Integrand size = 20, antiderivative size = 273 \[ \int \left (c-a^2 c x^2\right )^2 \arcsin (a x)^3 \, dx=-\frac {4144 c^2 \sqrt {1-a^2 x^2}}{1125 a}-\frac {272 c^2 \left (1-a^2 x^2\right )^{3/2}}{3375 a}-\frac {6 c^2 \left (1-a^2 x^2\right )^{5/2}}{625 a}-\frac {298}{75} c^2 x \arcsin (a x)+\frac {76}{225} a^2 c^2 x^3 \arcsin (a x)-\frac {6}{125} a^4 c^2 x^5 \arcsin (a x)+\frac {8 c^2 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{5 a}+\frac {4 c^2 \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2}{15 a}+\frac {3 c^2 \left (1-a^2 x^2\right )^{5/2} \arcsin (a x)^2}{25 a}+\frac {8}{15} c^2 x \arcsin (a x)^3+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \arcsin (a x)^3+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \arcsin (a x)^3 \]
-272/3375*c^2*(-a^2*x^2+1)^(3/2)/a-6/625*c^2*(-a^2*x^2+1)^(5/2)/a-298/75*c ^2*x*arcsin(a*x)+76/225*a^2*c^2*x^3*arcsin(a*x)-6/125*a^4*c^2*x^5*arcsin(a *x)+4/15*c^2*(-a^2*x^2+1)^(3/2)*arcsin(a*x)^2/a+3/25*c^2*(-a^2*x^2+1)^(5/2 )*arcsin(a*x)^2/a+8/15*c^2*x*arcsin(a*x)^3+4/15*c^2*x*(-a^2*x^2+1)*arcsin( a*x)^3+1/5*c^2*x*(-a^2*x^2+1)^2*arcsin(a*x)^3-4144/1125*c^2*(-a^2*x^2+1)^( 1/2)/a+8/5*c^2*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)/a
Time = 0.13 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.51 \[ \int \left (c-a^2 c x^2\right )^2 \arcsin (a x)^3 \, dx=\frac {c^2 \left (-2 \sqrt {1-a^2 x^2} \left (31841-842 a^2 x^2+81 a^4 x^4\right )-30 a x \left (2235-190 a^2 x^2+27 a^4 x^4\right ) \arcsin (a x)+225 \sqrt {1-a^2 x^2} \left (149-38 a^2 x^2+9 a^4 x^4\right ) \arcsin (a x)^2+1125 a x \left (15-10 a^2 x^2+3 a^4 x^4\right ) \arcsin (a x)^3\right )}{16875 a} \]
(c^2*(-2*Sqrt[1 - a^2*x^2]*(31841 - 842*a^2*x^2 + 81*a^4*x^4) - 30*a*x*(22 35 - 190*a^2*x^2 + 27*a^4*x^4)*ArcSin[a*x] + 225*Sqrt[1 - a^2*x^2]*(149 - 38*a^2*x^2 + 9*a^4*x^4)*ArcSin[a*x]^2 + 1125*a*x*(15 - 10*a^2*x^2 + 3*a^4* x^4)*ArcSin[a*x]^3))/(16875*a)
Time = 1.35 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.38, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5158, 27, 5158, 5130, 5182, 5130, 241, 5154, 27, 353, 53, 1576, 1140, 2009}
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.
