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

Optimal. Leaf size=32 \[ -\frac{x^2}{4}+\frac{1}{2} \sqrt{x^2+1} x \sinh ^{-1}(x)+\frac{1}{4} \sinh ^{-1}(x)^2 \]

[Out]

-x^2/4 + (x*Sqrt[1 + x^2]*ArcSinh[x])/2 + ArcSinh[x]^2/4

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Rubi [A]  time = 0.0290022, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5682, 5675, 30} \[ -\frac{x^2}{4}+\frac{1}{2} \sqrt{x^2+1} x \sinh ^{-1}(x)+\frac{1}{4} \sinh ^{-1}(x)^2 \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x^2]*ArcSinh[x],x]

[Out]

-x^2/4 + (x*Sqrt[1 + x^2]*ArcSinh[x])/2 + ArcSinh[x]^2/4

Rule 5682

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(x*Sqrt[d + e*x^2]*
(a + b*ArcSinh[c*x])^n)/2, x] + (Dist[Sqrt[d + e*x^2]/(2*Sqrt[1 + c^2*x^2]), Int[(a + b*ArcSinh[c*x])^n/Sqrt[1
+ c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(2*Sqrt[1 + c^2*x^2]), Int[x*(a + b*ArcSinh[c*x])^(n - 1),
x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0]

Rule 5675

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(a + b*ArcSinh[c*x]
)^(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && GtQ[d, 0] && NeQ[n, -1
]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \sqrt{1+x^2} \sinh ^{-1}(x) \, dx &=\frac{1}{2} x \sqrt{1+x^2} \sinh ^{-1}(x)-\frac{\int x \, dx}{2}+\frac{1}{2} \int \frac{\sinh ^{-1}(x)}{\sqrt{1+x^2}} \, dx\\ &=-\frac{x^2}{4}+\frac{1}{2} x \sqrt{1+x^2} \sinh ^{-1}(x)+\frac{1}{4} \sinh ^{-1}(x)^2\\ \end{align*}

Mathematica [A]  time = 0.0128693, size = 28, normalized size = 0.88 \[ \frac{1}{4} \left (-x^2+2 \sqrt{x^2+1} x \sinh ^{-1}(x)+\sinh ^{-1}(x)^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + x^2]*ArcSinh[x],x]

[Out]

(-x^2 + 2*x*Sqrt[1 + x^2]*ArcSinh[x] + ArcSinh[x]^2)/4

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Maple [A]  time = 0., size = 26, normalized size = 0.8 \begin{align*}{\frac{x{\it Arcsinh} \left ( x \right ) }{2}\sqrt{{x}^{2}+1}}+{\frac{ \left ({\it Arcsinh} \left ( x \right ) \right ) ^{2}}{4}}-{\frac{{x}^{2}}{4}}-{\frac{1}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x*arcsinh(x)*(x^2+1)^(1/2)+1/4*arcsinh(x)^2-1/4*x^2-1/4

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Maxima [A]  time = 1.65747, size = 38, normalized size = 1.19 \begin{align*} -\frac{1}{4} \, x^{2} + \frac{1}{2} \,{\left (\sqrt{x^{2} + 1} x + \operatorname{arsinh}\left (x\right )\right )} \operatorname{arsinh}\left (x\right ) - \frac{1}{4} \, \operatorname{arsinh}\left (x\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*x^2 + 1/2*(sqrt(x^2 + 1)*x + arcsinh(x))*arcsinh(x) - 1/4*arcsinh(x)^2

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Fricas [A]  time = 2.40515, size = 115, normalized size = 3.59 \begin{align*} \frac{1}{2} \, \sqrt{x^{2} + 1} x \log \left (x + \sqrt{x^{2} + 1}\right ) - \frac{1}{4} \, x^{2} + \frac{1}{4} \, \log \left (x + \sqrt{x^{2} + 1}\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(x^2 + 1)*x*log(x + sqrt(x^2 + 1)) - 1/4*x^2 + 1/4*log(x + sqrt(x^2 + 1))^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x**2 + 1)*asinh(x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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(arcsinh(x)*(x^2+1)^(1/2),x, algorithm="giac")

[Out]

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