Integrand size = 41, antiderivative size = 314 \[ \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\frac {(b B-3 a C) x}{b^4}+\frac {\left (2 A b^6-2 a^5 b B+5 a^3 b^3 B-6 a b^5 B+6 a^6 C-15 a^4 b^2 C+a^2 b^4 (A+12 C)\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} b^4 (a+b)^{5/2} d}+\frac {\left (A b^2-a b B+3 a^2 C-2 b^2 C\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac {a \left (2 A b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))} \]
(B*b-3*C*a)*x/b^4+(2*A*b^6-2*B*a^5*b+5*B*a^3*b^3-6*B*a*b^5+6*a^6*C-15*a^4* b^2*C+a^2*b^4*(A+12*C))*arctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2)) /(a-b)^(5/2)/b^4/(a+b)^(5/2)/d+1/2*(A*b^2-B*a*b+3*C*a^2-2*C*b^2)*sin(d*x+c )/b^3/(a^2-b^2)/d-1/2*(A*b^2-a*(B*b-C*a))*cos(d*x+c)^2*sin(d*x+c)/b/(a^2-b ^2)/d/(a+b*cos(d*x+c))^2-1/2*a*(2*A*b^4+B*a^3*b-4*B*a*b^3-3*a^4*C+a^2*b^2* (A+6*C))*sin(d*x+c)/b^3/(a^2-b^2)^2/d/(a+b*cos(d*x+c))
Time = 5.00 (sec) , antiderivative size = 573, normalized size of antiderivative = 1.82 \[ \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\frac {-\frac {4 \left (2 A b^6-2 a^5 b B+5 a^3 b^3 B-6 a b^5 B+6 a^6 C-15 a^4 b^2 C+a^2 b^4 (A+12 C)\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{5/2}}+\frac {4 a^6 b B c-6 a^4 b^3 B c+2 b^7 B c-12 a^7 c C+18 a^5 b^2 c C-6 a b^6 c C+4 a^6 b B d x-6 a^4 b^3 B d x+2 b^7 B d x-12 a^7 C d x+18 a^5 b^2 C d x-6 a b^6 C d x-8 a b \left (a^2-b^2\right )^2 (-b B+3 a C) (c+d x) \cos (c+d x)+2 \left (-a^2 b+b^3\right )^2 (b B-3 a C) (c+d x) \cos (2 (c+d x))-6 a^2 A b^5 \sin (c+d x)-4 a^5 b^2 B \sin (c+d x)+10 a^3 b^4 B \sin (c+d x)+12 a^6 b C \sin (c+d x)-21 a^4 b^3 C \sin (c+d x)+2 a^2 b^5 C \sin (c+d x)+b^7 C \sin (c+d x)+a^3 A b^4 \sin (2 (c+d x))-4 a A b^6 \sin (2 (c+d x))-3 a^4 b^3 B \sin (2 (c+d x))+6 a^2 b^5 B \sin (2 (c+d x))+9 a^5 b^2 C \sin (2 (c+d x))-16 a^3 b^4 C \sin (2 (c+d x))+4 a b^6 C \sin (2 (c+d x))+a^4 b^3 C \sin (3 (c+d x))-2 a^2 b^5 C \sin (3 (c+d x))+b^7 C \sin (3 (c+d x))}{\left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}}{4 b^4 d} \]
((-4*(2*A*b^6 - 2*a^5*b*B + 5*a^3*b^3*B - 6*a*b^5*B + 6*a^6*C - 15*a^4*b^2 *C + a^2*b^4*(A + 12*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^ 2]])/(-a^2 + b^2)^(5/2) + (4*a^6*b*B*c - 6*a^4*b^3*B*c + 2*b^7*B*c - 12*a^ 7*c*C + 18*a^5*b^2*c*C - 6*a*b^6*c*C + 4*a^6*b*B*d*x - 6*a^4*b^3*B*d*x + 2 *b^7*B*d*x - 12*a^7*C*d*x + 18*a^5*b^2*C*d*x - 6*a*b^6*C*d*x - 8*a*b*(a^2 - b^2)^2*(-(b*B) + 3*a*C)*(c + d*x)*Cos[c + d*x] + 2*(-(a^2*b) + b^3)^2*(b *B - 3*a*C)*(c + d*x)*Cos[2*(c + d*x)] - 6*a^2*A*b^5*Sin[c + d*x] - 4*a^5* b^2*B*Sin[c + d*x] + 10*a^3*b^4*B*Sin[c + d*x] + 12*a^6*b*C*Sin[c + d*x] - 21*a^4*b^3*C*Sin[c + d*x] + 2*a^2*b^5*C*Sin[c + d*x] + b^7*C*Sin[c + d*x] + a^3*A*b^4*Sin[2*(c + d*x)] - 4*a*A*b^6*Sin[2*(c + d*x)] - 3*a^4*b^3*B*S in[2*(c + d*x)] + 6*a^2*b^5*B*Sin[2*(c + d*x)] + 9*a^5*b^2*C*Sin[2*(c + d* x)] - 16*a^3*b^4*C*Sin[2*(c + d*x)] + 4*a*b^6*C*Sin[2*(c + d*x)] + a^4*b^3 *C*Sin[3*(c + d*x)] - 2*a^2*b^5*C*Sin[3*(c + d*x)] + b^7*C*Sin[3*(c + d*x) ])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2))/(4*b^4*d)
Time = 1.63 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.12, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.293, Rules used = {3042, 3526, 3042, 3510, 25, 3042, 3502, 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.
