Integrand size = 14, antiderivative size = 223 \[ \int \frac {1}{a+b \cos (x)+c \cos ^2(x)} \, dx=\frac {4 c \arctan \left (\frac {\sqrt {b-2 c-\sqrt {b^2-4 a c}} \tan \left (\frac {x}{2}\right )}{\sqrt {b+2 c-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-2 c-\sqrt {b^2-4 a c}} \sqrt {b+2 c-\sqrt {b^2-4 a c}}}-\frac {4 c \arctan \left (\frac {\sqrt {b-2 c+\sqrt {b^2-4 a c}} \tan \left (\frac {x}{2}\right )}{\sqrt {b+2 c+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-2 c+\sqrt {b^2-4 a c}} \sqrt {b+2 c+\sqrt {b^2-4 a c}}} \]
4*c*arctan((b-2*c-(-4*a*c+b^2)^(1/2))^(1/2)*tan(1/2*x)/(b+2*c-(-4*a*c+b^2) ^(1/2))^(1/2))/(-4*a*c+b^2)^(1/2)/(b-2*c-(-4*a*c+b^2)^(1/2))^(1/2)/(b+2*c- (-4*a*c+b^2)^(1/2))^(1/2)-4*c*arctan((b-2*c+(-4*a*c+b^2)^(1/2))^(1/2)*tan( 1/2*x)/(b+2*c+(-4*a*c+b^2)^(1/2))^(1/2))/(-4*a*c+b^2)^(1/2)/(b-2*c+(-4*a*c +b^2)^(1/2))^(1/2)/(b+2*c+(-4*a*c+b^2)^(1/2))^(1/2)
Time = 0.66 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.89 \[ \int \frac {1}{a+b \cos (x)+c \cos ^2(x)} \, dx=\frac {2 \sqrt {2} c \left (\frac {\text {arctanh}\left (\frac {\left (b-2 c+\sqrt {b^2-4 a c}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {-2 b^2+4 c (a+c)-2 b \sqrt {b^2-4 a c}}}\right )}{\sqrt {-b^2+2 c (a+c)-b \sqrt {b^2-4 a c}}}+\frac {\text {arctanh}\left (\frac {\left (-b+2 c+\sqrt {b^2-4 a c}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {-2 b^2+4 c (a+c)+2 b \sqrt {b^2-4 a c}}}\right )}{\sqrt {-b^2+2 c (a+c)+b \sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c}} \]
(2*Sqrt[2]*c*(ArcTanh[((b - 2*c + Sqrt[b^2 - 4*a*c])*Tan[x/2])/Sqrt[-2*b^2 + 4*c*(a + c) - 2*b*Sqrt[b^2 - 4*a*c]]]/Sqrt[-b^2 + 2*c*(a + c) - b*Sqrt[ b^2 - 4*a*c]] + ArcTanh[((-b + 2*c + Sqrt[b^2 - 4*a*c])*Tan[x/2])/Sqrt[-2* b^2 + 4*c*(a + c) + 2*b*Sqrt[b^2 - 4*a*c]]]/Sqrt[-b^2 + 2*c*(a + c) + b*Sq rt[b^2 - 4*a*c]]))/Sqrt[b^2 - 4*a*c]
Time = 0.46 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3042, 3730, 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.
