Integrand size = 31, antiderivative size = 144 \[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx=-\frac {2 a C x}{b^3}-\frac {2 \left (A b^4-2 a^4 C+3 a^2 b^2 C\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} b^3 (a+b)^{3/2} d}+\frac {C \sin (c+d x)}{b^2 d}+\frac {a \left (A b^2+a^2 C\right ) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))} \]
-2*a*C*x/b^3-2*(A*b^4-2*C*a^4+3*C*a^2*b^2)*arctan((a-b)^(1/2)*tan(1/2*d*x+ 1/2*c)/(a+b)^(1/2))/(a-b)^(3/2)/b^3/(a+b)^(3/2)/d+C*sin(d*x+c)/b^2/d+a*(A* b^2+C*a^2)*sin(d*x+c)/b^2/(a^2-b^2)/d/(a+b*cos(d*x+c))
Time = 1.84 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.94 \[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx=\frac {-2 a C (c+d x)-\frac {2 \left (A b^4-2 a^4 C+3 a^2 b^2 C\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{3/2}}+b C \sin (c+d x)+\frac {a b \left (A b^2+a^2 C\right ) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}}{b^3 d} \]
(-2*a*C*(c + d*x) - (2*(A*b^4 - 2*a^4*C + 3*a^2*b^2*C)*ArcTanh[((a - b)*Ta n[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + b*C*Sin[c + d*x] + (a*b*(A*b^2 + a^2*C)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])) )/(b^3*d)
Time = 0.82 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.22, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {3042, 3511, 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.
\(\displaystyle \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx\) |
\(\Big \downarrow \) 3511 |
\(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\int \frac {-b \left (a^2-b^2\right ) C \cos ^2(c+d x)+a \left (a^2-b^2\right ) C \cos (c+d x)+b \left (C a^2+A b^2\right )}{a+b \cos (c+d x)}dx}{b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\int \frac {-b \left (a^2-b^2\right ) C \sin \left (c+d x+\frac {\pi }{2}\right )^2+a \left (a^2-b^2\right ) C \sin \left (c+d x+\frac {\pi }{2}\right )+b \left (C a^2+A b^2\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3502 |
\(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\int \frac {\left (C a^2+A b^2\right ) b^2+2 a \left (a^2-b^2\right ) C \cos (c+d x) b}{a+b \cos (c+d x)}dx}{b}-\frac {C \left (a^2-b^2\right ) \sin (c+d x)}{d}}{b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\int \frac {\left (C a^2+A b^2\right ) b^2+2 a \left (a^2-b^2\right ) C \sin \left (c+d x+\frac {\pi }{2}\right ) b}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {C \left (a^2-b^2\right ) \sin (c+d x)}{d}}{b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3214 |
\(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\left (-2 a^4 C+3 a^2 b^2 C+A b^4\right ) \int \frac {1}{a+b \cos (c+d x)}dx+2 a C x \left (a^2-b^2\right )}{b}-\frac {C \left (a^2-b^2\right ) \sin (c+d x)}{d}}{b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\left (-2 a^4 C+3 a^2 b^2 C+A b^4\right ) \int \frac {1}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx+2 a C x \left (a^2-b^2\right )}{b}-\frac {C \left (a^2-b^2\right ) \sin (c+d x)}{d}}{b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 3138 |
\(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\frac {2 \left (-2 a^4 C+3 a^2 b^2 C+A b^4\right ) \int \frac {1}{(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{d}+2 a C x \left (a^2-b^2\right )}{b}-\frac {C \left (a^2-b^2\right ) \sin (c+d x)}{d}}{b^2 \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {a \left (a^2 C+A b^2\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {2 a C x \left (a^2-b^2\right )+\frac {2 \left (-2 a^4 C+3 a^2 b^2 C+A b^4\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d \sqrt {a-b} \sqrt {a+b}}}{b}-\frac {C \left (a^2-b^2\right ) \sin (c+d x)}{d}}{b^2 \left (a^2-b^2\right )}\) |
