\(\int \frac {-1-4 x^2-2 x^2 \log (4)}{x-2 e^7 x^2-2 x^3-2 x^3 \log (4)+2 x^2 \log (6 e^{-x})} \, dx\) [7909]

Optimal result

Integrand size = 52, antiderivative size = 30 \[ \int \frac {-1-4 x^2-2 x^2 \log (4)}{x-2 e^7 x^2-2 x^3-2 x^3 \log (4)+2 x^2 \log \left (6 e^{-x}\right )} \, dx=\log \left (x+\frac {e^7-\frac {1}{2 x}+x-\log \left (6 e^{-x}\right )}{\log (4)}\right ) \]

[Out]

ln(x+1/2*(x-1/2/x-ln(6/exp(x))+exp(7))/ln(2))

Rubi [F]

\[ \int \frac {-1-4 x^2-2 x^2 \log (4)}{x-2 e^7 x^2-2 x^3-2 x^3 \log (4)+2 x^2 \log \left (6 e^{-x}\right )} \, dx=\int \frac {-1-4 x^2-2 x^2 \log (4)}{x-2 e^7 x^2-2 x^3-2 x^3 \log (4)+2 x^2 \log \left (6 e^{-x}\right )} \, dx \]

[In]

Int[(-1 - 4*x^2 - 2*x^2*Log[4])/(x - 2*E^7*x^2 - 2*x^3 - 2*x^3*Log[4] + 2*x^2*Log[6/E^x]),x]

[Out]

Defer[Int][1/(x*(-1 + 2*E^7*x + 2*x^2*(1 + Log[4]) - 2*x*Log[6/E^x])), x] - 2*(2 + Log[4])*Defer[Int][x/(1 - 2
*E^7*x - 2*x^2*(1 + Log[4]) + 2*x*Log[6/E^x]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-1+x^2 (-4-2 \log (4))}{x-2 e^7 x^2-2 x^3-2 x^3 \log (4)+2 x^2 \log \left (6 e^{-x}\right )} \, dx \\ & = \int \frac {-1+x^2 (-4-2 \log (4))}{x-2 e^7 x^2+x^3 (-2-2 \log (4))+2 x^2 \log \left (6 e^{-x}\right )} \, dx \\ & = \int \left (\frac {1}{x \left (-1+2 e^7 x+2 x^2 (1+\log (4))-2 x \log \left (6 e^{-x}\right )\right )}+\frac {2 x (-2-\log (4))}{1-2 e^7 x-2 x^2 (1+\log (4))+2 x \log \left (6 e^{-x}\right )}\right ) \, dx \\ & = -\left ((2 (2+\log (4))) \int \frac {x}{1-2 e^7 x-2 x^2 (1+\log (4))+2 x \log \left (6 e^{-x}\right )} \, dx\right )+\int \frac {1}{x \left (-1+2 e^7 x+2 x^2 (1+\log (4))-2 x \log \left (6 e^{-x}\right )\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {-1-4 x^2-2 x^2 \log (4)}{x-2 e^7 x^2-2 x^3-2 x^3 \log (4)+2 x^2 \log \left (6 e^{-x}\right )} \, dx=-\log (x)+\log \left (1-2 e^7 x-2 x^2-x^2 \log (16)+2 x \log \left (6 e^{-x}\right )\right ) \]

[In]

Integrate[(-1 - 4*x^2 - 2*x^2*Log[4])/(x - 2*E^7*x^2 - 2*x^3 - 2*x^3*Log[4] + 2*x^2*Log[6/E^x]),x]

[Out]

-Log[x] + Log[1 - 2*E^7*x - 2*x^2 - x^2*Log[16] + 2*x*Log[6/E^x]]

