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]

Shi(arcsinh(x))

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Rubi [A]  time = 0.06, antiderivative size = 3, normalized size of antiderivative = 1.00, 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 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]

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])

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*}

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Mathematica [A]  time = 0.06, size = 3, normalized size = 1.00 \[ \text {Shi}\left (\sinh ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

SinhIntegral[ArcSinh[x]]

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fricas [F]  time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x}{\sqrt {x^{2} + 1} \operatorname {arsinh}\relax (x)}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {x^{2} + 1} \operatorname {arsinh}\relax (x)}\,{d x} \]

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)

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maple [A]  time = 0.13, size = 4, normalized size = 1.33 \[ \Shi \left (\arcsinh \relax (x )\right ) \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {x^{2} + 1} \operatorname {arsinh}\relax (x)}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.33 \[ \int \frac {x}{\mathrm {asinh}\relax (x)\,\sqrt {x^2+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(asinh(x)*(x^2 + 1)^(1/2)),x)

[Out]

int(x/(asinh(x)*(x^2 + 1)^(1/2)), x)

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

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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