Integrand size = 31, antiderivative size = 203 \[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\frac {C x}{b^3}-\frac {a \left (3 A b^4+\left (2 a^4-5 a^2 b^2+6 b^4\right ) C\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} b^3 (a+b)^{5/2} d}+\frac {a \left (A b^2+a^2 C\right ) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {\left (2 A b^4-3 a^4 C+a^2 b^2 (A+6 C)\right ) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))} \]
C*x/b^3-a*(3*A*b^4+(2*a^4-5*a^2*b^2+6*b^4)*C)*arctan((a-b)^(1/2)*tan(1/2*d *x+1/2*c)/(a+b)^(1/2))/(a-b)^(5/2)/b^3/(a+b)^(5/2)/d+1/2*a*(A*b^2+C*a^2)*s in(d*x+c)/b^2/(a^2-b^2)/d/(a+b*cos(d*x+c))^2+1/2*(2*A*b^4-3*a^4*C+a^2*b^2* (A+6*C))*sin(d*x+c)/b^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c))
Time = 2.20 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.96 \[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\frac {2 C (c+d x)+\frac {2 a \left (3 A b^4+\left (2 a^4-5 a^2 b^2+6 b^4\right ) C\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{5/2}}+\frac {a b \left (A b^2+a^2 C\right ) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^2}+\frac {b \left (2 A b^4-3 a^4 C+a^2 b^2 (A+6 C)\right ) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))}}{2 b^3 d} \]
(2*C*(c + d*x) + (2*a*(3*A*b^4 + (2*a^4 - 5*a^2*b^2 + 6*b^4)*C)*ArcTanh[(( a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + (a*b*(A*b ^2 + a^2*C)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])^2) + (b*(2 *A*b^4 - 3*a^4*C + a^2*b^2*(A + 6*C))*Sin[c + d*x])/((a - b)^2*(a + b)^2*( a + b*Cos[c + d*x])))/(2*b^3*d)
Time = 0.98 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.18, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {3042, 3511, 3042, 3500, 25, 25, 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.
| \(\displaystyle \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3511 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \sin (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\int \frac {-2 b \left (a^2-b^2\right ) C \cos ^2(c+d x)-a \left (-C a^2+A b^2+2 b^2 C\right ) \cos (c+d x)+2 b \left (C a^2+A b^2\right )}{(a+b \cos (c+d x))^2}dx}{2 b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \sin (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\int \frac {-2 b \left (a^2-b^2\right ) C \sin \left (c+d x+\frac {\pi }{2}\right )^2-a \left (-C a^2+A b^2+2 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 b \left (C a^2+A b^2\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \sin (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\int -\frac {a b^2 \left (3 A b^2-\left (a^2-4 b^2\right ) C\right )-2 b \left (a^2-b^2\right )^2 C \cos (c+d x)}{a+b \cos (c+d x)}dx}{b \left (a^2-b^2\right )}-\frac {\left (-3 a^4 C+a^2 b^2 (A+6 C)+2 A b^4\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \sin (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\int -\frac {a \left (a^2 C-b^2 (3 A+4 C)\right ) b^2+2 \left (a^2-b^2\right )^2 C \cos (c+d x) b}{a+b \cos (c+d x)}dx}{b \left (a^2-b^2\right )}-\frac {\left (-3 a^4 C+a^2 b^2 (A+6 C)+2 A b^4\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \sin (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\int \frac {a \left (a^2 C-b^2 (3 A+4 C)\right ) b^2+2 \left (a^2-b^2\right )^2 C \cos (c+d x) b}{a+b \cos (c+d x)}dx}{b \left (a^2-b^2\right )}-\frac {\left (-3 a^4 C+a^2 b^2 (A+6 C)+2 A b^4\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \sin (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\int \frac {a \left (a^2 C-b^2 (3 A+4 C)\right ) b^2+2 \left (a^2-b^2\right )^2 C \sin \left (c+d x+\frac {\pi }{2}\right ) b}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b \left (a^2-b^2\right )}-\frac {\left (-3 a^4 C+a^2 b^2 (A+6 C)+2 A b^4\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \sin (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {2 C x \left (a^2-b^2\right )^2-a \left (C \left (2 a^4-5 a^2 b^2+6 b^4\right )+3 A b^4\right ) \int \frac {1}{a+b \cos (c+d x)}dx}{b \left (a^2-b^2\right )}-\frac {\left (-3 a^4 C+a^2 b^2 (A+6 C)+2 A b^4\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \sin (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {2 C x \left (a^2-b^2\right )^2-a \left (C \left (2 a^4-5 a^2 b^2+6 b^4\right )+3 A b^4\right ) \int \frac {1}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b \left (a^2-b^2\right )}-\frac {\left (-3 a^4 C+a^2 b^2 (A+6 C)+2 A b^4\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \sin (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {2 C x \left (a^2-b^2\right )^2-\frac {2 a \left (C \left (2 a^4-5 a^2 b^2+6 b^4\right )+3 A b^4\right ) \int \frac {1}{(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{d}}{b \left (a^2-b^2\right )}-\frac {\left (-3 a^4 C+a^2 b^2 (A+6 C)+2 A b^4\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b^2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \sin (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {2 C x \left (a^2-b^2\right )^2-\frac {2 a \left (C \left (2 a^4-5 a^2 b^2+6 b^4\right )+3 A b^4\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d \sqrt {a-b} \sqrt {a+b}}}{b \left (a^2-b^2\right )}-\frac {\left (-3 a^4 C+a^2 b^2 (A+6 C)+2 A b^4\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b^2 \left (a^2-b^2\right )}\) |
