Integrand size = 33, antiderivative size = 182 \[ \int \frac {\cos ^2(c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\frac {3 a x}{b^4}-\frac {\left (6 a^4-9 a^2 b^2+2 b^4\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} b^4 (a+b)^{3/2} d}-\frac {3 \sin (c+d x)}{2 b^3 d}+\frac {\cos ^2(c+d x) \sin (c+d x)}{2 b d (a+b \cos (c+d x))^2}-\frac {a \left (3 a^2-2 b^2\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))} \]
3*a*x/b^4-(6*a^4-9*a^2*b^2+2*b^4)*arctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a +b)^(1/2))/(a-b)^(3/2)/b^4/(a+b)^(3/2)/d-3/2*sin(d*x+c)/b^3/d+1/2*cos(d*x+ c)^2*sin(d*x+c)/b/d/(a+b*cos(d*x+c))^2-1/2*a*(3*a^2-2*b^2)*sin(d*x+c)/b^3/ (a^2-b^2)/d/(a+b*cos(d*x+c))
Time = 1.45 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.87 \[ \int \frac {\cos ^2(c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\frac {6 a (c+d x)-\frac {2 \left (6 a^4-9 a^2 b^2+2 b^4\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{3/2}}-2 b \sin (c+d x)+\frac {a^2 b \sin (c+d x)}{(a+b \cos (c+d x))^2}+\frac {a b \left (-5 a^2+4 b^2\right ) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}}{2 b^4 d} \]
(6*a*(c + d*x) - (2*(6*a^4 - 9*a^2*b^2 + 2*b^4)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) - 2*b*Sin[c + d*x] + (a^2*b *Sin[c + d*x])/(a + b*Cos[c + d*x])^2 + (a*b*(-5*a^2 + 4*b^2)*Sin[c + d*x] )/((a - b)*(a + b)*(a + b*Cos[c + d*x])))/(2*b^4*d)
Time = 1.12 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.25, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.394, Rules used = {3042, 3527, 25, 3042, 3511, 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.
| \(\displaystyle \int \frac {\cos ^2(c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (1-\sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3527 |
| \(\displaystyle \frac {\sin (c+d x) \cos ^2(c+d x)}{2 b d (a+b \cos (c+d x))^2}-\frac {\int -\frac {\cos (c+d x) \left (2 \left (a^2-b^2\right )-3 \left (a^2-b^2\right ) \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2}dx}{2 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\cos (c+d x) \left (2 \left (a^2-b^2\right )-3 \left (a^2-b^2\right ) \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2}dx}{2 b \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^2(c+d x)}{2 b d (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (2 \left (a^2-b^2\right )-3 \left (a^2-b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 b \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^2(c+d x)}{2 b d (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3511 |
| \(\displaystyle \frac {-\frac {\int -\frac {-3 b \cos ^2(c+d x) \left (a^2-b^2\right )^2+3 a \cos (c+d x) \left (a^2-b^2\right )^2+b \left (3 a^4-5 b^2 a^2+2 b^4\right )}{a+b \cos (c+d x)}dx}{b^2 \left (a^2-b^2\right )}-\frac {a \left (3 a^2-2 b^2\right ) \sin (c+d x)}{b^2 d (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^2(c+d x)}{2 b d (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {-3 b \cos ^2(c+d x) \left (a^2-b^2\right )^2+3 a \cos (c+d x) \left (a^2-b^2\right )^2+b \left (3 a^4-5 b^2 a^2+2 b^4\right )}{a+b \cos (c+d x)}dx}{b^2 \left (a^2-b^2\right )}-\frac {a \left (3 a^2-2 b^2\right ) \sin (c+d x)}{b^2 d (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^2(c+d x)}{2 b d (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {-3 b \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a^2-b^2\right )^2+3 a \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a^2-b^2\right )^2+b \left (3 a^4-5 b^2 a^2+2 b^4\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b^2 \left (a^2-b^2\right )}-\frac {a \left (3 a^2-2 b^2\right ) \sin (c+d x)}{b^2 d (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^2(c+d x)}{2 b d (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\left (3 a^4-5 b^2 a^2+2 b^4\right ) b^2+6 a \left (a^2-b^2\right )^2 \cos (c+d x) b}{a+b \cos (c+d x)}dx}{b}-\frac {3 \left (a^2-b^2\right )^2 \sin (c+d x)}{d}}{b^2 \left (a^2-b^2\right )}-\frac {a \left (3 a^2-2 b^2\right ) \sin (c+d x)}{b^2 d (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^2(c+d x)}{2 b d (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\left (3 a^4-5 b^2 a^2+2 b^4\right ) b^2+6 a \left (a^2-b^2\right )^2 \sin \left (c+d x+\frac {\pi }{2}\right ) b}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {3 \left (a^2-b^2\right )^2 \sin (c+d x)}{d}}{b^2 \left (a^2-b^2\right )}-\frac {a \left (3 a^2-2 b^2\right ) \sin (c+d x)}{b^2 d (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^2(c+d x)}{2 b d (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {\frac {\frac {6 a x \left (a^2-b^2\right )^2-\left (6 a^6-15 a^4 b^2+11 a^2 b^4-2 b^6\right ) \int \frac {1}{a+b \cos (c+d x)}dx}{b}-\frac {3 \left (a^2-b^2\right )^2 \sin (c+d x)}{d}}{b^2 \left (a^2-b^2\right )}-\frac {a \left (3 a^2-2 b^2\right ) \sin (c+d x)}{b^2 d (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^2(c+d x)}{2 b d (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {6 a x \left (a^2-b^2\right )^2-\left (6 a^6-15 a^4 