### 3.231 $$\int \log (x+\sqrt{1+x^2}) \, dx$$

Optimal. Leaf size=26 $x \log \left (\sqrt{x^2+1}+x\right )-\sqrt{x^2+1}$

[Out]

-Sqrt[1 + x^2] + x*Log[x + Sqrt[1 + x^2]]

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Rubi [A]  time = 0.0040469, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.167, Rules used = {2534, 261} $x \log \left (\sqrt{x^2+1}+x\right )-\sqrt{x^2+1}$

Antiderivative was successfully veriﬁed.

[In]

Int[Log[x + Sqrt[1 + x^2]],x]

[Out]

-Sqrt[1 + x^2] + x*Log[x + Sqrt[1 + x^2]]

Rule 2534

Int[Log[(d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_.) + (c_.)*(x_)^2]], x_Symbol] :> Simp[x*Log[d + e*x + f*Sqrt[a + c
*x^2]], x] - Dist[a*c*f^2, Int[x/(d*e*(a + c*x^2) + f*(a*e - c*d*x)*Sqrt[a + c*x^2]), x], x] /; FreeQ[{a, c, d
, e, f}, x] && EqQ[e^2 - c*f^2, 0]

Rule 261

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]

Rubi steps

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

Mathematica [A]  time = 0.0157574, size = 26, normalized size = 1. $x \log \left (\sqrt{x^2+1}+x\right )-\sqrt{x^2+1}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Log[x + Sqrt[1 + x^2]],x]

[Out]

-Sqrt[1 + x^2] + x*Log[x + Sqrt[1 + x^2]]

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Maple [A]  time = 0.004, size = 23, normalized size = 0.9 \begin{align*} x\ln \left ( x+\sqrt{{x}^{2}+1} \right ) -\sqrt{{x}^{2}+1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} x \log \left (x + \sqrt{x^{2} + 1}\right ) - x + \arctan \left (x\right ) - \int \frac{x}{x^{3} +{\left (x^{2} + 1\right )}^{\frac{3}{2}} + x}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(x + sqrt(x^2 + 1)) - x + arctan(x) - integrate(x/(x^3 + (x^2 + 1)^(3/2) + x), x)

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Fricas [A]  time = 1.73915, size = 57, normalized size = 2.19 \begin{align*} x \log \left (x + \sqrt{x^{2} + 1}\right ) - \sqrt{x^{2} + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 5.46699, size = 20, normalized size = 0.77 \begin{align*} x \log{\left (x + \sqrt{x^{2} + 1} \right )} - \sqrt{x^{2} + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18404, size = 30, normalized size = 1.15 \begin{align*} x \log \left (x + \sqrt{x^{2} + 1}\right ) - \sqrt{x^{2} + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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