Integrand size = 18, antiderivative size = 158 \[ \int \left (c-a^2 c x^2\right ) \arcsin (a x)^3 \, dx=-\frac {40 c \sqrt {1-a^2 x^2}}{9 a}-\frac {2 c \left (1-a^2 x^2\right )^{3/2}}{27 a}-\frac {14}{3} c x \arcsin (a x)+\frac {2}{9} a^2 c x^3 \arcsin (a x)+\frac {2 c \sqrt {1-a^2 x^2} \arcsin (a x)^2}{a}+\frac {c \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2}{3 a}+\frac {2}{3} c x \arcsin (a x)^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \arcsin (a x)^3 \]
-2/27*c*(-a^2*x^2+1)^(3/2)/a-14/3*c*x*arcsin(a*x)+2/9*a^2*c*x^3*arcsin(a*x )+1/3*c*(-a^2*x^2+1)^(3/2)*arcsin(a*x)^2/a+2/3*c*x*arcsin(a*x)^3+1/3*c*x*( -a^2*x^2+1)*arcsin(a*x)^3-40/9*c*(-a^2*x^2+1)^(1/2)/a+2*c*arcsin(a*x)^2*(- a^2*x^2+1)^(1/2)/a
Time = 0.06 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.64 \[ \int \left (c-a^2 c x^2\right ) \arcsin (a x)^3 \, dx=\frac {c \left (2 \sqrt {1-a^2 x^2} \left (-61+a^2 x^2\right )+6 a x \left (-21+a^2 x^2\right ) \arcsin (a x)-9 \sqrt {1-a^2 x^2} \left (-7+a^2 x^2\right ) \arcsin (a x)^2-9 a x \left (-3+a^2 x^2\right ) \arcsin (a x)^3\right )}{27 a} \]
(c*(2*Sqrt[1 - a^2*x^2]*(-61 + a^2*x^2) + 6*a*x*(-21 + a^2*x^2)*ArcSin[a*x ] - 9*Sqrt[1 - a^2*x^2]*(-7 + a^2*x^2)*ArcSin[a*x]^2 - 9*a*x*(-3 + a^2*x^2 )*ArcSin[a*x]^3))/(27*a)
Time = 0.72 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.28, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {5158, 5130, 5182, 5130, 241, 5154, 27, 353, 53, 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.
| \(\displaystyle \int \arcsin (a x)^3 \left (c-a^2 c x^2\right ) \, dx\) |
| \(\Big \downarrow \) 5158 |
| \(\displaystyle -a c \int x \sqrt {1-a^2 x^2} \arcsin (a x)^2dx+\frac {2}{3} c \int \arcsin (a x)^3dx+\frac {1}{3} c x \left (1-a^2 x^2\right ) \arcsin (a x)^3\) |
| \(\Big \downarrow \) 5130 |
| \(\displaystyle \frac {2}{3} c \left (x \arcsin (a x)^3-3 a \int \frac {x \arcsin (a x)^2}{\sqrt {1-a^2 x^2}}dx\right )-a c \int x \sqrt {1-a^2 x^2} \arcsin (a x)^2dx+\frac {1}{3} c x \left (1-a^2 x^2\right ) \arcsin (a x)^3\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {2}{3} c \left (x \arcsin (a x)^3-3 a \left (\frac {2 \int \arcsin (a x)dx}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{a^2}\right )\right )-a c \left (\frac {2 \int \left (1-a^2 x^2\right ) \arcsin (a x)dx}{3 a}-\frac {\left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2}{3 a^2}\right )+\frac {1}{3} c x \left (1-a^2 x^2\right ) \arcsin (a x)^3\) |
| \(\Big \downarrow \) 5130 |
| \(\displaystyle \frac {2}{3} c \left (x \arcsin (a x)^3-3 a \left (\frac {2 \left (x \arcsin (a x)-a \int \frac {x}{\sqrt {1-a^2 x^2}}dx\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{a^2}\right )\right )-a c \left (\frac {2 \int \left (1-a^2 x^2\right ) \arcsin (a x)dx}{3 a}-\frac {\left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2}{3 a^2}\right )+\frac {1}{3} c x \left (1-a^2 x^2\right ) \arcsin (a x)^3\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle -a c \left (\frac {2 \int \left (1-a^2 x^2\right ) \arcsin (a x)dx}{3 a}-\frac {\left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2}{3 a^2}\right )+\frac {1}{3} c x \left (1-a^2 x^2\right ) \arcsin (a x)^3+\frac {2}{3} c \left (x \arcsin (a x)^3-3 a \left (\frac {2 \left (\frac {\sqrt {1-a^2 x^2}}{a}+x \arcsin (a x)\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 5154 |