(a*(A*b^2 + a^2*C)*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x]) ^2) - (-((2*(a^2 - b^2)^2*C*x - (2*a*(3*A*b^4 + (2*a^4 - 5*a^2*b^2 + 6*b^4 )*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))) - ((2*A*b^4 - 3*a^4*C + a^2*b^2*(A + 6*C))*Si n[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x])))/(2*b^2*(a^2 - b^2))
3.6.80.3.1 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[(-(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]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[ (-(b*c - a*d))*(A*b^2 + 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] + Simp[1/(b^2*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(m + 1)*(a*C*(b*c - a*d) + A*b*(a*c - b*d )) - ((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] /; FreeQ[{a, b, c, d, e , f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
Time = 1.84 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.45
| method | result | size |
| derivativedivides | \(\frac {\frac {2 C \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{3}}-\frac {2 \left (\frac {-\frac {\left (2 A \,a^{2} b^{2}+A a \,b^{3}+2 A \,b^{4}-2 C \,a^{4}+C \,a^{3} b +6 C \,a^{2} b^{2}\right ) b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {b \left (2 A \,a^{2} b^{2}-A a \,b^{3}+2 A \,b^{4}-2 C \,a^{4}-C \,a^{3} b +6 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}^{2}}+\frac {a \left (3 A \,b^{4}+2 C \,a^{4}-5 C \,a^{2} b^{2}+6 C \,b^{4}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{3}}}{d}\) | \(294\) |
| default | \(\frac {\frac {2 C \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{3}}-\frac {2 \left (\frac {-\frac {\left (2 A \,a^{2} b^{2}+A a \,b^{3}+2 A \,b^{4}-2 C \,a^{4}+C \,a^{3} b +6 C \,a^{2} b^{2}\right ) b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {b \left (2 A \,a^{2} b^{2}-A a \,b^{3}+2 A \,b^{4}-2 C \,a^{4}-C \,a^{3} b +6 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}^{2}}+\frac {a \left (3 A \,b^{4}+2 C \,a^{4}-5 C \,a^{2} b^{2}+6 C \,b^{4}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{3}}}{d}\) | \(294\) |
| risch | \(\frac {C x}{b^{3}}-\frac {i \left (-3 A a \,b^{5} {\mathrm e}^{3 i \left (d x +c \right )}+4 C \,a^{5} b \,{\mathrm e}^{3 i \left (d x +c \right )}-7 C \,a^{3} b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-2 A \,a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-5 A \,a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-2 A \,b^{6} {\mathrm e}^{2 i \left (d x +c \right )}+6 C \,a^{6} {\mathrm e}^{2 i \left (d x +c \right )}-9 C \,a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-6 C \,a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-4 A \,a^{3} b^{3} {\mathrm e}^{i \left (d x +c \right )}-5 A a \,b^{5} {\mathrm e}^{i \left (d x +c \right )}+8 C \,a^{5} b \,{\mathrm e}^{i \left (d x +c \right )}-17 C \,a^{3} b^{3} {\mathrm e}^{i \left (d x +c \right )}-A \,a^{2} b^{4}-2 A \,b^{6}+3 C \,a^{4} b^{2}-6 C \,a^{2} b^{4}\right )}{b^{3} \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}}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) A}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}-\frac {a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+a \sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) C}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,b^{3}}+\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+a \sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) C}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d b}-\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}}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) 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 ) A}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}+\frac {a^{5} \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}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,b^{3}}-\frac {5 a^{3} \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 b}+\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 ) C}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d}\) | \(988\) |
1/d*(2*C/b^3*arctan(tan(1/2*d*x+1/2*c))-2/b^3*((-1/2*(2*A*a^2*b^2+A*a*b^3+ 2*A*b^4-2*C*a^4+C*a^3*b+6*C*a^2*b^2)*b/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1 /2*c)^3-1/2*b*(2*A*a^2*b^2-A*a*b^3+2*A*b^4-2*C*a^4-C*a^3*b+6*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-b*tan(1/2*d*x+1/2* c)^2+a+b)^2+1/2*a*(3*A*b^4+2*C*a^4-5*C*a^2*b^2+6*C*b^4)/(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. 491 vs. \(2 (191) = 382\).
