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

Optimal. Leaf size=17 \[ \log \left (2+e^{2 e-2 e^4}-x\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.18, number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {12, 33, 31} \begin {gather*} \log \left (e^{2 e^4} (2-x)+e^{2 e}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[E^(2*E) + E^(2*E^4)*(2 - x)]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

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

Rule 33

Int[((a_.) + (b_.)*(u_))^(m_), x_Symbol] :> Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*x)^m, x], x, u], x]
/; FreeQ[{a, b, m}, x] && LinearQ[u, x] && NeQ[u, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=e^{-2 e \left (1-e^3\right )} \int \frac {1}{-1+e^{-2 e+2 e^4} (-2+x)} \, dx\\ &=e^{-2 e \left (1-e^3\right )} \operatorname {Subst}\left (\int \frac {1}{-1+e^{-2 e+2 e^4} x} \, dx,x,-2+x\right )\\ &=\log \left (e^{2 e}+e^{2 e^4} (2-x)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 1.18 \begin {gather*} \log \left (-e^{2 e}+e^{2 e^4} (-2+x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[-E^(2*E) + E^(2*E^4)*(-2 + x)]

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fricas [A]  time = 0.72, size = 17, normalized size = 1.00 \begin {gather*} \log \left ({\left (x - 2\right )} e^{\left (2 \, e^{4} - 2 \, e\right )} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((x - 2)*e^(2*e^4 - 2*e) - 1)

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giac [A]  time = 0.26, size = 18, normalized size = 1.06 \begin {gather*} \log \left ({\left | {\left (x - 2\right )} e^{\left (2 \, e^{4} - 2 \, e\right )} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs((x - 2)*e^(2*e^4 - 2*e) - 1))

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maple [A]  time = 0.46, size = 24, normalized size = 1.41

method	result	size
risch	\(\ln \left ({\mathrm e}^{2 \,{\mathrm e}^{4}} x -2 \,{\mathrm e}^{2 \,{\mathrm e}^{4}}-{\mathrm e}^{2 \,{\mathrm e}}\right )\)	\(24\)
default	\(\ln \left ({\mathrm e}^{2 \,{\mathrm e}^{4}-2 \,{\mathrm e}} x -2 \,{\mathrm e}^{2 \,{\mathrm e}^{4}-2 \,{\mathrm e}}-1\right )\)	\(32\)
norman	\(\ln \left ({\mathrm e}^{2 \,{\mathrm e}^{4}-2 \,{\mathrm e}} x -2 \,{\mathrm e}^{2 \,{\mathrm e}^{4}-2 \,{\mathrm e}}-1\right )\)	\(32\)
meijerg	\(-\frac {\left (1+2 \,{\mathrm e}^{2 \,{\mathrm e}^{4}-2 \,{\mathrm e}}\right ) \ln \left (1+\frac {x \,{\mathrm e}^{2 \,{\mathrm e}^{4}-2 \,{\mathrm e}}}{-2 \,{\mathrm e}^{2 \,{\mathrm e}^{4}-2 \,{\mathrm e}}-1}\right )}{-2 \,{\mathrm e}^{2 \,{\mathrm e}^{4}-2 \,{\mathrm e}}-1}\)	\(64\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.35, size = 17, normalized size = 1.00 \begin {gather*} \log \left ({\left (x - 2\right )} e^{\left (2 \, e^{4} - 2 \, e\right )} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((x - 2)*e^(2*e^4 - 2*e) - 1)

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mupad [B]  time = 0.27, size = 35, normalized size = 2.06 \begin {gather*} {\mathrm {e}}^{2\,\mathrm {e}-2\,{\mathrm {e}}^4}\,{\mathrm {e}}^{2\,\mathrm {e}\,\left ({\mathrm {e}}^3-1\right )}\,\ln \left (x-{\mathrm {e}}^{-2\,\mathrm {e}\,\left ({\mathrm {e}}^3-1\right )}-2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(2*exp(1) - 2*exp(4))*exp(2*exp(1)*(exp(3) - 1))*log(x - exp(-2*exp(1)*(exp(3) - 1)) - 2)

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sympy [A]  time = 0.08, size = 24, normalized size = 1.41 \begin {gather*} \log {\left (x e^{2 e^{4}} - 2 e^{2 e^{4}} - e^{2 e} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x*exp(2*exp(4)) - 2*exp(2*exp(4)) - exp(2*E))

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