| \(\displaystyle \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle -\frac {\int \frac {\cos (c+d x) \left (-\left (\left (3 C a^2-b B a+A b^2-2 b^2 C\right ) \cos ^2(c+d x)\right )+2 b (b B-a (A+C)) \cos (c+d x)+2 \left (A b^2-a (b B-a C)\right )\right )}{(a+b \cos (c+d x))^2}dx}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (\left (-3 C a^2+b B a-A b^2+2 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 b (b B-a (A+C)) \sin \left (c+d x+\frac {\pi }{2}\right )+2 \left (A b^2-a (b B-a C)\right )\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle -\frac {\frac {\int -\frac {b \left (a^2-b^2\right ) \left (3 C a^2-b B a+A b^2-2 b^2 C\right ) \cos ^2(c+d x)+\left (a^2-b^2\right ) \left (-3 C a^3+b B a^2+b^2 (A+4 C) a-2 b^3 B\right ) \cos (c+d x)+b \left (-3 C a^4+b B a^3+b^2 (A+6 C) a^2-4 b^3 B a+2 A b^4\right )}{a+b \cos (c+d x)}dx}{b^2 \left (a^2-b^2\right )}+\frac {a \sin (c+d x) \left (-3 a^4 C+a^3 b B+a^2 b^2 (A+6 C)-4 a b^3 B+2 A b^4\right )}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {a \sin (c+d x) \left (-3 a^4 C+a^3 b B+a^2 b^2 (A+6 C)-4 a b^3 B+2 A b^4\right )}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\int \frac {b \left (a^2-b^2\right ) \left (3 C a^2-b B a+A b^2-2 b^2 C\right ) \cos ^2(c+d x)+\left (a^2-b^2\right ) \left (-3 C a^3+b B a^2+b^2 (A+4 C) a-2 b^3 B\right ) \cos (c+d x)+b \left (-3 C a^4+b B a^3+b^2 (A+6 C) a^2-4 b^3 B a+2 A b^4\right )}{a+b \cos (c+d x)}dx}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {a \sin (c+d x) \left (-3 a^4 C+a^3 b B+a^2 b^2 (A+6 C)-4 a b^3 B+2 A b^4\right )}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\int \frac {b \left (a^2-b^2\right ) \left (3 C a^2-b B a+A b^2-2 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (a^2-b^2\right ) \left (-3 C a^3+b B a^2+b^2 (A+4 C) a-2 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+b \left (-3 C a^4+b B a^3+b^2 (A+6 C) a^2-4 b^3 B a+2 A b^4\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle -\frac {\frac {a \sin (c+d x) \left (-3 a^4 C+a^3 b B+a^2 b^2 (A+6 C)-4 a b^3 B+2 A b^4\right )}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\int \frac {\left (-3 C a^4+b B a^3+b^2 (A+6 C) a^2-4 b^3 B a+2 A b^4\right ) b^2+2 \left (a^2-b^2\right )^2 (b B-3 a C) \cos (c+d x) b}{a+b \cos (c+d x)}dx}{b}+\frac {\left (a^2-b^2\right ) \sin (c+d x) \left (3 a^2 C-a b B+A b^2-2 b^2 C\right )}{d}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {a \sin (c+d x) \left (-3 a^4 C+a^3 b B+a^2 b^2 (A+6 C)-4 a b^3 B+2 A b^4\right )}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\int \frac {\left (-3 C a^4+b B a^3+b^2 (A+6 C) a^2-4 b^3 B a+2 A b^4\right ) b^2+2 \left (a^2-b^2\right )^2 (b B-3 a C) \sin \left (c+d x+\frac {\pi }{2}\right ) b}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}+\frac {\left (a^2-b^2\right ) \sin (c+d x) \left (3 a^2 C-a b B+A b^2-2 b^2 C\right )}{d}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle -\frac {\frac {a \sin (c+d