3.2.42 \(\int \frac {8 x+4 x \log (\frac {5}{x^2})+(-8+e^{x/4} x) \log ^2(\frac {5}{x^2})+8 \log ^2(\frac {5}{x^2}) \log (\frac {x}{3})}{4 x^2 \log (\frac {5}{x^2})+(4 x+4 e^{x/4} x) \log ^2(\frac {5}{x^2})-8 x \log ^2(\frac {5}{x^2}) \log (\frac {x}{3})+4 x \log ^2(\frac {5}{x^2}) \log ^2(\frac {x}{3})} \, dx\)

Optimal. Leaf size=29 \[ \log \left (e^{x/4}+\frac {x}{\log \left (\frac {5}{x^2}\right )}+\left (-1+\log \left (\frac {x}{3}\right )\right )^2\right ) \]

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Rubi [F]  time = 17.88, 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 {8 x+4 x \log \left (\frac {5}{x^2}\right )+\left (-8+e^{x/4} x\right ) \log ^2\left (\frac {5}{x^2}\right )+8 \log ^2\left (\frac {5}{x^2}\right ) \log \left (\frac {x}{3}\right )}{4 x^2 \log \left (\frac {5}{x^2}\right )+\left (4 x+4 e^{x/4} x\right ) \log ^2\left (\frac {5}{x^2}\right )-8 x \log ^2\left (\frac {5}{x^2}\right ) \log \left (\frac {x}{3}\right )+4 x \log ^2\left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(8*x + 4*x*Log[5/x^2] + (-8 + E^(x/4)*x)*Log[5/x^2]^2 + 8*Log[5/x^2]^2*Log[x/3])/(4*x^2*Log[5/x^2] + (4*x
+ 4*E^(x/4)*x)*Log[5/x^2]^2 - 8*x*Log[5/x^2]^2*Log[x/3] + 4*x*Log[5/x^2]^2*Log[x/3]^2),x]

[Out]