| \(\displaystyle \int \arcsin (a x)^3 \left (c-a^2 c x^2\right )^2 \, dx\) |
| \(\Big \downarrow \) 5158 |
| \(\displaystyle -\frac {3}{5} a c^2 \int x \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2dx+\frac {4}{5} c \int c \left (1-a^2 x^2\right ) \arcsin (a x)^3dx+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \arcsin (a x)^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3}{5} a c^2 \int x \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2dx+\frac {4}{5} c^2 \int \left (1-a^2 x^2\right ) \arcsin (a x)^3dx+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \arcsin (a x)^3\) |
| \(\Big \downarrow \) 5158 |
| \(\displaystyle -\frac {3}{5} a c^2 \int x \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2dx+\frac {4}{5} c^2 \left (-a \int x \sqrt {1-a^2 x^2} \arcsin (a x)^2dx+\frac {2}{3} \int \arcsin (a x)^3dx+\frac {1}{3} x \left (1-a^2 x^2\right ) \arcsin (a x)^3\right )+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \arcsin (a x)^3\) |
| \(\Big \downarrow \) 5130 |
| \(\displaystyle \frac {4}{5} c^2 \left (\frac {2}{3} \left (x \arcsin (a x)^3-3 a \int \frac {x \arcsin (a x)^2}{\sqrt {1-a^2 x^2}}dx\right )-a \int x \sqrt {1-a^2 x^2} \arcsin (a x)^2dx+\frac {1}{3} x \left (1-a^2 x^2\right ) \arcsin (a x)^3\right )-\frac {3}{5} a c^2 \int x \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2dx+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \arcsin (a x)^3\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {4}{5} c^2 \left (\frac {2}{3} \left (x \arcsin (a x)^3-3 a \left (\frac {2 \int \arcsin (a x)dx}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{a^2}\right )\right )-a \left (\frac {2 \int \left (1-a^2 x^2\right ) \arcsin (a x)dx}{3 a}-\frac {\left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2}{3 a^2}\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \arcsin (a x)^3\right )-\frac {3}{5} a c^2 \left (\frac {2 \int \left (1-a^2 x^2\right )^2 \arcsin (a x)dx}{5 a}-\frac {\left (1-a^2 x^2\right )^{5/2} \arcsin (a x)^2}{5 a^2}\right )+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \arcsin (a x)^3\) |
| \(\Big \downarrow \) 5130 |
| \(\displaystyle \frac {4}{5} c^2 \left (\frac {2}{3} \left (x \arcsin (a x)^3-3 a \left (\frac {2 \left (x \arcsin (a x)-a \int \frac {x}{\sqrt {1-a^2 x^2}}dx\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{a^2}\right )\right )-a \left (\frac {2 \int \left (1-a^2 x^2\right ) \arcsin (a x)dx}{3 a}-\frac {\left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2}{3 a^2}\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \arcsin (a x)^3\right )-\frac {3}{5} a c^2 \left (\frac {2 \int \left (1-a^2 x^2\right )^2 \arcsin (a x)dx}{5 a}-\frac {\left (1-a^2 x^2\right )^{5/2} \arcsin (a x)^2}{5 a^2}\right )+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \arcsin (a x)^3\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {4}{5} c^2 \left (-a \left (\frac {2 \int \left (1-a^2 x^2\right ) \arcsin (a x)dx}{3 a}-\frac {\left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2}{3 a^2}\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \arcsin (a x)^3+\frac {2}{3} \left (x \arcsin (a x)^3-3 a \left (\frac {2 \left (\frac {\sqrt {1-a^2 x^2}}{a}+x \arcsin (a x)\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{a^2}\right )\right )\right )-\frac {3}{5} a c^2 \left (\frac {2 \int \left (1-a^2 x^2\right )^2 \arcsin (a x)dx}{5 a}-\frac {\left (1-a^2 x^2\right )^{5/2} \arcsin (a x)^2}{5 a^2}\right )+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \arcsin (a x)^3\) |
