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

Optimal. Leaf size=34 \[ \frac{1}{2} x^2 \sinh ^{-1}\left (a x^2\right )-\frac{\sqrt{a^2 x^4+1}}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.0224494, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6715, 5653, 261} \[ \frac{1}{2} x^2 \sinh ^{-1}\left (a x^2\right )-\frac{\sqrt{a^2 x^4+1}}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 5653

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSinh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcSinh[c*x])^(n - 1))/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 261

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]

Rubi steps

\begin{align*} \int x \sinh ^{-1}\left (a x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \sinh ^{-1}(a x) \, dx,x,x^2\right )\\ &=\frac{1}{2} x^2 \sinh ^{-1}\left (a x^2\right )-\frac{1}{2} a \operatorname{Subst}\left (\int \frac{x}{\sqrt{1+a^2 x^2}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{1+a^2 x^4}}{2 a}+\frac{1}{2} x^2 \sinh ^{-1}\left (a x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0146358, size = 34, normalized size = 1. \[ \frac{1}{2} x^2 \sinh ^{-1}\left (a x^2\right )-\frac{\sqrt{a^2 x^4+1}}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 31, normalized size = 0.9 \begin{align*}{\frac{1}{2\,a} \left ({x}^{2}a{\it Arcsinh} \left ( a{x}^{2} \right ) -\sqrt{{a}^{2}{x}^{4}+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.14811, size = 41, normalized size = 1.21 \begin{align*} \frac{a x^{2} \operatorname{arsinh}\left (a x^{2}\right ) - \sqrt{a^{2} x^{4} + 1}}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.65478, size = 89, normalized size = 2.62 \begin{align*} \frac{a x^{2} \log \left (a x^{2} + \sqrt{a^{2} x^{4} + 1}\right ) - \sqrt{a^{2} x^{4} + 1}}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.251971, size = 27, normalized size = 0.79 \begin{align*} \begin{cases} \frac{x^{2} \operatorname{asinh}{\left (a x^{2} \right )}}{2} - \frac{\sqrt{a^{2} x^{4} + 1}}{2 a} & \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*asinh(a*x**2),x)

[Out]

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

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Giac [A]  time = 1.29993, size = 54, normalized size = 1.59 \begin{align*} \frac{1}{2} \, x^{2} \log \left (a x^{2} + \sqrt{a^{2} x^{4} + 1}\right ) - \frac{\sqrt{a^{2} x^{4} + 1}}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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