Integrand size = 32, antiderivative size = 180 \[ \int \frac {B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx=-\frac {\left (3 a b B-a^2 C-2 b^2 C\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} (a+b)^{5/2} d}+\frac {a (b B-a C) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {\left (a^2 b B+2 b^3 B+a^3 C-4 a b^2 C\right ) \sin (c+d x)}{2 b \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))} \]
-(3*B*a*b-C*a^2-2*C*b^2)*arctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2) )/(a-b)^(5/2)/(a+b)^(5/2)/d+1/2*a*(B*b-C*a)*sin(d*x+c)/b/(a^2-b^2)/d/(a+b* cos(d*x+c))^2+1/2*(B*a^2*b+2*B*b^3+C*a^3-4*C*a*b^2)*sin(d*x+c)/b/(a^2-b^2) ^2/d/(a+b*cos(d*x+c))
Time = 1.52 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.96 \[ \int \frac {B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\frac {-\frac {2 \left (-3 a b B+a^2 C+2 b^2 C\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{5/2}}+\frac {a (b B-a C) \sin (c+d x)}{(a-b) b (a+b) (a+b \cos (c+d x))^2}+\frac {\left (a^2 b B+2 b^3 B+a^3 C-4 a b^2 C\right ) \sin (c+d x)}{(a-b)^2 b (a+b)^2 (a+b \cos (c+d x))}}{2 d} \]
((-2*(-3*a*b*B + a^2*C + 2*b^2*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[ -a^2 + b^2]])/(-a^2 + b^2)^(5/2) + (a*(b*B - a*C)*Sin[c + d*x])/((a - b)*b *(a + b)*(a + b*Cos[c + d*x])^2) + ((a^2*b*B + 2*b^3*B + a^3*C - 4*a*b^2*C )*Sin[c + d*x])/((a - b)^2*b*(a + b)^2*(a + b*Cos[c + d*x])))/(2*d)
Time = 0.61 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.14, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {3042, 3500, 3042, 3233, 25, 27, 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.
| \(\displaystyle \int \frac {B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {a (b B-a C) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\int \frac {2 b (b B-a C)-\left (C a^2+b B a-2 b^2 C\right ) \cos (c+d x)}{(a+b \cos (c+d x))^2}dx}{2 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (b B-a C) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\int \frac {2 b (b B-a C)+\left (-C a^2-b B a+2 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {a (b B-a C) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\int -\frac {b \left (-C a^2+3 b B a-2 b^2 C\right )}{a+b \cos (c+d x)}dx}{a^2-b^2}-\frac {\left (a^3 C+a^2 b B-4 a b^2 C+2 b^3 B\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a (b B-a C) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\int \frac {b \left (-C a^2+3 b B a-2 b^2 C\right )}{a+b \cos (c+d x)}dx}{a^2-b^2}-\frac {\left (a^3 C+a^2 b B-4 a b^2 C+2 b^3 B\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a (b B-a C) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {b \left (a^2 (-C)+3 a b B-2 b^2 C\right ) \int \frac {1}{a+b \cos (c+d x)}dx}{a^2-b^2}-\frac {\left (a^3 C+a^2 b B-4 a b^2 C+2 b^3 B\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (b B-a C) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {b \left (a^2 (-C)+3 a b B-2 b^2 C\right ) \int \frac {1}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2-b^2}-\frac {\left (a^3 C+a^2 b B-4 a b^2 C+2 b^3 B\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {a (b B-a C) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {2 b \left (a^2 (-C)+3 a b B-2 b^2 C\right ) \int \frac {1}{(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{d \left (a^2-b^2\right )}-\frac {\left (a^3 C+a^2 b B-4 a b^2 C+2 b^3 B\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {a (b B-a C) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {2 b \left (a^2 (-C)+3 a b B-2 b^2 C\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d \sqrt {a-b} \sqrt {a+b} \left (a^2-b^2\right )}-\frac {\left (a^3 C+a^2 b B-4 a b^2 C+2 b^3 B\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}\) |
(a*(b*B - a*C)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((2*b*(3*a*b*B - a^2*C - 2*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sq rt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*(a^2 - b^2)*d) - ((a^2*b*B + 2*b^3*B + a^3*C - 4*a*b^2*C)*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x])))/( 2*b*(a^2 - b^2))
3.9.10.3.1 Defintions of rubi rules used
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_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*( m + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((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)/(b*f*(m + 1)* (a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A *b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
