Integrand size = 37, antiderivative size = 312 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {a^{5/2} (1304 A+1015 C) \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{512 d}+\frac {a^3 (1304 A+1015 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{512 d \sqrt {a+a \cos (c+d x)}}+\frac {a^3 (1304 A+1015 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{768 d \sqrt {a+a \cos (c+d x)}}+\frac {a^3 (136 A+109 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{192 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (24 A+23 C) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{96 d}+\frac {a C \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {C \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{6 d} \]
1/512*a^(5/2)*(1304*A+1015*C)*arcsin(sin(d*x+c)*a^(1/2)/(a+a*cos(d*x+c))^( 1/2))/d+1/12*a*C*cos(d*x+c)^(5/2)*(a+a*cos(d*x+c))^(3/2)*sin(d*x+c)/d+1/6* C*cos(d*x+c)^(5/2)*(a+a*cos(d*x+c))^(5/2)*sin(d*x+c)/d+1/768*a^3*(1304*A+1 015*C)*cos(d*x+c)^(3/2)*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+1/192*a^3*(136 *A+109*C)*cos(d*x+c)^(5/2)*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+1/512*a^3*( 1304*A+1015*C)*sin(d*x+c)*cos(d*x+c)^(1/2)/d/(a+a*cos(d*x+c))^(1/2)+1/96*a ^2*(24*A+23*C)*cos(d*x+c)^(5/2)*sin(d*x+c)*(a+a*cos(d*x+c))^(1/2)/d
Time = 3.84 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.54 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {a^2 \sqrt {a (1+\cos (c+d x))} \sec \left (\frac {1}{2} (c+d x)\right ) \left (3 \sqrt {2} (1304 A+1015 C) \arcsin \left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right )+2 \sqrt {\cos (c+d x)} (4648 A+4193 C+(2896 A+3234 C) \cos (c+d x)+4 (184 A+315 C) \cos (2 (c+d x))+96 A \cos (3 (c+d x))+428 C \cos (3 (c+d x))+112 C \cos (4 (c+d x))+16 C \cos (5 (c+d x))) \sin \left (\frac {1}{2} (c+d x)\right )\right )}{3072 d} \]
(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(3*Sqrt[2]*(1304*A + 1015 *C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(4648*A + 4193 *C + (2896*A + 3234*C)*Cos[c + d*x] + 4*(184*A + 315*C)*Cos[2*(c + d*x)] + 96*A*Cos[3*(c + d*x)] + 428*C*Cos[3*(c + d*x)] + 112*C*Cos[4*(c + d*x)] + 16*C*Cos[5*(c + d*x)])*Sin[(c + d*x)/2]))/(3072*d)
Time = 1.87 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.01, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.486, Rules used = {3042, 3525, 27, 3042, 3455, 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.
| \(\displaystyle \int \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^{5/2} \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3525 |
| \(\displaystyle \frac {\int \frac {1}{2} \cos ^{\frac {3}{2}}(c+d x) (\cos (c+d x) a+a)^{5/2} (a (12 A+5 C)+5 a C \cos (c+d x))dx}{6 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \cos ^{\frac {3}{2}}(c+d x) (\cos (c+d x) a+a)^{5/2} (a (12 A+5 C)+5 a C \cos (c+d x))dx}{12 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2} \left (a (12 A+5 C)+5 a C \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{12 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{5} \int \frac {5}{2} \cos ^{\frac {3}{2}}(c+d x) (\cos (c+d x) a+a)^{3/2} \left (3 (8 A+5 C) a^2+(24 A+23 C) \cos (c+d x) a^2\right )dx+\frac {a^2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{d}}{12 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{2} \int \cos ^{\frac {3}{2}}(c+d x) (\cos (c+d x) a+a)^{3/2} \left (3 (8 A+5 C) a^2+(24 A+23 C) \cos (c+d x) a^2\right )dx+\frac {a^2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{d}}{12 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2} \left (3 (8 A+5 C) a^2+(24 A+23 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )dx+\frac {a^2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{d}}{12 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{4} \int \frac {1}{2} \cos ^{\frac {3}{2}}(c+d x) \sqrt {\cos (c+d x) a+a} \left ((312 A+235 C) a^3+3 (136 A+109 C) \cos (c+d x) a^3\right )dx+\frac {a^3 (24 A+23 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}\right )+\frac {a^2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{d}}{12 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{8} \int \cos ^{\frac {3}{2}}(c+d x) \sqrt {\cos (c+d x) a+a} \left ((312 A+235 C) a^3+3 (136 A+109 C) \cos (c+d x) a^3\right )dx+\frac {a^3 (24 A+23 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}\right )+\frac {a^2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{d}}{12 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{8} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a} \left ((312 A+235 C) a^3+3 (136 A+109 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )dx+\frac {a^3 (24 A+23 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}\right )+\frac {a^2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{d}}{12 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 3460 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{8} \left (\frac {1}{2} a^3 (1304 A+1015 C) \int \cos ^{\frac {3}{2}}(c+d x) \sqrt {\cos (c+d x) a+a}dx+\frac {a^4 (136 A+109 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^3 (24 A+23 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}\right )+\frac {a^2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{d}}{12 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{8} \left (\frac {1}{2} a^3 (1304 A+1015 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx+\frac {a^4 (136 A+109 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^3 (24 A+23 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}\right )+\frac {a^2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{d}}{12 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 3249 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{8} \left (\frac {1}{2} a^3 (1304 A+1015 C) \left (\frac {3}{4} \int \sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}dx+\frac {a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^4 (136 A+109 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^3 (24 A+23 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}\right )+\frac {a^2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{d}}{12 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{8} \left (\frac {1}{2} a^3 (1304 A+1015 C) \left (\frac {3}{4} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx+\frac {a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^4 (136 A+109 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^3 (24 A+23 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}\right )+\frac {a^2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{d}}{12 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 3249 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{8} \left (\frac {1}{2} a^3 (1304 A+1015 C) \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\cos (c+d x) a+a}}{\sqrt {\cos (c+d x)}}dx+\frac {a \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^4 (136 A+109 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^3 (24 A+23 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}\right )+\frac {a^2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{d}}{12 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{8} \left (\frac {1}{2} a^3 (1304 A+1015 C) \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {a \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^4 (136 A+109 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^3 (24 A+23 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}\right )+\frac {a^2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{d}}{12 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 3253 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{8} \left (\frac {1}{2} a^3 (1304 A+1015 C) \left (\frac {3}{4} \left (\frac {a \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}-\frac {\int \frac {1}{\sqrt {1-\frac {a \sin ^2(c+d x)}{\cos (c+d x) a+a}}}d\left (-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x) a+a}}\right )}{d}\right )+\frac {a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^4 (136 A+109 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a^3 (24 A+23 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}\right )+\frac {a^2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{d}}{12 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {\frac {a^2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{d}+\frac {1}{2} \left (\frac {a^3 (24 A+23 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}+\frac {1}{8} \left (\frac {a^4 (136 A+109 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{d \sqrt {a \cos (c+d x)+a}}+\frac {1}{2} a^3 (1304 A+1015 C) \left (\frac {3}{4} \left (\frac {\sqrt {a} \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{d}+\frac {a \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}\right )+\frac {a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\right )\right )\right )}{12 a}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d}\) |
(C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(6*d) + ((a ^2*C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/d + ((a^3 *(24*A + 23*C)*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/( 4*d) + ((a^4*(136*A + 109*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(1304*A + 1015*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))/2)/8)/2)/(12*a)
3.2.88.3.1 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_.) + (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))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[m, -2^(-1)] && NeQ[m + n + 2, 0]
Time = 24.50 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.47
| method | result | size |
| default | \(\frac {a^{2} \left (256 C \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+896 C \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+384 A \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+1392 C \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+1472 A \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+1624 C \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+2608 A \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+2030 C \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+3912 A \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+3045 C \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+3912 A \arctan \left (\tan \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right )+3045 C \arctan \left (\tan \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right )\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}}{1536 d \left (1+\cos \left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\) | \(459\) |