| \(\displaystyle -a c \left (\frac {2 \left (-a \int \frac {x \left (3-a^2 x^2\right )}{3 \sqrt {1-a^2 x^2}}dx-\frac {1}{3} a^2 x^3 \arcsin (a x)+x \arcsin (a x)\right )}{3 a}-\frac {\left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2}{3 a^2}\right )+\frac {1}{3} c x \left (1-a^2 x^2\right ) \arcsin (a x)^3+\frac {2}{3} c \left (x \arcsin (a x)^3-3 a \left (\frac {2 \left (\frac {\sqrt {1-a^2 x^2}}{a}+x \arcsin (a x)\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -a c \left (\frac {2 \left (-\frac {1}{3} a \int \frac {x \left (3-a^2 x^2\right )}{\sqrt {1-a^2 x^2}}dx-\frac {1}{3} a^2 x^3 \arcsin (a x)+x \arcsin (a x)\right )}{3 a}-\frac {\left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2}{3 a^2}\right )+\frac {1}{3} c x \left (1-a^2 x^2\right ) \arcsin (a x)^3+\frac {2}{3} c \left (x \arcsin (a x)^3-3 a \left (\frac {2 \left (\frac {\sqrt {1-a^2 x^2}}{a}+x \arcsin (a x)\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 353 |
| \(\displaystyle -a c \left (\frac {2 \left (-\frac {1}{6} a \int \frac {3-a^2 x^2}{\sqrt {1-a^2 x^2}}dx^2-\frac {1}{3} a^2 x^3 \arcsin (a x)+x \arcsin (a x)\right )}{3 a}-\frac {\left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2}{3 a^2}\right )+\frac {1}{3} c x \left (1-a^2 x^2\right ) \arcsin (a x)^3+\frac {2}{3} c \left (x \arcsin (a x)^3-3 a \left (\frac {2 \left (\frac {\sqrt {1-a^2 x^2}}{a}+x \arcsin (a x)\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle -a c \left (\frac {2 \left (-\frac {1}{6} a \int \left (\sqrt {1-a^2 x^2}+\frac {2}{\sqrt {1-a^2 x^2}}\right )dx^2-\frac {1}{3} a^2 x^3 \arcsin (a x)+x \arcsin (a x)\right )}{3 a}-\frac {\left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2}{3 a^2}\right )+\frac {1}{3} c x \left (1-a^2 x^2\right ) \arcsin (a x)^3+\frac {2}{3} c \left (x \arcsin (a x)^3-3 a \left (\frac {2 \left (\frac {\sqrt {1-a^2 x^2}}{a}+x \arcsin (a x)\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{a^2}\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} c x \left (1-a^2 x^2\right ) \arcsin (a x)^3+\frac {2}{3} c \left (x \arcsin (a x)^3-3 a \left (\frac {2 \left (\frac {\sqrt {1-a^2 x^2}}{a}+x \arcsin (a x)\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{a^2}\right )\right )-a c \left (\frac {2 \left (-\frac {1}{3} a^2 x^3 \arcsin (a x)-\frac {1}{6} a \left (-\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^2}-\frac {4 \sqrt {1-a^2 x^2}}{a^2}\right )+x \arcsin (a x)\right )}{3 a}-\frac {\left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2}{3 a^2}\right )\) |
(c*x*(1 - a^2*x^2)*ArcSin[a*x]^3)/3 - a*c*(-1/3*((1 - a^2*x^2)^(3/2)*ArcSi n[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* c*(x*ArcSin[a*x]^3 - 3*a*(-((Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/a^2) + (2*(S qrt[1 - a^2*x^2]/a + x*ArcSin[a*x]))/a)))/3
3.3.91.3.1 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[((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 = 0.07 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.84
| method | result | size |
| derivativedivides | \(-\frac {c \left (9 a^{3} x^{3} \arcsin \left (a x \right )^{3}+9 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-27 a x \arcsin \left (a x \right )^{3}-6 a^{3} x^{3} \arcsin \left (a x \right )-63 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}-2 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}+126 a x \arcsin \left (a x \right )+122 \sqrt {-a^{2} x^{2}+1}\right )}{27 a}\) | \(132\) |
| default | \(-\frac {c \left (9 a^{3} x^{3} \arcsin \left (a x \right )^{3}+9 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-27 a x \arcsin \left (a x \right )^{3}-6 a^{3} x^{3} \arcsin \left (a x \right )-63 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}-2 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}+126 a x \arcsin \left (a x \right )+122 \sqrt {-a^{2} x^{2}+1}\right )}{27 a}\) | \(132\) |
