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

Optimal. Leaf size=33 \[ -\frac{\sqrt{x^2+1}}{x}-\frac{\sqrt{x^2+1} \log (x)}{x}+\sinh ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.042239, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2335, 277, 215} \[ -\frac{\sqrt{x^2+1}}{x}-\frac{\sqrt{x^2+1} \log (x)}{x}+\sinh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[Log[x]/(x^2*Sqrt[1 + x^2]),x]

[Out]

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

Rule 2335

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp
[((f*x)^(m + 1)*(d + e*x^r)^(q + 1)*(a + b*Log[c*x^n]))/(d*f*(m + 1)), x] - Dist[(b*n)/(d*(m + 1)), Int[(f*x)^
m*(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[m + r*(q + 1) + 1, 0] && NeQ[
m, -1]

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

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

Mathematica [A]  time = 0.0246225, size = 21, normalized size = 0.64 \[ \sinh ^{-1}(x)-\frac{\sqrt{x^2+1} (\log (x)+1)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.029, size = 29, normalized size = 0.9 \begin{align*}{\it Arcsinh} \left ( x \right ) +{\frac{1}{x} \left ( -\ln \left ( x \right ) \sqrt{{x}^{2}+1}-\sqrt{{x}^{2}+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.42408, size = 39, normalized size = 1.18 \begin{align*} -\frac{\sqrt{x^{2} + 1} \log \left (x\right )}{x} - \frac{\sqrt{x^{2} + 1}}{x} + \operatorname{arsinh}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.07127, size = 88, normalized size = 2.67 \begin{align*} -\frac{x \log \left (-x + \sqrt{x^{2} + 1}\right ) + \sqrt{x^{2} + 1}{\left (\log \left (x\right ) + 1\right )} + x}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 6.69013, size = 26, normalized size = 0.79 \begin{align*} \operatorname{asinh}{\left (x \right )} - \frac{\sqrt{x^{2} + 1} \log{\left (x \right )}}{x} - \frac{\sqrt{x^{2} + 1}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.084, size = 78, normalized size = 2.36 \begin{align*} \frac{2 \, \log \left (x\right )}{{\left (x - \sqrt{x^{2} + 1}\right )}^{2} - 1} + \frac{2}{{\left (x - \sqrt{x^{2} + 1}\right )}^{2} - 1} - \log \left (-x + \sqrt{x^{2} + 1}\right ) + \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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