Time = 1.09 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.30
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 \left (-\frac {\left (2 B \,a^{2}+B a b +2 B \,b^{2}-a^{2} C -4 C a b \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a^{2}+2 a b +b^{2}\right ) \left (a -b \right )}-\frac {\left (2 B \,a^{2}-B a b +2 B \,b^{2}+a^{2} C -4 C a b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a^{2}-2 a b +b^{2}\right ) \left (a +b \right )}\right )}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}^{2}}-\frac {\left (3 B a b -a^{2} C -2 b^{2} C \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}}{d}\) | \(234\) |
| default | \(\frac {-\frac {2 \left (-\frac {\left (2 B \,a^{2}+B a b +2 B \,b^{2}-a^{2} C -4 C a b \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a^{2}+2 a b +b^{2}\right ) \left (a -b \right )}-\frac {\left (2 B \,a^{2}-B a b +2 B \,b^{2}+a^{2} C -4 C a b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a^{2}-2 a b +b^{2}\right ) \left (a +b \right )}\right )}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}^{2}}-\frac {\left (3 B a b -a^{2} C -2 b^{2} C \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}}{d}\) | \(234\) |
| risch | \(\frac {i \left (3 B a \,b^{4} {\mathrm e}^{3 i \left (d x +c \right )}+2 C \,a^{4} b \,{\mathrm e}^{3 i \left (d x +c \right )}-5 C \,a^{2} b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+2 B \,a^{4} b \,{\mathrm e}^{2 i \left (d x +c \right )}+5 B \,a^{2} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+2 B \,b^{5} {\mathrm e}^{2 i \left (d x +c \right )}+2 C \,a^{5} {\mathrm e}^{2 i \left (d x +c \right )}-7 \,{\mathrm e}^{2 i \left (d x +c \right )} C \,a^{3} b^{2}-4 C a \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+4 B \,a^{3} b^{2} {\mathrm e}^{i \left (d x +c \right )}+5 B a \,b^{4} {\mathrm e}^{i \left (d x +c \right )}+2 \,{\mathrm e}^{i \left (d x +c \right )} C \,a^{4} b -11 \,{\mathrm e}^{i \left (d x +c \right )} C \,a^{2} b^{3}+B \,a^{2} b^{3}+2 B \,b^{5}+C \,a^{3} b^{2}-4 C a \,b^{4}\right )}{b^{2} \left (a^{2}-b^{2}\right )^{2} d \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 a \,{\mathrm e}^{i \left (d x +c \right )}+b \right )^{2}}-\frac {3 a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right ) B}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right ) C}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right ) b^{2} C}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}+\frac {3 a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right ) B}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right ) C}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right ) b^{2} C}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}\) | \(802\) |
1/d*(-2*(-1/2*(2*B*a^2+B*a*b+2*B*b^2-C*a^2-4*C*a*b)/(a^2+2*a*b+b^2)/(a-b)* tan(1/2*d*x+1/2*c)^3-1/2*(2*B*a^2-B*a*b+2*B*b^2+C*a^2-4*C*a*b)/(a^2-2*a*b+ b^2)/(a+b)*tan(1/2*d*x+1/2*c))/(tan(1/2*d*x+1/2*c)^2*a-b*tan(1/2*d*x+1/2*c )^2+a+b)^2-(3*B*a*b-C*a^2-2*C*b^2)/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2) *arctan((a-b)*tan(1/2*d*x+1/2*c)/((a-b)*(a+b))^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (165) = 330\).
Time = 0.34 (sec) , antiderivative size = 740, normalized size of antiderivative = 4.11 \[ \int \frac {B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\left [-\frac {{\left (C a^{4} - 3 \, B a^{3} b + 2 \, C a^{2} b^{2} + {\left (C a^{2} b^{2} - 3 \, B a b^{3} + 2 \, C b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (C a^{3} b - 3 \, B a^{2} b^{2} + 2 \, C a b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}\right ) - 2 \, {\left (2 \, B a^{5} - 3 \, C a^{4} b - B a^{3} b^{2} + 3 \, C a^{2} b^{3} - B a b^{4} + {\left (C a^{5} + B a^{4} b - 5 \, C a^{3} b^{2} + B a^{2} b^{3} + 4 \, C a b^{4} - 2 \, B b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} d \cos \left (d x + c\right ) + {\left (a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}\right )} d\right )}}, \frac {{\left (C a^{4} - 3 \, B a^{3} b + 2 \, C a^{2} b^{2} + {\left (C a^{2} b^{2} - 3 \, B a b^{3} + 2 \, C b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (C a^{3} b - 3 \, B a^{2} b^{2} + 2 \, C a b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cos \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (d x + c\right )}\right ) + {\left (2 \, B a^{5} - 3 \, C a^{4} b - B a^{3} b^{2} + 3 \, C a^{2} b^{3} - B a b^{4} + {\left (C a^{5} + B a^{4} b - 5 \, C a^{3} b^{2} + B a^{2} b^{3} + 4 \, C a b^{4} - 2 \, B b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} d \cos \left (d x + c\right ) + {\left (a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}\right )} d\right )}}\right ] \]