x/4 + Defer[Int][x/(-x - Log[5/x^2]*(1 + E^(x/4) + Log[9] + Log[x/3]^2 - 2*Log[x])), x]/4 + ((1 + Log[9])*Defe
r[Int][Log[5/x^2]/(-x - Log[5/x^2]*(1 + E^(x/4) + Log[9] + Log[x/3]^2 - 2*Log[x])), x])/4 + 2*Defer[Int][Log[5
/x^2]/(x*(-x - Log[5/x^2]*(1 + E^(x/4) + Log[9] + Log[x/3]^2 - 2*Log[x]))), x] + Defer[Int][(Log[5/x^2]*Log[x/
3]^2)/(-x - Log[5/x^2]*(1 + E^(x/4) + Log[9] + Log[x/3]^2 - 2*Log[x])), x]/4 + Defer[Int][(x + Log[5/x^2]*(1 +
E^(x/4) + Log[9] + Log[x/3]^2 - 2*Log[x]))^(-1), x] + 2*Defer[Int][1/(Log[5/x^2]*(x + Log[5/x^2]*(1 + E^(x/4)
+ Log[9] + Log[x/3]^2 - 2*Log[x]))), x] + 2*Defer[Int][(Log[5/x^2]*Log[x/3])/(x*(x + Log[5/x^2]*(1 + E^(x/4)
+ Log[9] + Log[x/3]^2 - 2*Log[x]))), x] + Defer[Int][(Log[5/x^2]*Log[x])/(x + Log[5/x^2]*(1 + E^(x/4) + Log[9]
+ Log[x/3]^2 - 2*Log[x])), x]/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 x+4 x \log \left (\frac {5}{x^2}\right )+\log ^2\left (\frac {5}{x^2}\right ) \left (-8+e^{x/4} x+8 \log \left (\frac {x}{3}\right )\right )}{4 x \log \left (\frac {5}{x^2}\right ) \left (x+\log \left (\frac {5}{x^2}\right ) \left (1+e^{x/4}+\log (9)+\log ^2\left (\frac {x}{3}\right )-2 \log (x)\right )\right )} \, dx\\ &=\frac {1}{4} \int \frac {8 x+4 x \log \left (\frac {5}{x^2}\right )+\log ^2\left (\frac {5}{x^2}\right ) \left (-8+e^{x/4} x+8 \log \left (\frac {x}{3}\right )\right )}{x \log \left (\frac {5}{x^2}\right ) \left (x+\log \left (\frac {5}{x^2}\right ) \left (1+e^{x/4}+\log (9)+\log ^2\left (\frac {x}{3}\right )-2 \log (x)\right )\right )} \, dx\\ &=\frac {1}{4} \int \left (1+\frac {8 x+4 x \log \left (\frac {5}{x^2}\right )-x^2 \log \left (\frac {5}{x^2}\right )-8 \log ^2\left (\frac {5}{x^2}\right )-x (1+\log (9)) \log ^2\left (\frac {5}{x^2}\right )+8 \log ^2\left (\frac {5}{x^2}\right ) \log \left (\frac {x}{3}\right )-x \log ^2\left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )+2 x \log ^2\left (\frac {5}{x^2}\right ) \log (x)}{x \log \left (\frac {5}{x^2}\right ) \left (x+e^{x/4} \log \left (\frac {5}{x^2}\right )+(1+\log (9)) \log \left (\frac {5}{x^2}\right )+\log \left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )-2 \log \left (\frac {5}{x^2}\right ) \log (x)\right )}\right ) \, dx\\ &=\frac {x}{4}+\frac {1}{4} \int \frac {8 x+4 x \log \left (\frac {5}{x^2}\right )-x^2 \log \left (\frac {5}{x^2}\right )-8 \log ^2\left (\frac {5}{x^2}\right )-x (1+\log (9)) \log ^2\left (\frac {5}{x^2}\right )+8 \log ^2\left (\frac {5}{x^2}\right ) \log \left (\frac {x}{3}\right )-x \log ^2\left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )+2 x \log ^2\left (\frac {5}{x^2}\right ) \log (x)}{x \log \left (\frac {5}{x^2}\right ) \left (x+e^{x/4} \log \left (\frac {5}{x^2}\right )+(1+\log (9)) \log \left (\frac {5}{x^2}\right )+\log \left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )-2 \log \left (\frac {5}{x^2}\right ) \log (x)\right )} \, dx\\ &=\frac {x}{4}+\frac {1}{4} \int \frac {8 x-(-4+x) x \log \left (\frac {5}{x^2}\right )-\log ^2\left (\frac {5}{x^2}\right ) \left (8+x+x \log (9)-8 \log \left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right )-2 x \log (x)\right )}{x \log \left (\frac {5}{x^2}\right ) \left (x+\log \left (\frac {5}{x^2}\right ) \left (1+e^{x/4}+\log (9)+\log ^2\left (\frac {x}{3}\right )-2 \log (x)\right )\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.48, size = 45, normalized size = 1.55 \begin {gather*} -\log \left (\log \left (\frac {5}{x^2}\right )\right )+\log \left (4 \left (x+\log \left (\frac {5}{x^2}\right ) \left (1+e^{x/4}+\log (9)+\log ^2\left (\frac {x}{3}\right )-2 \log (x)\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8*x + 4*x*Log[5/x^2] + (-8 + E^(x/4)*x)*Log[5/x^2]^2 + 8*Log[5/x^2]^2*Log[x/3])/(4*x^2*Log[5/x^2] +
(4*x + 4*E^(x/4)*x)*Log[5/x^2]^2 - 8*x*Log[5/x^2]^2*Log[x/3] + 4*x*Log[5/x^2]^2*Log[x/3]^2),x]

[Out]

-Log[Log[5/x^2]] + Log[4*(x + Log[5/x^2]*(1 + E^(x/4) + Log[9] + Log[x/3]^2 - 2*Log[x]))]

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fricas [B]  time = 0.66, size = 60, normalized size = 2.07 \begin {gather*} \log \left (-2 \, {\left (\log \left (\frac {5}{9}\right ) - 2\right )} \log \left (\frac {5}{x^{2}}\right )^{2} + \log \left (\frac {5}{x^{2}}\right )^{3} + {\left (\log \left (\frac {5}{9}\right )^{2} + 4 \, e^{\left (\frac {1}{4} \, x\right )} - 4 \, \log \left (\frac {5}{9}\right ) + 4\right )} \log \left (\frac {5}{x^{2}}\right ) + 4 \, x\right ) - \log \left (\log \left (\frac {5}{x^{2}}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*log(5/x^2)^2*log(1/3*x)+(x*exp(1/4*x)-8)*log(5/x^2)^2+4*x*log(5/x^2)+8*x)/(4*x*log(5/x^2)^2*log(1
/3*x)^2-8*x*log(5/x^2)^2*log(1/3*x)+(4*x*exp(1/4*x)+4*x)*log(5/x^2)^2+4*x^2*log(5/x^2)),x, algorithm="fricas")

