3.180 \(\int \frac{x \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx\)

Optimal. Leaf size=28 \[ \frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{a^2}-\frac{x}{a} \]

[Out]

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

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Rubi [A]  time = 0.0419967, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {5717, 8} \[ \frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{a^2}-\frac{x}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0262278, size = 28, normalized size = 1. \[ \frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{a^2}-\frac{x}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0., size = 47, normalized size = 1.7 \begin{align*}{\frac{1}{{a}^{2}} \left ({a}^{2}{x}^{2}{\it Arcsinh} \left ( ax \right ) +{\it Arcsinh} \left ( ax \right ) -ax\sqrt{{a}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*arcsinh(a*x)/(a^2*x^2+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.24198, size = 35, normalized size = 1.25 \begin{align*} -\frac{x}{a} + \frac{\sqrt{a^{2} x^{2} + 1} \operatorname{arsinh}\left (a x\right )}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arcsinh(a*x)/(a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 3.10367, size = 82, normalized size = 2.93 \begin{align*} -\frac{a x - \sqrt{a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arcsinh(a*x)/(a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.538699, size = 24, normalized size = 0.86 \begin{align*} \begin{cases} - \frac{x}{a} + \frac{\sqrt{a^{2} x^{2} + 1} \operatorname{asinh}{\left (a x \right )}}{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*asinh(a*x)/(a**2*x**2+1)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.41754, size = 51, normalized size = 1.82 \begin{align*} -\frac{x}{a} + \frac{\sqrt{a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*arcsinh(a*x)/(a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

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