∫ 50 log2(3)+e8x log(−x 2 ) log3(5 log(−x 2 )) ________________________________________________________________________________________−25xlog 2 (3) log(−x 2 ) log(5 log(−x 2 ))+e8x2 log(−x 2 ) log3(5 log(−x 2 )) dx

Optimal. Leaf size=25 \[ \log \left (-x+\frac {25 \log ^2(3)}{e^8 \log ^2\left (5 \log \left (-\frac {x}{2}\right )\right )}\right ) \]

________________________________________________________________________________________

Rubi [F] time = 3.97, 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 {50 \log ^2(3)+e^8 x \log \left (-\frac {x}{2}\right ) \log ^3\left (5 \log \left (-\frac {x}{2}\right )\right )}{-25 x \log ^2(3) \log \left (-\frac {x}{2}\right ) \log \left (5 \log \left (-\frac {x}{2}\right )\right )+e^8 x^2 \log \left (-\frac {x}{2}\right ) \log ^3\left (5 \log \left (-\frac {x}{2}\right )\right )} \, dx \end {gather*}

Int[(50*Log[3]^2 + E^8*x*Log[-1/2*x]*Log[5*Log[-1/2*x]]^3)/(-25*x*Log[3]^2*Log[-1/2*x]*Log[5*Log[-1/2*x]] + E^

Log[x] - 2*Log[Log[5*Log[-1/2*x]]] - 2*E^8*Defer[Int][Log[5*Log[-1/2*x]]/(Log[-1/2*x]*(25*Log[3]^2 - E^8*x*Log

[5*Log[-1/2*x]]^2)), x] + 25*Log[3]^2*Defer[Int][1/(x*(-25*Log[3]^2 + E^8*x*Log[5*Log[-1/2*x]]^2)), x]

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-50 \log ^2(3)-e^8 x \log \left (-\frac {x}{2}\right ) \log ^3\left (5 \log \left (-\frac {x}{2}\right )\right )}{x \log \left (-\frac {x}{2}\right ) \log \left (5 \log \left (-\frac {x}{2}\right )\right ) \left (25 \log ^2(3)-e^8 x \log ^2\left (5 \log \left (-\frac {x}{2}\right )\right )\right )} \, dx\\ &=\int \left (\frac {1}{x}-\frac {2}{x \log \left (-\frac {x}{2}\right ) \log \left (5 \log \left (-\frac {x}{2}\right )\right )}+\frac {25 \log ^2(3) \log \left (-\frac {x}{2}\right )+2 e^8 x \log \left (5 \log \left (-\frac {x}{2}\right )\right )}{x \log \left (-\frac {x}{2}\right ) \left (-25 \log ^2(3)+e^8 x \log ^2\left (5 \log \left (-\frac {x}{2}\right )\right )\right )}\right ) \, dx\\ &=\log (x)-2 \int \frac {1}{x \log \left (-\frac {x}{2}\right ) \log \left (5 \log \left (-\frac {x}{2}\right )\right )} \, dx+\int \frac {25 \log ^2(3) \log \left (-\frac {x}{2}\right )+2 e^8 x \log \left (5 \log \left (-\frac {x}{2}\right )\right )}{x \log \left (-\frac {x}{2}\right ) \left (-25 \log ^2(3)+e^8 x \log ^2\left (5 \log \left (-\frac {x}{2}\right )\right )\right )} \, dx\\ &=\log (x)-2 \operatorname {Subst}\left (\int \frac {1}{x \log (5 x)} \, dx,x,\log \left (-\frac {x}{2}\right )\right )+\int \left (-\frac {2 e^8 \log \left (5 \log \left (-\frac {x}{2}\right )\right )}{\log \left (-\frac {x}{2}\right ) \left (25 \log ^2(3)-e^8 x \log ^2\left (5 \log \left (-\frac {x}{2}\right )\right )\right )}+\frac {25 \log ^2(3)}{x \left (-25 \log ^2(3)+e^8 x \log ^2\left (5 \log \left (-\frac {x}{2}\right )\right )\right )}\right ) \, dx\\ &=\log (x)-2 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (5 \log \left (-\frac {x}{2}\right )\right )\right )-\left (2 e^8\right ) \int \frac {\log \left (5 \log \left (-\frac {x}{2}\right )\right )}{\log \left (-\frac {x}{2}\right ) \left (25 \log ^2(3)-e^8 x \log ^2\left (5 \log \left (-\frac {x}{2}\right )\right )\right )} \, dx+\left (25 \log ^2(3)\right ) \int \frac {1}{x \left (-25 \log ^2(3)+e^8 x \log ^2\left (5 \log \left (-\frac {x}{2}\right )\right )\right )} \, dx\\ &=\log (x)-2 \log \left (\log \left (5 \log \left (-\frac {x}{2}\right )\right )\right )-\left (2 e^8\right ) \int \frac {\log \left (5 \log \left (-\frac {x}{2}\right )\right )}{\log \left (-\frac {x}{2}\right ) \left (25 \log ^2(3)-e^8 x \log ^2\left (5 \log \left (-\frac {x}{2}\right )\right )\right )} \, dx+\left (25 \log ^2(3)\right ) \int \frac {1}{x \left (-25 \log ^2(3)+e^8 x \log ^2\left (5 \log \left (-\frac {x}{2}\right )\right )\right )} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A] time = 0.40, size = 38, normalized size = 1.52 \begin {gather*} -2 \log \left (\log \left (5 \log \left (-\frac {x}{2}\right )\right )\right )+\log \left (25 \log ^2(3)-e^8 x \log ^2\left (5 \log \left (-\frac {x}{2}\right )\right )\right ) \end {gather*}