[Out]

log(-2*(log(5/9) - 2)*log(5/x^2)^2 + log(5/x^2)^3 + (log(5/9)^2 + 4*e^(1/4*x) - 4*log(5/9) + 4)*log(5/x^2) + 4
*x) - log(log(5/x^2))

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giac [B]  time = 1.18, size = 103, normalized size = 3.55 \begin {gather*} \log \left (\log \relax (5) \log \relax (3)^{2} - 2 \, \log \relax (5) \log \relax (3) \log \relax (x) - 2 \, \log \relax (3)^{2} \log \relax (x) + \log \relax (5) \log \relax (x)^{2} + 4 \, \log \relax (3) \log \relax (x)^{2} - 2 \, \log \relax (x)^{3} + e^{\left (\frac {1}{4} \, x\right )} \log \relax (5) + 2 \, \log \relax (5) \log \relax (3) - 2 \, e^{\left (\frac {1}{4} \, x\right )} \log \relax (x) - 2 \, \log \relax (5) \log \relax (x) - 4 \, \log \relax (3) \log \relax (x) + 4 \, \log \relax (x)^{2} + x + \log \relax (5) - 2 \, \log \relax (x)\right ) - \log \left (\log \relax (5) - 2 \, \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*log(5/x^2)^2*log(1/3*x)+(x*exp(1/4*x)-8)*log(5/x^2)^2+4*x*log(5/x^2)+8*x)/(4*x*log(5/x^2)^2*log(1
/3*x)^2-8*x*log(5/x^2)^2*log(1/3*x)+(4*x*exp(1/4*x)+4*x)*log(5/x^2)^2+4*x^2*log(5/x^2)),x, algorithm="giac")

[Out]

log(log(5)*log(3)^2 - 2*log(5)*log(3)*log(x) - 2*log(3)^2*log(x) + log(5)*log(x)^2 + 4*log(3)*log(x)^2 - 2*log
(x)^3 + e^(1/4*x)*log(5) + 2*log(5)*log(3) - 2*e^(1/4*x)*log(x) - 2*log(5)*log(x) - 4*log(3)*log(x) + 4*log(x)
^2 + x + log(5) - 2*log(x)) - log(log(5) - 2*log(x))

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {8 \ln \left (\frac {5}{x^{2}}\right )^{2} \ln \left (\frac {x}{3}\right )+\left (x \,{\mathrm e}^{\frac {x}{4}}-8\right ) \ln \left (\frac {5}{x^{2}}\right )^{2}+4 x \ln \left (\frac {5}{x^{2}}\right )+8 x}{4 x \ln \left (\frac {5}{x^{2}}\right )^{2} \ln \left (\frac {x}{3}\right )^{2}-8 x \ln \left (\frac {5}{x^{2}}\right )^{2} \ln \left (\frac {x}{3}\right )+\left (4 x \,{\mathrm e}^{\frac {x}{4}}+4 x \right ) \ln \left (\frac {5}{x^{2}}\right )^{2}+4 x^{2} \ln \left (\frac {5}{x^{2}}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*ln(5/x^2)^2*ln(1/3*x)+(x*exp(1/4*x)-8)*ln(5/x^2)^2+4*x*ln(5/x^2)+8*x)/(4*x*ln(5/x^2)^2*ln(1/3*x)^2-8*x*
ln(5/x^2)^2*ln(1/3*x)+(4*x*exp(1/4*x)+4*x)*ln(5/x^2)^2+4*x^2*ln(5/x^2)),x)

[Out]

int((8*ln(5/x^2)^2*ln(1/3*x)+(x*exp(1/4*x)-8)*ln(5/x^2)^2+4*x*ln(5/x^2)+8*x)/(4*x*ln(5/x^2)^2*ln(1/3*x)^2-8*x*
ln(5/x^2)^2*ln(1/3*x)+(4*x*exp(1/4*x)+4*x)*ln(5/x^2)^2+4*x^2*ln(5/x^2)),x)

