3.25 \(\int x \sin ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=99 \[ -\frac{3 x \sqrt{1-a^2 x^2}}{8 a}+\frac{3 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{4 a}-\frac{\sin ^{-1}(a x)^3}{4 a^2}+\frac{3 \sin ^{-1}(a x)}{8 a^2}+\frac{1}{2} x^2 \sin ^{-1}(a x)^3-\frac{3}{4} x^2 \sin ^{-1}(a x) \]

[Out]

(-3*x*Sqrt[1 - a^2*x^2])/(8*a) + (3*ArcSin[a*x])/(8*a^2) - (3*x^2*ArcSin[a*x])/4 + (3*x*Sqrt[1 - a^2*x^2]*ArcS
in[a*x]^2)/(4*a) - ArcSin[a*x]^3/(4*a^2) + (x^2*ArcSin[a*x]^3)/2

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Rubi [A]  time = 0.155988, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {4627, 4707, 4641, 321, 216} \[ -\frac{3 x \sqrt{1-a^2 x^2}}{8 a}+\frac{3 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{4 a}-\frac{\sin ^{-1}(a x)^3}{4 a^2}+\frac{3 \sin ^{-1}(a x)}{8 a^2}+\frac{1}{2} x^2 \sin ^{-1}(a x)^3-\frac{3}{4} x^2 \sin ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*x*Sqrt[1 - a^2*x^2])/(8*a) + (3*ArcSin[a*x])/(8*a^2) - (3*x^2*ArcSin[a*x])/4 + (3*x*Sqrt[1 - a^2*x^2]*ArcS
in[a*x]^2)/(4*a) - ArcSin[a*x]^3/(4*a^2) + (x^2*ArcSin[a*x]^3)/2

Rule 4627

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 4707

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

Rule 4641

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

Rule 321

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 216

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]

Rubi steps

\begin{align*} \int x \sin ^{-1}(a x)^3 \, dx &=\frac{1}{2} x^2 \sin ^{-1}(a x)^3-\frac{1}{2} (3 a) \int \frac{x^2 \sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{3 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{4 a}+\frac{1}{2} x^2 \sin ^{-1}(a x)^3-\frac{3}{2} \int x \sin ^{-1}(a x) \, dx-\frac{3 \int \frac{\sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx}{4 a}\\ &=-\frac{3}{4} x^2 \sin ^{-1}(a x)+\frac{3 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{4 a}-\frac{\sin ^{-1}(a x)^3}{4 a^2}+\frac{1}{2} x^2 \sin ^{-1}(a x)^3+\frac{1}{4} (3 a) \int \frac{x^2}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{3 x \sqrt{1-a^2 x^2}}{8 a}-\frac{3}{4} x^2 \sin ^{-1}(a x)+\frac{3 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{4 a}-\frac{\sin ^{-1}(a x)^3}{4 a^2}+\frac{1}{2} x^2 \sin ^{-1}(a x)^3+\frac{3 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{8 a}\\ &=-\frac{3 x \sqrt{1-a^2 x^2}}{8 a}+\frac{3 \sin ^{-1}(a x)}{8 a^2}-\frac{3}{4} x^2 \sin ^{-1}(a x)+\frac{3 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{4 a}-\frac{\sin ^{-1}(a x)^3}{4 a^2}+\frac{1}{2} x^2 \sin ^{-1}(a x)^3\\ \end{align*}

Mathematica [A]  time = 0.025192, size = 82, normalized size = 0.83 \[ \frac{-3 a x \sqrt{1-a^2 x^2}+\left (4 a^2 x^2-2\right ) \sin ^{-1}(a x)^3+6 a x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2+\left (3-6 a^2 x^2\right ) \sin ^{-1}(a x)}{8 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a*x*Sqrt[1 - a^2*x^2] + (3 - 6*a^2*x^2)*ArcSin[a*x] + 6*a*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2 + (-2 + 4*a^2*
x^2)*ArcSin[a*x]^3)/(8*a^2)

