frac {1}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx-2 x \left (a^2-b^2\right )^3 (b B-4 a C)\right )}{b}+\frac {\left (a^2-b^2\right ) \sin (c+d x) \left (-12 a^4 C+3 a^3 b B+23 a^2 b^2 C-8 a b^3 B+5 A b^4-6 b^4 C\right )}{d}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle -\frac {-\frac {\frac {3 a \sin (c+d x) \left (4 a^6 C-a^5 b B-11 a^4 b^2 C+2 a^3 b^3 B+3 a^2 b^4 (A+4 C)-6 a b^5 B+2 A b^6\right )}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \left (\frac {2 \left (-8 a^8 C+2 a^7 b B+28 a^6 b^2 C-7 a^5 b^3 B-35 a^4 b^4 C+8 a^3 b^5 B+a^2 b^6 (3 A+20 C)-8 a b^7 B+2 A b^8\right ) \int \frac {1}{(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{d}-2 x \left (a^2-b^2\right )^3 (b B-4 a C)\right )}{b}+\frac {\left (a^2-b^2\right ) \sin (c+d x) \left (-12 a^4 C+3 a^3 b B+23 a^2 b^2 C-8 a b^3 B+5 A b^4-6 b^4 C\right )}{d}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {-\frac {\sin (c+d x) \cos ^2(c+d x) \left (-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {3 a \sin (c+d x) \left (4 a^6 C-a^5 b B-11 a^4 b^2 C+2 a^3 b^3 B+3 a^2 b^4 (A+4 C)-6 a b^5 B+2 A b^6\right )}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\left (a^2-b^2\right ) \sin (c+d x) \left (-12 a^4 C+3 a^3 b B+23 a^2 b^2 C-8 a b^3 B+5 A b^4-6 b^4 C\right )}{d}+\frac {3 \left (\frac {2 \left (-8 a^8 C+2 a^7 b B+28 a^6 b^2 C-7 a^5 b^3 B-35 a^4 b^4 C+8 a^3 b^5 B+a^2 b^6 (3 A+20 C)-8 a b^7 B+2 A b^8\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d \sqrt {a-b} \sqrt {a+b}}-2 x \left (a^2-b^2\right )^3 (b B-4 a C)\right )}{b}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}\) |
Input:
Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^4,x]
Output:
-1/3*((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^3*Sin[c + d*x])/(b*(a^2 - b^2)* d*(a + b*Cos[c + d*x])^3) - (-1/2*((3*A*b^4 + a^3*b*B - 6*a*b^3*B - 4*a^4* C + a^2*b^2*(2*A + 9*C))*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((3*a*(2*A*b^6 - a^5*b*B + 2*a^3*b^3*B - 6*a*b^5*B + 4*a^6*C - 11*a^4*b^2*C + 3*a^2*b^4*(A + 4*C))*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])) - ((3*(-2*(a^2 - b^2)^3*(b*B - 4*a*C)*x + (2* (2*A*b^8 + 2*a^7*b*B - 7*a^5*b^3*B + 8*a^3*b^5*B - 8*a*b^7*B - 8*a^8*C + 2 8*a^6*b^2*C - 35*a^4*b^4*C + a^2*b^6*(3*A + 20*C))*ArcTan[(Sqrt[a - b]*Tan [(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*d)))/b + ((a^2 - b^2 )*(5*A*b^4 + 3*a^3*b*B - 8*a*b^3*B - 12*a^4*C + 23*a^2*b^2*C - 6*b^4*C)*Si n[c + d*x])/d)/(b^2*(a^2 - b^2)))/(2*b*(a^2 - b^2)))/(3*b*(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_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-(b*c - a*d))*(A*b^2 - a*b*B + a^2*C)*Cos[ e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - S imp[1/(b^2*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*( m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1)) ))*Sin[e + f*x] - b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; F reeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d , 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Time = 2.72 (sec) , antiderivative size = 640, normalized size of antiderivative = 1.39
