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

Optimal. Leaf size=13 \[ \log \left (\frac {x}{\log (-2+(-632+x) x)}\right ) \]

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Rubi [A]  time = 0.28, antiderivative size = 15, normalized size of antiderivative = 1.15, number of steps used = 4, number of rules used = 3, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {1594, 6688, 6684} \begin {gather*} \log (x)-\log \left (\log \left (x^2-632 x-2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[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 1594

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 6684

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

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.27, size = 15, normalized size = 1.15 \begin {gather*} \log (x)-\log \left (\log \left (-2-632 x+x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[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]]

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fricas [A]  time = 0.49, size = 15, normalized size = 1.15 \begin {gather*} \log \relax (x) - \log \left (\log \left (x^{2} - 632 \, x - 2\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 0.20, size = 15, normalized size = 1.15 \begin {gather*} \log \relax (x) - \log \left (\log \left (x^{2} - 632 \, x - 2\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.04, size = 16, normalized size = 1.23

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

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maxima [A]  time = 0.39, size = 15, normalized size = 1.15 \begin {gather*} \log \relax (x) - \log \left (\log \left (x^{2} - 632 \, x - 2\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 3.74, size = 15, normalized size = 1.15 \begin {gather*} \ln \relax (x)-\ln \left (\ln \left (x^2-632\,x-2\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 0.17, size = 14, normalized size = 1.08 \begin {gather*} \log {\relax (x )} - \log {\left (\log {\left (x^{2} - 632 x - 2 \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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