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Maple [A]  time = 0.041, size = 96, normalized size = 1. \begin{align*}{\frac{1}{{a}^{2}} \left ({\frac{ \left ({a}^{2}{x}^{2}-1 \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{3}}{2}}+{\frac{3\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}}{4} \left ( ax\sqrt{-{a}^{2}{x}^{2}+1}+\arcsin \left ( ax \right ) \right ) }-{\frac{ \left ( 3\,{a}^{2}{x}^{2}-3 \right ) \arcsin \left ( ax \right ) }{4}}-{\frac{3\,ax}{8}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{3\,\arcsin \left ( ax \right ) }{8}}-{\frac{ \left ( \arcsin \left ( ax \right ) \right ) ^{3}}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^2*(1/2*(a^2*x^2-1)*arcsin(a*x)^3+3/4*arcsin(a*x)^2*(a*x*(-a^2*x^2+1)^(1/2)+arcsin(a*x))-3/4*(a^2*x^2-1)*ar
csin(a*x)-3/8*a*x*(-a^2*x^2+1)^(1/2)-3/8*arcsin(a*x)-1/2*arcsin(a*x)^3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, x^{2} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )^{3} + 3 \, a \int \frac{\sqrt{a x + 1} \sqrt{-a x + 1} x^{2} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )^{2}}{2 \,{\left (a^{2} x^{2} - 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^3 + 3*a*integrate(1/2*sqrt(a*x + 1)*sqrt(-a*x + 1)*x^2*arct
an2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^2/(a^2*x^2 - 1), x)

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Fricas [A]  time = 2.03548, size = 170, normalized size = 1.72 \begin{align*} \frac{2 \,{\left (2 \, a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{3} - 3 \,{\left (2 \, a^{2} x^{2} - 1\right )} \arcsin \left (a x\right ) + 3 \, \sqrt{-a^{2} x^{2} + 1}{\left (2 \, a x \arcsin \left (a x\right )^{2} - a x\right )}}{8 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2*(2*a^2*x^2 - 1)*arcsin(a*x)^3 - 3*(2*a^2*x^2 - 1)*arcsin(a*x) + 3*sqrt(-a^2*x^2 + 1)*(2*a*x*arcsin(a*x)
^2 - a*x))/a^2

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Sympy [A]  time = 1.60898, size = 92, normalized size = 0.93 \begin{align*} \begin{cases} \frac{x^{2} \operatorname{asin}^{3}{\left (a x \right )}}{2} - \frac{3 x^{2} \operatorname{asin}{\left (a x \right )}}{4} + \frac{3 x \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a x \right )}}{4 a} - \frac{3 x \sqrt{- a^{2} x^{2} + 1}}{8 a} - \frac{\operatorname{asin}^{3}{\left (a x \right )}}{4 a^{2}} + \frac{3 \operatorname{asin}{\left (a x \right )}}{8 a^{2}} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**2*asin(a*x)**3/2 - 3*x**2*asin(a*x)/4 + 3*x*sqrt(-a**2*x**2 + 1)*asin(a*x)**2/(4*a) - 3*x*sqrt(-
a**2*x**2 + 1)/(8*a) - asin(a*x)**3/(4*a**2) + 3*asin(a*x)/(8*a**2), Ne(a, 0)), (0, True))

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Giac [A]  time = 1.30534, size = 136, normalized size = 1.37 \begin{align*} \frac{3 \, \sqrt{-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{2}}{4 \, a} + \frac{{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{3}}{2 \, a^{2}} + \frac{\arcsin \left (a x\right )^{3}}{4 \, a^{2}} - \frac{3 \, \sqrt{-a^{2} x^{2} + 1} x}{8 \, a} - \frac{3 \,{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )}{4 \, a^{2}} - \frac{3 \, \arcsin \left (a x\right )}{8 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*sqrt(-a^2*x^2 + 1)*x*arcsin(a*x)^2/a + 1/2*(a^2*x^2 - 1)*arcsin(a*x)^3/a^2 + 1/4*arcsin(a*x)^3/a^2 - 3/8*s
qrt(-a^2*x^2 + 1)*x/a - 3/4*(a^2*x^2 - 1)*arcsin(a*x)/a^2 - 3/8*arcsin(a*x)/a^2