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maxima [B]  time = 0.66, size = 78, normalized size = 2.69 \begin {gather*} \log \left (\frac {\log \relax (5) \log \relax (3)^{2} + {\left (\log \relax (5) + 4 \, \log \relax (3) + 4\right )} \log \relax (x)^{2} - 2 \, \log \relax (x)^{3} + {\left (\log \relax (5) - 2 \, \log \relax (x)\right )} e^{\left (\frac {1}{4} \, x\right )} + 2 \, \log \relax (5) \log \relax (3) - 2 \, {\left ({\left (\log \relax (5) + 2\right )} \log \relax (3) + \log \relax (3)^{2} + \log \relax (5) + 1\right )} \log \relax (x) + x + \log \relax (5)}{\log \relax (5) - 2 \, \log \relax (x)}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*log(5/x^2)^2*log(1/3*x)+(x*exp(1/4*x)-8)*log(5/x^2)^2+4*x*log(5/x^2)+8*x)/(4*x*log(5/x^2)^2*log(1
/3*x)^2-8*x*log(5/x^2)^2*log(1/3*x)+(4*x*exp(1/4*x)+4*x)*log(5/x^2)^2+4*x^2*log(5/x^2)),x, algorithm="maxima")

[Out]

log((log(5)*log(3)^2 + (log(5) + 4*log(3) + 4)*log(x)^2 - 2*log(x)^3 + (log(5) - 2*log(x))*e^(1/4*x) + 2*log(5
)*log(3) - 2*((log(5) + 2)*log(3) + log(3)^2 + log(5) + 1)*log(x) + x + log(5))/(log(5) - 2*log(x)))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {8\,x+4\,x\,\ln \left (\frac {5}{x^2}\right )+8\,\ln \left (\frac {x}{3}\right )\,{\ln \left (\frac {5}{x^2}\right )}^2+{\ln \left (\frac {5}{x^2}\right )}^2\,\left (x\,{\mathrm {e}}^{x/4}-8\right )}{{\ln \left (\frac {5}{x^2}\right )}^2\,\left (4\,x+4\,x\,{\mathrm {e}}^{x/4}\right )+4\,x^2\,\ln \left (\frac {5}{x^2}\right )-8\,x\,\ln \left (\frac {x}{3}\right )\,{\ln \left (\frac {5}{x^2}\right )}^2+4\,x\,{\ln \left (\frac {x}{3}\right )}^2\,{\ln \left (\frac {5}{x^2}\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x + 4*x*log(5/x^2) + 8*log(x/3)*log(5/x^2)^2 + log(5/x^2)^2*(x*exp(x/4) - 8))/(log(5/x^2)^2*(4*x + 4*x*
exp(x/4)) + 4*x^2*log(5/x^2) - 8*x*log(x/3)*log(5/x^2)^2 + 4*x*log(x/3)^2*log(5/x^2)^2),x)

[Out]

int((8*x + 4*x*log(5/x^2) + 8*log(x/3)*log(5/x^2)^2 + log(5/x^2)^2*(x*exp(x/4) - 8))/(log(5/x^2)^2*(4*x + 4*x*
exp(x/4)) + 4*x^2*log(5/x^2) - 8*x*log(x/3)*log(5/x^2)^2 + 4*x*log(x/3)^2*log(5/x^2)^2), x)

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sympy [B]  time = 6.10, size = 97, normalized size = 3.34 \begin {gather*} \log {\left (e^{\frac {x}{4}} + \frac {- x + 2 \log {\left (\frac {x}{3} \right )}^{3} - 4 \log {\left (\frac {x}{3} \right )}^{2} - \log {\relax (5 )} \log {\left (\frac {x}{3} \right )}^{2} + 2 \log {\relax (3 )} \log {\left (\frac {x}{3} \right )}^{2} - 4 \log {\relax (3 )} \log {\left (\frac {x}{3} \right )} + 2 \log {\left (\frac {x}{3} \right )} + 2 \log {\relax (5 )} \log {\left (\frac {x}{3} \right )} - \log {\relax (5 )} + 2 \log {\relax (3 )}}{2 \log {\left (\frac {x}{3} \right )} - \log {\relax (5 )} + 2 \log {\relax (3 )}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*ln(5/x**2)**2*ln(1/3*x)+(x*exp(1/4*x)-8)*ln(5/x**2)**2+4*x*ln(5/x**2)+8*x)/(4*x*ln(5/x**2)**2*ln(
1/3*x)**2-8*x*ln(5/x**2)**2*ln(1/3*x)+(4*x*exp(1/4*x)+4*x)*ln(5/x**2)**2+4*x**2*ln(5/x**2)),x)

[Out]

log(exp(x/4) + (-x + 2*log(x/3)**3 - 4*log(x/3)**2 - log(5)*log(x/3)**2 + 2*log(3)*log(x/3)**2 - 4*log(3)*log(
x/3) + 2*log(x/3) + 2*log(5)*log(x/3) - log(5) + 2*log(3))/(2*log(x/3) - log(5) + 2*log(3)))

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