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

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

[Out]

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

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Rubi [A]  time = 0.0451432, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5653, 5717, 8} \[ -\frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{a}+x \sinh ^{-1}(a x)^2+2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 5717

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [A]  time = 0.0135055, size = 34, normalized size = 1. \[ -\frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{a}+x \sinh ^{-1}(a x)^2+2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.023, size = 36, normalized size = 1.1 \begin{align*}{\frac{1}{a} \left ( \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}ax-2\,{\it Arcsinh} \left ( ax \right ) \sqrt{{a}^{2}{x}^{2}+1}+2\,ax \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.18164, size = 43, normalized size = 1.26 \begin{align*} x \operatorname{arsinh}\left (a x\right )^{2} + 2 \, x - \frac{2 \, \sqrt{a^{2} x^{2} + 1} \operatorname{arsinh}\left (a x\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.10122, size = 134, normalized size = 3.94 \begin{align*} \frac{a x \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2} + 2 \, a x - 2 \, \sqrt{a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.218243, size = 32, normalized size = 0.94 \begin{align*} \begin{cases} x \operatorname{asinh}^{2}{\left (a x \right )} + 2 x - \frac{2 \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}{\left (a x \right )}}{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(asinh(a*x)**2,x)

[Out]

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

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Giac [A]  time = 1.45534, size = 84, normalized size = 2.47 \begin{align*} x \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2} + 2 \, a{\left (\frac{x}{a} - \frac{\sqrt{a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}{a^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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