\(\int \frac {632 x-2 x^2+(-2-632 x+x^2) \log (-2-632 x+x^2)}{(-2 x-632 x^2+x^3) \log (-2-632 x+x^2)} \, dx\) [5650]

Optimal result

Integrand size = 53, antiderivative size = 13 \[ \int \frac {632 x-2 x^2+\left (-2-632 x+x^2\right ) \log \left (-2-632 x+x^2\right )}{\left (-2 x-632 x^2+x^3\right ) \log \left (-2-632 x+x^2\right )} \, dx=\log \left (\frac {x}{\log (-2+(-632+x) x)}\right ) \]

[Out]

ln(x/ln((-632+x)*x-2))

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {1608, 6820, 6816} \[ \int \frac {632 x-2 x^2+\left (-2-632 x+x^2\right ) \log \left (-2-632 x+x^2\right )}{\left (-2 x-632 x^2+x^3\right ) \log \left (-2-632 x+x^2\right )} \, dx=\log (x)-\log \left (\log \left (x^2-632 x-2\right )\right ) \]

[In]

Int[(632*x - 2*x^2 + (-2 - 632*x + x^2)*Log[-2 - 632*x + x^2])/((-2*x - 632*x^2 + x^3)*Log[-2 - 632*x + x^2]),
x]

[Out]

Log[x] - Log[Log[-2 - 632*x + x^2]]

Rule 1608

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 6816

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

Rule 6820

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {632 x-2 x^2+\left (-2-632 x+x^2\right ) \log \left (-2-632 x+x^2\right )}{x \left (-2-632 x+x^2\right ) \log \left (-2-632 x+x^2\right )} \, dx \\ & = \int \left (\frac {1}{x}-\frac {2 (-316+x)}{\left (-2-632 x+x^2\right ) \log \left (-2-632 x+x^2\right )}\right ) \, dx \\ & = \log (x)-2 \int \frac {-316+x}{\left (-2-632 x+x^2\right ) \log \left (-2-632 x+x^2\right )} \, dx \\ & = \log (x)-\log \left (\log \left (-2-632 x+x^2\right )\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \frac {632 x-2 x^2+\left (-2-632 x+x^2\right ) \log \left (-2-632 x+x^2\right )}{\left (-2 x-632 x^2+x^3\right ) \log \left (-2-632 x+x^2\right )} \, dx=\log (x)-\log \left (\log \left (-2-632 x+x^2\right )\right ) \]

[In]

Integrate[(632*x - 2*x^2 + (-2 - 632*x + x^2)*Log[-2 - 632*x + x^2])/((-2*x - 632*x^2 + x^3)*Log[-2 - 632*x +
x^2]),x]

[Out]

Log[x] - Log[Log[-2 - 632*x + x^2]]

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.23

method	result	size
default	\(\ln \left (x \right )-\ln \left (\ln \left (x^{2}-632 x -2\right )\right )\)	\(16\)
norman	\(\ln \left (x \right )-\ln \left (\ln \left (x^{2}-632 x -2\right )\right )\)	\(16\)
risch	\(\ln \left (x \right )-\ln \left (\ln \left (x^{2}-632 x -2\right )\right )\)	\(16\)
parallelrisch	\(\ln \left (x \right )-\ln \left (\ln \left (x^{2}-632 x -2\right )\right )\)	\(16\)
parts	\(\ln \left (x \right )-\ln \left (\ln \left (x^{2}-632 x -2\right )\right )\)	\(16\)

[In]

int(((x^2-632*x-2)*ln(x^2-632*x-2)-2*x^2+632*x)/(x^3-632*x^2-2*x)/ln(x^2-632*x-2),x,method=_RETURNVERBOSE)

[Out]

ln(x)-ln(ln(x^2-632*x-2))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \frac {632 x-2 x^2+\left (-2-632 x+x^2\right ) \log \left (-2-632 x+x^2\right )}{\left (-2 x-632 x^2+x^3\right ) \log \left (-2-632 x+x^2\right )} \, dx=\log \left (x\right ) - \log \left (\log \left (x^{2} - 632 \, x - 2\right )\right ) \]

[In]

integrate(((x^2-632*x-2)*log(x^2-632*x-2)-2*x^2+632*x)/(x^3-632*x^2-2*x)/log(x^2-632*x-2),x, algorithm="fricas
")

[Out]

log(x) - log(log(x^2 - 632*x - 2))

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {632 x-2 x^2+\left (-2-632 x+x^2\right ) \log \left (-2-632 x+x^2\right )}{\left (-2 x-632 x^2+x^3\right ) \log \left (-2-632 x+x^2\right )} \, dx=\log {\left (x \right )} - \log {\left (\log {\left (x^{2} - 632 x - 2 \right )} \right )} \]

[In]

integrate(((x**2-632*x-2)*ln(x**2-632*x-2)-2*x**2+632*x)/(x**3-632*x**2-2*x)/ln(x**2-632*x-2),x)

[Out]

log(x) - log(log(x**2 - 632*x - 2))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \frac {632 x-2 x^2+\left (-2-632 x+x^2\right ) \log \left (-2-632 x+x^2\right )}{\left (-2 x-632 x^2+x^3\right ) \log \left (-2-632 x+x^2\right )} \, dx=\log \left (x\right ) - \log \left (\log \left (x^{2} - 632 \, x - 2\right )\right ) \]

[In]

integrate(((x^2-632*x-2)*log(x^2-632*x-2)-2*x^2+632*x)/(x^3-632*x^2-2*x)/log(x^2-632*x-2),x, algorithm="maxima
")

[Out]

log(x) - log(log(x^2 - 632*x - 2))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \frac {632 x-2 x^2+\left (-2-632 x+x^2\right ) \log \left (-2-632 x+x^2\right )}{\left (-2 x-632 x^2+x^3\right ) \log \left (-2-632 x+x^2\right )} \, dx=\log \left (x\right ) - \log \left (\log \left (x^{2} - 632 \, x - 2\right )\right ) \]

[In]

integrate(((x^2-632*x-2)*log(x^2-632*x-2)-2*x^2+632*x)/(x^3-632*x^2-2*x)/log(x^2-632*x-2),x, algorithm="giac")

[Out]

log(x) - log(log(x^2 - 632*x - 2))

Mupad [B] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \frac {632 x-2 x^2+\left (-2-632 x+x^2\right ) \log \left (-2-632 x+x^2\right )}{\left (-2 x-632 x^2+x^3\right ) \log \left (-2-632 x+x^2\right )} \, dx=\ln \left (x\right )-\ln \left (\ln \left (x^2-632\,x-2\right )\right ) \]

[In]

int((log(x^2 - 632*x - 2)*(632*x - x^2 + 2) - 632*x + 2*x^2)/(log(x^2 - 632*x - 2)*(2*x + 632*x^2 - x^3)),x)

[Out]

log(x) - log(log(x^2 - 632*x - 2))