\(\int x^2 \arcsin (a x)^2 \, dx\) [14]

Optimal result

Integrand size = 10, antiderivative size = 82 \[ \int x^2 \arcsin (a x)^2 \, dx=-\frac {4 x}{9 a^2}-\frac {2 x^3}{27}+\frac {4 \sqrt {1-a^2 x^2} \arcsin (a x)}{9 a^3}+\frac {2 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)}{9 a}+\frac {1}{3} x^3 \arcsin (a x)^2 \]

[Out]

-4/9*x/a^2-2/27*x^3+1/3*x^3*arcsin(a*x)^2+4/9*arcsin(a*x)*(-a^2*x^2+1)^(1/2)/a^3+2/9*x^2*arcsin(a*x)*(-a^2*x^2
+1)^(1/2)/a

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4723, 4795, 4767, 8, 30} \[ \int x^2 \arcsin (a x)^2 \, dx=\frac {2 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)}{9 a}-\frac {4 x}{9 a^2}+\frac {4 \sqrt {1-a^2 x^2} \arcsin (a x)}{9 a^3}+\frac {1}{3} x^3 \arcsin (a x)^2-\frac {2 x^3}{27} \]

[In]

Int[x^2*ArcSin[a*x]^2,x]

[Out]

(-4*x)/(9*a^2) - (2*x^3)/27 + (4*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(9*a^3) + (2*x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]
)/(9*a) + (x^3*ArcSin[a*x]^2)/3

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 4723

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSi
n[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 -
c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 4767

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] + Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p
], Int[(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]

Rule 4795

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[f
*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m
+ 2*p + 1))), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + Dist[b*f*(n/(c*(m + 2*p + 1)))*S
imp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x],
x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \arcsin (a x)^2-\frac {1}{3} (2 a) \int \frac {x^3 \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx \\ & = \frac {2 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)}{9 a}+\frac {1}{3} x^3 \arcsin (a x)^2-\frac {2 \int x^2 \, dx}{9}-\frac {4 \int \frac {x \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx}{9 a} \\ & = -\frac {2 x^3}{27}+\frac {4 \sqrt {1-a^2 x^2} \arcsin (a x)}{9 a^3}+\frac {2 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)}{9 a}+\frac {1}{3} x^3 \arcsin (a x)^2-\frac {4 \int 1 \, dx}{9 a^2} \\ & = -\frac {4 x}{9 a^2}-\frac {2 x^3}{27}+\frac {4 \sqrt {1-a^2 x^2} \arcsin (a x)}{9 a^3}+\frac {2 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)}{9 a}+\frac {1}{3} x^3 \arcsin (a x)^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.78 \[ \int x^2 \arcsin (a x)^2 \, dx=\frac {-2 a x \left (6+a^2 x^2\right )+6 \sqrt {1-a^2 x^2} \left (2+a^2 x^2\right ) \arcsin (a x)+9 a^3 x^3 \arcsin (a x)^2}{27 a^3} \]

[In]

Integrate[x^2*ArcSin[a*x]^2,x]

[Out]

(-2*a*x*(6 + a^2*x^2) + 6*Sqrt[1 - a^2*x^2]*(2 + a^2*x^2)*ArcSin[a*x] + 9*a^3*x^3*ArcSin[a*x]^2)/(27*a^3)

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.72

method	result	size
derivativedivides	\(\frac {\frac {a^{3} x^{3} \arcsin \left (a x \right )^{2}}{3}+\frac {2 \arcsin \left (a x \right ) \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{9}-\frac {2 a^{3} x^{3}}{27}-\frac {4 a x}{9}}{a^{3}}\)	\(59\)
default	\(\frac {\frac {a^{3} x^{3} \arcsin \left (a x \right )^{2}}{3}+\frac {2 \arcsin \left (a x \right ) \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{9}-\frac {2 a^{3} x^{3}}{27}-\frac {4 a x}{9}}{a^{3}}\)	\(59\)

