\(\int \frac {(2-2 e^3-2 \log (x)) \log (\frac {2 e^7+x+2 e^4 \log (x)}{e^3+\log (x)})}{2 e^{10}+e^3 x+(4 e^7+x) \log (x)+2 e^4 \log ^2(x)} \, dx\) [7980]

Optimal result

Integrand size = 68, antiderivative size = 23 \[ \int \frac {\left (2-2 e^3-2 \log (x)\right ) \log \left (\frac {2 e^7+x+2 e^4 \log (x)}{e^3+\log (x)}\right )}{2 e^{10}+e^3 x+\left (4 e^7+x\right ) \log (x)+2 e^4 \log ^2(x)} \, dx=1-\log ^2\left (2 e^4+\frac {x}{e^3+\log (x)}\right ) \]

[Out]

1-ln(2*exp(4)+x/(ln(x)+exp(3)))^2

Rubi [F]

\[ \int \frac {\left (2-2 e^3-2 \log (x)\right ) \log \left (\frac {2 e^7+x+2 e^4 \log (x)}{e^3+\log (x)}\right )}{2 e^{10}+e^3 x+\left (4 e^7+x\right ) \log (x)+2 e^4 \log ^2(x)} \, dx=\int \frac {\left (2-2 e^3-2 \log (x)\right ) \log \left (\frac {2 e^7+x+2 e^4 \log (x)}{e^3+\log (x)}\right )}{2 e^{10}+e^3 x+\left (4 e^7+x\right ) \log (x)+2 e^4 \log ^2(x)} \, dx \]

[In]

Int[((2 - 2*E^3 - 2*Log[x])*Log[(2*E^7 + x + 2*E^4*Log[x])/(E^3 + Log[x])])/(2*E^10 + E^3*x + (4*E^7 + x)*Log[
x] + 2*E^4*Log[x]^2),x]

[Out]

2*(1 - E^3)*Defer[Int][Log[(2*E^7 + x + 2*E^4*Log[x])/(E^3 + Log[x])]/((E^3 + Log[x])*(2*E^7 + x + 2*E^4*Log[x
])), x] - 2*Defer[Int][(Log[x]*Log[(2*E^7 + x + 2*E^4*Log[x])/(E^3 + Log[x])])/((E^3 + Log[x])*(2*E^7 + x + 2*
E^4*Log[x])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \left (1-e^3-\log (x)\right ) \log \left (\frac {2 e^7+x+2 e^4 \log (x)}{e^3+\log (x)}\right )}{\left (e^3+\log (x)\right ) \left (2 e^7+x+2 e^4 \log (x)\right )} \, dx \\ & = 2 \int \frac {\left (1-e^3-\log (x)\right ) \log \left (\frac {2 e^7+x+2 e^4 \log (x)}{e^3+\log (x)}\right )}{\left (e^3+\log (x)\right ) \left (2 e^7+x+2 e^4 \log (x)\right )} \, dx \\ & = 2 \int \left (\frac {\left (1-e^3\right ) \log \left (\frac {2 e^7+x+2 e^4 \log (x)}{e^3+\log (x)}\right )}{\left (e^3+\log (x)\right ) \left (2 e^7+x+2 e^4 \log (x)\right )}-\frac {\log (x) \log \left (\frac {2 e^7+x+2 e^4 \log (x)}{e^3+\log (x)}\right )}{\left (e^3+\log (x)\right ) \left (2 e^7+x+2 e^4 \log (x)\right )}\right ) \, dx \\ & = -\left (2 \int \frac {\log (x) \log \left (\frac {2 e^7+x+2 e^4 \log (x)}{e^3+\log (x)}\right )}{\left (e^3+\log (x)\right ) \left (2 e^7+x+2 e^4 \log (x)\right )} \, dx\right )+\left (2 \left (1-e^3\right )\right ) \int \frac {\log \left (\frac {2 e^7+x+2 e^4 \log (x)}{e^3+\log (x)}\right )}{\left (e^3+\log (x)\right ) \left (2 e^7+x+2 e^4 \log (x)\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {\left (2-2 e^3-2 \log (x)\right ) \log \left (\frac {2 e^7+x+2 e^4 \log (x)}{e^3+\log (x)}\right )}{2 e^{10}+e^3 x+\left (4 e^7+x\right ) \log (x)+2 e^4 \log ^2(x)} \, dx=-\log ^2\left (\frac {2 e^7+x+2 e^4 \log (x)}{e^3+\log (x)}\right ) \]

