3.4.48 \(\int \frac {\log (\frac {a}{x})}{a x-x^2} \, dx\) [348]

Optimal. Leaf size=14 \[ \frac {\text {Li}_2\left (1-\frac {a}{x}\right )}{a} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 14, 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}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[a/x]/(a*x - x^2),x]

[Out]

PolyLog[2, 1 - a/x]/a

Rule 1607

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]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2370

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]

Rule 2378

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]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 16, normalized size = 1.14 \begin {gather*} \frac {\text {Li}_2\left (-\frac {a-x}{x}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[a/x]/(a*x - x^2),x]

[Out]

PolyLog[2, -((a - x)/x)]/a

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Maple [A]
time = 0.11, size = 11, normalized size = 0.79

method	result	size
derivativedivides	\(\frac {\dilog \left (\frac {a}{x}\right )}{a}\)	\(11\)
default	\(\frac {\dilog \left (\frac {a}{x}\right )}{a}\)	\(11\)
risch	\(\frac {\dilog \left (\frac {a}{x}\right )}{a}\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a*dilog(a/x)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (13) = 26\).
time = 0.28, size = 72, normalized size = 5.14 \begin {gather*} -{\left (\frac {\log \left (-a + x\right )}{a} - \frac {\log \left (x\right )}{a}\right )} \log \left (\frac {a}{x}\right ) - \frac {2 \, \log \left (-a + x\right ) \log \left (x\right ) - \log \left (x\right )^{2}}{2 \, a} + \frac {\log \left (x\right ) \log \left (-\frac {x}{a} + 1\right ) + {\rm Li}_2\left (\frac {x}{a}\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.36, size = 13, normalized size = 0.93 \begin {gather*} \frac {{\rm Li}_2\left (-\frac {a}{x} + 1\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

dilog(-a/x + 1)/a

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Sympy [C] Result contains complex when optimal does not.
time = 5.57, size = 82, normalized size = 5.86 \begin {gather*} - \left (\begin {cases} - \frac {1}{x} & \text {for}\: a = 0 \\\frac {\log {\left (\frac {a}{x} - 1 \right )}}{a} & \text {otherwise} \end {cases}\right ) \log {\left (\frac {a}{x} \right )} - \begin {cases} \frac {1}{x} & \text {for}\: a = 0 \\\frac {\begin {cases} \operatorname {Li}_{2}\left (\frac {a}{x}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\i \pi \log {\left (x \right )} + \operatorname {Li}_{2}\left (\frac {a}{x}\right ) & \text {for}\: \left |{x}\right | < 1 \\- i \pi \log {\left (\frac {1}{x} \right )} + \operatorname {Li}_{2}\left (\frac {a}{x}\right ) & \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 )} + \operatorname {Li}_{2}\left (\frac {a}{x}\right ) & \text {otherwise} \end {cases}}{a} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Piecewise((-1/x, Eq(a, 0)), (log(a/x - 1)/a, True))*log(a/x) - Piecewise((1/x, Eq(a, 0)), (Piecewise((polylog
(2, a/x), (Abs(x) < 1) & (1/Abs(x) < 1)), (I*pi*log(x) + polylog(2, a/x), Abs(x) < 1), (-I*pi*log(1/x) + polyl
og(2, a/x), 1/Abs(x) < 1), (-I*pi*meijerg(((), (1, 1)), ((0, 0), ()), x) + I*pi*meijerg(((1, 1), ()), ((), (0,
0)), x) + polylog(2, a/x), True))/a, True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log(a/x)/(a*x - x^2), x)

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Mupad [B]
time = 3.46, size = 10, normalized size = 0.71 \begin {gather*} \frac {{\mathrm {Li}}_{\mathrm {2}}\left (\frac {a}{x}\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(a/x)/(a*x - x^2),x)

[Out]

dilog(a/x)/a

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