b^2+11 a^2 b^4-2 b^6\right ) \int \frac {1}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {3 \left (a^2-b^2\right )^2 \sin (c+d x)}{d}}{b^2 \left (a^2-b^2\right )}-\frac {a \left (3 a^2-2 b^2\right ) \sin (c+d x)}{b^2 d (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^2(c+d x)}{2 b d (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {\frac {\frac {6 a x \left (a^2-b^2\right )^2-\frac {2 \left (6 a^6-15 a^4 b^2+11 a^2 b^4-2 b^6\right ) \int \frac {1}{(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{d}}{b}-\frac {3 \left (a^2-b^2\right )^2 \sin (c+d x)}{d}}{b^2 \left (a^2-b^2\right )}-\frac {a \left (3 a^2-2 b^2\right ) \sin (c+d x)}{b^2 d (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^2(c+d x)}{2 b d (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {6 a x \left (a^2-b^2\right )^2-\frac {2 \left (6 a^6-15 a^4 b^2+11 a^2 b^4-2 b^6\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d \sqrt {a-b} \sqrt {a+b}}}{b}-\frac {3 \left (a^2-b^2\right )^2 \sin (c+d x)}{d}}{b^2 \left (a^2-b^2\right )}-\frac {a \left (3 a^2-2 b^2\right ) \sin (c+d x)}{b^2 d (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^2(c+d x)}{2 b d (a+b \cos (c+d x))^2}\) |
(Cos[c + d*x]^2*Sin[c + d*x])/(2*b*d*(a + b*Cos[c + d*x])^2) + (-((a*(3*a^ 2 - 2*b^2)*Sin[c + d*x])/(b^2*d*(a + b*Cos[c + d*x]))) + ((6*a*(a^2 - b^2) ^2*x - (2*(6*a^6 - 15*a^4*b^2 + 11*a^2*b^4 - 2*b^6)*ArcTan[(Sqrt[a - b]*Ta n[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*d))/b - (3*(a^2 - b ^2)^2*Sin[c + d*x])/d)/(b^2*(a^2 - b^2)))/(2*b*(a^2 - b^2))
3.7.12.3.1 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]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + 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*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b *c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*( A*d^2*(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, 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 = 2.34 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.27
| method | result | size |
| derivativedivides | \(\frac {\frac {-\frac {2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+6 a \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{4}}-\frac {2 \left (\frac {\frac {\left (4 a^{2}-a b -4 b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a +2 b}+\frac {\left (4 a^{2}+a b -4 b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a -2 b}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}^{2}}+\frac {\left (6 a^{4}-9 a^{2} b^{2}+2 b^{4}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{2}-b^{2}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{4}}}{d}\) | \(231\) |
| default | \(\frac {\frac {-\frac {2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+6 a \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{4}}-\frac {2 \left (\frac {\frac {\left (4 a^{2}-a b -4 b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a +2 b}+\frac {\left (4 a^{2}+a b -4 b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a -2 b}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}^{2}}+\frac {\left (6 a^{4}-9 a^{2} b^{2}+2 b^{4}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{2}-b^{2}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{4}}}{d}\) | \(231\) |
| risch | \(\frac {3 a x}{b^{4}}+\frac {i {\mathrm e}^{i \left (d x +c \right )}}{2 b^{3} d}-\frac {i {\mathrm e}^{-i \left (d x +c \right )}}{2 b^{3} d}-\frac {i a \left (6 b \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-5 b^{3} a \,{\mathrm e}^{3 i \left (d x +c \right )}+10 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-3 b^{2} a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-4 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+14 b \,a^{3} {\mathrm e}^{i \left (d x +c \right )}-11 \,{\mathrm e}^{i \left (d x +c \right )} b^{3} a +5 a^{2} b^{2}-4 b^{4}\right )}{b^{4} \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 )^{2}}-\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 ) a^{4}}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{4}}+\frac {9 \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}}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{2}}-\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 )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}+\frac {3 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 )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{4}}-\frac {9 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 )}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{2}}+\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 )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}\) | \(721\) |
1/d*(2/b^4*(-b*tan(1/2*d*x+1/2*c)/(1+tan(1/2*d*x+1/2*c)^2)+3*a*arctan(tan( 1/2*d*x+1/2*c)))-2/b^4*((1/2*(4*a^2-a*b-4*b^2)*a*b/(a+b)*tan(1/2*d*x+1/2*c )^3+1/2*(4*a^2+a*b-4*b^2)*a*b/(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+1/2*(6*a^4-9*a^2*b^2+2*b^4)/(a^2-b^2)/ ((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. 394 vs. \(2 (167) = 334\).