| \(\Big \downarrow \) 5154 |
| \(\displaystyle \frac {4}{5} c^2 \left (-a \left (\frac {2 \left (-a \int \frac {x \left (3-a^2 x^2\right )}{3 \sqrt {1-a^2 x^2}}dx-\frac {1}{3} a^2 x^3 \arcsin (a x)+x \arcsin (a x)\right )}{3 a}-\frac {\left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2}{3 a^2}\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \arcsin (a x)^3+\frac {2}{3} \left (x \arcsin (a x)^3-3 a \left (\frac {2 \left (\frac {\sqrt {1-a^2 x^2}}{a}+x \arcsin (a x)\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{a^2}\right )\right )\right )-\frac {3}{5} a c^2 \left (\frac {2 \left (-a \int \frac {x \left (3 a^4 x^4-10 a^2 x^2+15\right )}{15 \sqrt {1-a^2 x^2}}dx+\frac {1}{5} a^4 x^5 \arcsin (a x)-\frac {2}{3} a^2 x^3 \arcsin (a x)+x \arcsin (a x)\right )}{5 a}-\frac {\left (1-a^2 x^2\right )^{5/2} \arcsin (a x)^2}{5 a^2}\right )+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \arcsin (a x)^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4}{5} c^2 \left (-a \left (\frac {2 \left (-\frac {1}{3} a \int \frac {x \left (3-a^2 x^2\right )}{\sqrt {1-a^2 x^2}}dx-\frac {1}{3} a^2 x^3 \arcsin (a x)+x \arcsin (a x)\right )}{3 a}-\frac {\left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2}{3 a^2}\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \arcsin (a x)^3+\frac {2}{3} \left (x \arcsin (a x)^3-3 a \left (\frac {2 \left (\frac {\sqrt {1-a^2 x^2}}{a}+x \arcsin (a x)\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{a^2}\right )\right )\right )-\frac {3}{5} a c^2 \left (\frac {2 \left (-\frac {1}{15} a \int \frac {x \left (3 a^4 x^4-10 a^2 x^2+15\right )}{\sqrt {1-a^2 x^2}}dx+\frac {1}{5} a^4 x^5 \arcsin (a x)-\frac {2}{3} a^2 x^3 \arcsin (a x)+x \arcsin (a x)\right )}{5 a}-\frac {\left (1-a^2 x^2\right )^{5/2} \arcsin (a x)^2}{5 a^2}\right )+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \arcsin (a x)^3\) |
| \(\Big \downarrow \) 353 |
| \(\displaystyle \frac {4}{5} c^2 \left (-a \left (\frac {2 \left (-\frac {1}{6} a \int \frac {3-a^2 x^2}{\sqrt {1-a^2 x^2}}dx^2-\frac {1}{3} a^2 x^3 \arcsin (a x)+x \arcsin (a x)\right )}{3 a}-\frac {\left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2}{3 a^2}\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \arcsin (a x)^3+\frac {2}{3} \left (x \arcsin (a x)^3-3 a \left (\frac {2 \left (\frac {\sqrt {1-a^2 x^2}}{a}+x \arcsin (a x)\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{a^2}\right )\right )\right )-\frac {3}{5} a c^2 \left (\frac {2 \left (-\frac {1}{15} a \int \frac {x \left (3 a^4 x^4-10 a^2 x^2+15\right )}{\sqrt {1-a^2 x^2}}dx+\frac {1}{5} a^4 x^5 \arcsin (a x)-\frac {2}{3} a^2 x^3 \arcsin (a x)+x \arcsin (a x)\right )}{5 a}-\frac {\left (1-a^2 x^2\right )^{5/2} \arcsin (a x)^2}{5 a^2}\right )+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \arcsin (a x)^3\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {4}{5} c^2 \left (-a \left (\frac {2 \left (-\frac {1}{6} a \int \left (\sqrt {1-a^2 x^2}+\frac {2}{\sqrt {1-a^2 x^2}}\right )dx^2-\frac {1}{3} a^2 x^3 \arcsin (a x)+x \arcsin (a x)\right )}{3 a}-\frac {\left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2}{3 a^2}\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \arcsin (a x)^3+\frac {2}{3} \left (x \arcsin (a x)^3-3 a \left (\frac {2 \left (\frac {\sqrt {1-a^2 x^2}}{a}+x \arcsin (a x)\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{a^2}\right )\right )\right )-\frac {3}{5} a c^2 \left (\frac {2 \left (-\frac {1}{15} a \int \frac {x \left (3 a^4 x^4-10 a^2 x^2+15\right )}{\sqrt {1-a^2 x^2}}dx+\frac {1}{5} a^4 x^5 \arcsin (a x)-\frac {2}{3} a^2 x^3 \arcsin (a x)+x \arcsin (a x)\right )}{5 a}-\frac {\left (1-a^2 x^2\right )^{5/2} \arcsin (a x)^2}{5 a^2}\right )+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \arcsin (a x)^3\) |
| \(\Big \downarrow \) 1576 |
| \(\displaystyle \frac {4}{5} c^2 \left (-a \left (\frac {2 \left (-\frac {1}{6} a \int \left (\sqrt {1-a^2 x^2}+\frac {2}{\sqrt {1-a^2 x^2}}\right )dx^2-\frac {1}{3} a^2 x^3 \arcsin (a x)+x \arcsin (a x)\right )}{3 a}-\frac {\left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2}{3 a^2}\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \arcsin (a x)^3+\frac {2}{3} \left (x \arcsin (a x)^3-3 a \left (\frac {2 \left (\frac {\sqrt {1-a^2 x^2}}{a}+x \arcsin (a x)\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{a^2}\right )\right )\right )-\frac {3}{5} a c^2 \left (\frac {2 \left (-\frac {1}{30} a \int \frac {3 a^4 x^4-10 a^2 x^2+15}{\sqrt {1-a^2 x^2}}dx^2+\frac {1}{5} a^4 x^5 \arcsin (a x)-\frac {2}{3} a^2 x^3 \arcsin (a x)+x \arcsin (a x)\right )}{5 a}-\frac {\left (1-a^2 x^2\right )^{5/2} \arcsin (a x)^2}{5 a^2}\right )+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \arcsin (a x)^3\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \frac {4}{5} c^2 \left (-a \left (\frac {2 \left (-\frac {1}{6} a \int \left (\sqrt {1-a^2 x^2}+\frac {2}{\sqrt {1-a^2 x^2}}\right )dx^2-\frac {1}{3} a^2 x^3 \arcsin (a x)+x \arcsin (a x)\right )}{3 a}-\frac {\left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2}{3 a^2}\right )+\frac {1}{3} x \left (1-a^2 x^2\right ) \arcsin (a x)^3+\frac {2}{3} \left (x \arcsin (a x)^3-3 a \left (\frac {2 \left (\frac {\sqrt {1-a^2 x^2}}{a}+x \arcsin (a x)\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{a^2}\right )\right )\right )-\frac {3}{5} a c^2 \left (\frac {2 \left (-\frac {1}{30} a \int \left (3 \left (1-a^2 x^2\right )^{3/2}+4 \sqrt {1-a^2 x^2}+\frac {8}{\sqrt {1-a^2 x^2}}\right )dx^2+\frac {1}{5} a^4 x^5 \arcsin (a x)-\frac {2}{3} a^2 x^3 \arcsin (a x)+x \arcsin (a x)\right )}{5 a}-\frac {\left (1-a^2 x^2\right )^{5/2} \arcsin (a x)^2}{5 a^2}\right )+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \arcsin (a x)^3\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \arcsin (a x)^3+\frac {4}{5} c^2 \left (\frac {1}{3} x \left (1-a^2 x^2\right ) \arcsin (a x)^3+\frac {2}{3} \left (x \arcsin (a x)^3-3 a \left (\frac {2 \left (\frac {\sqrt {1-a^2 x^2}}{a}+x \arcsin (a x)\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{a^2}\right )\right )-a \left (\frac {2 \left (-\frac {1}{3} a^2 x^3 \arcsin (a x)-\frac {1}{6} a \left (-\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^2}-\frac {4 \sqrt {1-a^2 x^2}}{a^2}\right )+x \arcsin (a x)\right )}{3 a}-\frac {\left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2}{3 a^2}\right )\right )-\frac {3}{5} a c^2 \left (\frac {2 \left (\frac {1}{5} a^4 x^5 \arcsin (a x)-\frac {2}{3} a^2 x^3 \arcsin (a x)-\frac {1}{30} a \left (-\frac {6 \left (1-a^2 x^2\right )^{5/2}}{5 a^2}-\frac {8 \left (1-a^2 x^2\right )^{3/2}}{3 a^2}-\frac {16 \sqrt {1-a^2 x^2}}{a^2}\right )+x \arcsin (a x)\right )}{5 a}-\frac {\left (1-a^2 x^2\right )^{5/2} \arcsin (a x)^2}{5 a^2}\right )\) |
(c^2*x*(1 - a^2*x^2)^2*ArcSin[a*x]^3)/5 - (3*a*c^2*(-1/5*((1 - a^2*x^2)^(5 /2)*ArcSin[a*x]^2)/a^2 + (2*(-1/30*(a*((-16*Sqrt[1 - a^2*x^2])/a^2 - (8*(1 - a^2*x^2)^(3/2))/(3*a^2) - (6*(1 - a^2*x^2)^(5/2))/(5*a^2))) + x*ArcSin[ a*x] - (2*a^2*x^3*ArcSin[a*x])/3 + (a^4*x^5*ArcSin[a*x])/5))/(5*a)))/5 + ( 4*c^2*((x*(1 - a^2*x^2)*ArcSin[a*x]^3)/3 - a*(-1/3*((1 - a^2*x^2)^(3/2)*Ar cSin[a*x]^2)/a^2 + (2*(-1/6*(a*((-4*Sqrt[1 - a^2*x^2])/a^2 - (2*(1 - a^2*x ^2)^(3/2))/(3*a^2))) + x*ArcSin[a*x] - (a^2*x^3*ArcSin[a*x])/3))/(3*a)) + (2*(x*ArcSin[a*x]^3 - 3*a*(-((Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/a^2) + (2*( Sqrt[1 - a^2*x^2]/a + x*ArcSin[a*x]))/a)))/3))/5
3.3.90.3.1 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[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ {a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] , x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Ar cSin[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbo l] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcSin[c*x]) u, x ] - Simp[b*c Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; Fr eeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcSin[c*x])^n/(2*p + 1)), x] + (S imp[2*d*(p/(2*p + 1)) Int[(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c , d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Time = 0.11 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.75
| method | result | size |
| derivativedivides | \(\frac {c^{2} \left (3375 a^{5} x^{5} \arcsin \left (a x \right )^{3}+2025 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-11250 a^{3} x^{3} \arcsin \left (a x \right )^{3}-810 a^{5} x^{5} \arcsin \left (a x \right )-8550 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-162 a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}+16875 a x \arcsin \left (a x \right )^{3}+5700 a^{3} x^{3} \arcsin \left (a x \right )+33525 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}+1684 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}-67050 a x \arcsin \left (a x \right )-63682 \sqrt {-a^{2} x^{2}+1}\right )}{16875 a}\) | \(206\) |
