Integrand size = 45, antiderivative size = 283 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {a^{3/2} (176 A+150 B+133 C) \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{128 d}+\frac {a^2 (176 A+150 B+133 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{128 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (176 A+150 B+133 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (80 A+90 B+67 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{240 d \sqrt {a+a \cos (c+d x)}}+\frac {a (10 B+3 C) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{40 d}+\frac {C \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d} \]
1/128*a^(3/2)*(176*A+150*B+133*C)*arcsin(sin(d*x+c)*a^(1/2)/(a+a*cos(d*x+c ))^(1/2))/d+1/5*C*cos(d*x+c)^(5/2)*(a+a*cos(d*x+c))^(3/2)*sin(d*x+c)/d+1/1 92*a^2*(176*A+150*B+133*C)*cos(d*x+c)^(3/2)*sin(d*x+c)/d/(a+a*cos(d*x+c))^ (1/2)+1/240*a^2*(80*A+90*B+67*C)*cos(d*x+c)^(5/2)*sin(d*x+c)/d/(a+a*cos(d* x+c))^(1/2)+1/128*a^2*(176*A+150*B+133*C)*sin(d*x+c)*cos(d*x+c)^(1/2)/d/(a +a*cos(d*x+c))^(1/2)+1/40*a*(10*B+3*C)*cos(d*x+c)^(5/2)*sin(d*x+c)*(a+a*co s(d*x+c))^(1/2)/d
Time = 1.71 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.60 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {a \sqrt {a (1+\cos (c+d x))} \sec \left (\frac {1}{2} (c+d x)\right ) \left (15 \sqrt {2} (176 A+150 B+133 C) \arcsin \left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right )+2 \sqrt {\cos (c+d x)} (2960 A+2850 B+2671 C+2 (880 A+930 B+1007 C) \cos (c+d x)+4 (80 A+150 B+181 C) \cos (2 (c+d x))+120 B \cos (3 (c+d x))+228 C \cos (3 (c+d x))+48 C \cos (4 (c+d x))) \sin \left (\frac {1}{2} (c+d x)\right )\right )}{3840 d} \]
Integrate[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x ] + C*Cos[c + d*x]^2),x]
(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(15*Sqrt[2]*(176*A + 150*B + 133*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(2960*A + 2850*B + 2671*C + 2*(880*A + 930*B + 1007*C)*Cos[c + d*x] + 4*(80*A + 150 *B + 181*C)*Cos[2*(c + d*x)] + 120*B*Cos[3*(c + d*x)] + 228*C*Cos[3*(c + d *x)] + 48*C*Cos[4*(c + d*x)])*Sin[(c + d*x)/2]))/(3840*d)
Time = 1.49 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.99, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3524, 27, 3042, 3455, 27, 3042, 3460, 3042, 3249, 3042, 3249, 3042, 3253, 223}
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.
| \(\displaystyle \int \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^{3/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3524 |
| \(\displaystyle \frac {\int \frac {1}{2} \cos ^{\frac {3}{2}}(c+d x) (\cos (c+d x) a+a)^{3/2} (5 a (2 A+C)+a (10 B+3 C) \cos (c+d x))dx}{5 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \cos ^{\frac {3}{2}}(c+d x) (\cos (c+d x) a+a)^{3/2} (5 a (2 A+C)+a (10 B+3 C) \cos (c+d x))dx}{10 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2} \left (5 a (2 A+C)+a (10 B+3 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{10 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{4} \int \frac {1}{2} \cos ^{\frac {3}{2}}(c+d x) \sqrt {\cos (c+d x) a+a} \left (5 (16 A+10 B+11 C) a^2+(80 A+90 B+67 C) \cos (c+d x) a^2\right )dx+\frac {a^2 (10 B+3 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}}{10 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{8} \int \cos ^{\frac {3}{2}}(c+d x) \sqrt {\cos (c+d x) a+a} \left (5 (16 A+10 B+11 C) a^2+(80 A+90 B+67 C) \cos (c+d x) a^2\right )dx+\frac {a^2 (10 B+3 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}}{10 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{8} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a} \left (5 (16 A+10 B+11 C) a^2+(80 A+90 B+67 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )dx+\frac {a^2 (10 B+3 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}}{10 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3460 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {5}{6} a^2 (176 A+150 B+133 C) \int \cos ^{\frac {3}{2}}(c+d x) \sqrt {\cos (c+d x) a+a}dx+\frac {a^3 (80 A+90 B+67 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^2 (10 B+3 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}}{10 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {5}{6} a^2 (176 A+150 B+133 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx+\frac {a^3 (80 A+90 B+67 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^2 (10 B+3 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}}{10 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3249 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {5}{6} a^2 (176 A+150 B+133 C) \left (\frac {3}{4} \int \sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}dx+\frac {a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^3 (80 A+90 B+67 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^2 (10 B+3 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}}{10 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {5}{6} a^2 (176 A+150 B+133 C) \left (\frac {3}{4} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx+\frac {a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^3 (80 A+90 B+67 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^2 (10 B+3 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}}{10 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3249 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {5}{6} a^2 (176 A+150 B+133 C) \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\cos (c+d x) a+a}}{\sqrt {\cos (c+d x)}}dx+\frac {a \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^3 (80 A+90 B+67 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^2 (10 B+3 