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

Optimal. Leaf size=25 \[ a \tanh ^{-1}\left (\sqrt{\frac{a^2}{x^2}+1}\right )+x \text{csch}^{-1}\left (\frac{x}{a}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0172984, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {5892, 6278, 266, 63, 208} \[ a \tanh ^{-1}\left (\sqrt{\frac{a^2}{x^2}+1}\right )+x \text{csch}^{-1}\left (\frac{x}{a}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a/x],x]

[Out]

x*ArcCsch[x/a] + a*ArcTanh[Sqrt[1 + a^2/x^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 6278

Int[ArcCsch[(c_.)*(x_)], x_Symbol] :> Simp[x*ArcCsch[c*x], x] + Dist[1/c, Int[1/(x*Sqrt[1 + 1/(c^2*x^2)]), x],
x] /; FreeQ[c, x]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

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

Mathematica [B]  time = 0.088855, size = 77, normalized size = 3.08 \[ \frac{a \sqrt{a^2+x^2} \left (\log \left (\frac{x}{\sqrt{a^2+x^2}}+1\right )-\log \left (1-\frac{x}{\sqrt{a^2+x^2}}\right )\right )}{2 x \sqrt{\frac{a^2}{x^2}+1}}+x \sinh ^{-1}\left (\frac{a}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSinh[a/x],x]

[Out]

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

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Maple [A]  time = 0.006, size = 31, normalized size = 1.2 \begin{align*} -a \left ( -{\frac{x}{a}{\it Arcsinh} \left ({\frac{a}{x}} \right ) }-{\it Artanh} \left ({\frac{1}{\sqrt{1+{\frac{{a}^{2}}{{x}^{2}}}}}} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(a/x),x)

[Out]

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

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Maxima [A]  time = 1.13453, size = 58, normalized size = 2.32 \begin{align*} \frac{1}{2} \, a{\left (\log \left (\sqrt{\frac{a^{2}}{x^{2}} + 1} + 1\right ) - \log \left (\sqrt{\frac{a^{2}}{x^{2}} + 1} - 1\right )\right )} + x \operatorname{arsinh}\left (\frac{a}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.75216, size = 219, normalized size = 8.76 \begin{align*} -a \log \left (x \sqrt{\frac{a^{2} + x^{2}}{x^{2}}} - x\right ) +{\left (x - 1\right )} \log \left (\frac{x \sqrt{\frac{a^{2} + x^{2}}{x^{2}}} + a}{x}\right ) + \log \left (x \sqrt{\frac{a^{2} + x^{2}}{x^{2}}} + a - x\right ) - \log \left (x \sqrt{\frac{a^{2} + x^{2}}{x^{2}}} - a - x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a/x),x)

[Out]

Integral(asinh(a/x), x)

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Giac [B]  time = 1.32074, size = 68, normalized size = 2.72 \begin{align*}{\left (\log \left ({\left | a \right |}\right ) \mathrm{sgn}\left (x\right ) - \frac{\log \left (-x + \sqrt{a^{2} + x^{2}}\right )}{\mathrm{sgn}\left (x\right )}\right )} a + x \log \left (\sqrt{\frac{a^{2}}{x^{2}} + 1} + \frac{a}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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