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

Optimal. Leaf size=37 \[ \frac{\text{Chi}\left (2 \sinh ^{-1}(a x)\right )}{a^2}-\frac{x \sqrt{a^2 x^2+1}}{a \sinh ^{-1}(a x)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0246061, antiderivative size = 37, 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 = {5665, 3301} \[ \frac{\text{Chi}\left (2 \sinh ^{-1}(a x)\right )}{a^2}-\frac{x \sqrt{a^2 x^2+1}}{a \sinh ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5665

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^m*Sqrt[1 + c^2*x^2]*(a + b*ArcSi
nh[c*x])^(n + 1))/(b*c*(n + 1)), x] - Dist[1/(b*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[(a + b*x)^(n +
1), Sinh[x]^(m - 1)*(m + (m + 1)*Sinh[x]^2), x], x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0]
&& GeQ[n, -2] && LtQ[n, -1]

Rule 3301

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[(c*f*fz)/d
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rubi steps

\begin{align*} \int \frac{x}{\sinh ^{-1}(a x)^2} \, dx &=-\frac{x \sqrt{1+a^2 x^2}}{a \sinh ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (2 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a^2}\\ &=-\frac{x \sqrt{1+a^2 x^2}}{a \sinh ^{-1}(a x)}+\frac{\text{Chi}\left (2 \sinh ^{-1}(a x)\right )}{a^2}\\ \end{align*}

Mathematica [A]  time = 0.0032678, size = 32, normalized size = 0.86 \[ \frac{\text{Chi}\left (2 \sinh ^{-1}(a x)\right )}{a^2}-\frac{\sinh \left (2 \sinh ^{-1}(a x)\right )}{2 a^2 \sinh ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

CoshIntegral[2*ArcSinh[a*x]]/a^2 - Sinh[2*ArcSinh[a*x]]/(2*a^2*ArcSinh[a*x])

________________________________________________________________________________________

Maple [A]  time = 0.026, size = 28, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{2}} \left ( -{\frac{\sinh \left ( 2\,{\it Arcsinh} \left ( ax \right ) \right ) }{2\,{\it Arcsinh} \left ( ax \right ) }}+{\it Chi} \left ( 2\,{\it Arcsinh} \left ( ax \right ) \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a^{3} x^{4} + a x^{2} +{\left (a^{2} x^{3} + x\right )} \sqrt{a^{2} x^{2} + 1}}{{\left (a^{3} x^{2} + \sqrt{a^{2} x^{2} + 1} a^{2} x + a\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )} + \int \frac{2 \, a^{5} x^{5} + 2 \,{\left (a^{2} x^{2} + 1\right )} a^{3} x^{3} + 4 \, a^{3} x^{3} + 2 \, a x +{\left (4 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 1\right )} \sqrt{a^{2} x^{2} + 1}}{{\left (a^{5} x^{4} +{\left (a^{2} x^{2} + 1\right )} a^{3} x^{2} + 2 \, a^{3} x^{2} + 2 \,{\left (a^{4} x^{3} + a^{2} x\right )} \sqrt{a^{2} x^{2} + 1} + a\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a^3*x^4 + a*x^2 + (a^2*x^3 + x)*sqrt(a^2*x^2 + 1))/((a^3*x^2 + sqrt(a^2*x^2 + 1)*a^2*x + a)*log(a*x + sqrt(a
^2*x^2 + 1))) + integrate((2*a^5*x^5 + 2*(a^2*x^2 + 1)*a^3*x^3 + 4*a^3*x^3 + 2*a*x + (4*a^4*x^4 + 4*a^2*x^2 +
1)*sqrt(a^2*x^2 + 1))/((a^5*x^4 + (a^2*x^2 + 1)*a^3*x^2 + 2*a^3*x^2 + 2*(a^4*x^3 + a^2*x)*sqrt(a^2*x^2 + 1) +
a)*log(a*x + sqrt(a^2*x^2 + 1))), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x}{\operatorname{arsinh}\left (a x\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x/arcsinh(a*x)^2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/asinh(a*x)**2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/arcsinh(a*x)^2, x)