Time = 0.34 (sec) , antiderivative size = 856, normalized size of antiderivative = 4.70 \[ \int \frac {\cos ^2(c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\left [\frac {12 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} d x \cos \left (d x + c\right )^{2} + 24 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d x \cos \left (d x + c\right ) + 12 \, {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d x + {\left (6 \, a^{6} - 9 \, a^{4} b^{2} + 2 \, a^{2} b^{4} + {\left (6 \, a^{4} b^{2} - 9 \, a^{2} b^{4} + 2 \, b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (6 \, a^{5} b - 9 \, a^{3} b^{3} + 2 \, 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 (6 \, a^{6} b - 11 \, a^{4} b^{3} + 5 \, a^{2} b^{5} + 2 \, {\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (9 \, a^{5} b^{2} - 17 \, a^{3} b^{4} + 8 \, a b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{4} b^{6} - 2 \, a^{2} b^{8} + b^{10}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{5} - 2 \, a^{3} b^{7} + a b^{9}\right )} d \cos \left (d x + c\right ) + {\left (a^{6} b^{4} - 2 \, a^{4} b^{6} + a^{2} b^{8}\right )} d\right )}}, \frac {6 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} d x \cos \left (d x + c\right )^{2} + 12 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d x \cos \left (d x + c\right ) + 6 \, {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d x - {\left (6 \, a^{6} - 9 \, a^{4} b^{2} + 2 \, a^{2} b^{4} + {\left (6 \, a^{4} b^{2} - 9 \, a^{2} b^{4} + 2 \, b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (6 \, a^{5} b - 9 \, a^{3} b^{3} + 2 \, 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 (6 \, a^{6} b - 11 \, a^{4} b^{3} + 5 \, a^{2} b^{5} + 2 \, {\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (9 \, a^{5} b^{2} - 17 \, a^{3} b^{4} + 8 \, a b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{4} b^{6} - 2 \, a^{2} b^{8} + b^{10}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{5} - 2 \, a^{3} b^{7} + a b^{9}\right )} d \cos \left (d x + c\right ) + {\left (a^{6} b^{4} - 2 \, a^{4} b^{6} + a^{2} b^{8}\right )} d\right )}}\right ] \]
[1/4*(12*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*d*x*cos(d*x + c)^2 + 24*(a^6*b - 2* a^4*b^3 + a^2*b^5)*d*x*cos(d*x + c) + 12*(a^7 - 2*a^5*b^2 + a^3*b^4)*d*x + (6*a^6 - 9*a^4*b^2 + 2*a^2*b^4 + (6*a^4*b^2 - 9*a^2*b^4 + 2*b^6)*cos(d*x + c)^2 + 2*(6*a^5*b - 9*a^3*b^3 + 2*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*(6*a^6*b - 11*a^4*b^3 + 5*a^2*b^5 + 2*(a^4*b ^3 - 2*a^2*b^5 + b^7)*cos(d*x + c)^2 + (9*a^5*b^2 - 17*a^3*b^4 + 8*a*b^6)* cos(d*x + c))*sin(d*x + c))/((a^4*b^6 - 2*a^2*b^8 + b^10)*d*cos(d*x + c)^2 + 2*(a^5*b^5 - 2*a^3*b^7 + a*b^9)*d*cos(d*x + c) + (a^6*b^4 - 2*a^4*b^6 + a^2*b^8)*d), 1/2*(6*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*d*x*cos(d*x + c)^2 + 12 *(a^6*b - 2*a^4*b^3 + a^2*b^5)*d*x*cos(d*x + c) + 6*(a^7 - 2*a^5*b^2 + a^3 *b^4)*d*x - (6*a^6 - 9*a^4*b^2 + 2*a^2*b^4 + (6*a^4*b^2 - 9*a^2*b^4 + 2*b^ 6)*cos(d*x + c)^2 + 2*(6*a^5*b - 9*a^3*b^3 + 2*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))) - ( 6*a^6*b - 11*a^4*b^3 + 5*a^2*b^5 + 2*(a^4*b^3 - 2*a^2*b^5 + b^7)*cos(d*x + c)^2 + (9*a^5*b^2 - 17*a^3*b^4 + 8*a*b^6)*cos(d*x + c))*sin(d*x + c))/((a ^4*b^6 - 2*a^2*b^8 + b^10)*d*cos(d*x + c)^2 + 2*(a^5*b^5 - 2*a^3*b^7 + a*b ^9)*d*cos(d*x + c) + (a^6*b^4 - 2*a^4*b^6 + a^2*b^8)*d)]
