3.12.44 \(\int -\frac {1}{e^{e^2} (-1+e^4)+x-e^4 x} \, dx\)

Optimal. Leaf size=18 \[ \frac {\log \left (e^{e^2}-x\right )}{-1+e^4} \]

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Rubi [A]  time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.17, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {6, 31} \begin {gather*} -\frac {\log \left (e^{e^2}-x\right )}{1-e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-(E^E^2*(-1 + E^4) + x - E^4*x)^(-1),x]

[Out]

-(Log[E^E^2 - x]/(1 - E^4))

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\int \frac {1}{e^{e^2} \left (-1+e^4\right )+\left (1-e^4\right ) x} \, dx\\ &=-\frac {\log \left (e^{e^2}-x\right )}{1-e^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} \frac {\log \left (e^{e^2}-x\right )}{-1+e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-(E^E^2*(-1 + E^4) + x - E^4*x)^(-1),x]

[Out]

Log[E^E^2 - x]/(-1 + E^4)

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fricas [A]  time = 0.90, size = 15, normalized size = 0.83 \begin {gather*} \frac {\log \left (x - e^{\left (e^{2}\right )}\right )}{e^{4} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/((exp(4)-1)*exp(exp(2))-x*exp(4)+x),x, algorithm="fricas")

[Out]

log(x - e^(e^2))/(e^4 - 1)

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giac [A]  time = 0.29, size = 26, normalized size = 1.44 \begin {gather*} \frac {\log \left ({\left | x e^{4} - {\left (e^{4} - 1\right )} e^{\left (e^{2}\right )} - x \right |}\right )}{e^{4} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/((exp(4)-1)*exp(exp(2))-x*exp(4)+x),x, algorithm="giac")

[Out]

log(abs(x*e^4 - (e^4 - 1)*e^(e^2) - x))/(e^4 - 1)

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maple [A]  time = 0.24, size = 16, normalized size = 0.89

method	result	size
norman	\(\frac {\ln \left ({\mathrm e}^{{\mathrm e}^{2}}-x \right )}{{\mathrm e}^{4}-1}\)	\(16\)
risch	\(\frac {\ln \left (-{\mathrm e}^{{\mathrm e}^{2}}+x \right )}{{\mathrm e}^{4}-1}\)	\(16\)
meijerg	\(\frac {\ln \left (1-x \,{\mathrm e}^{-{\mathrm e}^{2}}\right )}{{\mathrm e}^{4}-1}\)	\(19\)
default	\(-\frac {\ln \left (\left (1-{\mathrm e}^{4}\right ) x +\left ({\mathrm e}^{4}-1\right ) {\mathrm e}^{{\mathrm e}^{2}}\right )}{1-{\mathrm e}^{4}}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/((exp(4)-1)*exp(exp(2))-x*exp(4)+x),x,method=_RETURNVERBOSE)

[Out]

ln(exp(exp(2))-x)/(exp(4)-1)

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maxima [A]  time = 0.54, size = 25, normalized size = 1.39 \begin {gather*} \frac {\log \left (x e^{4} - {\left (e^{4} - 1\right )} e^{\left (e^{2}\right )} - x\right )}{e^{4} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/((exp(4)-1)*exp(exp(2))-x*exp(4)+x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.20, size = 15, normalized size = 0.83 \begin {gather*} \frac {\ln \left (x-{\mathrm {e}}^{{\mathrm {e}}^2}\right )}{{\mathrm {e}}^4-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(x - x*exp(4) + exp(exp(2))*(exp(4) - 1)),x)

[Out]

log(x - exp(exp(2)))/(exp(4) - 1)

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sympy [B]  time = 0.07, size = 26, normalized size = 1.44 \begin {gather*} \frac {\log {\left (x \left (-1 + e^{4}\right ) - e^{4} e^{e^{2}} + e^{e^{2}} \right )}}{-1 + e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/((exp(4)-1)*exp(exp(2))-x*exp(4)+x),x)

[Out]

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

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