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20

method	result	size
norman	\(-\ln \left (x \right )+\ln \left (4 x^{2} \ln \left (2\right )-2 x \ln \left (6 \,{\mathrm e}^{-x}\right )+2 x \,{\mathrm e}^{7}+2 x^{2}-1\right )\)	\(36\)
parallelrisch	\(-\ln \left (x \right )+\ln \left (\frac {4 x^{2} \ln \left (2\right )-2 x \ln \left (6 \,{\mathrm e}^{-x}\right )+2 x \,{\mathrm e}^{7}+2 x^{2}-1}{4 \ln \left (2\right )+2}\right )\)	\(46\)
risch	\(\ln \left (\ln \left ({\mathrm e}^{x}\right )+\frac {i \left (-4 i x^{2} \ln \left (2\right )+2 i x \ln \left (2\right )+2 i \ln \left (3\right ) x -2 i x \,{\mathrm e}^{7}-2 i x^{2}+i\right )}{2 x}\right )\)	\(47\)
default	\(-\ln \left (x \right )+\ln \left (4 x^{2} \ln \left (2\right )+2 x \,{\mathrm e}^{7}+4 x^{2}+2 x \left (\ln \left ({\mathrm e}^{x}\right )-x \right )-2 x \left (\ln \left (6 \,{\mathrm e}^{-x}\right )+\ln \left ({\mathrm e}^{x}\right )\right )-1\right )\)	\(50\)

[In]

int((-4*x^2*ln(2)-4*x^2-1)/(2*x^2*ln(6/exp(x))-4*x^3*ln(2)-2*x^2*exp(7)-2*x^3+x),x,method=_RETURNVERBOSE)

[Out]

-ln(x)+ln(4*x^2*ln(2)-2*x*ln(6/exp(x))+2*x*exp(7)+2*x^2-1)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {-1-4 x^2-2 x^2 \log (4)}{x-2 e^7 x^2-2 x^3-2 x^3 \log (4)+2 x^2 \log \left (6 e^{-x}\right )} \, dx=\log \left (4 \, x^{2} \log \left (2\right ) + 4 \, x^{2} + 2 \, x e^{7} - 2 \, x \log \left (6\right ) - 1\right ) - \log \left (x\right ) \]

[In]

integrate((-4*x^2*log(2)-4*x^2-1)/(2*x^2*log(6/exp(x))-4*x^3*log(2)-2*x^2*exp(7)-2*x^3+x),x, algorithm="fricas
")

[Out]

log(4*x^2*log(2) + 4*x^2 + 2*x*e^7 - 2*x*log(6) - 1) - log(x)

Sympy [A] (verification not implemented)

Time = 0.62 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {-1-4 x^2-2 x^2 \log (4)}{x-2 e^7 x^2-2 x^3-2 x^3 \log (4)+2 x^2 \log \left (6 e^{-x}\right )} \, dx=- \log {\left (x \right )} + \log {\left (x^{2} + \frac {x \left (- \log {\left (6 \right )} + e^{7}\right )}{2 \log {\left (2 \right )} + 2} - \frac {1}{4 \log {\left (2 \right )} + 4} \right )} \]

[In]

integrate((-4*x**2*ln(2)-4*x**2-1)/(2*x**2*ln(6/exp(x))-4*x**3*ln(2)-2*x**2*exp(7)-2*x**3+x),x)

[Out]

-log(x) + log(x**2 + x*(-log(6) + exp(7))/(2*log(2) + 2) - 1/(4*log(2) + 4))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 579 vs. \(2 (31) = 62\).

Time = 0.30 (sec) , antiderivative size = 579, normalized size of antiderivative = 19.30 \[ \int \frac {-1-4 x^2-2 x^2 \log (4)}{x-2 e^7 x^2-2 x^3-2 x^3 \log (4)+2 x^2 \log \left (6 e^{-x}\right )} \, dx=\text {Too large to display} \]

[In]

integrate((-4*x^2*log(2)-4*x^2-1)/(2*x^2*log(6/exp(x))-4*x^3*log(2)-2*x^2*exp(7)-2*x^3+x),x, algorithm="maxima
")

[Out]

