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

Optimal. Leaf size=33 \[ \frac{1}{2} a x \sqrt{\frac{a^2}{x^2}+1}+\frac{1}{2} x^2 \text{csch}^{-1}\left (\frac{x}{a}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0172917, antiderivative size = 33, 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 = {5892, 6284, 191} \[ \frac{1}{2} a x \sqrt{\frac{a^2}{x^2}+1}+\frac{1}{2} x^2 \text{csch}^{-1}\left (\frac{x}{a}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x*ArcSinh[a/x],x]

[Out]

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

Rule 5892

Int[ArcSinh[(c_.)/((a_.) + (b_.)*(x_)^(n_.))]^(m_.)*(u_.), x_Symbol] :> Int[u*ArcCsch[a/c + (b*x^n)/c]^m, x] /
; FreeQ[{a, b, c, n, m}, x]

Rule 6284

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcCsch[c*
x]))/(d*(m + 1)), x] + Dist[(b*d)/(c*(m + 1)), Int[(d*x)^(m - 1)/Sqrt[1 + 1/(c^2*x^2)], x], x] /; FreeQ[{a, b,
c, d, m}, x] && NeQ[m, -1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

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

Mathematica [A]  time = 0.0220711, size = 29, normalized size = 0.88 \[ \frac{1}{2} x \left (a \sqrt{\frac{a^2}{x^2}+1}+x \sinh ^{-1}\left (\frac{a}{x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*ArcSinh[a/x],x]

[Out]

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

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Maple [A]  time = 0.005, size = 38, normalized size = 1.2 \begin{align*} -{a}^{2} \left ( -{\frac{{x}^{2}}{2\,{a}^{2}}{\it Arcsinh} \left ({\frac{a}{x}} \right ) }-{\frac{x}{2\,a}\sqrt{1+{\frac{{a}^{2}}{{x}^{2}}}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*arcsinh(a/x),x)

[Out]

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

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Maxima [A]  time = 1.04402, size = 36, normalized size = 1.09 \begin{align*} \frac{1}{2} \, x^{2} \operatorname{arsinh}\left (\frac{a}{x}\right ) + \frac{1}{2} \, a x \sqrt{\frac{a^{2}}{x^{2}} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.67991, size = 105, normalized size = 3.18 \begin{align*} \frac{1}{2} \, x^{2} \log \left (\frac{x \sqrt{\frac{a^{2} + x^{2}}{x^{2}}} + a}{x}\right ) + \frac{1}{2} \, a x \sqrt{\frac{a^{2} + x^{2}}{x^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \operatorname{asinh}{\left (\frac{a}{x} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*asinh(a/x),x)

[Out]

Integral(x*asinh(a/x), x)

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Giac [A]  time = 1.31523, size = 65, normalized size = 1.97 \begin{align*} \frac{1}{2} \, x^{2} \log \left (\sqrt{\frac{a^{2}}{x^{2}} + 1} + \frac{a}{x}\right ) - \frac{1}{2} \,{\left ({\left | a \right |} \mathrm{sgn}\left (x\right ) - \frac{\sqrt{a^{2} + x^{2}}}{\mathrm{sgn}\left (x\right )}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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