Time = 0.34 (sec) , antiderivative size = 1051, normalized size of antiderivative = 5.18 \[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Too large to display} \]
[1/4*(4*(C*a^6*b^2 - 3*C*a^4*b^4 + 3*C*a^2*b^6 - C*b^8)*d*x*cos(d*x + c)^2 + 8*(C*a^7*b - 3*C*a^5*b^3 + 3*C*a^3*b^5 - C*a*b^7)*d*x*cos(d*x + c) + 4* (C*a^8 - 3*C*a^6*b^2 + 3*C*a^4*b^4 - C*a^2*b^6)*d*x - (2*C*a^7 - 5*C*a^5*b ^2 + 3*(A + 2*C)*a^3*b^4 + (2*C*a^5*b^2 - 5*C*a^3*b^4 + 3*(A + 2*C)*a*b^6) *cos(d*x + c)^2 + 2*(2*C*a^6*b - 5*C*a^4*b^3 + 3*(A + 2*C)*a^2*b^5)*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*(2*C*a^7*b - (2*A + 7*C)*a^5*b^3 + (A + 5*C)*a^3*b^5 + A*a*b^7 + (3*C*a^6*b^2 - (A + 9*C)*a^4 *b^4 - (A - 6*C)*a^2*b^6 + 2*A*b^8)*cos(d*x + c))*sin(d*x + c))/((a^6*b^5 - 3*a^4*b^7 + 3*a^2*b^9 - b^11)*d*cos(d*x + c)^2 + 2*(a^7*b^4 - 3*a^5*b^6 + 3*a^3*b^8 - a*b^10)*d*cos(d*x + c) + (a^8*b^3 - 3*a^6*b^5 + 3*a^4*b^7 - a^2*b^9)*d), 1/2*(2*(C*a^6*b^2 - 3*C*a^4*b^4 + 3*C*a^2*b^6 - C*b^8)*d*x*co s(d*x + c)^2 + 4*(C*a^7*b - 3*C*a^5*b^3 + 3*C*a^3*b^5 - C*a*b^7)*d*x*cos(d *x + c) + 2*(C*a^8 - 3*C*a^6*b^2 + 3*C*a^4*b^4 - C*a^2*b^6)*d*x - (2*C*a^7 - 5*C*a^5*b^2 + 3*(A + 2*C)*a^3*b^4 + (2*C*a^5*b^2 - 5*C*a^3*b^4 + 3*(A + 2*C)*a*b^6)*cos(d*x + c)^2 + 2*(2*C*a^6*b - 5*C*a^4*b^3 + 3*(A + 2*C)*a^2 *b^5)*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*C*a^7*b - (2*A + 7*C)*a^5*b^3 + (A + 5*C)*a^3* b^5 + A*a*b^7 + (3*C*a^6*b^2 - (A + 9*C)*a^4*b^4 - (A - 6*C)*a^2*b^6 + ...
Timed out. \[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
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. 479 vs. \(2 (191) = 382\).