-1/2*((e^7 - log(3) - log(2))*log((4*x*(log(2) + 1) - sqrt(-2*(e^7 - log(2))*log(3) + log(3)^2 - 2*(e^7 - 2)*l
og(2) + log(2)^2 + e^14 + 4) + e^7 - log(3) - log(2))/(4*x*(log(2) + 1) + sqrt(-2*(e^7 - log(2))*log(3) + log(
3)^2 - 2*(e^7 - 2)*log(2) + log(2)^2 + e^14 + 4) + e^7 - log(3) - log(2)))/(sqrt(-2*(e^7 - log(2))*log(3) + lo
g(3)^2 - 2*(e^7 - 2)*log(2) + log(2)^2 + e^14 + 4)*(log(2) + 1)) - log(4*x^2*(log(2) + 1) + 2*x*(e^7 - log(3)
- log(2)) - 1)/(log(2) + 1))*log(2) + 1/2*(e^7 - log(3) - log(2))*log((4*x*(log(2) + 1) - sqrt(-2*(e^7 - log(2
))*log(3) + log(3)^2 - 2*(e^7 - 2)*log(2) + log(2)^2 + e^14 + 4) + e^7 - log(3) - log(2))/(4*x*(log(2) + 1) +
sqrt(-2*(e^7 - log(2))*log(3) + log(3)^2 - 2*(e^7 - 2)*log(2) + log(2)^2 + e^14 + 4) + e^7 - log(3) - log(2)))
/sqrt(-2*(e^7 - log(2))*log(3) + log(3)^2 - 2*(e^7 - 2)*log(2) + log(2)^2 + e^14 + 4) - 1/2*(e^7 - log(3) - lo
g(2))*log((4*x*(log(2) + 1) - sqrt(-2*(e^7 - log(2))*log(3) + log(3)^2 - 2*(e^7 - 2)*log(2) + log(2)^2 + e^14
+ 4) + e^7 - log(3) - log(2))/(4*x*(log(2) + 1) + sqrt(-2*(e^7 - log(2))*log(3) + log(3)^2 - 2*(e^7 - 2)*log(2
) + log(2)^2 + e^14 + 4) + e^7 - log(3) - log(2)))/(sqrt(-2*(e^7 - log(2))*log(3) + log(3)^2 - 2*(e^7 - 2)*log
(2) + log(2)^2 + e^14 + 4)*(log(2) + 1)) + 1/2*log(4*x^2*(log(2) + 1) + 2*x*(e^7 - log(3) - log(2)) - 1)/(log(
2) + 1) + 1/2*log(4*x^2*(log(2) + 1) + 2*x*(e^7 - log(3) - log(2)) - 1) - log(x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {-1-4 x^2-2 x^2 \log (4)}{x-2 e^7 x^2-2 x^3-2 x^3 \log (4)+2 x^2 \log \left (6 e^{-x}\right )} \, dx=\log \left ({\left | 4 \, x^{2} \log \left (2\right ) + 4 \, x^{2} + 2 \, x e^{7} - 2 \, x \log \left (6\right ) - 1 \right |}\right ) - \log \left ({\left | x \right |}\right ) \]

[In]

integrate((-4*x^2*log(2)-4*x^2-1)/(2*x^2*log(6/exp(x))-4*x^3*log(2)-2*x^2*exp(7)-2*x^3+x),x, algorithm="giac")

[Out]

log(abs(4*x^2*log(2) + 4*x^2 + 2*x*e^7 - 2*x*log(6) - 1)) - log(abs(x))

Mupad [B] (verification not implemented)

Time = 1.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {-1-4 x^2-2 x^2 \log (4)}{x-2 e^7 x^2-2 x^3-2 x^3 \log (4)+2 x^2 \log \left (6 e^{-x}\right )} \, dx=\ln \left (4\,x\,\ln \left (6\right )-4\,x\,{\mathrm {e}}^7-8\,x^2\,\ln \left (2\right )-8\,x^2+2\right )-\ln \left (x\right ) \]

[In]

int((4*x^2*log(2) + 4*x^2 + 1)/(2*x^2*exp(7) - 2*x^2*log(6*exp(-x)) - x + 4*x^3*log(2) + 2*x^3),x)

[Out]

log(4*x*log(6) - 4*x*exp(7) - 8*x^2*log(2) - 8*x^2 + 2) - log(x)