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]

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

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Rubi [A]  time = 0.06, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 29

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

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*}

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Mathematica [A]  time = 0.04, 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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fricas [A]  time = 0.43, size = 39, normalized size = 1.44 \[ \frac {a x \log \relax (x) - \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{x} \]

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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giac [B]  time = 0.33, size = 71, normalized size = 2.63 \[ -a \log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right ) + a \log \left ({\left | x \right |}\right ) + \frac {2 \, {\left | a \right |} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{{\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{2} - 1} \]

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

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maple [B]  time = 0.08, size = 56, normalized size = 2.07 \[ -2 a \arcsinh \left (a x \right )+\frac {\left (a x -\sqrt {a^{2} x^{2}+1}\right ) \arcsinh \left (a x \right )}{x}+a \ln \left (\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}-1\right ) \]

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 = 0.34, size = 25, normalized size = 0.93 \[ a \log \relax (x) - \frac {\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )}{x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\mathrm {asinh}\left (a\,x\right )}{x^2\,\sqrt {a^2\,x^2+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {asinh}{\left (a x \right )}}{x^{2} \sqrt {a^{2} x^{2} + 1}}\, dx \]

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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