Time = 0.37 (sec) , antiderivative size = 479, normalized size of antiderivative = 2.36 \[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=-\frac {\frac {{\left (2 \, C a^{5} - 5 \, C a^{3} b^{2} + 3 \, A a b^{4} + 6 \, C a b^{4}\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^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} \sqrt {a^{2} - b^{2}}} - \frac {{\left (d x + c\right )} C}{b^{3}} + \frac {2 \, C a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, C a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, A a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + A a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - A a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, A b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, C a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, C a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, A a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - A a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - A a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, A b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\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} \]
-((2*C*a^5 - 5*C*a^3*b^2 + 3*A*a*b^4 + 6*C*a*b^4)*(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^3 - 2*a^2*b^5 + b^7)*sqrt(a^2 - b^2)) - (d*x + c)*C/b^3 + (2*C*a^5*tan(1/2*d*x + 1/2*c)^3 - 3*C*a^4*b*tan(1/2*d* x + 1/2*c)^3 - 2*A*a^3*b^2*tan(1/2*d*x + 1/2*c)^3 - 5*C*a^3*b^2*tan(1/2*d* x + 1/2*c)^3 + A*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 + 6*C*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 - A*a*b^4*tan(1/2*d*x + 1/2*c)^3 + 2*A*b^5*tan(1/2*d*x + 1/2*c) ^3 + 2*C*a^5*tan(1/2*d*x + 1/2*c) + 3*C*a^4*b*tan(1/2*d*x + 1/2*c) - 2*A*a ^3*b^2*tan(1/2*d*x + 1/2*c) - 5*C*a^3*b^2*tan(1/2*d*x + 1/2*c) - A*a^2*b^3 *tan(1/2*d*x + 1/2*c) - 6*C*a^2*b^3*tan(1/2*d*x + 1/2*c) - A*a*b^4*tan(1/2 *d*x + 1/2*c) - 2*A*b^5*tan(1/2*d*x + 1/2*c))/((a^4*b^2 - 2*a^2*b^4 + b^6) *(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 = 10.92 (sec) , antiderivative size = 6587, normalized size of antiderivative = 32.45 \[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Too large to display} \]
((tan(c/2 + (d*x)/2)^3*(2*A*b^4 - 2*C*a^4 + 2*A*a^2*b^2 + 6*C*a^2*b^2 + A* a*b^3 + C*a^3*b))/((a*b^2 - b^3)*(a + b)^2) + (tan(c/2 + (d*x)/2)*(2*A*b^4 - 2*C*a^4 + 2*A*a^2*b^2 + 6*C*a^2*b^2 - A*a*b^3 - C*a^3*b))/((a + b)*(b^4 - 2*a*b^3 + a^2*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)) - (2*C*atan(((C*(( C*((8*(4*C*b^15 + 6*A*a^2*b^13 + 12*A*a^3*b^12 - 12*A*a^4*b^11 - 6*A*a^5*b ^10 + 6*A*a^6*b^9 - 8*C*a^2*b^13 + 34*C*a^3*b^12 + 6*C*a^4*b^11 - 36*C*a^5 *b^10 - 4*C*a^6*b^9 + 18*C*a^7*b^8 + 2*C*a^8*b^7 - 4*C*a^9*b^6 - 6*A*a*b^1 4 - 12*C*a*b^14))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3 *a^5*b^8 - a^6*b^7 - a^7*b^6) - (C*tan(c/2 + (d*x)/2)*(8*a*b^15 - 8*a^2*b^ 14 - 32*a^3*b^13 + 32*a^4*b^12 + 48*a^5*b^11 - 48*a^6*b^10 - 32*a^7*b^9 + 32*a^8*b^8 + 8*a^9*b^7 - 8*a^10*b^6)*8i)/(b^3*(a*b^10 + b^11 - 3*a^2*b^9 - 3*a^3*b^8 + 3*a^4*b^7 + 3*a^5*b^6 - a^6*b^5 - a^7*b^4)))*1i)/b^3 + (8*tan (c/2 + (d*x)/2)*(8*C^2*a^10 + 4*C^2*b^10 - 8*C^2*a*b^9 - 8*C^2*a^9*b + 9*A ^2*a^2*b^8 + 24*C^2*a^2*b^8 + 32*C^2*a^3*b^7 - 52*C^2*a^4*b^6 - 48*C^2*a^5 *b^5 + 57*C^2*a^6*b^4 + 32*C^2*a^7*b^3 - 32*C^2*a^8*b^2 + 36*A*C*a^2*b^8 - 30*A*C*a^4*b^6 + 12*A*C*a^6*b^4))/(a*b^10 + b^11 - 3*a^2*b^9 - 3*a^3*b^8 + 3*a^4*b^7 + 3*a^5*b^6 - a^6*b^5 - a^7*b^4)))/b^3 - (C*((C*((8*(4*C*b^15 + 6*A*a^2*b^13 + 12*A*a^3*b^12 - 12*A*a^4*b^11 - 6*A*a^5*b^10 + 6*A*a^6*b^ 9 - 8*C*a^2*b^13 + 34*C*a^3*b^12 + 6*C*a^4*b^11 - 36*C*a^5*b^10 - 4*C*a...