| \(\displaystyle \int \frac {1}{a+b \cos (x)+c \cos ^2(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{a+b \cos (x)+c \cos (x)^2}dx\) |
| \(\Big \downarrow \) 3730 |
| \(\displaystyle \frac {2 c \int \frac {1}{b+2 c \cos (x)-\sqrt {b^2-4 a c}}dx}{\sqrt {b^2-4 a c}}-\frac {2 c \int \frac {1}{b+2 c \cos (x)+\sqrt {b^2-4 a c}}dx}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 c \int \frac {1}{b+2 c \sin \left (x+\frac {\pi }{2}\right )-\sqrt {b^2-4 a c}}dx}{\sqrt {b^2-4 a c}}-\frac {2 c \int \frac {1}{b+2 c \sin \left (x+\frac {\pi }{2}\right )+\sqrt {b^2-4 a c}}dx}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {4 c \int \frac {1}{\left (b-2 c-\sqrt {b^2-4 a c}\right ) \tan ^2\left (\frac {x}{2}\right )+b+2 c-\sqrt {b^2-4 a c}}d\tan \left (\frac {x}{2}\right )}{\sqrt {b^2-4 a c}}-\frac {4 c \int \frac {1}{\left (b-2 c+\sqrt {b^2-4 a c}\right ) \tan ^2\left (\frac {x}{2}\right )+b+2 c+\sqrt {b^2-4 a c}}d\tan \left (\frac {x}{2}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {4 c \arctan \left (\frac {\tan \left (\frac {x}{2}\right ) \sqrt {-\sqrt {b^2-4 a c}+b-2 c}}{\sqrt {-\sqrt {b^2-4 a c}+b+2 c}}\right )}{\sqrt {b^2-4 a c} \sqrt {-\sqrt {b^2-4 a c}+b-2 c} \sqrt {-\sqrt {b^2-4 a c}+b+2 c}}-\frac {4 c \arctan \left (\frac {\tan \left (\frac {x}{2}\right ) \sqrt {\sqrt {b^2-4 a c}+b-2 c}}{\sqrt {\sqrt {b^2-4 a c}+b+2 c}}\right )}{\sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b-2 c} \sqrt {\sqrt {b^2-4 a c}+b+2 c}}\) |
(4*c*ArcTan[(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c - Sq rt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Sq rt[b + 2*c - Sqrt[b^2 - 4*a*c]]) - (4*c*ArcTan[(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*S qrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]])
3.1.17.3.1 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_.) + cos[(d_.) + (e_.)*(x_)]^(n_.)*(b_.) + cos[(d_.) + (e_.)*(x_)]^ (n2_.)*(c_.))^(-1), x_Symbol] :> Module[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c /q) Int[1/(b - q + 2*c*Cos[d + e*x]^n), x], x] - Simp[2*(c/q) Int[1/(b + q + 2*c*Cos[d + e*x]^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n 2, 2*n] && NeQ[b^2 - 4*a*c, 0]
Time = 1.12 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.91
| method | result | size |
| default | \(2 \left (a -b +c \right ) \left (\frac {\left (\sqrt {-4 a c +b^{2}}+2 c -b \right ) \operatorname {arctanh}\left (\frac {\left (-a +b -c \right ) \tan \left (\frac {x}{2}\right )}{\sqrt {\left (\sqrt {-4 a c +b^{2}}-a +c \right ) \left (a -b +c \right )}}\right )}{2 \sqrt {-4 a c +b^{2}}\, \left (a -b +c \right ) \sqrt {\left (\sqrt {-4 a c +b^{2}}-a +c \right ) \left (a -b +c \right )}}+\frac {\left (b -2 c +\sqrt {-4 a c +b^{2}}\right ) \arctan \left (\frac {\left (a -b +c \right ) \tan \left (\frac {x}{2}\right )}{\sqrt {\left (\sqrt {-4 a c +b^{2}}+a -c \right ) \left (a -b +c \right )}}\right )}{2 \sqrt {-4 a c +b^{2}}\, \left (a -b +c \right ) \sqrt {\left (\sqrt {-4 a c +b^{2}}+a -c \right ) \left (a -b +c \right )}}\right )\) | \(204\) |
| risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (16 a^{4} c^{2}-8 a^{3} b^{2} c +32 c^{3} a^{3}+a^{2} b^{4}-32 a^{2} b^{2} c^{2}+16 a^{2} c^{4}+10 a \,b^{4} c -8 a \,b^{2} c^{3}-b^{6}+b^{4} c^{2}\right ) \textit {\_Z}^{4}+\left (8 a^{2} c^{2}-6 a \,b^{2} c +8 a \,c^{3}+b^{4}-2 b^{2} c^{2}\right ) \textit {\_Z}^{2}+c^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i x}+\left (\frac {i b^{3} a^{2}}{c^{2}}+\frac {24 i c^{2} a^{2}}{b}+\frac {8 i b^{3} a}{c}-18 i a b c +\frac {8 i c^{3} a}{b}-\frac {6 i b \,a^{3}}{c}+\frac {24 i c \,a^{3}}{b}-\frac {i b^{5}}{c^{2}}-2 i b \,c^{2}+\frac {8 i a^{4}}{b}-22 i b \,a^{2}+3 i b^{3}\right ) \textit {\_R}^{3}+\left (-\frac {4 a^{3}}{b}+\frac {a^{2} b}{c}-\frac {8 a^{2} c}{b}+6 a b -\frac {4 c^{2} a}{b}-\frac {b^{3}}{c}+c b \right ) \textit {\_R}^{2}+\left (-\frac {4 i b a}{c}+\frac {4 i a c}{b}+\frac {i b^{3}}{c^{2}}+\frac {2 i c^{2}}{b}+\frac {2 i a^{2}}{b}-2 i b \right ) \textit {\_R} +\frac {b}{c}-\frac {a}{b}-\frac {c}{b}\right )\) | \(373\) |
2*(a-b+c)*(1/2*((-4*a*c+b^2)^(1/2)+2*c-b)/(-4*a*c+b^2)^(1/2)/(a-b+c)/(((-4 *a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2)*arctanh((-a+b-c)*tan(1/2*x)/(((-4*a*c+ b^2)^(1/2)-a+c)*(a-b+c))^(1/2))+1/2*(b-2*c+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2 )^(1/2)/(a-b+c)/(((-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2)*arctan((a-b+c)*ta n(1/2*x)/(((-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 3493 vs. \(2 (183) = 366\).