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 \left (\frac {-\frac {\left (2 a^{2} A \,b^{4}+3 A a \,b^{5}+6 A \,b^{6}-2 B \,a^{5} b +B \,a^{4} b^{2}+6 B \,a^{3} b^{3}-4 B \,a^{2} b^{4}-12 B a \,b^{5}+6 a^{6} C -2 C \,a^{5} b -18 a^{4} b^{2} C +5 C \,a^{3} b^{3}+20 a^{2} C \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {2 \left (a^{2} A \,b^{4}+9 A \,b^{6}-3 B \,a^{5} b +11 B \,a^{3} b^{3}-18 B a \,b^{5}+9 a^{6} C -29 a^{4} b^{2} C +30 a^{2} C \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (2 a^{2} A \,b^{4}-3 A a \,b^{5}+6 A \,b^{6}-2 B \,a^{5} b -B \,a^{4} b^{2}+6 B \,a^{3} b^{3}+4 B \,a^{2} b^{4}-12 B a \,b^{5}+6 a^{6} C +2 C \,a^{5} b -18 a^{4} b^{2} C -5 C \,a^{3} b^{3}+20 a^{2} C \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{3}}+\frac {\left (3 a^{2} A \,b^{6}+2 A \,b^{8}+2 a^{7} b B -7 a^{5} b^{3} B +8 a^{3} b^{5} B -8 a \,b^{7} B -8 a^{8} C +28 a^{6} b^{2} C -35 a^{4} b^{4} C +20 C \,a^{2} b^{6}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{5}}+\frac {\frac {2 C b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+2 \left (B b -4 C a \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{5}}}{d}\) | \(640\) |
| default | \(\frac {-\frac {2 \left (\frac {-\frac {\left (2 a^{2} A \,b^{4}+3 A a \,b^{5}+6 A \,b^{6}-2 B \,a^{5} b +B \,a^{4} b^{2}+6 B \,a^{3} b^{3}-4 B \,a^{2} b^{4}-12 B a \,b^{5}+6 a^{6} C -2 C \,a^{5} b -18 a^{4} b^{2} C +5 C \,a^{3} b^{3}+20 a^{2} C \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {2 \left (a^{2} A \,b^{4}+9 A \,b^{6}-3 B \,a^{5} b +11 B \,a^{3} b^{3}-18 B a \,b^{5}+9 a^{6} C -29 a^{4} b^{2} C +30 a^{2} C \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (2 a^{2} A \,b^{4}-3 A a \,b^{5}+6 A \,b^{6}-2 B \,a^{5} b -B \,a^{4} b^{2}+6 B \,a^{3} b^{3}+4 B \,a^{2} b^{4}-12 B a \,b^{5}+6 a^{6} C +2 C \,a^{5} b -18 a^{4} b^{2} C -5 C \,a^{3} b^{3}+20 a^{2} C \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{3}}+\frac {\left (3 a^{2} A \,b^{6}+2 A \,b^{8}+2 a^{7} b B -7 a^{5} b^{3} B +8 a^{3} b^{5} B -8 a \,b^{7} B -8 a^{8} C +28 a^{6} b^{2} C -35 a^{4} b^{4} C +20 C \,a^{2} b^{6}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{5}}+\frac {\frac {2 C b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+2 \left (B b -4 C a \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{5}}}{d}\) | \(640\) |
| risch | \(\text {Expression too large to display}\) | \(2860\) |
Input:
int(cos(d*x+c)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^4,x,meth od=_RETURNVERBOSE)
Output:
1/d*(-2/b^5*((-1/2*(2*A*a^2*b^4+3*A*a*b^5+6*A*b^6-2*B*a^5*b+B*a^4*b^2+6*B* a^3*b^3-4*B*a^2*b^4-12*B*a*b^5+6*C*a^6-2*C*a^5*b-18*C*a^4*b^2+5*C*a^3*b^3+ 20*C*a^2*b^4)*a*b/(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 *(A*a^2*b^4+9*A*b^6-3*B*a^5*b+11*B*a^3*b^3-18*B*a*b^5+9*C*a^6-29*C*a^4*b^2 +30*C*a^2*b^4)*a*b/(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*(2*A*a^2*b^4-3*A*a*b^5+6*A*b^6-2*B*a^5*b-B*a^4*b^2+6*B*a^3*b^3+4*B*a^2*b ^4-12*B*a*b^5+6*C*a^6+2*C*a^5*b-18*C*a^4*b^2-5*C*a^3*b^3+20*C*a^2*b^4)*a*b /(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*(3*A*a^2*b^6+2*A*b^8+2*B*a^7*b-7*B*a^ 5*b^3+8*B*a^3*b^5-8*B*a*b^7-8*C*a^8+28*C*a^6*b^2-35*C*a^4*b^4+20*C*a^2*b^6 )/(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)))+2/b^5*(C*b*tan(1/2*d*x+1/2*c)/(1+tan(1/2*d *x+1/2*c)^2)+(B*b-4*C*a)*arctan(tan(1/2*d*x+1/2*c))))
Leaf count of result is larger than twice the leaf count of optimal. 1354 vs. \(2 (447) = 894\).
