\(\int x \text {arcsinh}(\frac {a}{x}) \, dx\) [301]

Optimal result

Integrand size = 8, antiderivative size = 33 \[ \int x \text {arcsinh}\left (\frac {a}{x}\right ) \, dx=\frac {1}{2} a \sqrt {1+\frac {a^2}{x^2}} x+\frac {1}{2} x^2 \text {csch}^{-1}\left (\frac {x}{a}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5870, 6419, 197} \[ \int x \text {arcsinh}\left (\frac {a}{x}\right ) \, dx=\frac {1}{2} a x \sqrt {\frac {a^2}{x^2}+1}+\frac {1}{2} x^2 \text {csch}^{-1}\left (\frac {x}{a}\right ) \]

[In]

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

[Out]

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

Rule 197

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 5870

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 6419

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*} \text {integral}& = \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] (verified)

Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int x \text {arcsinh}\left (\frac {a}{x}\right ) \, dx=\frac {1}{2} x \left (a \sqrt {1+\frac {a^2}{x^2}}+x \text {arcsinh}\left (\frac {a}{x}\right )\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.15

method	result	size
derivativedivides	\(-a^{2} \left (-\frac {x^{2} \operatorname {arcsinh}\left (\frac {a}{x}\right )}{2 a^{2}}-\frac {x \sqrt {\frac {a^{2}}{x^{2}}+1}}{2 a}\right )\)	\(38\)
default	\(-a^{2} \left (-\frac {x^{2} \operatorname {arcsinh}\left (\frac {a}{x}\right )}{2 a^{2}}-\frac {x \sqrt {\frac {a^{2}}{x^{2}}+1}}{2 a}\right )\)	\(38\)
parts	\(\frac {x^{2} \operatorname {arcsinh}\left (\frac {a}{x}\right )}{2}+\frac {a \left (a^{2}+x^{2}\right )}{2 \sqrt {\frac {a^{2}+x^{2}}{x^{2}}}\, x}\)	\(39\)

[In]

int(x*arcsinh(a/x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.36 \[ \int x \text {arcsinh}\left (\frac {a}{x}\right ) \, dx=\frac {1}{2} \, x^{2} \log \left (\frac {x \sqrt {\frac {a^{2} + x^{2}}{x^{2}}} + a}{x}\right ) + \frac {1}{2} \, a x \sqrt {\frac {a^{2} + x^{2}}{x^{2}}} \]

[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)

Sympy [F]

\[ \int x \text {arcsinh}\left (\frac {a}{x}\right ) \, dx=\int x \operatorname {asinh}{\left (\frac {a}{x} \right )}\, dx \]

[In]

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

[Out]

Integral(x*asinh(a/x), x)

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int x \text {arcsinh}\left (\frac {a}{x}\right ) \, dx=\frac {1}{2} \, x^{2} \operatorname {arsinh}\left (\frac {a}{x}\right ) + \frac {1}{2} \, a x \sqrt {\frac {a^{2}}{x^{2}} + 1} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.42 \[ \int x \text {arcsinh}\left (\frac {a}{x}\right ) \, dx=\frac {1}{2} \, x^{2} \log \left (\sqrt {\frac {a^{2}}{x^{2}} + 1} + \frac {a}{x}\right ) - \frac {1}{2} \, a {\left | a \right |} \mathrm {sgn}\left (x\right ) + \frac {\sqrt {a^{2} + x^{2}} a}{2 \, \mathrm {sgn}\left (x\right )} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int x \text {arcsinh}\left (\frac {a}{x}\right ) \, dx=\frac {x^2\,\mathrm {asinh}\left (\frac {a}{x}\right )}{2}+\frac {a\,x\,\sqrt {\frac {a^2}{x^2}+1}}{2} \]

[In]

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

[Out]

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