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

Optimal. Leaf size=3 \[ \text{Shi}\left (\sinh ^{-1}(x)\right ) \]

[Out]

SinhIntegral[ArcSinh[x]]

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Rubi [A]  time = 0.0595927, antiderivative size = 3, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {5779, 3298} \[ \text{Shi}\left (\sinh ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]

Int[x/(Sqrt[1 + x^2]*ArcSinh[x]),x]

[Out]

SinhIntegral[ArcSinh[x]]

Rule 5779

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d^p/c^
(m + 1), Subst[Int[(a + b*x)^n*Sinh[x]^m*Cosh[x]^(2*p + 1), x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e,
n}, x] && EqQ[e, c^2*d] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])

Rule 3298

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

Rubi steps

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

Mathematica [A]  time = 0.0583648, size = 3, normalized size = 1. \[ \text{Shi}\left (\sinh ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

SinhIntegral[ArcSinh[x]]

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Maple [A]  time = 0.047, size = 4, normalized size = 1.3 \begin{align*}{\it Shi} \left ({\it Arcsinh} \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Shi(arcsinh(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/(sqrt(x^2 + 1)*arcsinh(x)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x}{\sqrt{x^{2} + 1} \operatorname{arsinh}\left (x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x/(sqrt(x^2 + 1)*arcsinh(x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/(sqrt(x^2 + 1)*arcsinh(x)), x)