x) \left (-3 a^4 C+a^3 b B+a^2 b^2 (A+6 C)-4 a b^3 B+2 A b^4\right )}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\left (6 a^6 C-2 a^5 b B-15 a^4 b^2 C+5 a^3 b^3 B+a^2 b^4 (A+12 C)-6 a b^5 B+2 A b^6\right ) \int \frac {1}{a+b \cos (c+d x)}dx+2 x \left (a^2-b^2\right )^2 (b B-3 a C)}{b}+\frac {\left (a^2-b^2\right ) \sin (c+d x) \left (3 a^2 C-a b B+A b^2-2 b^2 C\right )}{d}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {a \sin (c+d x) \left (-3 a^4 C+a^3 b B+a^2 b^2 (A+6 C)-4 a b^3 B+2 A b^4\right )}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\left (6 a^6 C-2 a^5 b B-15 a^4 b^2 C+5 a^3 b^3 B+a^2 b^4 (A+12 C)-6 a b^5 B+2 A b^6\right ) \int \frac {1}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx+2 x \left (a^2-b^2\right )^2 (b B-3 a C)}{b}+\frac {\left (a^2-b^2\right ) \sin (c+d x) \left (3 a^2 C-a b B+A b^2-2 b^2 C\right )}{d}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle -\frac {\frac {a \sin (c+d x) \left (-3 a^4 C+a^3 b B+a^2 b^2 (A+6 C)-4 a b^3 B+2 A b^4\right )}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\frac {2 \left (6 a^6 C-2 a^5 b B-15 a^4 b^2 C+5 a^3 b^3 B+a^2 b^4 (A+12 C)-6 a b^5 B+2 A b^6\right ) \int \frac {1}{(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{d}+2 x \left (a^2-b^2\right )^2 (b B-3 a C)}{b}+\frac {\left (a^2-b^2\right ) \sin (c+d x) \left (3 a^2 C-a b B+A b^2-2 b^2 C\right )}{d}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {a \sin (c+d x) \left (-3 a^4 C+a^3 b B+a^2 b^2 (A+6 C)-4 a b^3 B+2 A b^4\right )}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\left (a^2-b^2\right ) \sin (c+d x) \left (3 a^2 C-a b B+A b^2-2 b^2 C\right )}{d}+\frac {2 x \left (a^2-b^2\right )^2 (b B-3 a C)+\frac {2 \left (6 a^6 C-2 a^5 b B-15 a^4 b^2 C+5 a^3 b^3 B+a^2 b^4 (A+12 C)-6 a b^5 B+2 A b^6\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d \sqrt {a-b} \sqrt {a+b}}}{b}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}\) |
-1/2*((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)* d*(a + b*Cos[c + d*x])^2) - ((a*(2*A*b^4 + a^3*b*B - 4*a*b^3*B - 3*a^4*C + a^2*b^2*(A + 6*C))*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])) - ((2*(a^2 - b^2)^2*(b*B - 3*a*C)*x + (2*(2*A*b^6 - 2*a^5*b*B + 5*a^3*b^3 *B - 6*a*b^5*B + 6*a^6*C - 15*a^4*b^2*C + a^2*b^4*(A + 12*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)*(A*b^2 - a*b*B + 3*a^2*C - 2*b^2*C)*Sin[c + d*x])/d)/(b^2*(a^2 - b^2)))/(2*b*(a^2 - b^2))
3.10.95.3.1 Defintions of rubi rules used
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 = 0.78 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.25
