3.637 \(\int \log (x-\sqrt {1+x^2}) \, dx\)

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

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Rubi [A]  time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2534, 261} \[ \sqrt {x^2+1}+x \log \left (x-\sqrt {x^2+1}\right ) \]

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 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]

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]

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*}

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Mathematica [A]  time = 0.02, size = 26, normalized size = 1.00 \[ \sqrt {x^2+1}+x \log \left (x-\sqrt {x^2+1}\right ) \]

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \log \left (x-\sqrt {1+x^2}\right ) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[Log[x - Sqrt[1 + x^2]],x]

[Out]

Could not integrate

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fricas [A]  time = 1.06, size = 22, normalized size = 0.85 \[ x \log \left (x - \sqrt {x^{2} + 1}\right ) + \sqrt {x^{2} + 1} \]

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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giac [A]  time = 1.13, size = 22, normalized size = 0.85 \[ x \log \left (x - \sqrt {x^{2} + 1}\right ) + \sqrt {x^{2} + 1} \]

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)

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maple [A]  time = 0.01, size = 23, normalized size = 0.88

method	result	size
default	\(x \ln \left (x -\sqrt {x^{2}+1}\right )+\sqrt {x^{2}+1}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(x-(x^2+1)^(1/2)),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ x \log \left (x - \sqrt {x^{2} + 1}\right ) - x + \arctan \relax (x) + \int -\frac {x}{x^{3} - {\left (x^{2} + 1\right )}^{\frac {3}{2}} + x}\,{d x} \]

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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mupad [B]  time = 0.08, size = 22, normalized size = 0.85 \[ x\,\ln \left (x-\sqrt {x^2+1}\right )+\sqrt {x^2+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(x - (x^2 + 1)^(1/2)),x)

[Out]

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

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sympy [A]  time = 6.26, size = 20, normalized size = 0.77 \[ x \log {\left (x - \sqrt {x^{2} + 1} \right )} + \sqrt {x^{2} + 1} \]

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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