\(\int \frac {e^{-2+x-x^6} (-1+6 x^5)}{-1+e^{-2+x-x^6}} \, dx\) [1780]

Optimal result

Integrand size = 32, antiderivative size = 15 \[ \int \frac {e^{-2+x-x^6} \left (-1+6 x^5\right )}{-1+e^{-2+x-x^6}} \, dx=-\log \left (-1+e^{-2+x-x^6}\right ) \]

[Out]

-ln(exp(-x^6+x-2)-1)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.80, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {6816} \[ \int \frac {e^{-2+x-x^6} \left (-1+6 x^5\right )}{-1+e^{-2+x-x^6}} \, dx=-\log \left (-e^{-x^6-2} \left (e^x-e^{x^6+2}\right )\right ) \]

[In]

Int[(E^(-2 + x - x^6)*(-1 + 6*x^5))/(-1 + E^(-2 + x - x^6)),x]

[Out]

-Log[-(E^(-2 - x^6)*(E^x - E^(2 + x^6)))]

Rule 6816

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps \begin{align*} \text {integral}& = -\log \left (-e^{-2-x^6} \left (e^x-e^{2+x^6}\right )\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.33 \[ \int \frac {e^{-2+x-x^6} \left (-1+6 x^5\right )}{-1+e^{-2+x-x^6}} \, dx=x^6-\log \left (e^x-e^{2+x^6}\right ) \]

[In]

Integrate[(E^(-2 + x - x^6)*(-1 + 6*x^5))/(-1 + E^(-2 + x - x^6)),x]

[Out]

x^6 - Log[E^x - E^(2 + x^6)]

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00

method	result	size
norman	\(-\ln \left ({\mathrm e}^{-x^{6}+x -2}-1\right )\)	\(15\)
parallelrisch	\(-\ln \left ({\mathrm e}^{-x^{6}+x -2}-1\right )\)	\(15\)
risch	\(-2-\ln \left ({\mathrm e}^{-x^{6}+x -2}-1\right )\)	\(17\)

[In]

int((6*x^5-1)*exp(-x^6+x-2)/(exp(-x^6+x-2)-1),x,method=_RETURNVERBOSE)

[Out]

-ln(exp(-x^6+x-2)-1)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {e^{-2+x-x^6} \left (-1+6 x^5\right )}{-1+e^{-2+x-x^6}} \, dx=-\log \left (e^{\left (-x^{6} + x - 2\right )} - 1\right ) \]

[In]

integrate((6*x^5-1)*exp(-x^6+x-2)/(exp(-x^6+x-2)-1),x, algorithm="fricas")

[Out]

-log(e^(-x^6 + x - 2) - 1)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {e^{-2+x-x^6} \left (-1+6 x^5\right )}{-1+e^{-2+x-x^6}} \, dx=- \log {\left (e^{- x^{6} + x - 2} - 1 \right )} \]

[In]

integrate((6*x**5-1)*exp(-x**6+x-2)/(exp(-x**6+x-2)-1),x)

[Out]

-log(exp(-x**6 + x - 2) - 1)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.40 \[ \int \frac {e^{-2+x-x^6} \left (-1+6 x^5\right )}{-1+e^{-2+x-x^6}} \, dx=x^{6} - \log \left ({\left (e^{\left (x^{6} + 2\right )} - e^{x}\right )} e^{\left (-2\right )}\right ) \]

[In]

integrate((6*x^5-1)*exp(-x^6+x-2)/(exp(-x^6+x-2)-1),x, algorithm="maxima")

[Out]

x^6 - log((e^(x^6 + 2) - e^x)*e^(-2))

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-2+x-x^6} \left (-1+6 x^5\right )}{-1+e^{-2+x-x^6}} \, dx=-\log \left (-e^{\left (-x^{6} + x - 2\right )} + 1\right ) \]

[In]

integrate((6*x^5-1)*exp(-x^6+x-2)/(exp(-x^6+x-2)-1),x, algorithm="giac")

[Out]

-log(-e^(-x^6 + x - 2) + 1)

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {e^{-2+x-x^6} \left (-1+6 x^5\right )}{-1+e^{-2+x-x^6}} \, dx=-\ln \left ({\mathrm {e}}^{-x^6+x-2}-1\right ) \]

[In]

int((exp(x - x^6 - 2)*(6*x^5 - 1))/(exp(x - x^6 - 2) - 1),x)

[Out]

-log(exp(x - x^6 - 2) - 1)