3.1.18 \(\int \frac {\log (x)}{x^2 \sqrt {-1+x^2}} \, dx\) [18]

Optimal. Leaf size=43 \[ \frac {\sqrt {-1+x^2}}{x}-\tanh ^{-1}\left (\frac {x}{\sqrt {-1+x^2}}\right )+\frac {\sqrt {-1+x^2} \log (x)}{x} \]

[Out]

-arctanh(x/(x^2-1)^(1/2))+(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 = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2373, 283, 223, 212} \begin {gather*} \frac {\sqrt {x^2-1}}{x}+\frac {\sqrt {x^2-1} \log (x)}{x}-\tanh ^{-1}\left (\frac {x}{\sqrt {x^2-1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[x]/(x^2*Sqrt[-1 + x^2]),x]

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

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

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}+\frac {\sqrt {-1+x^2} \log (x)}{x}-\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-1+x^2}}\right )\\ &=\frac {\sqrt {-1+x^2}}{x}-\tanh ^{-1}\left (\frac {x}{\sqrt {-1+x^2}}\right )+\frac {\sqrt {-1+x^2} \log (x)}{x}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 43, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^2}}{x}+\frac {\sqrt {-1+x^2} \log (x)}{x}-\log \left (x+\sqrt {-1+x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: } \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

cought exception:

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.03, size = 89, normalized size = 2.07

method	result	size
meijerg	\(-\frac {\sqrt {-\mathrm {signum}\left (x^{2}-1\right )}\, \arcsin \left (x \right )}{\sqrt {\mathrm {signum}\left (x^{2}-1\right )}}+\frac {-\frac {\sqrt {-\mathrm {signum}\left (x^{2}-1\right )}\, \sqrt {-x^{2}+1}}{\sqrt {\mathrm {signum}\left (x^{2}-1\right )}}-\frac {\sqrt {-\mathrm {signum}\left (x^{2}-1\right )}\, \ln \left (x \right ) \sqrt {-x^{2}+1}}{\sqrt {\mathrm {signum}\left (x^{2}-1\right )}}}{x}\)	\(89\)

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]

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

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Maxima [A]
time = 0.34, size = 41, normalized size = 0.95 \begin {gather*} \frac {\sqrt {x^{2} - 1} \log \left (x\right )}{x} + \frac {\sqrt {x^{2} - 1}}{x} - \log \left (2 \, x + 2 \, \sqrt {x^{2} - 1}\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 - log(2*x + 2*sqrt(x^2 - 1))

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Fricas [A]
time = 0.41, size = 32, normalized size = 0.74 \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 = 2.39, size = 34, normalized size = 0.79 \begin {gather*} \left (\begin {cases} \frac {\sqrt {x^{2} - 1}}{x} & \text {for}\: x > -1 \wedge x < 1 \end {cases}\right ) \log {\left (x \right )} - \begin {cases} \text {NaN} & \text {for}\: x < -1 \\\operatorname {acosh}{\left (x \right )} - \frac {\sqrt {x^{2} - 1}}{x} & \text {for}\: x < 1 \\\text {NaN} & \text {otherwise} \end {cases} \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]

Piecewise((sqrt(x**2 - 1)/x, (x > -1) & (x < 1)))*log(x) - Piecewise((nan, x < -1), (acosh(x) - sqrt(x**2 - 1)
/x, x < 1), (nan, True))

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Giac [A]
time = 0.01, size = 67, normalized size = 1.56 \begin {gather*} -\ln \left |x\right |-2 \left (-\frac 1{\left (\sqrt {x^{2}-1}-x\right )^{2}+1}-\frac {\ln \left (\left (\sqrt {x^{2}-1}-x\right )^{2}\right )}{4}\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) + 1/2*log((x - sqrt(x^2 - 1))^2) - log(ab
s(x))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \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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