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

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

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 26, normalized size of antiderivative = 1.00, 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 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 \[ x \log \left (\sqrt {x^2-1}+x\right )-\sqrt {x^2-1} \]

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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fricas [A]  time = 0.45, 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 = 0.16, 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 = 0.07, size = 23, normalized size = 0.88 \[ x \ln \left (x +\sqrt {x^{2}-1}\right )-\sqrt {x^{2}-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

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mupad [B]  time = 0.55, 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 = 5.78, 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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