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}}{10 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {5}{6} a^2 (176 A+150 B+133 C) \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {a \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^3 (80 A+90 B+67 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^2 (10 B+3 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}}{10 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3253 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {5}{6} a^2 (176 A+150 B+133 C) \left (\frac {3}{4} \left (\frac {a \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}-\frac {\int \frac {1}{\sqrt {1-\frac {a \sin ^2(c+d x)}{\cos (c+d x) a+a}}}d\left (-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x) a+a}}\right )}{d}\right )+\frac {a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^3 (80 A+90 B+67 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^2 (10 B+3 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}}{10 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {\frac {a^2 (10 B+3 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}+\frac {1}{8} \left (\frac {a^3 (80 A+90 B+67 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}+\frac {5}{6} a^2 (176 A+150 B+133 C) \left (\frac {3}{4} \left (\frac {\sqrt {a} \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{d}+\frac {a \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\right )\right )}{10 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}\) |
(C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + ((a ^2*(10*B + 3*C)*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/ (4*d) + ((a^3*(80*A + 90*B + 67*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*S qrt[a + a*Cos[c + d*x]]) + (5*a^2*(176*A + 150*B + 133*C)*((a*Cos[c + d*x] ^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]]) + (3*((Sqrt[a]*ArcSin[ (Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (a*Sqrt[Cos[c + d*x] ]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])))/4))/6)/8)/(10*a)
3.5.84.3.1 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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[-2*b*Cos[e + f*x]*((c + d*Sin[e + f*x]) ^n/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[2*n*((b*c + a*d)/(b*( 2*n + 1))) Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 0] && IntegerQ[2*n]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(d_.)*sin[(e_.) + (f_.) *(x_)]], x_Symbol] :> Simp[-2/f Subst[Int[1/Sqrt[1 - x^2/a], x], x, b*(Co s[e + f*x]/Sqrt[a + b*Sin[e + f*x]])], x] /; FreeQ[{a, b, d, e, f}, x] && E qQ[a^2 - b^2, 0] && EqQ[d, a/b]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-b)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1 ) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && !LtQ[n, -1 ] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + ( f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp [-2*b*B*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(2*n + 3)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(b *d*(2*n + 3)) Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[n, -1]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_. ) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x] )^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(b*d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n} , x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !Lt Q[m, -2^(-1)] && NeQ[m + n + 2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(548\) vs. \(2(245)=490\).
Time = 29.40 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.94
| method | result | size |
| default | \(\frac {a \left (384 C \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+480 B \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+912 C \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+640 A \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+1200 B \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+1064 C \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+1760 A \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+1500 B \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+1330 C \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+2640 A \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+2250 B \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+1995 C \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+2640 A \arctan \left (\tan \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right )+2250 B \arctan \left (\tan \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right )+1995 C \arctan \left (\tan \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right )\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}}{1920 d \left (1+\cos \left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\) | \(549\) |
| parts | \(\frac {A \left (8 \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+22 \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+33 \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+33 \arctan \left (\tan \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right )\right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) a}{24 d \left (1+\cos \left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}+\frac {B \left (16 \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+40 \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+50 \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+75 \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+75 \arctan \left (\tan \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right )\right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) a}{64 d \left (1+\cos \left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}+\frac {C \left (384 \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+912 \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+1064 \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+1330 \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+1995 \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+1995 \arctan \left (\tan \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right )\right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) a}{1920 d \left (1+\cos \left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\) | \(650\) |