| default | \(\frac {c^{2} \left (3375 a^{5} x^{5} \arcsin \left (a x \right )^{3}+2025 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-11250 a^{3} x^{3} \arcsin \left (a x \right )^{3}-810 a^{5} x^{5} \arcsin \left (a x \right )-8550 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-162 a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}+16875 a x \arcsin \left (a x \right )^{3}+5700 a^{3} x^{3} \arcsin \left (a x \right )+33525 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}+1684 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}-67050 a x \arcsin \left (a x \right )-63682 \sqrt {-a^{2} x^{2}+1}\right )}{16875 a}\) | \(206\) |
1/16875/a*c^2*(3375*a^5*x^5*arcsin(a*x)^3+2025*arcsin(a*x)^2*(-a^2*x^2+1)^ (1/2)*a^4*x^4-11250*a^3*x^3*arcsin(a*x)^3-810*a^5*x^5*arcsin(a*x)-8550*arc sin(a*x)^2*(-a^2*x^2+1)^(1/2)*a^2*x^2-162*a^4*x^4*(-a^2*x^2+1)^(1/2)+16875 *a*x*arcsin(a*x)^3+5700*a^3*x^3*arcsin(a*x)+33525*arcsin(a*x)^2*(-a^2*x^2+ 1)^(1/2)+1684*a^2*x^2*(-a^2*x^2+1)^(1/2)-67050*a*x*arcsin(a*x)-63682*(-a^2 *x^2+1)^(1/2))
Time = 0.26 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.58 \[ \int \left (c-a^2 c x^2\right )^2 \arcsin (a x)^3 \, dx=\frac {1125 \, {\left (3 \, a^{5} c^{2} x^{5} - 10 \, a^{3} c^{2} x^{3} + 15 \, a c^{2} x\right )} \arcsin \left (a x\right )^{3} - 30 \, {\left (27 \, a^{5} c^{2} x^{5} - 190 \, a^{3} c^{2} x^{3} + 2235 \, a c^{2} x\right )} \arcsin \left (a x\right ) - {\left (162 \, a^{4} c^{2} x^{4} - 1684 \, a^{2} c^{2} x^{2} - 225 \, {\left (9 \, a^{4} c^{2} x^{4} - 38 \, a^{2} c^{2} x^{2} + 149 \, c^{2}\right )} \arcsin \left (a x\right )^{2} + 63682 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{16875 \, a} \]
1/16875*(1125*(3*a^5*c^2*x^5 - 10*a^3*c^2*x^3 + 15*a*c^2*x)*arcsin(a*x)^3 - 30*(27*a^5*c^2*x^5 - 190*a^3*c^2*x^3 + 2235*a*c^2*x)*arcsin(a*x) - (162* a^4*c^2*x^4 - 1684*a^2*c^2*x^2 - 225*(9*a^4*c^2*x^4 - 38*a^2*c^2*x^2 + 149 *c^2)*arcsin(a*x)^2 + 63682*c^2)*sqrt(-a^2*x^2 + 1))/a
Time = 0.58 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.96 \[ \int \left (c-a^2 c x^2\right )^2 \arcsin (a x)^3 \, dx=\begin {cases} \frac {a^{4} c^{2} x^{5} \operatorname {asin}^{3}{\left (a x \right )}}{5} - \frac {6 a^{4} c^{2} x^{5} \operatorname {asin}{\left (a x \right )}}{125} + \frac {3 a^{3} c^{2} x^{4} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{25} - \frac {6 a^{3} c^{2} x^{4} \sqrt {- a^{2} x^{2} + 1}}{625} - \frac {2 a^{2} c^{2} x^{3} \operatorname {asin}^{3}{\left (a x \right )}}{3} + \frac {76 a^{2} c^{2} x^{3} \operatorname {asin}{\left (a x \right )}}{225} - \frac {38 a c^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{75} + \frac {1684 a c^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{16875} + c^{2} x \operatorname {asin}^{3}{\left (a x \right )} - \frac {298 c^{2} x \operatorname {asin}{\left (a x \right )}}{75} + \frac {149 c^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{75 a} - \frac {63682 c^{2} \sqrt {- a^{2} x^{2} + 1}}{16875 a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((a**4*c**2*x**5*asin(a*x)**3/5 - 6*a**4*c**2*x**5*asin(a*x)/125 + 3*a**3*c**2*x**4*sqrt(-a**2*x**2 + 1)*asin(a*x)**2/25 - 6*a**3*c**2*x**4 *sqrt(-a**2*x**2 + 1)/625 - 2*a**2*c**2*x**3*asin(a*x)**3/3 + 76*a**2*c**2 *x**3*asin(a*x)/225 - 38*a*c**2*x**2*sqrt(-a**2*x**2 + 1)*asin(a*x)**2/75 + 1684*a*c**2*x**2*sqrt(-a**2*x**2 + 1)/16875 + c**2*x*asin(a*x)**3 - 298* c**2*x*asin(a*x)/75 + 149*c**2*sqrt(-a**2*x**2 + 1)*asin(a*x)**2/(75*a) - 63682*c**2*sqrt(-a**2*x**2 + 1)/(16875*a), Ne(a, 0)), (0, True))