[-1/4*((C*a^4 - 3*B*a^3*b + 2*C*a^2*b^2 + (C*a^2*b^2 - 3*B*a*b^3 + 2*C*b^4 )*cos(d*x + c)^2 + 2*(C*a^3*b - 3*B*a^2*b^2 + 2*C*a*b^3)*cos(d*x + c))*sqr t(-a^2 + b^2)*log((2*a*b*cos(d*x + c) + (2*a^2 - b^2)*cos(d*x + c)^2 + 2*s qrt(-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*(2*B*a^5 - 3*C*a^4*b - B*a^3*b ^2 + 3*C*a^2*b^3 - B*a*b^4 + (C*a^5 + B*a^4*b - 5*C*a^3*b^2 + B*a^2*b^3 + 4*C*a*b^4 - 2*B*b^5)*cos(d*x + c))*sin(d*x + c))/((a^6*b^2 - 3*a^4*b^4 + 3 *a^2*b^6 - b^8)*d*cos(d*x + c)^2 + 2*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^ 7)*d*cos(d*x + c) + (a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6)*d), 1/2*((C*a^ 4 - 3*B*a^3*b + 2*C*a^2*b^2 + (C*a^2*b^2 - 3*B*a*b^3 + 2*C*b^4)*cos(d*x + c)^2 + 2*(C*a^3*b - 3*B*a^2*b^2 + 2*C*a*b^3)*cos(d*x + c))*sqrt(a^2 - b^2) *arctan(-(a*cos(d*x + c) + b)/(sqrt(a^2 - b^2)*sin(d*x + c))) + (2*B*a^5 - 3*C*a^4*b - B*a^3*b^2 + 3*C*a^2*b^3 - B*a*b^4 + (C*a^5 + B*a^4*b - 5*C*a^ 3*b^2 + B*a^2*b^3 + 4*C*a*b^4 - 2*B*b^5)*cos(d*x + c))*sin(d*x + c))/((a^6 *b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*d*cos(d*x + c)^2 + 2*(a^7*b - 3*a^5*b^ 3 + 3*a^3*b^5 - a*b^7)*d*cos(d*x + c) + (a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2 *b^6)*d)]
Timed out. \[ \int \frac {B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
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. 391 vs. \(2 (165) = 330\).
Time = 0.36 (sec) , antiderivative size = 391, normalized size of antiderivative = 2.17 \[ \int \frac {B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\frac {\frac {{\left (C a^{2} - 3 \, B a b + 2 \, C b^{2}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a - 2 \, b\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} - b^{2}}} + \frac {2 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b\right )}^{2}}}{d} \]
((C*a^2 - 3*B*a*b + 2*C*b^2)*(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 - 2*a^2*b^2 + b^4)*sqrt(a^2 - b^2)) + (2*B*a^3*tan(1/2*d*x + 1/2*c)^3 - C*a^3*tan(1/2*d*x + 1/2*c)^3 - B*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 3*C*a^2*b*tan(1/2*d*x + 1/2*c)^3 + B*a*b^2*tan(1/2*d*x + 1/2*c)^3 + 4*C*a *b^2*tan(1/2*d*x + 1/2*c)^3 - 2*B*b^3*tan(1/2*d*x + 1/2*c)^3 + 2*B*a^3*tan (1/2*d*x + 1/2*c) + C*a^3*tan(1/2*d*x + 1/2*c) + B*a^2*b*tan(1/2*d*x + 1/2 *c) - 3*C*a^2*b*tan(1/2*d*x + 1/2*c) + B*a*b^2*tan(1/2*d*x + 1/2*c) - 4*C* a*b^2*tan(1/2*d*x + 1/2*c) + 2*B*b^3*tan(1/2*d*x + 1/2*c))/((a^4 - 2*a^2*b ^2 + b^4)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 + a + b)^2) )/d
Time = 5.42 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.38 \[ \int \frac {B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (2\,B\,a^2+2\,B\,b^2-C\,a^2+B\,a\,b-4\,C\,a\,b\right )}{{\left (a+b\right )}^2\,\left (a-b\right )}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,B\,a^2+2\,B\,b^2+C\,a^2-B\,a\,b-4\,C\,a\,b\right )}{\left (a+b\right )\,\left (a^2-2\,a\,b+b^2\right )}}{d\,\left (2\,a\,b+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (2\,a^2-2\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (a^2-2\,a\,b+b^2\right )+a^2+b^2\right )}+\frac {\mathrm {atan}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a-2\,b\right )\,\left (a^2-2\,a\,b+b^2\right )}{2\,\sqrt {a+b}\,{\left (a-b\right )}^{5/2}}\right )\,\left (C\,a^2-3\,B\,a\,b+2\,C\,b^2\right )}{d\,{\left (a+b\right )}^{5/2}\,{\left (a-b\right )}^{5/2}} \]
((tan(c/2 + (d*x)/2)^3*(2*B*a^2 + 2*B*b^2 - C*a^2 + B*a*b - 4*C*a*b))/((a + b)^2*(a - b)) + (tan(c/2 + (d*x)/2)*(2*B*a^2 + 2*B*b^2 + C*a^2 - B*a*b - 4*C*a*b))/((a + b)*(a^2 - 2*a*b + b^2)))/(d*(2*a*b + tan(c/2 + (d*x)/2)^2 *(2*a^2 - 2*b^2) + tan(c/2 + (d*x)/2)^4*(a^2 - 2*a*b + b^2) + a^2 + b^2)) + (atan((tan(c/2 + (d*x)/2)*(2*a - 2*b)*(a^2 - 2*a*b + b^2))/(2*(a + b)^(1 /2)*(a - b)^(5/2)))*(C*a^2 + 2*C*b^2 - 3*B*a*b))/(d*(a + b)^(5/2)*(a - b)^ (5/2))