Integrate[(50*Log[3]^2 + E^8*x*Log[-1/2*x]*Log[5*Log[-1/2*x]]^3)/(-25*x*Log[3]^2*Log[-1/2*x]*Log[5*Log[-1/2*x]

-2*Log[Log[5*Log[-1/2*x]]] + Log[25*Log[3]^2 - E^8*x*Log[5*Log[-1/2*x]]^2]

________________________________________________________________________________________

fricas [A] time = 0.70, size = 40, normalized size = 1.60 \begin {gather*} \log \left (-\frac {1}{2} \, x\right ) + \log \left (\frac {x e^{8} \log \left (5 \, \log \left (-\frac {1}{2} \, x\right )\right )^{2} - 25 \, \log \relax (3)^{2}}{x}\right ) - 2 \, \log \left (\log \left (5 \, \log \left (-\frac {1}{2} \, x\right )\right )\right ) \end {gather*}

integrate((x*exp(4)^2*log(-1/2*x)*log(5*log(-1/2*x))^3+50*log(3)^2)/(x^2*exp(4)^2*log(-1/2*x)*log(5*log(-1/2*x

))^3-25*x*log(3)^2*log(-1/2*x)*log(5*log(-1/2*x))),x, algorithm="fricas")

log(-1/2*x) + log((x*e^8*log(5*log(-1/2*x))^2 - 25*log(3)^2)/x) - 2*log(log(5*log(-1/2*x)))

________________________________________________________________________________________

giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x e^{8} \log \left (-\frac {1}{2} \, x\right ) \log \left (5 \, \log \left (-\frac {1}{2} \, x\right )\right )^{3} + 50 \, \log \relax (3)^{2}}{x^{2} e^{8} \log \left (-\frac {1}{2} \, x\right ) \log \left (5 \, \log \left (-\frac {1}{2} \, x\right )\right )^{3} - 25 \, x \log \relax (3)^{2} \log \left (-\frac {1}{2} \, x\right ) \log \left (5 \, \log \left (-\frac {1}{2} \, x\right )\right )}\,{d x} \end {gather*}

integrate((x*exp(4)^2*log(-1/2*x)*log(5*log(-1/2*x))^3+50*log(3)^2)/(x^2*exp(4)^2*log(-1/2*x)*log(5*log(-1/2*x

))^3-25*x*log(3)^2*log(-1/2*x)*log(5*log(-1/2*x))),x, algorithm="giac")

integrate((x*e^8*log(-1/2*x)*log(5*log(-1/2*x))^3 + 50*log(3)^2)/(x^2*e^8*log(-1/2*x)*log(5*log(-1/2*x))^3 - 2

________________________________________________________________________________________


method	result	size

risch	\(\ln \relax (x )-2 \ln \left (\ln \left (5 \ln \left (-\frac {x}{2}\right )\right )\right )+\ln \left (\ln \left (5 \ln \left (-\frac {x}{2}\right )\right )^{2}-\frac {25 \ln \relax (3)^{2} {\mathrm e}^{-8}}{x}\right )\)	\(36\)

int((x*exp(4)^2*ln(-1/2*x)*ln(5*ln(-1/2*x))^3+50*ln(3)^2)/(x^2*exp(4)^2*ln(-1/2*x)*ln(5*ln(-1/2*x))^3-25*x*ln(

3)^2*ln(-1/2*x)*ln(5*ln(-1/2*x))),x,method=_RETURNVERBOSE)

ln(x)-2*ln(ln(5*ln(-1/2*x)))+ln(ln(5*ln(-1/2*x))^2-25*ln(3)^2*exp(-8)/x)