Time = 0.52 (sec) , antiderivative size = 3493, normalized size of antiderivative = 15.66 \[ \int \frac {1}{a+b \cos (x)+c \cos ^2(x)} \, dx=\text {Too large to display} \]
1/4*sqrt(2)*sqrt(-(b^2 - 2*a*c - 2*c^2 - (a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c)*sqrt(b^2/(a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2 *b^2 + b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c)))/(a^2*b^2 - b^4 - 4*a* c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c))*log(b^2*c*cos(x) + 2*b*c ^2 - (4*a*c^4 + (8*a^2 - b^2)*c^3 + 2*(2*a^3 - 3*a*b^2)*c^2 - (a^2*b^2 - b ^4)*c)*sqrt(b^2/(a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^2 - 4*(a^5 - 3*a ^3*b^2 + 2*a*b^4)*c))*cos(x) + 1/2*sqrt(2)*((a^2*b^4 - b^6 + 8*a*c^5 + 2*( 12*a^2 - b^2)*c^4 + 6*(4*a^3 - 3*a*b^2)*c^3 + (8*a^4 - 22*a^2*b^2 + 3*b^4) *c^2 - 2*(3*a^3*b^2 - 4*a*b^4)*c)*sqrt(b^2/(a^4*b^2 - 2*a^2*b^4 + b^6 - 4* a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^ 2 + b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c))*sin(x) + (b^4 - 4*a*b^2*c )*sin(x))*sqrt(-(b^2 - 2*a*c - 2*c^2 - (a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c)*sqrt(b^2/(a^4*b^2 - 2*a^2*b^4 + b^6 - 4 *a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b ^2 + b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c)))/(a^2*b^2 - b^4 - 4*a*c^ 3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c))) - 1/4*sqrt(2)*sqrt(-(b^2 - 2*a*c - 2*c^2 - (a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c)*sqrt(b^2/(a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 -...
Timed out. \[ \int \frac {1}{a+b \cos (x)+c \cos ^2(x)} \, dx=\text {Timed out} \]
\[ \int \frac {1}{a+b \cos (x)+c \cos ^2(x)} \, dx=\int { \frac {1}{c \cos \left (x\right )^{2} + b \cos \left (x\right ) + a} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 2994 vs. \(2 (183) = 366\).
Time = 1.46 (sec) , antiderivative size = 2994, normalized size of antiderivative = 13.43 \[ \int \frac {1}{a+b \cos (x)+c \cos ^2(x)} \, dx=\text {Too large to display} \]
-(2*a^2*b^3 - 8*a*b^4 + 6*b^5 - 8*a^3*b*c + 52*a^2*b^2*c - 44*a*b^3*c - 4* b^4*c - 80*a^3*c^2 + 80*a^2*b*c^2 + 40*a*b^2*c^2 - 6*b^3*c^2 - 96*a^2*c^3 + 24*a*b*c^3 + 4*b^2*c^3 - 16*a*c^4 - 3*sqrt(a^2 - a*b + b*c - c^2 + sqrt( b^2 - 4*a*c)*(a - b + c))*a^2*b^2 + 2*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^ 2 - 4*a*c)*(a - b + c))*a*b^3 + 5*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*b^4 + 12*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c )*(a - b + c))*a^3*c - 8*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a^2*b*c - 34*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a*b^2*c - 6*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*b^3*c + 56*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a^2*c^2 + 24*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a*b*c^2 + 5*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c) )*b^2*c^2 - 20*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c)) *a*c^3 + 3*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*sqr t(b^2 - 4*a*c)*a^2*b - 2*(b^2 - 4*a*c)*a^2*b - 2*sqrt(a^2 - a*b + b*c - c^ 2 + sqrt(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*a*b^2 + 8*(b^2 - 4*a* c)*a*b^2 - 5*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*s qrt(b^2 - 4*a*c)*b^3 - 6*(b^2 - 4*a*c)*b^3 + 6*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*a^2*c - 20*(b^2 - 4*a*c )*a^2*c + 10*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c)...
