3.53.32 \(\int \frac {-160-2 x^2+(320-4 x^2) \log (-\frac {x}{-80+x^2})+(-160+2 x^2) \log (-\frac {x}{-80+x^2}) \log (x \log (-\frac {x}{-80+x^2}))}{(240 x-3 x^3) \log (-\frac {x}{-80+x^2})+(-80 x+x^3) \log (-\frac {x}{-80+x^2}) \log (x \log (-\frac {x}{-80+x^2}))} \, dx\)

Optimal. Leaf size=25 \[ \log \left (5 x^2 \left (-3+\log \left (x \log \left (\frac {x}{80-x^2}\right )\right )\right )^2\right ) \]

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Rubi [A]  time = 0.29, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 117, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {6688, 12, 6685} \begin {gather*} 2 \log \left (-x \left (3-\log \left (x \log \left (\frac {x}{80-x^2}\right )\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-160 - 2*x^2 + (320 - 4*x^2)*Log[-(x/(-80 + x^2))] + (-160 + 2*x^2)*Log[-(x/(-80 + x^2))]*Log[x*Log[-(x/(
-80 + x^2))]])/((240*x - 3*x^3)*Log[-(x/(-80 + x^2))] + (-80*x + x^3)*Log[-(x/(-80 + x^2))]*Log[x*Log[-(x/(-80
+ x^2))]]),x]

[Out]

2*Log[-(x*(3 - Log[x*Log[x/(80 - x^2)]]))]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6685

Int[(u_)/((w_)*(y_)), x_Symbol] :> With[{q = DerivativeDivides[y*w, u, x]}, Simp[q*Log[RemoveContent[y*w, x]],
x] /;  !FalseQ[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 {2 \left (-80-x^2+\left (-80+x^2\right ) \log \left (-\frac {x}{-80+x^2}\right ) \left (-2+\log \left (x \log \left (-\frac {x}{-80+x^2}\right )\right )\right )\right )}{x \left (80-x^2\right ) \log \left (-\frac {x}{-80+x^2}\right ) \left (3-\log \left (x \log \left (-\frac {x}{-80+x^2}\right )\right )\right )} \, dx\\ &=2 \int \frac {-80-x^2+\left (-80+x^2\right ) \log \left (-\frac {x}{-80+x^2}\right ) \left (-2+\log \left (x \log \left (-\frac {x}{-80+x^2}\right )\right )\right )}{x \left (80-x^2\right ) \log \left (-\frac {x}{-80+x^2}\right ) \left (3-\log \left (x \log \left (-\frac {x}{-80+x^2}\right )\right )\right )} \, dx\\ &=2 \log \left (-x \left (3-\log \left (x \log \left (\frac {x}{80-x^2}\right )\right )\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 24, normalized size = 0.96 \begin {gather*} 2 \left (\log (x)+\log \left (3-\log \left (x \log \left (-\frac {x}{-80+x^2}\right )\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-160 - 2*x^2 + (320 - 4*x^2)*Log[-(x/(-80 + x^2))] + (-160 + 2*x^2)*Log[-(x/(-80 + x^2))]*Log[x*Log
[-(x/(-80 + x^2))]])/((240*x - 3*x^3)*Log[-(x/(-80 + x^2))] + (-80*x + x^3)*Log[-(x/(-80 + x^2))]*Log[x*Log[-(
x/(-80 + x^2))]]),x]

[Out]

2*(Log[x] + Log[3 - Log[x*Log[-(x/(-80 + x^2))]]])

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fricas [A]  time = 0.55, size = 24, normalized size = 0.96 \begin {gather*} 2 \, \log \relax (x) + 2 \, \log \left (\log \left (x \log \left (-\frac {x}{x^{2} - 80}\right )\right ) - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-160)*log(-x/(x^2-80))*log(x*log(-x/(x^2-80)))+(-4*x^2+320)*log(-x/(x^2-80))-2*x^2-160)/((x^3
-80*x)*log(-x/(x^2-80))*log(x*log(-x/(x^2-80)))+(-3*x^3+240*x)*log(-x/(x^2-80))),x, algorithm="fricas")

[Out]

