\(\int x \arcsin (a x)^2 \, dx\) [15]

Optimal result

Integrand size = 8, antiderivative size = 60 \[ \int x \arcsin (a x)^2 \, dx=-\frac {x^2}{4}+\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a}-\frac {\arcsin (a x)^2}{4 a^2}+\frac {1}{2} x^2 \arcsin (a x)^2 \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4723, 4795, 4737, 30} \[ \int x \arcsin (a x)^2 \, dx=\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a}-\frac {\arcsin (a x)^2}{4 a^2}+\frac {1}{2} x^2 \arcsin (a x)^2-\frac {x^2}{4} \]

[In]

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

[Out]

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

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 4737

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Si
mp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c
^2*d + e, 0] && NeQ[n, -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}{2} x^2 \arcsin (a x)^2-a \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx \\ & = \frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a}+\frac {1}{2} x^2 \arcsin (a x)^2-\frac {\int x \, dx}{2}-\frac {\int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx}{2 a} \\ & = -\frac {x^2}{4}+\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a}-\frac {\arcsin (a x)^2}{4 a^2}+\frac {1}{2} x^2 \arcsin (a x)^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.92 \[ \int x \arcsin (a x)^2 \, dx=\frac {-a^2 x^2+2 a x \sqrt {1-a^2 x^2} \arcsin (a x)+\left (-1+2 a^2 x^2\right ) \arcsin (a x)^2}{4 a^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.08

method	result	size
derivativedivides	\(\frac {\frac {\arcsin \left (a x \right )^{2} \left (a^{2} x^{2}-1\right )}{2}+\frac {\arcsin \left (a x \right ) \left (a x \sqrt {-a^{2} x^{2}+1}+\arcsin \left (a x \right )\right )}{2}-\frac {\arcsin \left (a x \right )^{2}}{4}-\frac {a^{2} x^{2}}{4}}{a^{2}}\)	\(65\)
default	\(\frac {\frac {\arcsin \left (a x \right )^{2} \left (a^{2} x^{2}-1\right )}{2}+\frac {\arcsin \left (a x \right ) \left (a x \sqrt {-a^{2} x^{2}+1}+\arcsin \left (a x \right )\right )}{2}-\frac {\arcsin \left (a x \right )^{2}}{4}-\frac {a^{2} x^{2}}{4}}{a^{2}}\)	\(65\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.85 \[ \int x \arcsin (a x)^2 \, dx=-\frac {a^{2} x^{2} - 2 \, \sqrt {-a^{2} x^{2} + 1} a x \arcsin \left (a x\right ) - {\left (2 \, a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{2}}{4 \, a^{2}} \]

[In]

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

[Out]

-1/4*(a^2*x^2 - 2*sqrt(-a^2*x^2 + 1)*a*x*arcsin(a*x) - (2*a^2*x^2 - 1)*arcsin(a*x)^2)/a^2

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.85 \[ \int x \arcsin (a x)^2 \, dx=\begin {cases} \frac {x^{2} \operatorname {asin}^{2}{\left (a x \right )}}{2} - \frac {x^{2}}{4} + \frac {x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{2 a} - \frac {\operatorname {asin}^{2}{\left (a x \right )}}{4 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((x**2*asin(a*x)**2/2 - x**2/4 + x*sqrt(-a**2*x**2 + 1)*asin(a*x)/(2*a) - asin(a*x)**2/(4*a**2), Ne(a
, 0)), (0, True))

Maxima [F]

\[ \int x \arcsin (a x)^2 \, dx=\int { x \arcsin \left (a x\right )^{2} \,d x } \]

[In]

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

[Out]

1/2*x^2*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^2 + a*integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)*x^2*arctan2(a*
x, sqrt(a*x + 1)*sqrt(-a*x + 1))/(a^2*x^2 - 1), x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.22 \[ \int x \arcsin (a x)^2 \, dx=\frac {\sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )}{2 \, a} + \frac {{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{2}}{2 \, a^{2}} + \frac {\arcsin \left (a x\right )^{2}}{4 \, a^{2}} - \frac {a^{2} x^{2} - 1}{4 \, a^{2}} - \frac {1}{8 \, a^{2}} \]

[In]

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

[Out]

1/2*sqrt(-a^2*x^2 + 1)*x*arcsin(a*x)/a + 1/2*(a^2*x^2 - 1)*arcsin(a*x)^2/a^2 + 1/4*arcsin(a*x)^2/a^2 - 1/4*(a^
2*x^2 - 1)/a^2 - 1/8/a^2

Mupad [F(-1)]

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

[In]

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

[Out]

int(x*asin(a*x)^2, x)