Time = 0.29 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.79 \[ \int \left (c-a^2 c x^2\right )^2 \arcsin (a x)^3 \, dx=\frac {1}{75} \, {\left (9 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{2} x^{4} - 38 \, \sqrt {-a^{2} x^{2} + 1} c^{2} x^{2} + \frac {149 \, \sqrt {-a^{2} x^{2} + 1} c^{2}}{a^{2}}\right )} a \arcsin \left (a x\right )^{2} + \frac {1}{15} \, {\left (3 \, a^{4} c^{2} x^{5} - 10 \, a^{2} c^{2} x^{3} + 15 \, c^{2} x\right )} \arcsin \left (a x\right )^{3} - \frac {2}{16875} \, {\left (81 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{2} x^{4} - 842 \, \sqrt {-a^{2} x^{2} + 1} c^{2} x^{2} + \frac {15 \, {\left (27 \, a^{4} c^{2} x^{5} - 190 \, a^{2} c^{2} x^{3} + 2235 \, c^{2} x\right )} \arcsin \left (a x\right )}{a} + \frac {31841 \, \sqrt {-a^{2} x^{2} + 1} c^{2}}{a^{2}}\right )} a \]
1/75*(9*sqrt(-a^2*x^2 + 1)*a^2*c^2*x^4 - 38*sqrt(-a^2*x^2 + 1)*c^2*x^2 + 1 49*sqrt(-a^2*x^2 + 1)*c^2/a^2)*a*arcsin(a*x)^2 + 1/15*(3*a^4*c^2*x^5 - 10* a^2*c^2*x^3 + 15*c^2*x)*arcsin(a*x)^3 - 2/16875*(81*sqrt(-a^2*x^2 + 1)*a^2 *c^2*x^4 - 842*sqrt(-a^2*x^2 + 1)*c^2*x^2 + 15*(27*a^4*c^2*x^5 - 190*a^2*c ^2*x^3 + 2235*c^2*x)*arcsin(a*x)/a + 31841*sqrt(-a^2*x^2 + 1)*c^2/a^2)*a
Time = 0.31 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.98 \[ \int \left (c-a^2 c x^2\right )^2 \arcsin (a x)^3 \, dx=\frac {1}{5} \, {\left (a^{2} x^{2} - 1\right )}^{2} c^{2} x \arcsin \left (a x\right )^{3} - \frac {4}{15} \, {\left (a^{2} x^{2} - 1\right )} c^{2} x \arcsin \left (a x\right )^{3} - \frac {6}{125} \, {\left (a^{2} x^{2} - 1\right )}^{2} c^{2} x \arcsin \left (a x\right ) + \frac {8}{15} \, c^{2} x \arcsin \left (a x\right )^{3} + \frac {3 \, {\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1} c^{2} \arcsin \left (a x\right )^{2}}{25 \, a} + \frac {272}{1125} \, {\left (a^{2} x^{2} - 1\right )} c^{2} x \arcsin \left (a x\right ) + \frac {4 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{2} \arcsin \left (a x\right )^{2}}{15 \, a} - \frac {4144}{1125} \, c^{2} x \arcsin \left (a x\right ) - \frac {6 \, {\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1} c^{2}}{625 \, a} + \frac {8 \, \sqrt {-a^{2} x^{2} + 1} c^{2} \arcsin \left (a x\right )^{2}}{5 \, a} - \frac {272 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{2}}{3375 \, a} - \frac {4144 \, \sqrt {-a^{2} x^{2} + 1} c^{2}}{1125 \, a} \]
1/5*(a^2*x^2 - 1)^2*c^2*x*arcsin(a*x)^3 - 4/15*(a^2*x^2 - 1)*c^2*x*arcsin( a*x)^3 - 6/125*(a^2*x^2 - 1)^2*c^2*x*arcsin(a*x) + 8/15*c^2*x*arcsin(a*x)^ 3 + 3/25*(a^2*x^2 - 1)^2*sqrt(-a^2*x^2 + 1)*c^2*arcsin(a*x)^2/a + 272/1125 *(a^2*x^2 - 1)*c^2*x*arcsin(a*x) + 4/15*(-a^2*x^2 + 1)^(3/2)*c^2*arcsin(a* x)^2/a - 4144/1125*c^2*x*arcsin(a*x) - 6/625*(a^2*x^2 - 1)^2*sqrt(-a^2*x^2 + 1)*c^2/a + 8/5*sqrt(-a^2*x^2 + 1)*c^2*arcsin(a*x)^2/a - 272/3375*(-a^2* x^2 + 1)^(3/2)*c^2/a - 4144/1125*sqrt(-a^2*x^2 + 1)*c^2/a
Timed out. \[ \int \left (c-a^2 c x^2\right )^2 \arcsin (a x)^3 \, dx=\int {\mathrm {asin}\left (a\,x\right )}^3\,{\left (c-a^2\,c\,x^2\right )}^2 \,d x \]