[In]

int(x^2*arcsin(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

1/a^3*(1/3*a^3*x^3*arcsin(a*x)^2+2/9*arcsin(a*x)*(a^2*x^2+2)*(-a^2*x^2+1)^(1/2)-2/27*a^3*x^3-4/9*a*x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.72 \[ \int x^2 \arcsin (a x)^2 \, dx=\frac {9 \, a^{3} x^{3} \arcsin \left (a x\right )^{2} - 2 \, a^{3} x^{3} + 6 \, {\left (a^{2} x^{2} + 2\right )} \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right ) - 12 \, a x}{27 \, a^{3}} \]

[In]

integrate(x^2*arcsin(a*x)^2,x, algorithm="fricas")

[Out]

1/27*(9*a^3*x^3*arcsin(a*x)^2 - 2*a^3*x^3 + 6*(a^2*x^2 + 2)*sqrt(-a^2*x^2 + 1)*arcsin(a*x) - 12*a*x)/a^3

Sympy [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.93 \[ \int x^2 \arcsin (a x)^2 \, dx=\begin {cases} \frac {x^{3} \operatorname {asin}^{2}{\left (a x \right )}}{3} - \frac {2 x^{3}}{27} + \frac {2 x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{9 a} - \frac {4 x}{9 a^{2}} + \frac {4 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{9 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

integrate(x**2*asin(a*x)**2,x)

[Out]

Piecewise((x**3*asin(a*x)**2/3 - 2*x**3/27 + 2*x**2*sqrt(-a**2*x**2 + 1)*asin(a*x)/(9*a) - 4*x/(9*a**2) + 4*sq
rt(-a**2*x**2 + 1)*asin(a*x)/(9*a**3), Ne(a, 0)), (0, True))

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.88 \[ \int x^2 \arcsin (a x)^2 \, dx=\frac {1}{3} \, x^{3} \arcsin \left (a x\right )^{2} + \frac {2}{9} \, a {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{4}}\right )} \arcsin \left (a x\right ) - \frac {2 \, {\left (a^{2} x^{3} + 6 \, x\right )}}{27 \, a^{2}} \]

[In]

integrate(x^2*arcsin(a*x)^2,x, algorithm="maxima")

[Out]

1/3*x^3*arcsin(a*x)^2 + 2/9*a*(sqrt(-a^2*x^2 + 1)*x^2/a^2 + 2*sqrt(-a^2*x^2 + 1)/a^4)*arcsin(a*x) - 2/27*(a^2*
x^3 + 6*x)/a^2

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.18 \[ \int x^2 \arcsin (a x)^2 \, dx=\frac {{\left (a^{2} x^{2} - 1\right )} x \arcsin \left (a x\right )^{2}}{3 \, a^{2}} + \frac {x \arcsin \left (a x\right )^{2}}{3 \, a^{2}} - \frac {2 \, {\left (a^{2} x^{2} - 1\right )} x}{27 \, a^{2}} - \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \arcsin \left (a x\right )}{9 \, a^{3}} - \frac {14 \, x}{27 \, a^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{3 \, a^{3}} \]

[In]

integrate(x^2*arcsin(a*x)^2,x, algorithm="giac")

[Out]

1/3*(a^2*x^2 - 1)*x*arcsin(a*x)^2/a^2 + 1/3*x*arcsin(a*x)^2/a^2 - 2/27*(a^2*x^2 - 1)*x/a^2 - 2/9*(-a^2*x^2 + 1
)^(3/2)*arcsin(a*x)/a^3 - 14/27*x/a^2 + 2/3*sqrt(-a^2*x^2 + 1)*arcsin(a*x)/a^3

Mupad [F(-1)]

Timed out. \[ \int x^2 \arcsin (a x)^2 \, dx=\int x^2\,{\mathrm {asin}\left (a\,x\right )}^2 \,d x \]

[In]

int(x^2*asin(a*x)^2,x)

[Out]

int(x^2*asin(a*x)^2, x)