\(\int x \arcsin (a x) \, dx\) [4]

Optimal result

Integrand size = 6, antiderivative size = 45 \[ \int x \arcsin (a x) \, dx=\frac {x \sqrt {1-a^2 x^2}}{4 a}-\frac {\arcsin (a x)}{4 a^2}+\frac {1}{2} x^2 \arcsin (a x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4723, 327, 222} \[ \int x \arcsin (a x) \, dx=-\frac {\arcsin (a x)}{4 a^2}+\frac {x \sqrt {1-a^2 x^2}}{4 a}+\frac {1}{2} x^2 \arcsin (a x) \]

[In]

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

[Out]

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

Rule 222

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]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.89 \[ \int x \arcsin (a x) \, dx=\frac {a x \sqrt {1-a^2 x^2}+\left (-1+2 a^2 x^2\right ) \arcsin (a x)}{4 a^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.89

method	result	size
derivativedivides	\(\frac {\frac {a^{2} x^{2} \arcsin \left (a x \right )}{2}+\frac {a x \sqrt {-a^{2} x^{2}+1}}{4}-\frac {\arcsin \left (a x \right )}{4}}{a^{2}}\)	\(40\)
default	\(\frac {\frac {a^{2} x^{2} \arcsin \left (a x \right )}{2}+\frac {a x \sqrt {-a^{2} x^{2}+1}}{4}-\frac {\arcsin \left (a x \right )}{4}}{a^{2}}\)	\(40\)
parts	\(\frac {x^{2} \arcsin \left (a x \right )}{2}-\frac {a \left (-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )}{2}\)	\(63\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.80 \[ \int x \arcsin (a x) \, dx=\frac {\sqrt {-a^{2} x^{2} + 1} a x + {\left (2 \, a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )}{4 \, a^{2}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.82 \[ \int x \arcsin (a x) \, dx=\begin {cases} \frac {x^{2} \operatorname {asin}{\left (a x \right )}}{2} + \frac {x \sqrt {- a^{2} x^{2} + 1}}{4 a} - \frac {\operatorname {asin}{\left (a x \right )}}{4 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.89 \[ \int x \arcsin (a x) \, dx=\frac {1}{2} \, x^{2} \arcsin \left (a x\right ) + \frac {1}{4} \, a {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x}{a^{2}} - \frac {\arcsin \left (a x\right )}{a^{3}}\right )} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.02 \[ \int x \arcsin (a x) \, dx=\frac {\sqrt {-a^{2} x^{2} + 1} x}{4 \, a} + \frac {{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )}{2 \, a^{2}} + \frac {\arcsin \left (a x\right )}{4 \, a^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.84 \[ \int x \arcsin (a x) \, dx=\frac {\mathrm {asin}\left (a\,x\right )\,\left (2\,a^2\,x^2-1\right )}{4\,a^2}+\frac {x\,\sqrt {1-a^2\,x^2}}{4\,a} \]

[In]

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

[Out]

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