3.4.55 \(\int \frac {\log (x)}{x (a+b x+c x^2)} \, dx\) [355]

Optimal. Leaf size=204 \[ \frac {\log ^2(x)}{2 a}-\frac {\left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 a}-\frac {\left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 a}-\frac {\left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \text {Li}_2\left (-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 a}-\frac {\left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \text {Li}_2\left (-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 a} \]

[Out]

1/2*ln(x)^2/a-1/2*ln(x)*ln(1+2*c*x/(b+(-4*a*c+b^2)^(1/2)))*(1-b/(-4*a*c+b^2)^(1/2))/a-1/2*polylog(2,-2*c*x/(b+
(-4*a*c+b^2)^(1/2)))*(1-b/(-4*a*c+b^2)^(1/2))/a-1/2*ln(x)*ln(1+2*c*x/(b-(-4*a*c+b^2)^(1/2)))*(1+b/(-4*a*c+b^2)
^(1/2))/a-1/2*polylog(2,-2*c*x/(b-(-4*a*c+b^2)^(1/2)))*(1+b/(-4*a*c+b^2)^(1/2))/a

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Rubi [A]
time = 0.20, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2404, 2338, 2354, 2438} \begin {gather*} -\frac {\left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \text {PolyLog}\left (2,-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 a}-\frac {\left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \text {PolyLog}\left (2,-\frac {2 c x}{\sqrt {b^2-4 a c}+b}\right )}{2 a}-\frac {\log (x) \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \log \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )}{2 a}-\frac {\log (x) \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \log \left (\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1\right )}{2 a}+\frac {\log ^2(x)}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[x]/(x*(a + b*x + c*x^2)),x]

[Out]

Log[x]^2/(2*a) - ((1 + b/Sqrt[b^2 - 4*a*c])*Log[x]*Log[1 + (2*c*x)/(b - Sqrt[b^2 - 4*a*c])])/(2*a) - ((1 - b/S
qrt[b^2 - 4*a*c])*Log[x]*Log[1 + (2*c*x)/(b + Sqrt[b^2 - 4*a*c])])/(2*a) - ((1 + b/Sqrt[b^2 - 4*a*c])*PolyLog[
2, (-2*c*x)/(b - Sqrt[b^2 - 4*a*c])])/(2*a) - ((1 - b/Sqrt[b^2 - 4*a*c])*PolyLog[2, (-2*c*x)/(b + Sqrt[b^2 - 4
*a*c])])/(2*a)

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2354

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[Log[1 + e*(x/d)]*((a +
b*Log[c*x^n])^p/e), x] - Dist[b*n*(p/e), Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2404

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

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]

Rubi steps

\begin {align*} \int \frac {\log (x)}{x \left (a+b x+c x^2\right )} \, dx &=\int \left (\frac {\log (x)}{a x}+\frac {(-b-c x) \log (x)}{a \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {\log (x)}{x} \, dx}{a}+\frac {\int \frac {(-b-c x) \log (x)}{a+b x+c x^2} \, dx}{a}\\ &=\frac {\log ^2(x)}{2 a}+\frac {\int \left (\frac {\left (-c-\frac {b c}{\sqrt {b^2-4 a c}}\right ) \log (x)}{b-\sqrt {b^2-4 a c}+2 c x}+\frac {\left (-c+\frac {b c}{\sqrt {b^2-4 a c}}\right ) \log (x)}{b+\sqrt {b^2-4 a c}+2 c x}\right ) \, dx}{a}\\ &=\frac {\log ^2(x)}{2 a}-\frac {\left (c \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {\log (x)}{b+\sqrt {b^2-4 a c}+2 c x} \, dx}{a}-\frac {\left (c \left (1+\frac {b}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {\log (x)}{b-\sqrt {b^2-4 a c}+2 c x} \, dx}{a}\\ &=\frac {\log ^2(x)}{2 a}-\frac {\left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 a}-\frac {\left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 a}+\frac {\left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \int \frac {\log \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{x} \, dx}{2 a}+\frac {\left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \int \frac {\log \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{x} \, dx}{2 a}\\ &=\frac {\log ^2(x)}{2 a}-\frac {\left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 a}-\frac {\left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 a}-\frac {\left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \text {Li}_2\left (-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 a}-\frac {\left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \text {Li}_2\left (-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 a}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 227, normalized size = 1.11 \begin {gather*} \frac {\log (x) \left (\sqrt {b^2-4 a c} \log (x)-\left (b+\sqrt {b^2-4 a c}\right ) \log \left (\frac {b-\sqrt {b^2-4 a c}+2 c x}{b-\sqrt {b^2-4 a c}}\right )+\left (b-\sqrt {b^2-4 a c}\right ) \log \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{b+\sqrt {b^2-4 a c}}\right )\right )-\left (b+\sqrt {b^2-4 a c}\right ) \text {Li}_2\left (\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )+\left (b-\sqrt {b^2-4 a c}\right ) \text {Li}_2\left (-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 a \sqrt {b^2-4 a c}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[x]/(x*(a + b*x + c*x^2)),x]

