\(\int \frac {\log (\frac {a+x^2}{x^2})}{x} \, dx\) [391]

Optimal result

Integrand size = 14, antiderivative size = 12 \[ \int \frac {\log \left (\frac {a+x^2}{x^2}\right )}{x} \, dx=\frac {1}{2} \operatorname {PolyLog}\left (2,-\frac {a}{x^2}\right ) \]

[Out]

1/2*polylog(2,-a/x^2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2511, 2438} \[ \int \frac {\log \left (\frac {a+x^2}{x^2}\right )}{x} \, dx=\frac {1}{2} \operatorname {PolyLog}\left (2,-\frac {a}{x^2}\right ) \]

[In]

Int[Log[(a + x^2)/x^2]/x,x]

[Out]

PolyLog[2, -(a/x^2)]/2

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2511

Int[((a_.) + Log[(c_.)*(v_)^(p_.)]*(b_.))^(q_.)*((f_.)*(x_))^(m_.), x_Symbol] :> Int[(f*x)^m*(a + b*Log[c*Expa
ndToSum[v, x]^p])^q, x] /; FreeQ[{a, b, c, f, m, p, q}, x] && BinomialQ[v, x] &&  !BinomialMatchQ[v, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\log \left (1+\frac {a}{x^2}\right )}{x} \, dx \\ & = \frac {1}{2} \text {Li}_2\left (-\frac {a}{x^2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (\frac {a+x^2}{x^2}\right )}{x} \, dx=\frac {1}{2} \operatorname {PolyLog}\left (2,-\frac {a}{x^2}\right ) \]

[In]

Integrate[Log[(a + x^2)/x^2]/x,x]

[Out]

PolyLog[2, -(a/x^2)]/2

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(75\) vs. \(2(10)=20\).

Time = 0.20 (sec) , antiderivative size = 76, normalized size of antiderivative = 6.33

method	result	size
risch	\(-\ln \left (\frac {1}{x}\right ) \ln \left (1+\frac {a}{x^{2}}\right )+\ln \left (\frac {1}{x}\right ) \ln \left (1+\frac {\sqrt {-a}}{x}\right )+\ln \left (\frac {1}{x}\right ) \ln \left (1-\frac {\sqrt {-a}}{x}\right )+\operatorname {dilog}\left (1+\frac {\sqrt {-a}}{x}\right )+\operatorname {dilog}\left (1-\frac {\sqrt {-a}}{x}\right )\)	\(76\)
derivativedivides	\(-\ln \left (\frac {1}{x}\right ) \ln \left (1+\frac {a}{x^{2}}\right )+2 a \left (\frac {\ln \left (\frac {1}{x}\right ) \left (\ln \left (1+\frac {\sqrt {-a}}{x}\right )+\ln \left (1-\frac {\sqrt {-a}}{x}\right )\right )}{2 a}+\frac {\operatorname {dilog}\left (1+\frac {\sqrt {-a}}{x}\right )+\operatorname {dilog}\left (1-\frac {\sqrt {-a}}{x}\right )}{2 a}\right )\)	\(86\)
default	\(-\ln \left (\frac {1}{x}\right ) \ln \left (1+\frac {a}{x^{2}}\right )+2 a \left (\frac {\ln \left (\frac {1}{x}\right ) \left (\ln \left (1+\frac {\sqrt {-a}}{x}\right )+\ln \left (1-\frac {\sqrt {-a}}{x}\right )\right )}{2 a}+\frac {\operatorname {dilog}\left (1+\frac {\sqrt {-a}}{x}\right )+\operatorname {dilog}\left (1-\frac {\sqrt {-a}}{x}\right )}{2 a}\right )\)	\(86\)
parts	\(\ln \left (\frac {x^{2}+a}{x^{2}}\right ) \ln \left (x \right )+\ln \left (x \right )^{2}-\ln \left (x \right ) \ln \left (\frac {\sqrt {-a}-x}{\sqrt {-a}}\right )-\ln \left (x \right ) \ln \left (\frac {\sqrt {-a}+x}{\sqrt {-a}}\right )-\operatorname {dilog}\left (\frac {\sqrt {-a}-x}{\sqrt {-a}}\right )-\operatorname {dilog}\left (\frac {\sqrt {-a}+x}{\sqrt {-a}}\right )\)	\(91\)

[In]

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

[Out]

-ln(1/x)*ln(1+1/x^2*a)+ln(1/x)*ln(1+1/x*(-a)^(1/2))+ln(1/x)*ln(1-1/x*(-a)^(1/2))+dilog(1+1/x*(-a)^(1/2))+dilog
(1-1/x*(-a)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.25 \[ \int \frac {\log \left (\frac {a+x^2}{x^2}\right )}{x} \, dx=\frac {1}{2} \, {\rm Li}_2\left (-\frac {x^{2} + a}{x^{2}} + 1\right ) \]

[In]

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

[Out]

1/2*dilog(-(x^2 + a)/x^2 + 1)

Sympy [F]

\[ \int \frac {\log \left (\frac {a+x^2}{x^2}\right )}{x} \, dx=\int \frac {\log {\left (\frac {a}{x^{2}} + 1 \right )}}{x}\, dx \]

[In]

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

[Out]

Integral(log(a/x**2 + 1)/x, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (9) = 18\).

Time = 0.19 (sec) , antiderivative size = 69, normalized size of antiderivative = 5.75 \[ \int \frac {\log \left (\frac {a+x^2}{x^2}\right )}{x} \, dx=-{\left (\log \left (x^{2} + a\right ) - 2 \, \log \left (x\right )\right )} \log \left (x\right ) + \log \left (x^{2} + a\right ) \log \left (x\right ) - \log \left (x\right )^{2} - \log \left (x\right ) \log \left (\frac {x^{2}}{a} + 1\right ) + \log \left (x\right ) \log \left (\frac {x^{2} + a}{x^{2}}\right ) - \frac {1}{2} \, {\rm Li}_2\left (-\frac {x^{2}}{a}\right ) \]

[In]

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

[Out]

-(log(x^2 + a) - 2*log(x))*log(x) + log(x^2 + a)*log(x) - log(x)^2 - log(x)*log(x^2/a + 1) + log(x)*log((x^2 +
a)/x^2) - 1/2*dilog(-x^2/a)

Giac [F]

\[ \int \frac {\log \left (\frac {a+x^2}{x^2}\right )}{x} \, dx=\int { \frac {\log \left (\frac {x^{2} + a}{x^{2}}\right )}{x} \,d x } \]

[In]

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

[Out]

integrate(log((x^2 + a)/x^2)/x, x)

Mupad [B] (verification not implemented)

Time = 1.50 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {\log \left (\frac {a+x^2}{x^2}\right )}{x} \, dx=\frac {\mathrm {polylog}\left (2,-\frac {a}{x^2}\right )}{2} \]

[In]

int(log((a + x^2)/x^2)/x,x)

[Out]

polylog(2, -a/x^2)/2