\(\int \frac {\text {arcsinh}(\frac {a}{x})}{x^2} \, dx\) [304]

Optimal result

Integrand size = 10, antiderivative size = 29 \[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x^2} \, dx=\frac {\sqrt {1+\frac {a^2}{x^2}}}{a}-\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5870, 6419, 267} \[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x^2} \, dx=\frac {\sqrt {\frac {a^2}{x^2}+1}}{a}-\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{x} \]

[In]

Int[ArcSinh[a/x]/x^2,x]

[Out]

Sqrt[1 + a^2/x^2]/a - ArcCsch[x/a]/x

Rule 267

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

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 \frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{x^2} \, dx \\ & = -\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{x}-a \int \frac {1}{\sqrt {1+\frac {a^2}{x^2}} x^3} \, dx \\ & = \frac {\sqrt {1+\frac {a^2}{x^2}}}{a}-\frac {\text {csch}^{-1}\left (\frac {x}{a}\right )}{x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x^2} \, dx=\frac {\sqrt {1+\frac {a^2}{x^2}}}{a}-\frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x} \]

[In]

Integrate[ArcSinh[a/x]/x^2,x]

[Out]

Sqrt[1 + a^2/x^2]/a - ArcSinh[a/x]/x

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07

method	result	size
derivativedivides	\(-\frac {\frac {a \,\operatorname {arcsinh}\left (\frac {a}{x}\right )}{x}-\sqrt {\frac {a^{2}}{x^{2}}+1}}{a}\)	\(31\)
default	\(-\frac {\frac {a \,\operatorname {arcsinh}\left (\frac {a}{x}\right )}{x}-\sqrt {\frac {a^{2}}{x^{2}}+1}}{a}\)	\(31\)
parts	\(-\frac {\operatorname {arcsinh}\left (\frac {a}{x}\right )}{x}+\frac {a^{2}+x^{2}}{a \sqrt {\frac {a^{2}+x^{2}}{x^{2}}}\, x^{2}}\)	\(40\)

[In]

int(arcsinh(a/x)/x^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x^2} \, dx=-\frac {a \log \left (\frac {x \sqrt {\frac {a^{2} + x^{2}}{x^{2}}} + a}{x}\right ) - x \sqrt {\frac {a^{2} + x^{2}}{x^{2}}}}{a x} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.69 \[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x^2} \, dx=\begin {cases} - \frac {\operatorname {asinh}{\left (\frac {a}{x} \right )}}{x} + \frac {\sqrt {\frac {a^{2}}{x^{2}} + 1}}{a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

integrate(asinh(a/x)/x**2,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x^2} \, dx=-\frac {\frac {a \operatorname {arsinh}\left (\frac {a}{x}\right )}{x} - \sqrt {\frac {a^{2}}{x^{2}} + 1}}{a} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x^2} \, dx=-\frac {\log \left (\sqrt {\frac {a^{2}}{x^{2}} + 1} + \frac {a}{x}\right )}{x} + \frac {\sqrt {\frac {a^{2}}{x^{2}} + 1}}{a} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 2.65 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {\text {arcsinh}\left (\frac {a}{x}\right )}{x^2} \, dx=\frac {\sqrt {\frac {a^2}{x^2}+1}}{a}-\frac {\mathrm {asinh}\left (\frac {a}{x}\right )}{x} \]

[In]

int(asinh(a/x)/x^2,x)

[Out]

(a^2/x^2 + 1)^(1/2)/a - asinh(a/x)/x