[In]

Integrate[((2 - 2*E^3 - 2*Log[x])*Log[(2*E^7 + x + 2*E^4*Log[x])/(E^3 + Log[x])])/(2*E^10 + E^3*x + (4*E^7 + x
)*Log[x] + 2*E^4*Log[x]^2),x]

[Out]

-Log[(2*E^7 + x + 2*E^4*Log[x])/(E^3 + Log[x])]^2

Maple [A] (verified)

Time = 1.73 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22

method	result	size
default	\(-\ln \left (\frac {2 \,{\mathrm e}^{4} \ln \left (x \right )+2 \,{\mathrm e}^{3} {\mathrm e}^{4}+x}{\ln \left (x \right )+{\mathrm e}^{3}}\right )^{2}\)	\(28\)
norman	\(-\ln \left (\frac {2 \,{\mathrm e}^{4} \ln \left (x \right )+2 \,{\mathrm e}^{3} {\mathrm e}^{4}+x}{\ln \left (x \right )+{\mathrm e}^{3}}\right )^{2}\)	\(28\)
risch	\(-\ln \left ({\mathrm e}^{4} \ln \left (x \right )+{\mathrm e}^{7}+\frac {x}{2}\right )^{2}+2 \ln \left (\ln \left (x \right )+{\mathrm e}^{3}\right ) \ln \left ({\mathrm e}^{4} \ln \left (x \right )+{\mathrm e}^{7}+\frac {x}{2}\right )-\ln \left (\ln \left (x \right )+{\mathrm e}^{3}\right )^{2}-i \pi \ln \left (\ln \left (x \right )+{\mathrm e}^{3}+\frac {x \,{\mathrm e}^{-4}}{2}\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{4} \ln \left (x \right )+{\mathrm e}^{7}+\frac {x}{2}\right )\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4} \ln \left (x \right )+{\mathrm e}^{7}+\frac {x}{2}\right )}{\ln \left (x \right )+{\mathrm e}^{3}}\right )}^{2}+i \pi \ln \left (\ln \left (x \right )+{\mathrm e}^{3}+\frac {x \,{\mathrm e}^{-4}}{2}\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4} \ln \left (x \right )+{\mathrm e}^{7}+\frac {x}{2}\right )}{\ln \left (x \right )+{\mathrm e}^{3}}\right )}^{3}+i \pi \ln \left (\ln \left (x \right )+{\mathrm e}^{3}\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )+{\mathrm e}^{3}}\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4} \ln \left (x \right )+{\mathrm e}^{7}+\frac {x}{2}\right )}{\ln \left (x \right )+{\mathrm e}^{3}}\right )}^{2}-i \pi \ln \left (\ln \left (x \right )+{\mathrm e}^{3}+\frac {x \,{\mathrm e}^{-4}}{2}\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )+{\mathrm e}^{3}}\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4} \ln \left (x \right )+{\mathrm e}^{7}+\frac {x}{2}\right )}{\ln \left (x \right )+{\mathrm e}^{3}}\right )}^{2}-i \pi \ln \left (\ln \left (x \right )+{\mathrm e}^{3}\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{4} \ln \left (x \right )+{\mathrm e}^{7}+\frac {x}{2}\right )\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )+{\mathrm e}^{3}}\right ) \operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4} \ln \left (x \right )+{\mathrm e}^{7}+\frac {x}{2}\right )}{\ln \left (x \right )+{\mathrm e}^{3}}\right )-i \pi \ln \left (\ln \left (x \right )+{\mathrm e}^{3}\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4} \ln \left (x \right )+{\mathrm e}^{7}+\frac {x}{2}\right )}{\ln \left (x \right )+{\mathrm e}^{3}}\right )}^{3}+i \pi \ln \left (\ln \left (x \right )+{\mathrm e}^{3}+\frac {x \,{\mathrm e}^{-4}}{2}\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{4} \ln \left (x \right )+{\mathrm e}^{7}+\frac {x}{2}\right )\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )+{\mathrm e}^{3}}\right ) \operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4} \ln \left (x \right )+{\mathrm e}^{7}+\frac {x}{2}\right )}{\ln \left (x \right )+{\mathrm e}^{3}}\right )+i \pi \ln \left (\ln \left (x \right )+{\mathrm e}^{3}\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{4} \ln \left (x \right )+{\mathrm e}^{7}+\frac {x}{2}\right )\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4} \ln \left (x \right )+{\mathrm e}^{7}+\frac {x}{2}\right )}{\ln \left (x \right )+{\mathrm e}^{3}}\right )}^{2}+2 \ln \left (2\right ) \ln \left (\ln \left (x \right )+{\mathrm e}^{3}\right )-2 \ln \left (2\right ) \ln \left (\ln \left (x \right )+{\mathrm e}^{3}+\frac {x \,{\mathrm e}^{-4}}{2}\right )\)	\(465\)