Timed out. \[ \int \frac {\cos ^2(c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\cos ^2(c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
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
Time = 0.35 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.83 \[ \int \frac {\cos ^2(c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\frac {\frac {{\left (6 \, a^{4} - 9 \, a^{2} b^{2} + 2 \, b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{2} b^{4} - b^{6}\right )} \sqrt {a^{2} - b^{2}}} - \frac {4 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{2} b^{3} - b^{5}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b\right )}^{2}} + \frac {3 \, {\left (d x + c\right )} a}{b^{4}} - \frac {2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} b^{3}}}{d} \]
((6*a^4 - 9*a^2*b^2 + 2*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^2*b^4 - b^6)*sqrt(a^2 - b^2)) - (4*a^4*tan(1/2*d*x + 1/2*c)^3 - 5*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 3*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 + 4*a *b^3*tan(1/2*d*x + 1/2*c)^3 + 4*a^4*tan(1/2*d*x + 1/2*c) + 5*a^3*b*tan(1/2 *d*x + 1/2*c) - 3*a^2*b^2*tan(1/2*d*x + 1/2*c) - 4*a*b^3*tan(1/2*d*x + 1/2 *c))/((a^2*b^3 - b^5)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 + a + b)^2) + 3*(d*x + c)*a/b^4 - 2*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2*c)^2 + 1)*b^3))/d
Time = 9.55 (sec) , antiderivative size = 3380, normalized size of antiderivative = 18.57 \[ \int \frac {\cos ^2(c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Too large to display} \]
((tan(c/2 + (d*x)/2)*(6*a*b^2 - 3*a^2*b - 6*a^3 + 2*b^3))/(a*b^3 - b^4) + (tan(c/2 + (d*x)/2)^5*(6*a*b^2 + 3*a^2*b - 6*a^3 - 2*b^3))/(b^3*(a + b)) - (2*tan(c/2 + (d*x)/2)^3*(6*a^4 + 2*b^4 - 7*a^2*b^2))/(b^3*(a + b)*(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))) + (6*a*atan(((3*a*((8*tan(c/2 + (d*x)/2)*(72*a^8 - 72*a^7*b + 4*b^8 - 72*a^3*b^5 + 69*a^4*b^4 + 144*a^5*b^3 - 144*a^6*b^2))/(a*b^8 + b^9 - a^2*b^7 - a^3*b^6) + (a*((8*(8*a*b^13 + 4*b^14 - 22*a^2*b^12 - 14*a^3*b ^11 + 30*a^4*b^10 + 6*a^5*b^9 - 12*a^6*b^8))/(a*b^11 + b^12 - a^2*b^10 - a ^3*b^9) - (a*tan(c/2 + (d*x)/2)*(8*a*b^13 - 8*a^2*b^12 - 16*a^3*b^11 + 16* a^4*b^10 + 8*a^5*b^9 - 8*a^6*b^8)*24i)/(b^4*(a*b^8 + b^9 - a^2*b^7 - a^3*b ^6)))*3i)/b^4))/b^4 + (3*a*((8*tan(c/2 + (d*x)/2)*(72*a^8 - 72*a^7*b + 4*b ^8 - 72*a^3*b^5 + 69*a^4*b^4 + 144*a^5*b^3 - 144*a^6*b^2))/(a*b^8 + b^9 - a^2*b^7 - a^3*b^6) - (a*((8*(8*a*b^13 + 4*b^14 - 22*a^2*b^12 - 14*a^3*b^11 + 30*a^4*b^10 + 6*a^5*b^9 - 12*a^6*b^8))/(a*b^11 + b^12 - a^2*b^10 - a^3* b^9) + (a*tan(c/2 + (d*x)/2)*(8*a*b^13 - 8*a^2*b^12 - 16*a^3*b^11 + 16*a^4 *b^10 + 8*a^5*b^9 - 8*a^6*b^8)*24i)/(b^4*(a*b^8 + b^9 - a^2*b^7 - a^3*b^6) ))*3i)/b^4))/b^4)/((16*(12*a*b^7 - 54*a^7*b + 108*a^8 - 36*a^2*b^6 - 72*a^ 3*b^5 + 198*a^4*b^4 + 117*a^5*b^3 - 270*a^6*b^2))/(a*b^11 + b^12 - a^2*b^1 0 - a^3*b^9) - (a*((8*tan(c/2 + (d*x)/2)*(72*a^8 - 72*a^7*b + 4*b^8 - 7...