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {2 \left (-\frac {\left (a A \,b^{3}+4 A \,b^{4}+2 B \,a^{3} b -B \,a^{2} b^{2}-6 B a \,b^{3}-4 a^{4} C +a^{3} b C +8 C \,a^{2} b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {b a \left (a A \,b^{3}-4 A \,b^{4}-2 B \,a^{3} b -B \,a^{2} b^{2}+6 B a \,b^{3}+4 a^{4} C +a^{3} b C -8 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}\right )}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right )}^{2}}+\frac {\left (a^{2} A \,b^{4}+2 A \,b^{6}-2 B \,a^{5} b +5 B \,a^{3} b^{3}-6 B a \,b^{5}+6 a^{6} C -15 a^{4} b^{2} C +12 a^{2} C \,b^{4}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a^{4}-2 b^{2} a^{2}+b^{4}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}}{b^{4}}+\frac {\frac {2 C b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+2 \left (B b -3 C a \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{4}}}{d}\) | \(391\) |
| default | \(\frac {\frac {\frac {2 \left (-\frac {\left (a A \,b^{3}+4 A \,b^{4}+2 B \,a^{3} b -B \,a^{2} b^{2}-6 B a \,b^{3}-4 a^{4} C +a^{3} b C +8 C \,a^{2} b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {b a \left (a A \,b^{3}-4 A \,b^{4}-2 B \,a^{3} b -B \,a^{2} b^{2}+6 B a \,b^{3}+4 a^{4} C +a^{3} b C -8 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}\right )}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right )}^{2}}+\frac {\left (a^{2} A \,b^{4}+2 A \,b^{6}-2 B \,a^{5} b +5 B \,a^{3} b^{3}-6 B a \,b^{5}+6 a^{6} C -15 a^{4} b^{2} C +12 a^{2} C \,b^{4}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a^{4}-2 b^{2} a^{2}+b^{4}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}}{b^{4}}+\frac {\frac {2 C b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+2 \left (B b -3 C a \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{4}}}{d}\) | \(391\) |
| risch | \(\text {Expression too large to display}\) | \(1875\) |
1/d*(2/b^4*((-1/2*(A*a*b^3+4*A*b^4+2*B*a^3*b-B*a^2*b^2-6*B*a*b^3-4*C*a^4+C *a^3*b+8*C*a^2*b^2)*a*b/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3+1/2*b*a *(A*a*b^3-4*A*b^4-2*B*a^3*b-B*a^2*b^2+6*B*a*b^3+4*C*a^4+C*a^3*b-8*C*a^2*b^ 2)/(a+b)/(a-b)^2*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)^2+1/2*(A*a^2*b^4+2*A*b^6-2*B*a^5*b+5*B*a^3*b^3-6*B*a*b^5+6* C*a^6-15*C*a^4*b^2+12*C*a^2*b^4)/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*a rctan((a-b)*tan(1/2*d*x+1/2*c)/((a-b)*(a+b))^(1/2)))+2/b^4*(C*b*tan(1/2*d* x+1/2*c)/(1+tan(1/2*d*x+1/2*c)^2)+(B*b-3*C*a)*arctan(tan(1/2*d*x+1/2*c))))
Leaf count of result is larger than twice the leaf count of optimal. 799 vs. \(2 (302) = 604\).
Time = 0.40 (sec) , antiderivative size = 1666, normalized size of antiderivative = 5.31 \[ \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Too large to display} \]
integrate(cos(d*x+c)^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^3, x, algorithm="fricas")
[-1/4*(4*(3*C*a^7*b^2 - B*a^6*b^3 - 9*C*a^5*b^4 + 3*B*a^4*b^5 + 9*C*a^3*b^ 6 - 3*B*a^2*b^7 - 3*C*a*b^8 + B*b^9)*d*x*cos(d*x + c)^2 + 8*(3*C*a^8*b - B *a^7*b^2 - 9*C*a^6*b^3 + 3*B*a^5*b^4 + 9*C*a^4*b^5 - 3*B*a^3*b^6 - 3*C*a^2 *b^7 + B*a*b^8)*d*x*cos(d*x + c) + 4*(3*C*a^9 - B*a^8*b - 9*C*a^7*b^2 + 3* B*a^6*b^3 + 9*C*a^5*b^4 - 3*B*a^4*b^5 - 3*C*a^3*b^6 + B*a^2*b^7)*d*x + (6* C*a^8 - 2*B*a^7*b - 15*C*a^6*b^2 + 5*B*a^5*b^3 + (A + 12*C)*a^4*b^4 - 6*B* a^3*b^5 + 2*A*a^2*b^6 + (6*C*a^6*b^2 - 2*B*a^5*b^3 - 15*C*a^4*b^4 + 5*B*a^ 3*b^5 + (A + 12*C)*a^2*b^6 - 6*B*a*b^7 + 2*A*b^8)*cos(d*x + c)^2 + 2*(6*C* a^7*b - 2*B*a^6*b^2 - 15*C*a^5*b^3 + 5*B*a^4*b^4 + (A + 12*C)*a^3*b^5 - 6* B*a^2*b^6 + 2*A*a*b^7)*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*(6*C*a^8*b - 2*B*a^7*b^2 - 17*C*a^6*b^3 + 7*B*a^5*b^4 - (3*A - 1 3*C)*a^4*b^5 - 5*B*a^3*b^6 + (3*A - 2*C)*a^2*b^7 + 2*(C*a^6*b^3 - 3*C*a^4* b^5 + 3*C*a^2*b^7 - C*b^9)*cos(d*x + c)^2 + (9*C*a^7*b^2 - 3*B*a^6*b^3 + ( A - 25*C)*a^5*b^4 + 9*B*a^4*b^5 - 5*(A - 4*C)*a^3*b^6 - 6*B*a^2*b^7 + 4*(A - C)*a*b^8)*cos(d*x + c))*sin(d*x + c))/((a^6*b^6 - 3*a^4*b^8 + 3*a^2*b^1 0 - b^12)*d*cos(d*x + c)^2 + 2*(a^7*b^5 - 3*a^5*b^7 + 3*a^3*b^9 - a*b^11)* d*cos(d*x + c) + (a^8*b^4 - 3*a^6*b^6 + 3*a^4*b^8 - a^2*b^10)*d), -1/2*(2* (3*C*a^7*b^2 - B*a^6*b^3 - 9*C*a^5*b^4 + 3*B*a^4*b^5 + 9*C*a^3*b^6 - 3*...
Timed out. \[ \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
integrate(cos(d*x+c)^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^3, x, algorithm="maxima")
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. 666 vs. \(2 (302) = 604\).
Time = 0.37 (sec) , antiderivative size = 666, normalized size of antiderivative = 2.12 \[ \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=-\frac {\frac {{\left (6 \, C a^{6} - 2 \, B a^{5} b - 15 \, C a^{4} b^{2} + 5 \, B a^{3} b^{3} + A a^{2} b^{4} + 12 \, C a^{2} b^{4} - 6 \, B a b^{5} + 2 \, A b^{6}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} \sqrt {a^{2} - b^{2}}} - \frac {4 \, C a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, C a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, B a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, C a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - A a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5 \, B a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, C a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, A a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, B a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, A a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, C a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, B a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, C a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, B a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 7 \, C a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + A a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, B a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, C a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, A a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, B a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, A a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b\right )}^{2}} + \frac {{\left (3 \, C a - B b\right )} {\left (d x + c\right )}}{b^{4}} - \frac {2 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} b^{3}}}{d} \]