int(cos(d*x+c)^(3/2)*(a+cos(d*x+c)*a)^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2 ),x,method=_RETURNVERBOSE)
1/1920*a/d*(384*C*sin(d*x+c)*cos(d*x+c)^4*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2 )+480*B*cos(d*x+c)^3*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+912*C*co s(d*x+c)^3*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+640*A*cos(d*x+c)^2 *sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+1200*B*cos(d*x+c)^2*sin(d*x+ c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+1064*C*cos(d*x+c)^2*sin(d*x+c)*(cos(d *x+c)/(1+cos(d*x+c)))^(1/2)+1760*A*cos(d*x+c)*sin(d*x+c)*(cos(d*x+c)/(1+co s(d*x+c)))^(1/2)+1500*B*cos(d*x+c)*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^ (1/2)+1330*C*cos(d*x+c)*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+2640* A*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+2250*B*sin(d*x+c)*(cos(d*x+ c)/(1+cos(d*x+c)))^(1/2)+1995*C*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/ 2)+2640*A*arctan(tan(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2))+2250*B*arct an(tan(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2))+1995*C*arctan(tan(d*x+c)* (cos(d*x+c)/(1+cos(d*x+c)))^(1/2)))*cos(d*x+c)^(1/2)*(a*(1+cos(d*x+c)))^(1 /2)/(1+cos(d*x+c))/(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)
Time = 0.57 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.69 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {{\left (384 \, C a \cos \left (d x + c\right )^{4} + 48 \, {\left (10 \, B + 19 \, C\right )} a \cos \left (d x + c\right )^{3} + 8 \, {\left (80 \, A + 150 \, B + 133 \, C\right )} a \cos \left (d x + c\right )^{2} + 10 \, {\left (176 \, A + 150 \, B + 133 \, C\right )} a \cos \left (d x + c\right ) + 15 \, {\left (176 \, A + 150 \, B + 133 \, C\right )} a\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 15 \, {\left ({\left (176 \, A + 150 \, B + 133 \, C\right )} a \cos \left (d x + c\right ) + {\left (176 \, A + 150 \, B + 133 \, C\right )} a\right )} \sqrt {a} \arctan \left (\frac {\sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right )}{1920 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
integrate(cos(d*x+c)^(3/2)*(a+a*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d* x+c)^2),x, algorithm="fricas")
1/1920*((384*C*a*cos(d*x + c)^4 + 48*(10*B + 19*C)*a*cos(d*x + c)^3 + 8*(8 0*A + 150*B + 133*C)*a*cos(d*x + c)^2 + 10*(176*A + 150*B + 133*C)*a*cos(d *x + c) + 15*(176*A + 150*B + 133*C)*a)*sqrt(a*cos(d*x + c) + a)*sqrt(cos( d*x + c))*sin(d*x + c) - 15*((176*A + 150*B + 133*C)*a*cos(d*x + c) + (176 *A + 150*B + 133*C)*a)*sqrt(a)*arctan(sqrt(a*cos(d*x + c) + a)*sqrt(cos(d* x + c))/(sqrt(a)*sin(d*x + c))))/(d*cos(d*x + c) + d)
Timed out. \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 11428 vs. \(2 (245) = 490\).
Time = 1.20 (sec) , antiderivative size = 11428, normalized size of antiderivative = 40.38 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
integrate(cos(d*x+c)^(3/2)*(a+a*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d* x+c)^2),x, algorithm="maxima")
1/7680*(80*(4*(a*cos(3/2*arctan2(sin(2/3*arctan2(sin(3*d*x + 3*c), cos(3*d *x + 3*c))), cos(2/3*arctan2(sin(3*d*x + 3*c), cos(3*d*x + 3*c))) + 1))*si n(3*d*x + 3*c) - (a*cos(3*d*x + 3*c) - a)*sin(3/2*arctan2(sin(2/3*arctan2( sin(3*d*x + 3*c), cos(3*d*x + 3*c))), cos(2/3*arctan2(sin(3*d*x + 3*c), co s(3*d*x + 3*c))) + 1)))*(cos(2/3*arctan2(sin(3*d*x + 3*c), cos(3*d*x + 3*c )))^2 + sin(2/3*arctan2(sin(3*d*x + 3*c), cos(3*d*x + 3*c)))^2 + 2*cos(2/3 *arctan2(sin(3*d*x + 3*c), cos(3*d*x + 3*c))) + 1)^(3/4)*sqrt(a) + 6*(cos( 2/3*arctan2(sin(3*d*x + 3*c), cos(3*d*x + 3*c)))^2 + sin(2/3*arctan2(sin(3 *d*x + 3*c), cos(3*d*x + 3*c)))^2 + 2*cos(2/3*arctan2(sin(3*d*x + 3*c), co s(3*d*x + 3*c))) + 1)^(1/4)*((3*a*sin(2/3*arctan2(sin(3*d*x + 3*c), cos(3* d*x + 3*c))) + 11*a*sin(1/3*arctan2(sin(3*d*x + 3*c), cos(3*d*x + 3*c))))* cos(1/2*arctan2(sin(2/3*arctan2(sin(3*d*x + 3*c), cos(3*d*x + 3*c))), cos( 2/3*arctan2(sin(3*d*x + 3*c), cos(3*d*x + 3*c))) + 1)) - (3*a*cos(2/3*arct an2(sin(3*d*x + 3*c), cos(3*d*x + 3*c))) + 5*a*cos(1/3*arctan2(sin(3*d*x + 3*c), cos(3*d*x + 3*c))) - 8*a)*sin(1/2*arctan2(sin(2/3*arctan2(sin(3*d*x + 3*c), cos(3*d*x + 3*c))), cos(2/3*arctan2(sin(3*d*x + 3*c), cos(3*d*x + 3*c))) + 1)))*sqrt(a) + 33*(a*arctan2(-(cos(2/3*arctan2(sin(3*d*x + 3*c), cos(3*d*x + 3*c)))^2 + sin(2/3*arctan2(sin(3*d*x + 3*c), cos(3*d*x + 3*c) ))^2 + 2*cos(2/3*arctan2(sin(3*d*x + 3*c), cos(3*d*x + 3*c))) + 1)^(1/4)*( cos(1/2*arctan2(sin(2/3*arctan2(sin(3*d*x + 3*c), cos(3*d*x + 3*c))), c...
\[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
integrate(cos(d*x+c)^(3/2)*(a+a*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d* x+c)^2),x, algorithm="giac")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(a*cos(d*x + c) + a)^(3/ 2)*cos(d*x + c)^(3/2), x)
Timed out. \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^{3/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]