3.118 \(\int \frac{\sinh ^{-1}(a x)}{x^2 \sqrt{1+a^2 x^2}} \, dx\)

Optimal. Leaf size=27 \[ a \log (x)-\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{x} \]

[Out]

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

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Rubi [A]  time = 0.0636175, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {5723, 29} \[ a \log (x)-\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5723

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
((f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n)/(d*f*(m + 1)), x] - Dist[(b*c*n*d^IntPart[p]*(d + e
*x^2)^FracPart[p])/(f*(m + 1)*(1 + c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*Arc
Sinh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && EqQ[m + 2*p
+ 3, 0] && NeQ[m, -1]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

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

Mathematica [A]  time = 0.0364905, size = 29, normalized size = 1.07 \[ a \log (a x)-\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.056, size = 56, normalized size = 2.1 \begin{align*} -2\,a{\it Arcsinh} \left ( ax \right ) +{\frac{{\it Arcsinh} \left ( ax \right ) }{x} \left ( ax-\sqrt{{a}^{2}{x}^{2}+1} \right ) }+a\ln \left ( \left ( ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) ^{2}-1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(a*x)/x^2/(a^2*x^2+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.16535, size = 34, normalized size = 1.26 \begin{align*} a \log \left (x\right ) - \frac{\sqrt{a^{2} x^{2} + 1} \operatorname{arsinh}\left (a x\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)/x^2/(a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.65206, size = 88, normalized size = 3.26 \begin{align*} \frac{a x \log \left (x\right ) - \sqrt{a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)/x^2/(a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asinh}{\left (a x \right )}}{x^{2} \sqrt{a^{2} x^{2} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a*x)/x**2/(a**2*x**2+1)**(1/2),x)

[Out]

Integral(asinh(a*x)/(x**2*sqrt(a**2*x**2 + 1)), x)

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Giac [B]  time = 1.38914, size = 113, normalized size = 4.19 \begin{align*} -a{\left (\frac{\log \left (-x{\left | a \right |} + \sqrt{a^{2} x^{2} + 1}\right )}{{\left | a \right |}} - \frac{{\left | a \right |} \log \left ({\left | x \right |}\right )}{a^{2}}\right )}{\left | a \right |} + \frac{2 \,{\left | a \right |} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}{{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )}^{2} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)/x^2/(a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

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