[In]

int((-2*ln(x)-2*exp(3)+2)*ln((2*exp(4)*ln(x)+2*exp(3)*exp(4)+x)/(ln(x)+exp(3)))/(2*exp(4)*ln(x)^2+(4*exp(3)*ex
p(4)+x)*ln(x)+2*exp(3)^2*exp(4)+x*exp(3)),x,method=_RETURNVERBOSE)

[Out]

-ln((2*exp(4)*ln(x)+2*exp(3)*exp(4)+x)/(ln(x)+exp(3)))^2

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\left (2-2 e^3-2 \log (x)\right ) \log \left (\frac {2 e^7+x+2 e^4 \log (x)}{e^3+\log (x)}\right )}{2 e^{10}+e^3 x+\left (4 e^7+x\right ) \log (x)+2 e^4 \log ^2(x)} \, dx=-\log \left (\frac {2 \, e^{4} \log \left (x\right ) + x + 2 \, e^{7}}{e^{3} + \log \left (x\right )}\right )^{2} \]

[In]

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

[Out]

-log((2*e^4*log(x) + x + 2*e^7)/(e^3 + log(x)))^2

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {\left (2-2 e^3-2 \log (x)\right ) \log \left (\frac {2 e^7+x+2 e^4 \log (x)}{e^3+\log (x)}\right )}{2 e^{10}+e^3 x+\left (4 e^7+x\right ) \log (x)+2 e^4 \log ^2(x)} \, dx=- \log {\left (\frac {x + 2 e^{4} \log {\left (x \right )} + 2 e^{7}}{\log {\left (x \right )} + e^{3}} \right )}^{2} \]

[In]

integrate((-2*ln(x)-2*exp(3)+2)*ln((2*exp(4)*ln(x)+2*exp(3)*exp(4)+x)/(ln(x)+exp(3)))/(2*exp(4)*ln(x)**2+(4*ex
p(3)*exp(4)+x)*ln(x)+2*exp(3)**2*exp(4)+x*exp(3)),x)

[Out]

-log((x + 2*exp(4)*log(x) + 2*exp(7))/(log(x) + exp(3)))**2

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (21) = 42\).

