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]

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

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Rubi [A]  time = 0.04, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 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]

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 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]

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*}

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Mathematica [A]  time = 0.03, 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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fricas [A]  time = 0.44, size = 33, normalized size = 1.00 \[ -\frac {x \log \left (-x + \sqrt {x^{2} + 1}\right ) + \sqrt {x^{2} + 1} {\left (\log \relax (x) + 1\right )} + x}{x} \]

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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giac [A]  time = 0.97, size = 58, normalized size = 1.76 \[ \frac {2 \, \log \relax (x)}{{\left (x - \sqrt {x^{2} + 1}\right )}^{2} - 1} + \frac {2}{{\left (x - \sqrt {x^{2} + 1}\right )}^{2} - 1} - \log \left (-x + \sqrt {x^{2} + 1}\right ) + \log \left ({\left | x \right |}\right ) \]

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))

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maple [A]  time = 0.10, size = 29, normalized size = 0.88 \[ \arcsinh \relax (x )+\frac {-\sqrt {x^{2}+1}\, \ln \relax (x )-\sqrt {x^{2}+1}}{x} \]

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 = 0.97, size = 29, normalized size = 0.88 \[ -\frac {\sqrt {x^{2} + 1} \log \relax (x)}{x} - \frac {\sqrt {x^{2} + 1}}{x} + \operatorname {arsinh}\relax (x) \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\ln \relax (x)}{x^2\,\sqrt {x^2+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(x)/(x^2*(x^2 + 1)^(1/2)),x)

[Out]

int(log(x)/(x^2*(x^2 + 1)^(1/2)), x)

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sympy [A]  time = 5.25, size = 26, normalized size = 0.79 \[ \operatorname {asinh}{\relax (x )} - \frac {\sqrt {x^{2} + 1} \log {\relax (x )}}{x} - \frac {\sqrt {x^{2} + 1}}{x} \]

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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