3.19 \(\int \frac{\sinh ^{-1}(a x)^2}{x^3} \, dx\)

Optimal. Leaf size=43 \[ -\frac{a \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{x}+a^2 \log (x)-\frac{\sinh ^{-1}(a x)^2}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0804655, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5661, 5723, 29} \[ -\frac{a \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{x}+a^2 \log (x)-\frac{\sinh ^{-1}(a x)^2}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a*x]^2/x^3,x]

[Out]

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

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

Mathematica [A]  time = 0.0310021, size = 43, normalized size = 1. \[ -\frac{a \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{x}+a^2 \log (x)-\frac{\sinh ^{-1}(a x)^2}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSinh[a*x]^2/x^3,x]

[Out]

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

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Maple [A]  time = 0.076, size = 67, normalized size = 1.6 \begin{align*} -{a}^{2}{\it Arcsinh} \left ( ax \right ) -{\frac{a{\it Arcsinh} \left ( ax \right ) }{x}\sqrt{{a}^{2}{x}^{2}+1}}-{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}{2\,{x}^{2}}}+{a}^{2}\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)^2/x^3,x)

[Out]

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

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Maxima [A]  time = 1.1918, size = 53, normalized size = 1.23 \begin{align*} a^{2} \log \left (x\right ) - \frac{\sqrt{a^{2} x^{2} + 1} a \operatorname{arsinh}\left (a x\right )}{x} - \frac{\operatorname{arsinh}\left (a x\right )^{2}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^2/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.17422, size = 157, normalized size = 3.65 \begin{align*} \frac{2 \, a^{2} x^{2} \log \left (x\right ) - 2 \, \sqrt{a^{2} x^{2} + 1} a x \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^2/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asinh}^{2}{\left (a x \right )}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a*x)**2/x**3,x)

[Out]

Integral(asinh(a*x)**2/x**3, x)

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Giac [B]  time = 1.48154, size = 149, normalized size = 3.47 \begin{align*} -{\left (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}\right )} a - \frac{\log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^2/x^3,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))*a - 1/2*log(a*x + sqrt(a^2*x^2 + 1))^2/x^2