Time = 0.29 (sec) , antiderivative size = 110, normalized size of antiderivative = 4.78 \[ \int \frac {\left (2-2 e^3-2 \log (x)\right ) \log \left (\frac {2 e^7+x+2 e^4 \log (x)}{e^3+\log (x)}\right )}{2 e^{10}+e^3 x+\left (4 e^7+x\right ) \log (x)+2 e^4 \log ^2(x)} \, dx=-2 \, {\left (\log \left (2\right ) + \log \left (e^{3} + \log \left (x\right )\right ) + 4\right )} \log \left (2 \, e^{4} \log \left (x\right ) + x + 2 \, e^{7}\right ) + \log \left (2 \, e^{4} \log \left (x\right ) + x + 2 \, e^{7}\right )^{2} - 2 \, {\left (\log \left (\frac {1}{2} \, {\left (2 \, e^{4} \log \left (x\right ) + x + 2 \, e^{7}\right )} e^{\left (-4\right )}\right ) - \log \left (e^{3} + \log \left (x\right )\right )\right )} \log \left (\frac {2 \, e^{4} \log \left (x\right ) + x + 2 \, e^{7}}{e^{3} + \log \left (x\right )}\right ) + 2 \, {\left (\log \left (2\right ) + 4\right )} \log \left (e^{3} + \log \left (x\right )\right ) + \log \left (e^{3} + \log \left (x\right )\right )^{2} \]

[In]

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

[Out]

-2*(log(2) + log(e^3 + log(x)) + 4)*log(2*e^4*log(x) + x + 2*e^7) + log(2*e^4*log(x) + x + 2*e^7)^2 - 2*(log(1
/2*(2*e^4*log(x) + x + 2*e^7)*e^(-4)) - log(e^3 + log(x)))*log((2*e^4*log(x) + x + 2*e^7)/(e^3 + log(x))) + 2*
(log(2) + 4)*log(e^3 + log(x)) + log(e^3 + log(x))^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (21) = 42\).

Time = 0.38 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.13 \[ \int \frac {\left (2-2 e^3-2 \log (x)\right ) \log \left (\frac {2 e^7+x+2 e^4 \log (x)}{e^3+\log (x)}\right )}{2 e^{10}+e^3 x+\left (4 e^7+x\right ) \log (x)+2 e^4 \log ^2(x)} \, dx=-\log \left (2 \, e^{4} \log \left (x\right ) + x + 2 \, e^{7}\right )^{2} + 2 \, \log \left (2 \, e^{4} \log \left (x\right ) + x + 2 \, e^{7}\right ) \log \left (e^{3} + \log \left (x\right )\right ) - \log \left (e^{3} + \log \left (x\right )\right )^{2} \]

[In]

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

[Out]

-log(2*e^4*log(x) + x + 2*e^7)^2 + 2*log(2*e^4*log(x) + x + 2*e^7)*log(e^3 + log(x)) - log(e^3 + log(x))^2

Mupad [B] (verification not implemented)

Time = 19.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\left (2-2 e^3-2 \log (x)\right ) \log \left (\frac {2 e^7+x+2 e^4 \log (x)}{e^3+\log (x)}\right )}{2 e^{10}+e^3 x+\left (4 e^7+x\right ) \log (x)+2 e^4 \log ^2(x)} \, dx=-{\ln \left (\frac {x+2\,{\mathrm {e}}^7+2\,{\mathrm {e}}^4\,\ln \left (x\right )}{{\mathrm {e}}^3+\ln \left (x\right )}\right )}^2 \]

[In]

int(-(log((x + 2*exp(7) + 2*exp(4)*log(x))/(exp(3) + log(x)))*(2*exp(3) + 2*log(x) - 2))/(2*exp(10) + 2*exp(4)
*log(x)^2 + x*exp(3) + log(x)*(x + 4*exp(7))),x)

[Out]

-log((x + 2*exp(7) + 2*exp(4)*log(x))/(exp(3) + log(x)))^2