Time = 0.34 (sec) , antiderivative size = 2777, normalized size of antiderivative = 6.02 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^4, x, algorithm="fricas")
Output:
[-1/12*(12*(4*C*a^9*b^3 - B*a^8*b^4 - 16*C*a^7*b^5 + 4*B*a^6*b^6 + 24*C*a^ 5*b^7 - 6*B*a^4*b^8 - 16*C*a^3*b^9 + 4*B*a^2*b^10 + 4*C*a*b^11 - B*b^12)*d *x*cos(d*x + c)^3 + 36*(4*C*a^10*b^2 - B*a^9*b^3 - 16*C*a^8*b^4 + 4*B*a^7* b^5 + 24*C*a^6*b^6 - 6*B*a^5*b^7 - 16*C*a^4*b^8 + 4*B*a^3*b^9 + 4*C*a^2*b^ 10 - B*a*b^11)*d*x*cos(d*x + c)^2 + 36*(4*C*a^11*b - B*a^10*b^2 - 16*C*a^9 *b^3 + 4*B*a^8*b^4 + 24*C*a^7*b^5 - 6*B*a^6*b^6 - 16*C*a^5*b^7 + 4*B*a^4*b ^8 + 4*C*a^3*b^9 - B*a^2*b^10)*d*x*cos(d*x + c) + 12*(4*C*a^12 - B*a^11*b - 16*C*a^10*b^2 + 4*B*a^9*b^3 + 24*C*a^8*b^4 - 6*B*a^7*b^5 - 16*C*a^6*b^6 + 4*B*a^5*b^7 + 4*C*a^4*b^8 - B*a^3*b^9)*d*x + 3*(8*C*a^11 - 2*B*a^10*b - 28*C*a^9*b^2 + 7*B*a^8*b^3 + 35*C*a^7*b^4 - 8*B*a^6*b^5 - (3*A + 20*C)*a^5 *b^6 + 8*B*a^4*b^7 - 2*A*a^3*b^8 + (8*C*a^8*b^3 - 2*B*a^7*b^4 - 28*C*a^6*b ^5 + 7*B*a^5*b^6 + 35*C*a^4*b^7 - 8*B*a^3*b^8 - (3*A + 20*C)*a^2*b^9 + 8*B *a*b^10 - 2*A*b^11)*cos(d*x + c)^3 + 3*(8*C*a^9*b^2 - 2*B*a^8*b^3 - 28*C*a ^7*b^4 + 7*B*a^6*b^5 + 35*C*a^5*b^6 - 8*B*a^4*b^7 - (3*A + 20*C)*a^3*b^8 + 8*B*a^2*b^9 - 2*A*a*b^10)*cos(d*x + c)^2 + 3*(8*C*a^10*b - 2*B*a^9*b^2 - 28*C*a^8*b^3 + 7*B*a^7*b^4 + 35*C*a^6*b^5 - 8*B*a^5*b^6 - (3*A + 20*C)*a^4 *b^7 + 8*B*a^3*b^8 - 2*A*a^2*b^9)*cos(d*x + c))*sqrt(-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)) - 2*(24*C*a^11*b - 6*B*a^10*b^2 - 92*C*a^9*b^3 + 23*B*...
