3.8 \(\int \frac{\sinh ^{-1}(a x)}{x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0136569, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5661, 264} \[ -\frac{a \sqrt{a^2 x^2+1}}{2 x}-\frac{\sinh ^{-1}(a x)}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a*x]/x^3,x]

[Out]

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

Rule 5661

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcS
inh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1))/Sqrt
[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 264

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

Rubi steps

\begin{align*} \int \frac{\sinh ^{-1}(a x)}{x^3} \, dx &=-\frac{\sinh ^{-1}(a x)}{2 x^2}+\frac{1}{2} a \int \frac{1}{x^2 \sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{a \sqrt{1+a^2 x^2}}{2 x}-\frac{\sinh ^{-1}(a x)}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0063373, size = 28, normalized size = 0.85 \[ -\frac{a x \sqrt{a^2 x^2+1}+\sinh ^{-1}(a x)}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSinh[a*x]/x^3,x]

[Out]

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

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Maple [A]  time = 0.004, size = 37, normalized size = 1.1 \begin{align*}{a}^{2} \left ( -{\frac{{\it Arcsinh} \left ( ax \right ) }{2\,{a}^{2}{x}^{2}}}-{\frac{1}{2\,ax}\sqrt{{a}^{2}{x}^{2}+1}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(a*x)/x^3,x)

[Out]

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

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Maxima [A]  time = 1.18445, size = 36, normalized size = 1.09 \begin{align*} -\frac{\sqrt{a^{2} x^{2} + 1} a}{2 \, x} - \frac{\operatorname{arsinh}\left (a x\right )}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.94808, size = 88, normalized size = 2.67 \begin{align*} -\frac{\sqrt{a^{2} x^{2} + 1} a x + \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asinh}{\left (a x \right )}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a*x)/x**3,x)

[Out]

Integral(asinh(a*x)/x**3, x)

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Giac [A]  time = 1.42686, size = 68, normalized size = 2.06 \begin{align*} \frac{a{\left | a \right |}}{{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )}^{2} - 1} - \frac{\log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)/x^3,x, algorithm="giac")

[Out]

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