2*log(x) + 2*log(log(x*log(-x/(x^2 - 80))) - 3)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left ({\left (x^{2} - 80\right )} \log \left (x \log \left (-\frac {x}{x^{2} - 80}\right )\right ) \log \left (-\frac {x}{x^{2} - 80}\right ) - x^{2} - 2 \, {\left (x^{2} - 80\right )} \log \left (-\frac {x}{x^{2} - 80}\right ) - 80\right )}}{{\left (x^{3} - 80 \, x\right )} \log \left (x \log \left (-\frac {x}{x^{2} - 80}\right )\right ) \log \left (-\frac {x}{x^{2} - 80}\right ) - 3 \, {\left (x^{3} - 80 \, x\right )} \log \left (-\frac {x}{x^{2} - 80}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-160)*log(-x/(x^2-80))*log(x*log(-x/(x^2-80)))+(-4*x^2+320)*log(-x/(x^2-80))-2*x^2-160)/((x^3
-80*x)*log(-x/(x^2-80))*log(x*log(-x/(x^2-80)))+(-3*x^3+240*x)*log(-x/(x^2-80))),x, algorithm="giac")

[Out]

integrate(2*((x^2 - 80)*log(x*log(-x/(x^2 - 80)))*log(-x/(x^2 - 80)) - x^2 - 2*(x^2 - 80)*log(-x/(x^2 - 80)) -
80)/((x^3 - 80*x)*log(x*log(-x/(x^2 - 80)))*log(-x/(x^2 - 80)) - 3*(x^3 - 80*x)*log(-x/(x^2 - 80))), x)

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (2 x^{2}-160\right ) \ln \left (-\frac {x}{x^{2}-80}\right ) \ln \left (x \ln \left (-\frac {x}{x^{2}-80}\right )\right )+\left (-4 x^{2}+320\right ) \ln \left (-\frac {x}{x^{2}-80}\right )-2 x^{2}-160}{\left (x^{3}-80 x \right ) \ln \left (-\frac {x}{x^{2}-80}\right ) \ln \left (x \ln \left (-\frac {x}{x^{2}-80}\right )\right )+\left (-3 x^{3}+240 x \right ) \ln \left (-\frac {x}{x^{2}-80}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2-160)*ln(-x/(x^2-80))*ln(x*ln(-x/(x^2-80)))+(-4*x^2+320)*ln(-x/(x^2-80))-2*x^2-160)/((x^3-80*x)*ln(
-x/(x^2-80))*ln(x*ln(-x/(x^2-80)))+(-3*x^3+240*x)*ln(-x/(x^2-80))),x)

[Out]

int(((2*x^2-160)*ln(-x/(x^2-80))*ln(x*ln(-x/(x^2-80)))+(-4*x^2+320)*ln(-x/(x^2-80))-2*x^2-160)/((x^3-80*x)*ln(
-x/(x^2-80))*ln(x*ln(-x/(x^2-80)))+(-3*x^3+240*x)*ln(-x/(x^2-80))),x)

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maxima [A]  time = 0.40, size = 26, normalized size = 1.04 \begin {gather*} 2 \, \log \relax (x) + 2 \, \log \left (\log \relax (x) + \log \left (-\log \left (-x^{2} + 80\right ) + \log \relax (x)\right ) - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-160)*log(-x/(x^2-80))*log(x*log(-x/(x^2-80)))+(-4*x^2+320)*log(-x/(x^2-80))-2*x^2-160)/((x^3
-80*x)*log(-x/(x^2-80))*log(x*log(-x/(x^2-80)))+(-3*x^3+240*x)*log(-x/(x^2-80))),x, algorithm="maxima")

[Out]

2*log(x) + 2*log(log(x) + log(-log(-x^2 + 80) + log(x)) - 3)

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mupad [B]  time = 4.48, size = 24, normalized size = 0.96 \begin {gather*} 2\,\ln \relax (x)+2\,\ln \left (\ln \left (x\,\ln \left (-\frac {x}{x^2-80}\right )\right )-3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(-x/(x^2 - 80))*(4*x^2 - 320) + 2*x^2 - log(x*log(-x/(x^2 - 80)))*log(-x/(x^2 - 80))*(2*x^2 - 160) +
160)/(log(-x/(x^2 - 80))*(240*x - 3*x^3) - log(x*log(-x/(x^2 - 80)))*log(-x/(x^2 - 80))*(80*x - x^3)),x)

[Out]

2*log(x) + 2*log(log(x*log(-x/(x^2 - 80))) - 3)

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sympy [A]  time = 0.59, size = 22, normalized size = 0.88 \begin {gather*} 2 \log {\relax (x )} + 2 \log {\left (\log {\left (x \log {\left (- \frac {x}{x^{2} - 80} \right )} \right )} - 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2-160)*ln(-x/(x**2-80))*ln(x*ln(-x/(x**2-80)))+(-4*x**2+320)*ln(-x/(x**2-80))-2*x**2-160)/((x
**3-80*x)*ln(-x/(x**2-80))*ln(x*ln(-x/(x**2-80)))+(-3*x**3+240*x)*ln(-x/(x**2-80))),x)

[Out]

2*log(x) + 2*log(log(x*log(-x/(x**2 - 80))) - 3)

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