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

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

[Out]

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

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Rubi [A]  time = 0.0360943, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, 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 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]

Rule 8

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

Mathematica [A]  time = 0.0214885, size = 26, normalized size = 1. \[ \sqrt{x^2-1} \log \left (\sqrt{x^2-1}+x\right )-x \]

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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Maple [F]  time = 0.024, size = 0, normalized size = 0. \begin{align*} \int{x\ln \left ( x+\sqrt{{x}^{2}-1} \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}\, dx \end{align*}

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 = 1.0574, size = 30, normalized size = 1.15 \begin{align*} \sqrt{x^{2} - 1} \log \left (x + \sqrt{x^{2} - 1}\right ) - x \end{align*}

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 = 1.99764, size = 57, normalized size = 2.19 \begin{align*} \sqrt{x^{2} - 1} \log \left (x + \sqrt{x^{2} - 1}\right ) - x \end{align*}

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 = 23.102, size = 20, normalized size = 0.77 \begin{align*} - x + \sqrt{x^{2} - 1} \log{\left (x + \sqrt{x^{2} - 1} \right )} \end{align*}

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 = 1.09971, size = 30, normalized size = 1.15 \begin{align*} \sqrt{x^{2} - 1} \log \left (x + \sqrt{x^{2} - 1}\right ) - x \end{align*}

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