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]

1/2*x^2*arccsch(x/a)+1/2*a*x*(1+a^2/x^2)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 0.87, size = 45, normalized size = 1.36 \[ \frac {1}{2} \, x^{2} \log \left (\frac {x \sqrt {\frac {a^{2} + x^{2}}{x^{2}}} + a}{x}\right ) + \frac {1}{2} \, a x \sqrt {\frac {a^{2} + x^{2}}{x^{2}}} \]

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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giac [A]  time = 0.49, size = 47, normalized size = 1.42 \[ \frac {1}{2} \, x^{2} \log \left (\sqrt {\frac {a^{2}}{x^{2}} + 1} + \frac {a}{x}\right ) - \frac {1}{2} \, a {\left | a \right |} \mathrm {sgn}\relax (x) + \frac {\sqrt {a^{2} + x^{2}} a}{2 \, \mathrm {sgn}\relax (x)} \]

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

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maple [A]  time = 0.00, size = 38, normalized size = 1.15 \[ -a^{2} \left (-\frac {x^{2} \arcsinh \left (\frac {a}{x}\right )}{2 a^{2}}-\frac {x \sqrt {1+\frac {a^{2}}{x^{2}}}}{2 a}\right ) \]

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 = 0.32, size = 27, normalized size = 0.82 \[ \frac {1}{2} \, x^{2} \operatorname {arsinh}\left (\frac {a}{x}\right ) + \frac {1}{2} \, a x \sqrt {\frac {a^{2}}{x^{2}} + 1} \]

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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mupad [B]  time = 0.03, size = 27, normalized size = 0.82 \[ \frac {x^2\,\mathrm {asinh}\left (\frac {a}{x}\right )}{2}+\frac {a\,x\,\sqrt {\frac {a^2}{x^2}+1}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2*asinh(a/x))/2 + (a*x*(a^2/x^2 + 1)^(1/2))/2

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {asinh}{\left (\frac {a}{x} \right )}\, dx \]

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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