3.97.32 \(\int \frac {-25+25 x+x^2-x^3+(-2 x^3+2 x^2 \log (x)) \log (x-\log (x))}{-x^2+x \log (x)} \, dx\)

Optimal. Leaf size=23 \[ -5-(5+x+(4-x) (5+x)) \log (x-\log (x)) \]

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Rubi [F]  time = 0.45, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-25+25 x+x^2-x^3+\left (-2 x^3+2 x^2 \log (x)\right ) \log (x-\log (x))}{-x^2+x \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-25 + 25*x + x^2 - x^3 + (-2*x^3 + 2*x^2*Log[x])*Log[x - Log[x]])/(-x^2 + x*Log[x]),x]

[Out]

-25*Defer[Int][(x - Log[x])^(-1), x] + 25*Defer[Int][1/(x*(x - Log[x])), x] - Defer[Int][x/(x - Log[x]), x] +
Defer[Int][x^2/(x - Log[x]), x] + 2*Defer[Int][x*Log[x - Log[x]], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-25+25 x+x^2-x^3+\left (-2 x^3+2 x^2 \log (x)\right ) \log (x-\log (x))}{x (-x+\log (x))} \, dx\\ &=\int \left (\frac {25-25 x-x^2+x^3}{x (x-\log (x))}+2 x \log (x-\log (x))\right ) \, dx\\ &=2 \int x \log (x-\log (x)) \, dx+\int \frac {25-25 x-x^2+x^3}{x (x-\log (x))} \, dx\\ &=2 \int x \log (x-\log (x)) \, dx+\int \left (-\frac {25}{x-\log (x)}+\frac {25}{x (x-\log (x))}-\frac {x}{x-\log (x)}+\frac {x^2}{x-\log (x)}\right ) \, dx\\ &=2 \int x \log (x-\log (x)) \, dx-25 \int \frac {1}{x-\log (x)} \, dx+25 \int \frac {1}{x (x-\log (x))} \, dx-\int \frac {x}{x-\log (x)} \, dx+\int \frac {x^2}{x-\log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 21, normalized size = 0.91 \begin {gather*} -25 \log (x-\log (x))+x^2 \log (x-\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-25 + 25*x + x^2 - x^3 + (-2*x^3 + 2*x^2*Log[x])*Log[x - Log[x]])/(-x^2 + x*Log[x]),x]

[Out]

-25*Log[x - Log[x]] + x^2*Log[x - Log[x]]

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fricas [A]  time = 0.52, size = 13, normalized size = 0.57 \begin {gather*} {\left (x^{2} - 25\right )} \log \left (x - \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2*log(x)-2*x^3)*log(x-log(x))-x^3+x^2+25*x-25)/(x*log(x)-x^2),x, algorithm="fricas")

[Out]

(x^2 - 25)*log(x - log(x))

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giac [A]  time = 0.15, size = 21, normalized size = 0.91 \begin {gather*} x^{2} \log \left (x - \log \relax (x)\right ) - 25 \, \log \left (x - \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2*log(x)-2*x^3)*log(x-log(x))-x^3+x^2+25*x-25)/(x*log(x)-x^2),x, algorithm="giac")

[Out]

x^2*log(x - log(x)) - 25*log(x - log(x))

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maple [A]  time = 0.03, size = 22, normalized size = 0.96

method	result	size
risch	\(x^{2} \ln \left (x -\ln \relax (x )\right )-25 \ln \left (\ln \relax (x )-x \right )\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2*ln(x)-2*x^3)*ln(x-ln(x))-x^3+x^2+25*x-25)/(x*ln(x)-x^2),x,method=_RETURNVERBOSE)

[Out]

x^2*ln(x-ln(x))-25*ln(ln(x)-x)

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maxima [A]  time = 0.38, size = 13, normalized size = 0.57 \begin {gather*} {\left (x^{2} - 25\right )} \log \left (x - \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2*log(x)-2*x^3)*log(x-log(x))-x^3+x^2+25*x-25)/(x*log(x)-x^2),x, algorithm="maxima")

[Out]

(x^2 - 25)*log(x - log(x))

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mupad [B]  time = 5.71, size = 21, normalized size = 0.91 \begin {gather*} x^2\,\ln \left (x-\ln \relax (x)\right )-25\,\ln \left (\ln \relax (x)-x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((25*x + log(x - log(x))*(2*x^2*log(x) - 2*x^3) + x^2 - x^3 - 25)/(x*log(x) - x^2),x)

[Out]

x^2*log(x - log(x)) - 25*log(log(x) - x)

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sympy [A]  time = 0.40, size = 17, normalized size = 0.74 \begin {gather*} x^{2} \log {\left (x - \log {\relax (x )} \right )} - 25 \log {\left (- x + \log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2*ln(x)-2*x**3)*ln(x-ln(x))-x**3+x**2+25*x-25)/(x*ln(x)-x**2),x)

[Out]

x**2*log(x - log(x)) - 25*log(-x + log(x))

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