3.3 \(\int x^2 \sinh ^{-1}(a x) \, dx\)

Optimal. Leaf size=52 \[ -\frac{\left (a^2 x^2+1\right )^{3/2}}{9 a^3}+\frac{\sqrt{a^2 x^2+1}}{3 a^3}+\frac{1}{3} x^3 \sinh ^{-1}(a x) \]

[Out]

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

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Rubi [A]  time = 0.0345017, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {5661, 266, 43} \[ -\frac{\left (a^2 x^2+1\right )^{3/2}}{9 a^3}+\frac{\sqrt{a^2 x^2+1}}{3 a^3}+\frac{1}{3} x^3 \sinh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5661

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcS
inh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSinh[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 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^2 \sinh ^{-1}(a x) \, dx &=\frac{1}{3} x^3 \sinh ^{-1}(a x)-\frac{1}{3} a \int \frac{x^3}{\sqrt{1+a^2 x^2}} \, dx\\ &=\frac{1}{3} x^3 \sinh ^{-1}(a x)-\frac{1}{6} a \operatorname{Subst}\left (\int \frac{x}{\sqrt{1+a^2 x}} \, dx,x,x^2\right )\\ &=\frac{1}{3} x^3 \sinh ^{-1}(a x)-\frac{1}{6} a \operatorname{Subst}\left (\int \left (-\frac{1}{a^2 \sqrt{1+a^2 x}}+\frac{\sqrt{1+a^2 x}}{a^2}\right ) \, dx,x,x^2\right )\\ &=\frac{\sqrt{1+a^2 x^2}}{3 a^3}-\frac{\left (1+a^2 x^2\right )^{3/2}}{9 a^3}+\frac{1}{3} x^3 \sinh ^{-1}(a x)\\ \end{align*}

Mathematica [A]  time = 0.0236022, size = 41, normalized size = 0.79 \[ \frac{1}{9} \left (\frac{\left (2-a^2 x^2\right ) \sqrt{a^2 x^2+1}}{a^3}+3 x^3 \sinh ^{-1}(a x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 50, normalized size = 1. \begin{align*}{\frac{1}{{a}^{3}} \left ({\frac{{a}^{3}{x}^{3}{\it Arcsinh} \left ( ax \right ) }{3}}-{\frac{{a}^{2}{x}^{2}}{9}\sqrt{{a}^{2}{x}^{2}+1}}+{\frac{2}{9}\sqrt{{a}^{2}{x}^{2}+1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^3*(1/3*a^3*x^3*arcsinh(a*x)-1/9*a^2*x^2*(a^2*x^2+1)^(1/2)+2/9*(a^2*x^2+1)^(1/2))

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Maxima [A]  time = 1.16436, size = 65, normalized size = 1.25 \begin{align*} \frac{1}{3} \, x^{3} \operatorname{arsinh}\left (a x\right ) - \frac{1}{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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*arcsinh(a*x) - 1/9*a*(sqrt(a^2*x^2 + 1)*x^2/a^2 - 2*sqrt(a^2*x^2 + 1)/a^4)

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Fricas [A]  time = 1.77979, size = 113, normalized size = 2.17 \begin{align*} \frac{3 \, a^{3} x^{3} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - \sqrt{a^{2} x^{2} + 1}{\left (a^{2} x^{2} - 2\right )}}{9 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(3*a^3*x^3*log(a*x + sqrt(a^2*x^2 + 1)) - sqrt(a^2*x^2 + 1)*(a^2*x^2 - 2))/a^3

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Sympy [A]  time = 0.550648, size = 48, normalized size = 0.92 \begin{align*} \begin{cases} \frac{x^{3} \operatorname{asinh}{\left (a x \right )}}{3} - \frac{x^{2} \sqrt{a^{2} x^{2} + 1}}{9 a} + \frac{2 \sqrt{a^{2} x^{2} + 1}}{9 a^{3}} & \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**2*asinh(a*x),x)

[Out]

Piecewise((x**3*asinh(a*x)/3 - x**2*sqrt(a**2*x**2 + 1)/(9*a) + 2*sqrt(a**2*x**2 + 1)/(9*a**3), Ne(a, 0)), (0,
True))

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Giac [A]  time = 1.31731, size = 70, normalized size = 1.35 \begin{align*} \frac{1}{3} \, x^{3} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - \frac{{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}} - 3 \, \sqrt{a^{2} x^{2} + 1}}{9 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*log(a*x + sqrt(a^2*x^2 + 1)) - 1/9*((a^2*x^2 + 1)^(3/2) - 3*sqrt(a^2*x^2 + 1))/a^3