(a*(A*b^2 + a^2*C)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])) - ((2*a*(a^2 - b^2)*C*x + (2*(A*b^4 - 2*a^4*C + 3*a^2*b^2*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)*C*Sin[c + d*x])/d)/(b^2*(a^2 - b^2))
3.6.72.3.1 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_.) + (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 = 2.20 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.38
method | result | size |
derivativedivides | \(\frac {-\frac {2 C \left (-\frac {b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+2 a \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{b^{3}}-\frac {2 \left (-\frac {a \left (A \,b^{2}+a^{2} C \right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}+\frac {\left (A \,b^{4}-2 C \,a^{4}+3 C \,a^{2} b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{3}}}{d}\) | \(198\) |
default | \(\frac {-\frac {2 C \left (-\frac {b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+2 a \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{b^{3}}-\frac {2 \left (-\frac {a \left (A \,b^{2}+a^{2} C \right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}+\frac {\left (A \,b^{4}-2 C \,a^{4}+3 C \,a^{2} b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{3}}}{d}\) | \(198\) |
risch | \(-\frac {2 a C x}{b^{3}}-\frac {i {\mathrm e}^{i \left (d x +c \right )} C}{2 b^{2} d}+\frac {i C \,{\mathrm e}^{-i \left (d x +c \right )}}{2 b^{2} d}-\frac {2 i a \left (A \,b^{2}+a^{2} C \right ) \left (a \,{\mathrm e}^{i \left (d x +c \right )}+b \right )}{b^{3} \left (-a^{2}+b^{2}\right ) d \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 a \,{\mathrm e}^{i \left (d x +c \right )}+b \right )}-\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+a \sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) A}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}+\frac {2 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+a \sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) C}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{3}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+a \sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) C \,a^{2}}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d b}+\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right ) A}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}-\frac {2 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right ) C}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{3}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right ) C \,a^{2}}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d b}\) | \(631\) |
1/d*(-2*C/b^3*(-b*tan(1/2*d*x+1/2*c)/(1+tan(1/2*d*x+1/2*c)^2)+2*a*arctan(t an(1/2*d*x+1/2*c)))-2/b^3*(-a*(A*b^2+C*a^2)*b/(a^2-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)+(A*b^4-2*C*a^4+3*C*a^ 2*b^2)/(a-b)/(a+b)/((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. 282 vs. \(2 (136) = 272\).
Time = 0.33 (sec) , antiderivative size = 632, normalized size of antiderivative = 4.39 \[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx=\left [-\frac {4 \, {\left (C a^{5} b - 2 \, C a^{3} b^{3} + C a b^{5}\right )} d x \cos \left (d x + c\right ) + 4 \, {\left (C a^{6} - 2 \, C a^{4} b^{2} + C a^{2} b^{4}\right )} d x + {\left (2 \, C a^{5} - 3 \, C a^{3} b^{2} - A a b^{4} + {\left (2 \, C a^{4} b - 3 \, C a^{2} b^{3} - A b^{5}\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 \, C a^{5} b + {\left (A - 3 \, C\right )} a^{3} b^{3} - {\left (A - C\right )} a b^{5} + {\left (C a^{4} b^{2} - 2 \, C a^{2} b^{4} + C b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} d \cos \left (d x + c\right ) + {\left (a^{5} b^{3} - 2 \, a^{3} b^{5} + a b^{7}\right )} d\right )}}, -\frac {2 \, {\left (C a^{5} b - 2 \, C a^{3} b^{3} + C a b^{5}\right )} d x \cos \left (d x + c\right ) + 2 \, {\left (C a^{6} - 2 \, C a^{4} b^{2} + C a^{2} b^{4}\right )} d x - {\left (2 \, C a^{5} - 3 \, C a^{3} b^{2} - A a b^{4} + {\left (2 \, C a^{4} b - 3 \, C a^{2} b^{3} - A b^{5}\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 \, C a^{5} b + {\left (A - 3 \, C\right )} a^{3} b^{3} - {\left (A - C\right )} a b^{5} + {\left (C a^{4} b^{2} - 2 \, C a^{2} b^{4} + C b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{{\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} d \cos \left (d x + c\right ) + {\left (a^{5} b^{3} - 2 \, a^{3} b^{5} + a b^{7}\right )} d}\right ] \]
[-1/2*(4*(C*a^5*b - 2*C*a^3*b^3 + C*a*b^5)*d*x*cos(d*x + c) + 4*(C*a^6 - 2 *C*a^4*b^2 + C*a^2*b^4)*d*x + (2*C*a^5 - 3*C*a^3*b^2 - A*a*b^4 + (2*C*a^4* b - 3*C*a^2*b^3 - A*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^5*b + (A - 3*C)*a^3*b^3 - (A - C)*a*b^5 + (C*a^4*b^2 - 2*C*a^2*b^4 + C*b^6)*cos(d*x + c))*sin(d*x + c))/((a^4*b^4 - 2*a^2*b^6 + b ^8)*d*cos(d*x + c) + (a^5*b^3 - 2*a^3*b^5 + a*b^7)*d), -(2*(C*a^5*b - 2*C* a^3*b^3 + C*a*b^5)*d*x*cos(d*x + c) + 2*(C*a^6 - 2*C*a^4*b^2 + C*a^2*b^4)* d*x - (2*C*a^5 - 3*C*a^3*b^2 - A*a*b^4 + (2*C*a^4*b - 3*C*a^2*b^3 - A*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^5*b + (A - 3*C)*a^3*b^3 - (A - C)*a*b^5 + (C*a^ 4*b^2 - 2*C*a^2*b^4 + C*b^6)*cos(d*x + c))*sin(d*x + c))/((a^4*b^4 - 2*a^2 *b^6 + b^8)*d*cos(d*x + c) + (a^5*b^3 - 2*a^3*b^5 + a*b^7)*d)]
Timed out. \[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]
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. 998 vs. \(2 (136) = 272\).
Time = 0.42 (sec) , antiderivative size = 998, normalized size of antiderivative = 6.93 \[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx=\text {Too large to display} \]
((4*C*a^6*b^2 - 2*C*a^5*b^3 - 9*C*a^4*b^4 + 4*C*a^3*b^5 - A*a^2*b^6 + 5*C* a^2*b^6 - 2*C*a*b^7 + A*b^8 + 2*C*a^3*abs(-a^2*b^3 + b^5) - C*a^2*b*abs(-a ^2*b^3 + b^5) - 2*C*a*b^2*abs(-a^2*b^3 + b^5) - A*b^3*abs(-a^2*b^3 + b^5)) *(pi*floor(1/2*(d*x + c)/pi + 1/2) + arctan(2*sqrt(1/2)*tan(1/2*d*x + 1/2* c)/sqrt((2*a^3*b^2 - 2*a*b^4 + sqrt(-4*(a^3*b^2 + a^2*b^3 - a*b^4 - b^5)*( a^3*b^2 - a^2*b^3 - a*b^4 + b^5) + 4*(a^3*b^2 - a*b^4)^2))/(a^3*b^2 - a^2* b^3 - a*b^4 + b^5))))/(a^3*b^2*abs(-a^2*b^3 + b^5) - a*b^4*abs(-a^2*b^3 + b^5) + (a^2*b^3 - b^5)^2) + (sqrt(a^2 - b^2)*A*b^3*abs(-a^2*b^3 + b^5)*abs (-a + b) - (2*a^3 - a^2*b - 2*a*b^2)*sqrt(a^2 - b^2)*C*abs(-a^2*b^3 + b^5) *abs(-a + b) - (a^2*b^6 - b^8)*sqrt(a^2 - b^2)*A*abs(-a + b) + (4*a^6*b^2 - 2*a^5*b^3 - 9*a^4*b^4 + 4*a^3*b^5 + 5*a^2*b^6 - 2*a*b^7)*sqrt(a^2 - b^2) *C*abs(-a + b))*(pi*floor(1/2*(d*x + c)/pi + 1/2) + arctan(2*sqrt(1/2)*tan (1/2*d*x + 1/2*c)/sqrt((2*a^3*b^2 - 2*a*b^4 - sqrt(-4*(a^3*b^2 + a^2*b^3 - a*b^4 - b^5)*(a^3*b^2 - a^2*b^3 - a*b^4 + b^5) + 4*(a^3*b^2 - a*b^4)^2))/ (a^3*b^2 - a^2*b^3 - a*b^4 + b^5))))/((a^2*b^3 - b^5)^2*(a^2 - 2*a*b + b^2 ) - (a^5*b^2 - 2*a^4*b^3 + 2*a^2*b^5 - a*b^6)*abs(-a^2*b^3 + b^5)) + 2*(2* C*a^3*tan(1/2*d*x + 1/2*c)^3 - C*a^2*b*tan(1/2*d*x + 1/2*c)^3 + A*a*b^2*ta n(1/2*d*x + 1/2*c)^3 - C*a*b^2*tan(1/2*d*x + 1/2*c)^3 + C*b^3*tan(1/2*d*x + 1/2*c)^3 + 2*C*a^3*tan(1/2*d*x + 1/2*c) + C*a^2*b*tan(1/2*d*x + 1/2*c) + A*a*b^2*tan(1/2*d*x + 1/2*c) - C*a*b^2*tan(1/2*d*x + 1/2*c) - C*b^3*ta...
