∫ log ( a_ x2 ) _________

Optimal. Leaf size=17 \[ \frac {\text {Li}_2\left (1-\frac {a}{x^2}\right )}{2 a} \]

________________________________________________________________________________________

Rubi [A]
time = 0.06, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1607, 2378, 2370, 2352} \begin {gather*} \frac {\text {PolyLog}\left (2,1-\frac {a}{x^2}\right )}{2 a} \end {gather*}

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F

reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)/(x_))^(q_.)*(x_)^(m_.), x_Symbol] :> Int[(e + d*

x)^q*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[m, q] && IntegerQ[q]

Int[((a_.) + Log[(c_.)*(x_)^(n_)]*(b_.))/((x_)*((d_) + (e_.)*(x_)^(r_.))), x_Symbol] :> Dist[1/n, Subst[Int[(a

+ b*Log[c*x])/(x*(d + e*x^(r/n))), x], x, x^n], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IntegerQ[r/n]

\begin {align*} \int \frac {\log \left (\frac {a}{x^2}\right )}{a x-x^3} \, dx &=\int \frac {\log \left (\frac {a}{x^2}\right )}{x \left (a-x^2\right )} \, dx\\ &=-\left (\frac {1}{2} \text {Subst}\left (\int \frac {\log (a x)}{\left (a-\frac {1}{x}\right ) x} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\left (\frac {1}{2} \text {Subst}\left (\int \frac {\log (a x)}{-1+a x} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {\text {Li}_2\left (1-\frac {a}{x^2}\right )}{2 a}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 21, normalized size = 1.24 \begin {gather*} \frac {\text {Li}_2\left (-\frac {a-x^2}{x^2}\right )}{2 a} \end {gather*}

________________________________________________________________________________________


method	result	size

derivativedivides	\(\frac {\dilog \left (\frac {a}{x^{2}}\right )}{2 a}\)	\(12\)
default	\(\frac {\dilog \left (\frac {a}{x^{2}}\right )}{2 a}\)	\(12\)
risch	\(\frac {\dilog \left (\frac {a}{x^{2}}\right )}{2 a}\)	\(12\)

int(ln(a/x^2)/(-x^3+a*x),x,method=_RETURNVERBOSE)

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (14) = 28\).
time = 0.32, size = 81, normalized size = 4.76 \begin {gather*} -\frac {1}{2} \, {\left (\frac {\log \left (x^{2} - a\right )}{a} - \frac {2 \, \log \left (x\right )}{a}\right )} \log \left (\frac {a}{x^{2}}\right ) - \frac {\log \left (x^{2} - a\right ) \log \left (x\right ) - \log \left (x\right )^{2}}{a} + \frac {2 \, \log \left (x\right ) \log \left (-\frac {x^{2}}{a} + 1\right ) + {\rm Li}_2\left (\frac {x^{2}}{a}\right )}{2 \, a} \end {gather*}

integrate(log(a/x^2)/(-x^3+a*x),x, algorithm="maxima")

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

________________________________________________________________________________________

Fricas [A]
time = 0.37, size = 14, normalized size = 0.82 \begin {gather*} \frac {{\rm Li}_2\left (-\frac {a}{x^{2}} + 1\right )}{2 \, a} \end {gather*}

integrate(log(a/x^2)/(-x^3+a*x),x, algorithm="fricas")

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 6.54, size = 92, normalized size = 5.41 \begin {gather*} - \frac {\begin {cases} \frac {\operatorname {Li}_{2}\left (\frac {a}{x^{2}}\right )}{2} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\i \pi \log {\left (x \right )} + \frac {\operatorname {Li}_{2}\left (\frac {a}{x^{2}}\right )}{2} & \text {for}\: \left |{x}\right | < 1 \\- i \pi \log {\left (\frac {1}{x} \right )} + \frac {\operatorname {Li}_{2}\left (\frac {a}{x^{2}}\right )}{2} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- i \pi {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} + i \pi {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} + \frac {\operatorname {Li}_{2}\left (\frac {a}{x^{2}}\right )}{2} & \text {otherwise} \end {cases}}{a} - \frac {\log {\left (\frac {a}{x^{2}} \right )} \log {\left (\frac {a}{x^{2}} - 1 \right )}}{2 a} \end {gather*}

-Piecewise((polylog(2, a/x**2)/2, (Abs(x) < 1) & (1/Abs(x) < 1)), (I*pi*log(x) + polylog(2, a/x**2)/2, Abs(x)

< 1), (-I*pi*log(1/x) + polylog(2, a/x**2)/2, 1/Abs(x) < 1), (-I*pi*meijerg(((), (1, 1)), ((0, 0), ()), x) + I

*pi*meijerg(((1, 1), ()), ((), (0, 0)), x) + polylog(2, a/x**2)/2, True))/a - log(a/x**2)*log(a/x**2 - 1)/(2*a

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

integrate(log(a/x^2)/(-x^3+a*x),x, algorithm="giac")

________________________________________________________________________________________

Mupad [B]
time = 3.50, size = 11, normalized size = 0.65 \begin {gather*} \frac {{\mathrm {Li}}_{\mathrm {2}}\left (\frac {a}{x^2}\right )}{2\,a} \end {gather*}

________________________________________________________________________________________

3.4.49 \(\int \frac {\log (\frac {a}{x^2})}{a x-x^3} \, dx\) [349]