Time = 13.76 (sec) , antiderivative size = 5514, normalized size of antiderivative = 24.73 \[ \int \frac {1}{a+b \cos (x)+c \cos ^2(x)} \, dx=\text {Too large to display} \]
- atan(((tan(x/2)*(32*a*b^2 - 64*a^2*c - 128*b*c^2 + 96*b^2*c - 32*b^3 + 6 4*c^3) + (-(8*a*c^3 + b*(-(4*a*c - b^2)^3)^(1/2) + b^4 + 8*a^2*c^2 - 2*b^2 *c^2 - 6*a*b^2*c)/(2*(a^2*b^4 - b^6 + 16*a^2*c^4 + 32*a^3*c^3 + 16*a^4*c^2 + b^4*c^2 - 8*a*b^2*c^3 - 8*a^3*b^2*c - 32*a^2*b^2*c^2 + 10*a*b^4*c)))^(1 /2)*(64*a*b^3 + 128*a*c^3 + 128*a^3*c + 64*b^3*c - 32*b^4 - 32*a^2*b^2 + 2 56*a^2*c^2 - 32*b^2*c^2 + tan(x/2)*(-(8*a*c^3 + b*(-(4*a*c - b^2)^3)^(1/2) + b^4 + 8*a^2*c^2 - 2*b^2*c^2 - 6*a*b^2*c)/(2*(a^2*b^4 - b^6 + 16*a^2*c^4 + 32*a^3*c^3 + 16*a^4*c^2 + b^4*c^2 - 8*a*b^2*c^3 - 8*a^3*b^2*c - 32*a^2* b^2*c^2 + 10*a*b^4*c)))^(1/2)*(64*a*b^4 + 256*a*c^4 - 256*a^4*c - 64*b^4*c - 128*a^2*b^3 + 64*a^3*b^2 + 256*a^2*c^3 - 256*a^3*c^2 - 64*b^2*c^3 + 128 *b^3*c^2 + 192*a*b^2*c^2 - 192*a^2*b^2*c - 512*a*b*c^3 + 512*a^3*b*c) - 25 6*a*b*c^2 + 64*a*b^2*c - 256*a^2*b*c))*(-(8*a*c^3 + b*(-(4*a*c - b^2)^3)^( 1/2) + b^4 + 8*a^2*c^2 - 2*b^2*c^2 - 6*a*b^2*c)/(2*(a^2*b^4 - b^6 + 16*a^2 *c^4 + 32*a^3*c^3 + 16*a^4*c^2 + b^4*c^2 - 8*a*b^2*c^3 - 8*a^3*b^2*c - 32* a^2*b^2*c^2 + 10*a*b^4*c)))^(1/2)*1i + (tan(x/2)*(32*a*b^2 - 64*a^2*c - 12 8*b*c^2 + 96*b^2*c - 32*b^3 + 64*c^3) - (-(8*a*c^3 + b*(-(4*a*c - b^2)^3)^ (1/2) + b^4 + 8*a^2*c^2 - 2*b^2*c^2 - 6*a*b^2*c)/(2*(a^2*b^4 - b^6 + 16*a^ 2*c^4 + 32*a^3*c^3 + 16*a^4*c^2 + b^4*c^2 - 8*a*b^2*c^3 - 8*a^3*b^2*c - 32 *a^2*b^2*c^2 + 10*a*b^4*c)))^(1/2)*(64*a*b^3 + 128*a*c^3 + 128*a^3*c + 64* b^3*c - 32*b^4 - 32*a^2*b^2 + 256*a^2*c^2 - 32*b^2*c^2 - tan(x/2)*(-(8*...