________________________________________________________________________________________

maxima [B] time = 0.52, size = 74, normalized size = 2.96 \begin {gather*} \log \relax (x) + \log \left (\frac {{\left (x e^{8} \log \relax (5)^{2} + 2 \, x e^{8} \log \relax (5) \log \left (-\log \relax (2) + \log \left (-x\right )\right ) + x e^{8} \log \left (-\log \relax (2) + \log \left (-x\right )\right )^{2} - 25 \, \log \relax (3)^{2}\right )} e^{\left (-8\right )}}{x}\right ) - 2 \, \log \left (\log \relax (5) + \log \left (-\log \relax (2) + \log \left (-x\right )\right )\right ) \end {gather*}

integrate((x*exp(4)^2*log(-1/2*x)*log(5*log(-1/2*x))^3+50*log(3)^2)/(x^2*exp(4)^2*log(-1/2*x)*log(5*log(-1/2*x

))^3-25*x*log(3)^2*log(-1/2*x)*log(5*log(-1/2*x))),x, algorithm="maxima")

log(x) + log((x*e^8*log(5)^2 + 2*x*e^8*log(5)*log(-log(2) + log(-x)) + x*e^8*log(-log(2) + log(-x))^2 - 25*log

(3)^2)*e^(-8)/x) - 2*log(log(5) + log(-log(2) + log(-x)))

________________________________________________________________________________________

mupad [B] time = 1.63, size = 138, normalized size = 5.52 \begin {gather*} 2\,\ln \left (\frac {4\,x\,{\mathrm {e}}^8-25\,{\ln \left (-\frac {x}{2}\right )}^2\,{\ln \relax (3)}^2}{x}\right )+\ln \left (16\,x\,\ln \left (-\frac {x}{2}\right )\,{\mathrm {e}}^{16}-400\,\ln \left (-\frac {x}{2}\right )\,{\mathrm {e}}^8\,{\ln \relax (3)}^2\right )-2\,\ln \left (5625\,\ln \left (5\,\ln \left (-\frac {x}{2}\right )\right )\,{\ln \left (-\frac {x}{2}\right )}^2\,{\ln \relax (3)}^4-900\,x\,\ln \left (5\,\ln \left (-\frac {x}{2}\right )\right )\,{\mathrm {e}}^8\,{\ln \relax (3)}^2\right )-\ln \left (x-25\,{\mathrm {e}}^{-8}\,{\ln \relax (3)}^2\right )+4\,\ln \relax (x)+\ln \left (\frac {25\,{\ln \relax (3)}^2-x\,{\mathrm {e}}^8\,{\left (\ln \relax (5)+\ln \left (\ln \left (-\frac {x}{2}\right )\right )\right )}^2}{x^2\,\ln \left (-\frac {x}{2}\right )}\right ) \end {gather*}

int((50*log(3)^2 + x*log(5*log(-x/2))^3*log(-x/2)*exp(8))/(x^2*log(5*log(-x/2))^3*log(-x/2)*exp(8) - 25*x*log(

2*log((4*x*exp(8) - 25*log(-x/2)^2*log(3)^2)/x) + log(16*x*log(-x/2)*exp(16) - 400*log(-x/2)*exp(8)*log(3)^2)

- 2*log(5625*log(5*log(-x/2))*log(-x/2)^2*log(3)^4 - 900*x*log(5*log(-x/2))*exp(8)*log(3)^2) - log(x - 25*exp(

-8)*log(3)^2) + 4*log(x) + log((25*log(3)^2 - x*exp(8)*(log(5) + log(log(-x/2)))^2)/(x^2*log(-x/2)))

________________________________________________________________________________________

sympy [A] time = 0.51, size = 41, normalized size = 1.64 \begin {gather*} \log {\relax (x )} + \log {\left (\log {\left (5 \log {\left (- \frac {x}{2} \right )} \right )}^{2} - \frac {25 \log {\relax (3 )}^{2}}{x e^{8}} \right )} - 2 \log {\left (\log {\left (5 \log {\left (- \frac {x}{2} \right )} \right )} \right )} \end {gather*}

integrate((x*exp(4)**2*ln(-1/2*x)*ln(5*ln(-1/2*x))**3+50*ln(3)**2)/(x**2*exp(4)**2*ln(-1/2*x)*ln(5*ln(-1/2*x))

log(x) + log(log(5*log(-x/2))**2 - 25*exp(-8)*log(3)**2/x) - 2*log(log(5*log(-x/2)))

________________________________________________________________________________________

3.14.17 \(\int \frac {50 \log ^2(3)+e^8 x \log (-\frac {x}{2}) \log ^3(5 \log (-\frac {x}{2}))}{-25 x \log ^2(3) \log (-\frac {x}{2}) \log (5 \log (-\frac {x}{2}))+e^8 x^2 \log (-\frac {x}{2}) \log ^3(5 \log (-\frac {x}{2}))} \, dx\)