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

Optimal. Leaf size=25 \[ x \sinh ^{-1}(a x)-\frac{\sqrt{a^2 x^2+1}}{a} \]

[Out]

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

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Rubi [A]  time = 0.0075172, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5653, 261} \[ x \sinh ^{-1}(a x)-\frac{\sqrt{a^2 x^2+1}}{a} \]

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a*x],x]

[Out]

-(Sqrt[1 + a^2*x^2]/a) + x*ArcSinh[a*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 \sinh ^{-1}(a x) \, dx &=x \sinh ^{-1}(a x)-a \int \frac{x}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1+a^2 x^2}}{a}+x \sinh ^{-1}(a x)\\ \end{align*}

Mathematica [A]  time = 0.0072242, size = 25, normalized size = 1. \[ x \sinh ^{-1}(a x)-\frac{\sqrt{a^2 x^2+1}}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSinh[a*x],x]

[Out]

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

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Maple [A]  time = 0.002, size = 26, normalized size = 1. \begin{align*}{\frac{1}{a} \left ( ax{\it Arcsinh} \left ( ax \right ) -\sqrt{{a}^{2}{x}^{2}+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(a*x),x)

[Out]

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

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Maxima [A]  time = 1.10788, size = 34, normalized size = 1.36 \begin{align*} \frac{a x \operatorname{arsinh}\left (a x\right ) - \sqrt{a^{2} x^{2} + 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.75271, size = 78, normalized size = 3.12 \begin{align*} \frac{a x \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - \sqrt{a^{2} x^{2} + 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.152813, size = 20, normalized size = 0.8 \begin{align*} \begin{cases} x \operatorname{asinh}{\left (a x \right )} - \frac{\sqrt{a^{2} x^{2} + 1}}{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),x)

[Out]

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

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Giac [A]  time = 1.35222, size = 47, normalized size = 1.88 \begin{align*} x \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - \frac{\sqrt{a^{2} x^{2} + 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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