\(\int \arcsin (a x)^3 \, dx\) [26]

Optimal result

Integrand size = 6, antiderivative size = 60 \[ \int \arcsin (a x)^3 \, dx=-\frac {6 \sqrt {1-a^2 x^2}}{a}-6 x \arcsin (a x)+\frac {3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{a}+x \arcsin (a x)^3 \]

[Out]

-6*x*arcsin(a*x)+x*arcsin(a*x)^3-6*(-a^2*x^2+1)^(1/2)/a+3*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)/a

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4715, 4767, 267} \[ \int \arcsin (a x)^3 \, dx=\frac {3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{a}-\frac {6 \sqrt {1-a^2 x^2}}{a}+x \arcsin (a x)^3-6 x \arcsin (a x) \]

[In]

Int[ArcSin[a*x]^3,x]

[Out]

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

Rule 267

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

Rule 4715

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSin[c*x])^n, x] - Dist[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]

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]

Rubi steps \begin{align*} \text {integral}& = x \arcsin (a x)^3-(3 a) \int \frac {x \arcsin (a x)^2}{\sqrt {1-a^2 x^2}} \, dx \\ & = \frac {3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{a}+x \arcsin (a x)^3-6 \int \arcsin (a x) \, dx \\ & = -6 x \arcsin (a x)+\frac {3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{a}+x \arcsin (a x)^3+(6 a) \int \frac {x}{\sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {6 \sqrt {1-a^2 x^2}}{a}-6 x \arcsin (a x)+\frac {3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{a}+x \arcsin (a x)^3 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00 \[ \int \arcsin (a x)^3 \, dx=-\frac {6 \sqrt {1-a^2 x^2}}{a}-6 x \arcsin (a x)+\frac {3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{a}+x \arcsin (a x)^3 \]

[In]

Integrate[ArcSin[a*x]^3,x]

[Out]

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

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.95

method	result	size
derivativedivides	\(\frac {a x \arcsin \left (a x \right )^{3}+3 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}-6 \sqrt {-a^{2} x^{2}+1}-6 a x \arcsin \left (a x \right )}{a}\)	\(57\)
default	\(\frac {a x \arcsin \left (a x \right )^{3}+3 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}-6 \sqrt {-a^{2} x^{2}+1}-6 a x \arcsin \left (a x \right )}{a}\)	\(57\)

[In]

int(arcsin(a*x)^3,x,method=_RETURNVERBOSE)

[Out]

1/a*(a*x*arcsin(a*x)^3+3*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)-6*(-a^2*x^2+1)^(1/2)-6*a*x*arcsin(a*x))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.73 \[ \int \arcsin (a x)^3 \, dx=\frac {a x \arcsin \left (a x\right )^{3} - 6 \, a x \arcsin \left (a x\right ) + 3 \, \sqrt {-a^{2} x^{2} + 1} {\left (\arcsin \left (a x\right )^{2} - 2\right )}}{a} \]

[In]

integrate(arcsin(a*x)^3,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.90 \[ \int \arcsin (a x)^3 \, dx=\begin {cases} x \operatorname {asin}^{3}{\left (a x \right )} - 6 x \operatorname {asin}{\left (a x \right )} + \frac {3 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{a} - \frac {6 \sqrt {- a^{2} x^{2} + 1}}{a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

integrate(asin(a*x)**3,x)

[Out]

Piecewise((x*asin(a*x)**3 - 6*x*asin(a*x) + 3*sqrt(-a**2*x**2 + 1)*asin(a*x)**2/a - 6*sqrt(-a**2*x**2 + 1)/a,
Ne(a, 0)), (0, True))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.95 \[ \int \arcsin (a x)^3 \, dx=x \arcsin \left (a x\right )^{3} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{2}}{a} - \frac {6 \, {\left (a x \arcsin \left (a x\right ) + \sqrt {-a^{2} x^{2} + 1}\right )}}{a} \]

[In]

integrate(arcsin(a*x)^3,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.93 \[ \int \arcsin (a x)^3 \, dx=x \arcsin \left (a x\right )^{3} - 6 \, x \arcsin \left (a x\right ) + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{2}}{a} - \frac {6 \, \sqrt {-a^{2} x^{2} + 1}}{a} \]

[In]

integrate(arcsin(a*x)^3,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.67 \[ \int \arcsin (a x)^3 \, dx=\frac {3\,\sqrt {1-a^2\,x^2}\,\left ({\mathrm {asin}\left (a\,x\right )}^2-2\right )}{a}+x\,\mathrm {asin}\left (a\,x\right )\,\left ({\mathrm {asin}\left (a\,x\right )}^2-6\right ) \]

[In]

int(asin(a*x)^3,x)

[Out]

(3*(1 - a^2*x^2)^(1/2)*(asin(a*x)^2 - 2))/a + x*asin(a*x)*(asin(a*x)^2 - 6)