Timed out. \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**3*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)/(a+b*cos(d*x+c))* *4,x)
Output:
Timed out
Exception generated. \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cos(d*x+c)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(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. 1225 vs. \(2 (447) = 894\).
Time = 0.21 (sec) , antiderivative size = 1225, normalized size of antiderivative = 2.66 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^4, x, algorithm="giac")
Output:
-1/3*(3*(8*C*a^8 - 2*B*a^7*b - 28*C*a^6*b^2 + 7*B*a^5*b^3 + 35*C*a^4*b^4 - 8*B*a^3*b^5 - 3*A*a^2*b^6 - 20*C*a^2*b^6 + 8*B*a*b^7 - 2*A*b^8)*(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^6*b^5 - 3*a^4*b^7 + 3*a^2 *b^9 - b^11)*sqrt(a^2 - b^2)) - (18*C*a^9*tan(1/2*d*x + 1/2*c)^5 - 6*B*a^8 *b*tan(1/2*d*x + 1/2*c)^5 - 42*C*a^8*b*tan(1/2*d*x + 1/2*c)^5 + 15*B*a^7*b ^2*tan(1/2*d*x + 1/2*c)^5 - 24*C*a^7*b^2*tan(1/2*d*x + 1/2*c)^5 + 6*B*a^6* b^3*tan(1/2*d*x + 1/2*c)^5 + 117*C*a^6*b^3*tan(1/2*d*x + 1/2*c)^5 + 6*A*a^ 5*b^4*tan(1/2*d*x + 1/2*c)^5 - 45*B*a^5*b^4*tan(1/2*d*x + 1/2*c)^5 - 24*C* a^5*b^4*tan(1/2*d*x + 1/2*c)^5 - 3*A*a^4*b^5*tan(1/2*d*x + 1/2*c)^5 + 6*B* a^4*b^5*tan(1/2*d*x + 1/2*c)^5 - 105*C*a^4*b^5*tan(1/2*d*x + 1/2*c)^5 + 6* A*a^3*b^6*tan(1/2*d*x + 1/2*c)^5 + 60*B*a^3*b^6*tan(1/2*d*x + 1/2*c)^5 + 6 0*C*a^3*b^6*tan(1/2*d*x + 1/2*c)^5 - 27*A*a^2*b^7*tan(1/2*d*x + 1/2*c)^5 - 36*B*a^2*b^7*tan(1/2*d*x + 1/2*c)^5 + 18*A*a*b^8*tan(1/2*d*x + 1/2*c)^5 + 36*C*a^9*tan(1/2*d*x + 1/2*c)^3 - 12*B*a^8*b*tan(1/2*d*x + 1/2*c)^3 - 152 *C*a^7*b^2*tan(1/2*d*x + 1/2*c)^3 + 56*B*a^6*b^3*tan(1/2*d*x + 1/2*c)^3 + 4*A*a^5*b^4*tan(1/2*d*x + 1/2*c)^3 + 236*C*a^5*b^4*tan(1/2*d*x + 1/2*c)^3 - 116*B*a^4*b^5*tan(1/2*d*x + 1/2*c)^3 + 32*A*a^3*b^6*tan(1/2*d*x + 1/2*c) ^3 - 120*C*a^3*b^6*tan(1/2*d*x + 1/2*c)^3 + 72*B*a^2*b^7*tan(1/2*d*x + 1/2 *c)^3 - 36*A*a*b^8*tan(1/2*d*x + 1/2*c)^3 + 18*C*a^9*tan(1/2*d*x + 1/2*...