integrate(cos(d*x+c)^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^3, x, algorithm="giac")
-((6*C*a^6 - 2*B*a^5*b - 15*C*a^4*b^2 + 5*B*a^3*b^3 + A*a^2*b^4 + 12*C*a^2 *b^4 - 6*B*a*b^5 + 2*A*b^6)*(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^4*b^4 - 2*a^2*b^6 + b^8)*sqrt(a^2 - b^2)) - (4*C*a^6*tan(1/2*d *x + 1/2*c)^3 - 2*B*a^5*b*tan(1/2*d*x + 1/2*c)^3 - 5*C*a^5*b*tan(1/2*d*x + 1/2*c)^3 + 3*B*a^4*b^2*tan(1/2*d*x + 1/2*c)^3 - 7*C*a^4*b^2*tan(1/2*d*x + 1/2*c)^3 - A*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 + 5*B*a^3*b^3*tan(1/2*d*x + 1 /2*c)^3 + 8*C*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 - 3*A*a^2*b^4*tan(1/2*d*x + 1 /2*c)^3 - 6*B*a^2*b^4*tan(1/2*d*x + 1/2*c)^3 + 4*A*a*b^5*tan(1/2*d*x + 1/2 *c)^3 + 4*C*a^6*tan(1/2*d*x + 1/2*c) - 2*B*a^5*b*tan(1/2*d*x + 1/2*c) + 5* C*a^5*b*tan(1/2*d*x + 1/2*c) - 3*B*a^4*b^2*tan(1/2*d*x + 1/2*c) - 7*C*a^4* b^2*tan(1/2*d*x + 1/2*c) + A*a^3*b^3*tan(1/2*d*x + 1/2*c) + 5*B*a^3*b^3*ta n(1/2*d*x + 1/2*c) - 8*C*a^3*b^3*tan(1/2*d*x + 1/2*c) - 3*A*a^2*b^4*tan(1/ 2*d*x + 1/2*c) + 6*B*a^2*b^4*tan(1/2*d*x + 1/2*c) - 4*A*a*b^5*tan(1/2*d*x + 1/2*c))/((a^4*b^3 - 2*a^2*b^5 + b^7)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1 /2*d*x + 1/2*c)^2 + a + b)^2) + (3*C*a - B*b)*(d*x + c)/b^4 - 2*C*tan(1/2* d*x + 1/2*c)/((tan(1/2*d*x + 1/2*c)^2 + 1)*b^3))/d
Time = 9.35 (sec) , antiderivative size = 6721, normalized size of antiderivative = 21.40 \[ \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Too large to display} \]
(log(tan(c/2 + (d*x)/2) + 1i)*(B*b - 3*C*a)*1i)/(b^4*d) - ((tan(c/2 + (d*x )/2)^5*(2*C*b^5 - 6*C*a^5 + A*a^2*b^3 - 6*B*a^2*b^3 - B*a^3*b^2 - 4*C*a^2* b^3 + 12*C*a^3*b^2 + 4*A*a*b^4 + 2*B*a^4*b - 2*C*a*b^4 + 3*C*a^4*b))/((a*b ^3 - b^4)*(a + b)^2) - (tan(c/2 + (d*x)/2)*(6*C*a^5 + 2*C*b^5 + A*a^2*b^3 + 6*B*a^2*b^3 - B*a^3*b^2 - 4*C*a^2*b^3 - 12*C*a^3*b^2 - 4*A*a*b^4 - 2*B*a ^4*b + 2*C*a*b^4 + 3*C*a^4*b))/((a + b)*(b^5 - 2*a*b^4 + a^2*b^3)) + (2*ta n(c/2 + (d*x)/2)^3*(2*C*b^6 - 6*C*a^6 + 3*A*a^2*b^4 - 5*B*a^3*b^3 - 6*C*a^ 2*b^4 + 13*C*a^4*b^2 + 2*B*a^5*b))/(b*(a*b^2 - b^3)*(a + b)^2*(a - b)))/(d *(2*a*b + tan(c/2 + (d*x)/2)^2*(2*a*b + 3*a^2 - b^2) + tan(c/2 + (d*x)/2)^ 6*(a^2 - 2*a*b + b^2) + a^2 + b^2 - tan(c/2 + (d*x)/2)^4*(2*a*b - 3*a^2 + b^2))) - (log(tan(c/2 + (d*x)/2) - 1i)*(B*b*1i - C*a*3i))/(b^4*d) - (atan( ((((8*tan(c/2 + (d*x)/2)*(4*A^2*b^12 + 4*B^2*b^12 + 72*C^2*a^12 - 8*B^2*a* b^11 - 72*C^2*a^11*b + 4*A^2*a^2*b^10 + A^2*a^4*b^8 + 24*B^2*a^2*b^10 + 32 *B^2*a^3*b^9 - 52*B^2*a^4*b^8 - 48*B^2*a^5*b^7 + 57*B^2*a^6*b^6 + 32*B^2*a ^7*b^5 - 32*B^2*a^8*b^4 - 8*B^2*a^9*b^3 + 8*B^2*a^10*b^2 + 36*C^2*a^2*b^10 - 72*C^2*a^3*b^9 + 36*C^2*a^4*b^8 + 288*C^2*a^5*b^7 - 288*C^2*a^6*b^6 - 4 32*C^2*a^7*b^5 + 441*C^2*a^8*b^4 + 288*C^2*a^9*b^3 - 288*C^2*a^10*b^2 - 24 *A*B*a*b^11 - 24*B*C*a*b^11 - 48*B*C*a^11*b + 8*A*B*a^3*b^9 + 2*A*B*a^5*b^ 7 - 4*A*B*a^7*b^5 + 48*A*C*a^2*b^10 - 36*A*C*a^4*b^8 - 6*A*C*a^6*b^6 + 12* A*C*a^8*b^4 + 48*B*C*a^2*b^10 - 72*B*C*a^3*b^9 - 192*B*C*a^4*b^8 + 252*...