Time = 9.41 (sec) , antiderivative size = 4124, normalized size of antiderivative = 28.64 \[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx=\text {Too large to display} \]
((2*tan(c/2 + (d*x)/2)^3*(2*C*a^3 + C*b^3 + A*a*b^2 - C*a*b^2 - C*a^2*b))/ (b^2*(a + b)*(a - b)) + (2*tan(c/2 + (d*x)/2)*(2*C*a^3 - C*b^3 + A*a*b^2 - C*a*b^2 + C*a^2*b))/(b^2*(a + b)*(a - b)))/(d*(a + b + tan(c/2 + (d*x)/2) ^4*(a - b) + 2*a*tan(c/2 + (d*x)/2)^2)) - (4*C*a*atan(((2*C*a*((32*tan(c/2 + (d*x)/2)*(A^2*b^8 + 8*C^2*a^8 - 8*C^2*a^7*b + 4*C^2*a^2*b^6 - 8*C^2*a^3 *b^5 + 5*C^2*a^4*b^4 + 16*C^2*a^5*b^3 - 16*C^2*a^6*b^2 + 6*A*C*a^2*b^6 - 4 *A*C*a^4*b^4))/(a*b^6 + b^7 - a^2*b^5 - a^3*b^4) + (C*a*((32*(A*b^12 - A*a ^2*b^10 + A*a^3*b^9 + 3*C*a^2*b^10 + 3*C*a^3*b^9 - 5*C*a^4*b^8 - C*a^5*b^7 + 2*C*a^6*b^6 - A*a*b^11 - 2*C*a*b^11))/(a*b^8 + b^9 - a^2*b^7 - a^3*b^6) - (C*a*tan(c/2 + (d*x)/2)*(2*a*b^11 - 2*a^2*b^10 - 4*a^3*b^9 + 4*a^4*b^8 + 2*a^5*b^7 - 2*a^6*b^6)*64i)/(b^3*(a*b^6 + b^7 - a^2*b^5 - a^3*b^4)))*2i) /b^3))/b^3 + (2*C*a*((32*tan(c/2 + (d*x)/2)*(A^2*b^8 + 8*C^2*a^8 - 8*C^2*a ^7*b + 4*C^2*a^2*b^6 - 8*C^2*a^3*b^5 + 5*C^2*a^4*b^4 + 16*C^2*a^5*b^3 - 16 *C^2*a^6*b^2 + 6*A*C*a^2*b^6 - 4*A*C*a^4*b^4))/(a*b^6 + b^7 - a^2*b^5 - a^ 3*b^4) - (C*a*((32*(A*b^12 - A*a^2*b^10 + A*a^3*b^9 + 3*C*a^2*b^10 + 3*C*a ^3*b^9 - 5*C*a^4*b^8 - C*a^5*b^7 + 2*C*a^6*b^6 - A*a*b^11 - 2*C*a*b^11))/( a*b^8 + b^9 - a^2*b^7 - a^3*b^6) + (C*a*tan(c/2 + (d*x)/2)*(2*a*b^11 - 2*a ^2*b^10 - 4*a^3*b^9 + 4*a^4*b^8 + 2*a^5*b^7 - 2*a^6*b^6)*64i)/(b^3*(a*b^6 + b^7 - a^2*b^5 - a^3*b^4)))*2i)/b^3))/b^3)/((64*(8*C^3*a^8 - 4*C^3*a^7*b + 12*C^3*a^4*b^4 + 6*C^3*a^5*b^3 - 20*C^3*a^6*b^2 + 2*A^2*C*a*b^7 + 4*A...