Time = 5.64 (sec) , antiderivative size = 9423, normalized size of antiderivative = 20.44 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:
int((cos(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^4,x)
Output:
((tan(c/2 + (d*x)/2)^7*(8*C*a^7 + 2*C*b^7 + 3*A*a^2*b^5 + 2*A*a^3*b^4 - 12 *B*a^2*b^5 - 4*B*a^3*b^4 + 6*B*a^4*b^3 + B*a^5*b^2 - 6*C*a^2*b^5 + 26*C*a^ 3*b^4 + 11*C*a^4*b^3 - 24*C*a^5*b^2 + 6*A*a*b^6 - 2*B*a^6*b - 2*C*a*b^6 - 4*C*a^6*b))/(b^4*(a + b)^3*(a - b)) + (tan(c/2 + (d*x)/2)*(8*C*a^7 - 2*C*b ^7 - 3*A*a^2*b^5 + 2*A*a^3*b^4 - 12*B*a^2*b^5 + 4*B*a^3*b^4 + 6*B*a^4*b^3 - B*a^5*b^2 + 6*C*a^2*b^5 + 26*C*a^3*b^4 - 11*C*a^4*b^3 - 24*C*a^5*b^2 + 6 *A*a*b^6 - 2*B*a^6*b - 2*C*a*b^6 + 4*C*a^6*b))/(b^4*(a + b)*(a - b)^3) + ( tan(c/2 + (d*x)/2)^3*(72*C*a^8 + 18*C*b^8 + 45*A*a^2*b^6 - 7*A*a^3*b^5 + 1 0*A*a^4*b^4 + 36*B*a^2*b^6 - 96*B*a^3*b^5 - 14*B*a^4*b^4 + 59*B*a^5*b^3 + 3*B*a^6*b^2 - 72*C*a^2*b^6 - 60*C*a^3*b^5 + 273*C*a^4*b^4 + 47*C*a^5*b^3 - 236*C*a^6*b^2 - 18*A*a*b^7 - 18*B*a^7*b - 12*C*a^7*b))/(3*b^4*(a + b)^2*( a - b)^3) + (tan(c/2 + (d*x)/2)^5*(72*C*a^8 + 18*C*b^8 + 45*A*a^2*b^6 + 7* A*a^3*b^5 + 10*A*a^4*b^4 - 36*B*a^2*b^6 - 96*B*a^3*b^5 + 14*B*a^4*b^4 + 59 *B*a^5*b^3 - 3*B*a^6*b^2 - 72*C*a^2*b^6 + 60*C*a^3*b^5 + 273*C*a^4*b^4 - 4 7*C*a^5*b^3 - 236*C*a^6*b^2 + 18*A*a*b^7 - 18*B*a^7*b + 12*C*a^7*b))/(3*b^ 4*(a + b)^3*(a - b)^2))/(d*(3*a*b^2 + 3*a^2*b - tan(c/2 + (d*x)/2)^4*(6*a* b^2 - 6*a^3) + tan(c/2 + (d*x)/2)^2*(6*a^2*b + 4*a^3 - 2*b^3) + tan(c/2 + (d*x)/2)^6*(4*a^3 - 6*a^2*b + 2*b^3) + a^3 + b^3 + tan(c/2 + (d*x)/2)^8*(3 *a*b^2 - 3*a^2*b + a^3 - b^3))) + (log(tan(c/2 + (d*x)/2) + 1i)*(B*b - 4*C *a)*1i)/(b^5*d) - (log(tan(c/2 + (d*x)/2) - 1i)*(B*b*1i - C*a*4i))/(b^5...
Time = 0.20 (sec) , antiderivative size = 4628, normalized size of antiderivative = 10.04 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx =\text {Too large to display} \] Input:
int(cos(d*x+c)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^4,x)
Output:
(48*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**8*b**3*c - 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*a**7*b**5 - 168*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**6*b**5*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**5*b**7 + 210*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**7*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**3*b**9 - 120*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**9*c + 36*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*b**11 - 14 4*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**10*b*c + 36*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**9*b**3 + 456*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**3*c - 114*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*...