| parts | \(\frac {A \left (48 \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 )}}+184 \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 )}}+326 \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 )}}+489 \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+489 \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^{2}}{192 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 (256 \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (\cos ^{5}\left (d x +c \right )\right )+896 \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+1392 \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 )}}+1624 \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 )}}+2030 \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 )}}+3045 \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+3045 \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^{2}}{1536 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 )}}}\) | \(508\) |
1/1536*a^2/d*(256*C*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1 /2)+896*C*sin(d*x+c)*cos(d*x+c)^4*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+384*A* sin(d*x+c)*cos(d*x+c)^3*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+1392*C*cos(d*x+c )^3*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+1472*A*cos(d*x+c)^2*sin(d *x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+1624*C*cos(d*x+c)^2*sin(d*x+c)*(co s(d*x+c)/(1+cos(d*x+c)))^(1/2)+2608*A*cos(d*x+c)*sin(d*x+c)*(cos(d*x+c)/(1 +cos(d*x+c)))^(1/2)+2030*C*cos(d*x+c)*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c) ))^(1/2)+3912*A*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+3045*C*sin(d* x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+3912*A*arctan(tan(d*x+c)*(cos(d*x+c )/(1+cos(d*x+c)))^(1/2))+3045*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 = 0.37 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.67 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {{\left (256 \, C a^{2} \cos \left (d x + c\right )^{5} + 896 \, C a^{2} \cos \left (d x + c\right )^{4} + 48 \, {\left (8 \, A + 29 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 8 \, {\left (184 \, A + 203 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 2 \, {\left (1304 \, A + 1015 \, C\right )} a^{2} \cos \left (d x + c\right ) + 3 \, {\left (1304 \, A + 1015 \, C\right )} a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 3 \, {\left ({\left (1304 \, A + 1015 \, C\right )} a^{2} \cos \left (d x + c\right ) + {\left (1304 \, A + 1015 \, C\right )} a^{2}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right )}{1536 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
1/1536*((256*C*a^2*cos(d*x + c)^5 + 896*C*a^2*cos(d*x + c)^4 + 48*(8*A + 2 9*C)*a^2*cos(d*x + c)^3 + 8*(184*A + 203*C)*a^2*cos(d*x + c)^2 + 2*(1304*A + 1015*C)*a^2*cos(d*x + c) + 3*(1304*A + 1015*C)*a^2)*sqrt(a*cos(d*x + c) + a)*sqrt(cos(d*x + c))*sin(d*x + c) - 3*((1304*A + 1015*C)*a^2*cos(d*x + c) + (1304*A + 1015*C)*a^2)*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. \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 42699 vs. \(2 (268) = 536\).
Time = 2.94 (sec) , antiderivative size = 42699, normalized size of antiderivative = 136.86 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
1/6144*(8*(10*(cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c)))^2 + si n(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c)))^2 + 2*cos(1/2*arctan2(s in(4*d*x + 4*c), cos(4*d*x + 4*c))) + 1)^(3/4)*((3*a^2*cos(4*d*x + 4*c)^2* sin(4*d*x + 4*c) + 3*a^2*sin(4*d*x + 4*c)^3 + 12*(a^2*sin(4*d*x + 4*c)^3 + (a^2*cos(4*d*x + 4*c)^2 - 2*a^2*cos(4*d*x + 4*c) + a^2)*sin(4*d*x + 4*c)) *cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c)))^2 + 12*(a^2*sin(4*d* x + 4*c)^3 + (a^2*cos(4*d*x + 4*c)^2 + 2*a^2*cos(4*d*x + 4*c) + a^2)*sin(4 *d*x + 4*c))*sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c)))^2 + 3*(2 *a^2*cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c)))*sin(4*d*x + 4*c) + a^2*sin(4*d*x + 4*c) - 2*(a^2*cos(4*d*x + 4*c) + a^2)*sin(1/2*arctan2(s in(4*d*x + 4*c), cos(4*d*x + 4*c))))*cos(3/4*arctan2(sin(4*d*x + 4*c), cos (4*d*x + 4*c))) + 12*(a^2*sin(4*d*x + 4*c)^3 + (a^2*cos(4*d*x + 4*c)^2 - a ^2*cos(4*d*x + 4*c))*sin(4*d*x + 4*c))*cos(1/2*arctan2(sin(4*d*x + 4*c), c os(4*d*x + 4*c))) + (8*a^2*cos(4*d*x + 4*c)^2 + 8*a^2*sin(4*d*x + 4*c)^2 - 3*a^2*cos(4*d*x + 4*c) + 32*(a^2*cos(4*d*x + 4*c)^2 + a^2*sin(4*d*x + 4*c )^2 - 2*a^2*cos(4*d*x + 4*c) + a^2)*cos(1/2*arctan2(sin(4*d*x + 4*c), cos( 4*d*x + 4*c)))^2 + 32*(a^2*cos(4*d*x + 4*c)^2 + a^2*sin(4*d*x + 4*c)^2 + 2 *a^2*cos(4*d*x + 4*c) + a^2)*sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c)))^2 + 2*(16*a^2*cos(4*d*x + 4*c)^2 + 16*a^2*sin(4*d*x + 4*c)^2 - 19* a^2*cos(4*d*x + 4*c) + 3*a^2)*cos(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d...
\[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^{3/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]