3.2.6 \(\int \text {arcsinh}(x) \, dx\) [106]

Optimal. Leaf size=16 \[ -\sqrt {1+x^2}+x \text {arcsinh}(x) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5772, 267} \begin {gather*} x \text {arcsinh}(x)-\sqrt {x^2+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcSinh[x],x]

[Out]

-Sqrt[1 + x^2] + x*ArcSinh[x]

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 5772

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

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=x \text {arcsinh}(x)-\int \frac {x}{\sqrt {1+x^2}} \, dx\\ &=-\sqrt {1+x^2}+x \text {arcsinh}(x)\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} -\sqrt {1+x^2}+x \text {arcsinh}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcSinh[x],x]

[Out]

-Sqrt[1 + x^2] + x*ArcSinh[x]

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Maple [A]
time = 0.01, size = 15, normalized size = 0.94

method	result	size
lookup	\(x \,\arcsinh \left (x \right )-\sqrt {x^{2}+1}\)	\(15\)
default	\(x \,\arcsinh \left (x \right )-\sqrt {x^{2}+1}\)	\(15\)
parts	\(x \,\arcsinh \left (x \right )-\sqrt {x^{2}+1}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.34, size = 14, normalized size = 0.88 \begin {gather*} x \operatorname {arsinh}\left (x\right ) - \sqrt {x^{2} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(x),x, algorithm="maxima")

[Out]

x*arcsinh(x) - sqrt(x^2 + 1)

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Fricas [A]
time = 0.59, size = 22, normalized size = 1.38 \begin {gather*} x \log \left (x + \sqrt {x^{2} + 1}\right ) - \sqrt {x^{2} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(x),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.05, size = 12, normalized size = 0.75 \begin {gather*} x \operatorname {asinh}{\left (x \right )} - \sqrt {x^{2} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(x),x)

[Out]

x*asinh(x) - sqrt(x**2 + 1)

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Giac [A]
time = 0.47, size = 22, normalized size = 1.38 \begin {gather*} x \log \left (x + \sqrt {x^{2} + 1}\right ) - \sqrt {x^{2} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(x),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.06, size = 14, normalized size = 0.88 \begin {gather*} x\,\mathrm {asinh}\left (x\right )-\sqrt {x^2+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asinh(x),x)

[Out]

x*asinh(x) - (x^2 + 1)^(1/2)

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Chatgpt [A]
time = 1.00, size = 14, normalized size = 0.88 \begin {gather*} x \,\arcsinh \left (x \right )-\sqrt {x^{2}+1} \end {gather*}

Antiderivative was successfully verified.

[In]

int(arcsinh(x),x)

[Out]

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

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