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]

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

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Rubi [A]  time = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 30

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

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 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]

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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.60, size = 40, normalized size = 1.25 \[ \frac {1}{2} \, \sqrt {x^{2} + 1} x \log \left (x + \sqrt {x^{2} + 1}\right ) - \frac {1}{4} \, x^{2} + \frac {1}{4} \, \log \left (x + \sqrt {x^{2} + 1}\right )^{2} \]

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

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)

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maple [A]  time = 0.00, size = 26, normalized size = 0.81 \[ \frac {x \arcsinh \relax (x ) \sqrt {x^{2}+1}}{2}+\frac {\arcsinh \relax (x )^{2}}{4}-\frac {x^{2}}{4}-\frac {1}{4} \]

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 = 0.47, size = 28, normalized size = 0.88 \[ -\frac {1}{4} \, x^{2} + \frac {1}{2} \, {\left (\sqrt {x^{2} + 1} x + \operatorname {arsinh}\relax (x)\right )} \operatorname {arsinh}\relax (x) - \frac {1}{4} \, \operatorname {arsinh}\relax (x)^{2} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \mathrm {asinh}\relax (x)\,\sqrt {x^2+1} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asinh(x)*(x^2 + 1)^(1/2),x)

[Out]

int(asinh(x)*(x^2 + 1)^(1/2), x)

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

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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