3.1.20 \(\int \frac {x \log (x+\sqrt {-1+x^2})}{\sqrt {-1+x^2}} \, dx\) [20]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {267, 2634, 8} \begin {gather*} \sqrt {x^2-1} \log \left (\sqrt {x^2-1}+x\right )-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x*Log[x + Sqrt[-1 + x^2]])/Sqrt[-1 + x^2],x]

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 2634

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[w*(D[u, x]
/u), x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rubi steps

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

Antiderivative was successfully verified.

[In]

Integrate[(x*Log[x + Sqrt[-1 + x^2]])/Sqrt[-1 + x^2],x]

[Out]

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

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x \ln \left (x +\sqrt {x^{2}-1}\right )}{\sqrt {x^{2}-1}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 10.71, size = 22, normalized size = 0.85 \begin {gather*} \sqrt {x^{2} - 1} \log \left (x + \sqrt {x^{2} - 1}\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.46, size = 22, normalized size = 0.85 \begin {gather*} \sqrt {x^{2} - 1} \log \left (x + \sqrt {x^{2} - 1}\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 3.05, size = 20, normalized size = 0.77 \begin {gather*} - x + \sqrt {x^{2} - 1} \log {\left (x + \sqrt {x^{2} - 1} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.45, size = 22, normalized size = 0.85 \begin {gather*} \sqrt {x^{2} - 1} \log \left (x + \sqrt {x^{2} - 1}\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x\,\ln \left (x+\sqrt {x^2-1}\right )}{\sqrt {x^2-1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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