3.5.57 \(\int \frac {(-56+2 x+56 x^3) \log (\frac {-28+3 x-14 x^3-x \log (x)}{x})}{28 x-3 x^2+14 x^4+x^2 \log (x)} \, dx\)

Optimal. Leaf size=21 \[ \log ^2\left (3-7 x \left (\frac {4}{x^2}+2 x\right )-\log (x)\right ) \]

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Rubi [A]  time = 0.33, antiderivative size = 22, normalized size of antiderivative = 1.05, number of steps used = 1, number of rules used = 2, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {6742, 6686} \begin {gather*} \log ^2\left (-\frac {14 x^3-3 x+x \log (x)+28}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-56 + 2*x + 56*x^3)*Log[(-28 + 3*x - 14*x^3 - x*Log[x])/x])/(28*x - 3*x^2 + 14*x^4 + x^2*Log[x]),x]

[Out]

Log[-((28 - 3*x + 14*x^3 + x*Log[x])/x)]^2

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log ^2\left (-\frac {28-3 x+14 x^3+x \log (x)}{x}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 19, normalized size = 0.90 \begin {gather*} \log ^2\left (3-\frac {28}{x}-14 x^2-\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-56 + 2*x + 56*x^3)*Log[(-28 + 3*x - 14*x^3 - x*Log[x])/x])/(28*x - 3*x^2 + 14*x^4 + x^2*Log[x]),x
]

[Out]

Log[3 - 28/x - 14*x^2 - Log[x]]^2

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fricas [A]  time = 0.94, size = 22, normalized size = 1.05 \begin {gather*} \log \left (-\frac {14 \, x^{3} + x \log \relax (x) - 3 \, x + 28}{x}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((56*x^3+2*x-56)*log((-x*log(x)-14*x^3+3*x-28)/x)/(x^2*log(x)+14*x^4-3*x^2+28*x),x, algorithm="fricas
")

[Out]

log(-(14*x^3 + x*log(x) - 3*x + 28)/x)^2

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left (28 \, x^{3} + x - 28\right )} \log \left (-\frac {14 \, x^{3} + x \log \relax (x) - 3 \, x + 28}{x}\right )}{14 \, x^{4} + x^{2} \log \relax (x) - 3 \, x^{2} + 28 \, x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((56*x^3+2*x-56)*log((-x*log(x)-14*x^3+3*x-28)/x)/(x^2*log(x)+14*x^4-3*x^2+28*x),x, algorithm="giac")

[Out]

integrate(2*(28*x^3 + x - 28)*log(-(14*x^3 + x*log(x) - 3*x + 28)/x)/(14*x^4 + x^2*log(x) - 3*x^2 + 28*x), x)

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (56 x^{3}+2 x -56\right ) \ln \left (\frac {-x \ln \relax (x )-14 x^{3}+3 x -28}{x}\right )}{x^{2} \ln \relax (x )+14 x^{4}-3 x^{2}+28 x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((56*x^3+2*x-56)*ln((-x*ln(x)-14*x^3+3*x-28)/x)/(x^2*ln(x)+14*x^4-3*x^2+28*x),x)

[Out]

int((56*x^3+2*x-56)*ln((-x*ln(x)-14*x^3+3*x-28)/x)/(x^2*ln(x)+14*x^4-3*x^2+28*x),x)

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maxima [B]  time = 0.44, size = 65, normalized size = 3.10 \begin {gather*} -\log \left (\frac {14 \, x^{3} + x \log \relax (x) - 3 \, x + 28}{x}\right )^{2} + 2 \, \log \left (\frac {14 \, x^{3} + x \log \relax (x) - 3 \, x + 28}{x}\right ) \log \left (-\frac {14 \, x^{3} + x \log \relax (x) - 3 \, x + 28}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((56*x^3+2*x-56)*log((-x*log(x)-14*x^3+3*x-28)/x)/(x^2*log(x)+14*x^4-3*x^2+28*x),x, algorithm="maxima
")

[Out]

-log((14*x^3 + x*log(x) - 3*x + 28)/x)^2 + 2*log((14*x^3 + x*log(x) - 3*x + 28)/x)*log(-(14*x^3 + x*log(x) - 3
*x + 28)/x)

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mupad [B]  time = 1.45, size = 19, normalized size = 0.90 \begin {gather*} {\ln \left (3-\frac {28}{x}-14\,x^2-\ln \relax (x)\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(-(x*log(x) - 3*x + 14*x^3 + 28)/x)*(2*x + 56*x^3 - 56))/(28*x + x^2*log(x) - 3*x^2 + 14*x^4),x)

[Out]

log(3 - 28/x - 14*x^2 - log(x))^2

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sympy [A]  time = 0.37, size = 19, normalized size = 0.90 \begin {gather*} \log {\left (\frac {- 14 x^{3} - x \log {\relax (x )} + 3 x - 28}{x} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((56*x**3+2*x-56)*ln((-x*ln(x)-14*x**3+3*x-28)/x)/(x**2*ln(x)+14*x**4-3*x**2+28*x),x)

[Out]

log((-14*x**3 - x*log(x) + 3*x - 28)/x)**2

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