-1/27/a*c*(9*a^3*x^3*arcsin(a*x)^3+9*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)*a^2* x^2-27*a*x*arcsin(a*x)^3-6*a^3*x^3*arcsin(a*x)-63*arcsin(a*x)^2*(-a^2*x^2+ 1)^(1/2)-2*a^2*x^2*(-a^2*x^2+1)^(1/2)+126*a*x*arcsin(a*x)+122*(-a^2*x^2+1) ^(1/2))
Time = 0.25 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.60 \[ \int \left (c-a^2 c x^2\right ) \arcsin (a x)^3 \, dx=-\frac {9 \, {\left (a^{3} c x^{3} - 3 \, a c x\right )} \arcsin \left (a x\right )^{3} - 6 \, {\left (a^{3} c x^{3} - 21 \, a c x\right )} \arcsin \left (a x\right ) - {\left (2 \, a^{2} c x^{2} - 9 \, {\left (a^{2} c x^{2} - 7 \, c\right )} \arcsin \left (a x\right )^{2} - 122 \, c\right )} \sqrt {-a^{2} x^{2} + 1}}{27 \, a} \]
-1/27*(9*(a^3*c*x^3 - 3*a*c*x)*arcsin(a*x)^3 - 6*(a^3*c*x^3 - 21*a*c*x)*ar csin(a*x) - (2*a^2*c*x^2 - 9*(a^2*c*x^2 - 7*c)*arcsin(a*x)^2 - 122*c)*sqrt (-a^2*x^2 + 1))/a
Time = 0.29 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.95 \[ \int \left (c-a^2 c x^2\right ) \arcsin (a x)^3 \, dx=\begin {cases} - \frac {a^{2} c x^{3} \operatorname {asin}^{3}{\left (a x \right )}}{3} + \frac {2 a^{2} c x^{3} \operatorname {asin}{\left (a x \right )}}{9} - \frac {a c x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{3} + \frac {2 a c x^{2} \sqrt {- a^{2} x^{2} + 1}}{27} + c x \operatorname {asin}^{3}{\left (a x \right )} - \frac {14 c x \operatorname {asin}{\left (a x \right )}}{3} + \frac {7 c \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{3 a} - \frac {122 c \sqrt {- a^{2} x^{2} + 1}}{27 a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((-a**2*c*x**3*asin(a*x)**3/3 + 2*a**2*c*x**3*asin(a*x)/9 - a*c*x **2*sqrt(-a**2*x**2 + 1)*asin(a*x)**2/3 + 2*a*c*x**2*sqrt(-a**2*x**2 + 1)/ 27 + c*x*asin(a*x)**3 - 14*c*x*asin(a*x)/3 + 7*c*sqrt(-a**2*x**2 + 1)*asin (a*x)**2/(3*a) - 122*c*sqrt(-a**2*x**2 + 1)/(27*a), Ne(a, 0)), (0, True))
Time = 0.29 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.81 \[ \int \left (c-a^2 c x^2\right ) \arcsin (a x)^3 \, dx=-\frac {1}{3} \, {\left (\sqrt {-a^{2} x^{2} + 1} c x^{2} - \frac {7 \, \sqrt {-a^{2} x^{2} + 1} c}{a^{2}}\right )} a \arcsin \left (a x\right )^{2} - \frac {1}{3} \, {\left (a^{2} c x^{3} - 3 \, c x\right )} \arcsin \left (a x\right )^{3} + \frac {2}{27} \, {\left (\sqrt {-a^{2} x^{2} + 1} c x^{2} + \frac {3 \, {\left (a^{2} c x^{3} - 21 \, c x\right )} \arcsin \left (a x\right )}{a} - \frac {61 \, \sqrt {-a^{2} x^{2} + 1} c}{a^{2}}\right )} a \]
-1/3*(sqrt(-a^2*x^2 + 1)*c*x^2 - 7*sqrt(-a^2*x^2 + 1)*c/a^2)*a*arcsin(a*x) ^2 - 1/3*(a^2*c*x^3 - 3*c*x)*arcsin(a*x)^3 + 2/27*(sqrt(-a^2*x^2 + 1)*c*x^ 2 + 3*(a^2*c*x^3 - 21*c*x)*arcsin(a*x)/a - 61*sqrt(-a^2*x^2 + 1)*c/a^2)*a
Time = 0.31 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.88 \[ \int \left (c-a^2 c x^2\right ) \arcsin (a x)^3 \, dx=-\frac {1}{3} \, {\left (a^{2} x^{2} - 1\right )} c x \arcsin \left (a x\right )^{3} + \frac {2}{3} \, c x \arcsin \left (a x\right )^{3} + \frac {2}{9} \, {\left (a^{2} x^{2} - 1\right )} c x \arcsin \left (a x\right ) + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c \arcsin \left (a x\right )^{2}}{3 \, a} - \frac {40}{9} \, c x \arcsin \left (a x\right ) + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} c \arcsin \left (a x\right )^{2}}{a} - \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c}{27 \, a} - \frac {40 \, \sqrt {-a^{2} x^{2} + 1} c}{9 \, a} \]
-1/3*(a^2*x^2 - 1)*c*x*arcsin(a*x)^3 + 2/3*c*x*arcsin(a*x)^3 + 2/9*(a^2*x^ 2 - 1)*c*x*arcsin(a*x) + 1/3*(-a^2*x^2 + 1)^(3/2)*c*arcsin(a*x)^2/a - 40/9 *c*x*arcsin(a*x) + 2*sqrt(-a^2*x^2 + 1)*c*arcsin(a*x)^2/a - 2/27*(-a^2*x^2 + 1)^(3/2)*c/a - 40/9*sqrt(-a^2*x^2 + 1)*c/a
Timed out. \[ \int \left (c-a^2 c x^2\right ) \arcsin (a x)^3 \, dx=\int {\mathrm {asin}\left (a\,x\right )}^3\,\left (c-a^2\,c\,x^2\right ) \,d x \]