3.61.67 \(\int \frac {-180+31 x+x^2+(-36 x+4 x^2+x^2 \log (x)) \log (\frac {-36+4 x+x \log (x)}{x})}{(180 x-56 x^2+4 x^3+(-5 x^2+x^3) \log (x)) \log (\frac {-36+4 x+x \log (x)}{x})} \, dx\)

Optimal. Leaf size=18 \[ \log \left (e^{25} (-5+x) \log \left (4-\frac {36}{x}+\log (x)\right )\right ) \]

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Rubi [A]  time = 1.28, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 83, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {6741, 6742, 6684} \begin {gather*} \log (5-x)+\log \left (\log \left (-\frac {36}{x}+\log (x)+4\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-180 + 31*x + x^2 + (-36*x + 4*x^2 + x^2*Log[x])*Log[(-36 + 4*x + x*Log[x])/x])/((180*x - 56*x^2 + 4*x^3
+ (-5*x^2 + x^3)*Log[x])*Log[(-36 + 4*x + x*Log[x])/x]),x]

[Out]

Log[5 - x] + Log[Log[4 - 36/x + Log[x]]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6741

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-180+31 x+x^2+\left (-36 x+4 x^2+x^2 \log (x)\right ) \log \left (\frac {-36+4 x+x \log (x)}{x}\right )}{(5-x) x (36-4 x-x \log (x)) \log \left (4-\frac {36}{x}+\log (x)\right )} \, dx\\ &=\int \left (\frac {1}{-5+x}+\frac {36+x}{x (-36+4 x+x \log (x)) \log \left (4-\frac {36}{x}+\log (x)\right )}\right ) \, dx\\ &=\log (5-x)+\int \frac {36+x}{x (-36+4 x+x \log (x)) \log \left (4-\frac {36}{x}+\log (x)\right )} \, dx\\ &=\log (5-x)+\log \left (\log \left (4-\frac {36}{x}+\log (x)\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 18, normalized size = 1.00 \begin {gather*} \log (5-x)+\log \left (\log \left (4-\frac {36}{x}+\log (x)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-180 + 31*x + x^2 + (-36*x + 4*x^2 + x^2*Log[x])*Log[(-36 + 4*x + x*Log[x])/x])/((180*x - 56*x^2 +
4*x^3 + (-5*x^2 + x^3)*Log[x])*Log[(-36 + 4*x + x*Log[x])/x]),x]

[Out]

Log[5 - x] + Log[Log[4 - 36/x + Log[x]]]

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fricas [A]  time = 0.54, size = 20, normalized size = 1.11 \begin {gather*} \log \left (x - 5\right ) + \log \left (\log \left (\frac {x \log \relax (x) + 4 \, x - 36}{x}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2*log(x)+4*x^2-36*x)*log((x*log(x)+4*x-36)/x)+x^2+31*x-180)/((x^3-5*x^2)*log(x)+4*x^3-56*x^2+180
*x)/log((x*log(x)+4*x-36)/x),x, algorithm="fricas")

[Out]

log(x - 5) + log(log((x*log(x) + 4*x - 36)/x))

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giac [A]  time = 0.24, size = 21, normalized size = 1.17 \begin {gather*} \log \left (x - 5\right ) + \log \left (-\log \left (x \log \relax (x) + 4 \, x - 36\right ) + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2*log(x)+4*x^2-36*x)*log((x*log(x)+4*x-36)/x)+x^2+31*x-180)/((x^3-5*x^2)*log(x)+4*x^3-56*x^2+180
*x)/log((x*log(x)+4*x-36)/x),x, algorithm="giac")

[Out]

log(x - 5) + log(-log(x*log(x) + 4*x - 36) + log(x))

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maple [A]  time = 0.12, size = 21, normalized size = 1.17

method	result	size
default	\(\ln \left (x -5\right )+\ln \left (\ln \left (\frac {x \ln \relax (x )+4 x -36}{x}\right )\right )\)	\(21\)
risch	\(\ln \left (x -5\right )+\ln \left (\ln \left (-36+\left (\ln \relax (x )+4\right ) x \right )+\frac {i \left (\pi \,\mathrm {csgn}\left (i \left (-36+\left (\ln \relax (x )+4\right ) x \right )\right ) \mathrm {csgn}\left (\frac {i \left (-36+\left (\ln \relax (x )+4\right ) x \right )}{x}\right )^{2}-\pi \,\mathrm {csgn}\left (i \left (-36+\left (\ln \relax (x )+4\right ) x \right )\right ) \mathrm {csgn}\left (\frac {i \left (-36+\left (\ln \relax (x )+4\right ) x \right )}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right )-\pi \mathrm {csgn}\left (\frac {i \left (-36+\left (\ln \relax (x )+4\right ) x \right )}{x}\right )^{3}+\pi \mathrm {csgn}\left (\frac {i \left (-36+\left (\ln \relax (x )+4\right ) x \right )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )+2 i \ln \relax (x )\right )}{2}\right )\)	\(140\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2*ln(x)+4*x^2-36*x)*ln((x*ln(x)+4*x-36)/x)+x^2+31*x-180)/((x^3-5*x^2)*ln(x)+4*x^3-56*x^2+180*x)/ln((x*
ln(x)+4*x-36)/x),x,method=_RETURNVERBOSE)

[Out]

ln(x-5)+ln(ln((x*ln(x)+4*x-36)/x))

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maxima [A]  time = 0.40, size = 21, normalized size = 1.17 \begin {gather*} \log \left (x - 5\right ) + \log \left (\log \left (x \log \relax (x) + 4 \, x - 36\right ) - \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2*log(x)+4*x^2-36*x)*log((x*log(x)+4*x-36)/x)+x^2+31*x-180)/((x^3-5*x^2)*log(x)+4*x^3-56*x^2+180
*x)/log((x*log(x)+4*x-36)/x),x, algorithm="maxima")

[Out]

log(x - 5) + log(log(x*log(x) + 4*x - 36) - log(x))

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mupad [B]  time = 5.27, size = 20, normalized size = 1.11 \begin {gather*} \ln \left (\ln \left (\frac {4\,x+x\,\ln \relax (x)-36}{x}\right )\right )+\ln \left (x-5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((31*x + log((4*x + x*log(x) - 36)/x)*(x^2*log(x) - 36*x + 4*x^2) + x^2 - 180)/(log((4*x + x*log(x) - 36)/x
)*(180*x - log(x)*(5*x^2 - x^3) - 56*x^2 + 4*x^3)),x)

[Out]

log(log((4*x + x*log(x) - 36)/x)) + log(x - 5)

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sympy [A]  time = 0.39, size = 19, normalized size = 1.06 \begin {gather*} \log {\left (x - 5 \right )} + \log {\left (\log {\left (\frac {x \log {\relax (x )} + 4 x - 36}{x} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2*ln(x)+4*x**2-36*x)*ln((x*ln(x)+4*x-36)/x)+x**2+31*x-180)/((x**3-5*x**2)*ln(x)+4*x**3-56*x**2+
180*x)/ln((x*ln(x)+4*x-36)/x),x)

[Out]

log(x - 5) + log(log((x*log(x) + 4*x - 36)/x))

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