3.25 \(\int \frac {x \log (x+\sqrt {1+x^2})}{\sqrt {1+x^2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.03, 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 = {261, 2554, 8} \[ \sqrt {x^2+1} \log \left (\sqrt {x^2+1}+x\right )-x \]

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 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 2554

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 \[ \sqrt {x^2+1} \log \left (\sqrt {x^2+1}+x\right )-x \]

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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fricas [A]  time = 0.42, size = 22, normalized size = 0.85 \[ \sqrt {x^{2} + 1} \log \left (x + \sqrt {x^{2} + 1}\right ) - x \]

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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giac [A]  time = 1.00, size = 22, normalized size = 0.85 \[ \sqrt {x^{2} + 1} \log \left (x + \sqrt {x^{2} + 1}\right ) - x \]

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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maple [F]  time = 0.08, 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 = 1.06, size = 22, normalized size = 0.85 \[ \sqrt {x^{2} + 1} \log \left (x + \sqrt {x^{2} + 1}\right ) - x \]

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

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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sympy [A]  time = 4.33, size = 20, normalized size = 0.77 \[ - x + \sqrt {x^{2} + 1} \log {\left (x + \sqrt {x^{2} + 1} \right )} \]

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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