[Out]

(Log[x]*(Sqrt[b^2 - 4*a*c]*Log[x] - (b + Sqrt[b^2 - 4*a*c])*Log[(b - Sqrt[b^2 - 4*a*c] + 2*c*x)/(b - Sqrt[b^2
- 4*a*c])] + (b - Sqrt[b^2 - 4*a*c])*Log[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/(b + Sqrt[b^2 - 4*a*c])]) - (b + Sqrt
[b^2 - 4*a*c])*PolyLog[2, (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])] + (b - Sqrt[b^2 - 4*a*c])*PolyLog[2, (-2*c*x)/(b +
Sqrt[b^2 - 4*a*c])])/(2*a*Sqrt[b^2 - 4*a*c])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(369\) vs. \(2(178)=356\).
time = 0.83, size = 370, normalized size = 1.81

method	result	size
default	\(\frac {\ln \left (x \right )^{2}}{2 a}+\frac {-\frac {\ln \left (x \right ) \left (\ln \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}+\ln \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b +\ln \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}-\ln \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) b \right )}{2 \sqrt {-4 c a +b^{2}}}-\frac {\dilog \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}+\dilog \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b +\dilog \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}-\dilog \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) b}{2 \sqrt {-4 c a +b^{2}}}}{a}\)	\(370\)
risch	\(\frac {\ln \left (x \right )^{2}}{2 a}-\frac {\ln \left (x \right ) \ln \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right )}{2 a}-\frac {\ln \left (x \right ) \ln \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b}{2 a \sqrt {-4 c a +b^{2}}}-\frac {\ln \left (x \right ) \ln \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right )}{2 a}+\frac {\ln \left (x \right ) \ln \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) b}{2 a \sqrt {-4 c a +b^{2}}}-\frac {\dilog \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right )}{2 a}-\frac {\dilog \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b}{2 a \sqrt {-4 c a +b^{2}}}-\frac {\dilog \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right )}{2 a}+\frac {\dilog \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) b}{2 a \sqrt {-4 c a +b^{2}}}\)	\(375\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*ln(x)^2/a+(-1/2*ln(x)*(ln((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-b+(-4*a*c+b^2)^(1/2)))*(-4*a*c+b^2)^(1/2)+ln((-2
*c*x+(-4*a*c+b^2)^(1/2)-b)/(-b+(-4*a*c+b^2)^(1/2)))*b+ln((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))*
(-4*a*c+b^2)^(1/2)-ln((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))*b)/(-4*a*c+b^2)^(1/2)-1/2*(dilog((-
2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-b+(-4*a*c+b^2)^(1/2)))*(-4*a*c+b^2)^(1/2)+dilog((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-
b+(-4*a*c+b^2)^(1/2)))*b+dilog((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))*(-4*a*c+b^2)^(1/2)-dilog((
b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))*b)/(-4*a*c+b^2)^(1/2))/a

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

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Fricas [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(x)/x/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

integral(log(x)/(c*x^3 + b*x^2 + a*x), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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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(x)/x/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

integrate(log(x)/((c*x^2 + b*x + a)*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(x)/(x*(a + b*x + c*x^2)),x)

[Out]

int(log(x)/(x*(a + b*x + c*x^2)), x)

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