3.1.34 \(\int \frac {\log (x)}{x^2 \sqrt {1+x^2}} \, dx\) [34]

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

[Out]

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

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Rubi [A]
time = 0.03, 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 = {2373, 283, 221} \begin {gather*} -\frac {\sqrt {x^2+1}}{x}-\frac {\sqrt {x^2+1} \log (x)}{x}+\sinh ^{-1}(x) \end {gather*}

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 221

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 283

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 2373

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 = 0.02, size = 21, normalized size = 0.64 \begin {gather*} \sinh ^{-1}(x)-\frac {\sqrt {1+x^2} (1+\log (x))}{x} \end {gather*}

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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Mathics [A]
time = 4.32, size = 29, normalized size = 0.88 \begin {gather*} \frac {x \text {ArcSinh}\left [x\right ]-\text {Log}\left [x\right ] \sqrt {1+x^2}-\sqrt {1+x^2}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.02, size = 29, normalized size = 0.88

method	result	size
meijerg	\(\arcsinh \left (x \right )+\frac {-\ln \left (x \right ) \sqrt {x^{2}+1}-\sqrt {x^{2}+1}}{x}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(x)/x^2/(x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.37, size = 29, normalized size = 0.88 \begin {gather*} -\frac {\sqrt {x^{2} + 1} \log \left (x\right )}{x} - \frac {\sqrt {x^{2} + 1}}{x} + \operatorname {arsinh}\left (x\right ) \end {gather*}

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 = 0.33, size = 33, normalized size = 1.00 \begin {gather*} -\frac {x \log \left (-x + \sqrt {x^{2} + 1}\right ) + \sqrt {x^{2} + 1} {\left (\log \left (x\right ) + 1\right )} + x}{x} \end {gather*}

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 = 2.71, size = 26, normalized size = 0.79 \begin {gather*} \operatorname {asinh}{\left (x \right )} - \frac {\sqrt {x^{2} + 1} \log {\left (x \right )}}{x} - \frac {\sqrt {x^{2} + 1}}{x} \end {gather*}

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 = 0.01, size = 63, normalized size = 1.91 \begin {gather*} \ln \left |x\right |-2 \left (-\frac 1{\left (\sqrt {x^{2}+1}-x\right )^{2}-1}+\frac {\ln \left (\sqrt {x^{2}+1}-x\right )}{2}\right )+\frac {2 \ln x}{\left (\sqrt {x^{2}+1}-x\right )^{2}-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x)/x^2/(x^2+1)^(1/2),x)

[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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\ln \left (x\right )}{x^2\,\sqrt {x^2+1}} \,d x \end {gather*}

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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