3.96.15 \(\int \frac {3 x^2+3 x^3-x^4+e^{e^5} (-3-3 x-6 x^3-6 x^4+2 x^5)+e^{2 e^5} (6 x+3 x^2+3 x^4+3 x^5-x^6)}{-4 x^2-x^3+x^4+e^{e^5} (3 x+8 x^3+2 x^4-2 x^5)+e^{2 e^5} (-3 x^2-4 x^4-x^5+x^6)} \, dx\)

Optimal. Leaf size=32 \[ -x+\log \left (4+x-x^2-\frac {3}{e^{-e^5} x-x^2}\right ) \]

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Rubi [B]  time = 0.32, antiderivative size = 68, normalized size of antiderivative = 2.12, number of steps used = 3, number of rules used = 2, integrand size = 143, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {2074, 1587} \begin {gather*} \log \left (-e^{e^5} x^4+\left (1+e^{e^5}\right ) x^3-\left (1-4 e^{e^5}\right ) x^2-4 x+3 e^{e^5}\right )-x-\log (x)-\log \left (1-e^{e^5} x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3*x^2 + 3*x^3 - x^4 + E^E^5*(-3 - 3*x - 6*x^3 - 6*x^4 + 2*x^5) + E^(2*E^5)*(6*x + 3*x^2 + 3*x^4 + 3*x^5 -
x^6))/(-4*x^2 - x^3 + x^4 + E^E^5*(3*x + 8*x^3 + 2*x^4 - 2*x^5) + E^(2*E^5)*(-3*x^2 - 4*x^4 - x^5 + x^6)),x]

[Out]

-x - Log[x] - Log[1 - E^E^5*x] + Log[3*E^E^5 - 4*x - (1 - 4*E^E^5)*x^2 + (1 + E^E^5)*x^3 - E^E^5*x^4]

Rule 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-1-\frac {1}{x}-\frac {e^{e^5}}{-1+e^{e^5} x}+\frac {-4-2 \left (1-4 e^{e^5}\right ) x+3 \left (1+e^{e^5}\right ) x^2-4 e^{e^5} x^3}{3 e^{e^5}-4 x-\left (1-4 e^{e^5}\right ) x^2+\left (1+e^{e^5}\right ) x^3-e^{e^5} x^4}\right ) \, dx\\ &=-x-\log (x)-\log \left (1-e^{e^5} x\right )+\int \frac {-4-2 \left (1-4 e^{e^5}\right ) x+3 \left (1+e^{e^5}\right ) x^2-4 e^{e^5} x^3}{3 e^{e^5}-4 x-\left (1-4 e^{e^5}\right ) x^2+\left (1+e^{e^5}\right ) x^3-e^{e^5} x^4} \, dx\\ &=-x-\log (x)-\log \left (1-e^{e^5} x\right )+\log \left (3 e^{e^5}-4 x-\left (1-4 e^{e^5}\right ) x^2+\left (1+e^{e^5}\right ) x^3-e^{e^5} x^4\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.10, size = 70, normalized size = 2.19 \begin {gather*} -x-\log (x)-\log \left (1-e^{e^5} x\right )+\log \left (-3 e^{e^5}+4 x+x^2-4 e^{e^5} x^2-x^3-e^{e^5} x^3+e^{e^5} x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3*x^2 + 3*x^3 - x^4 + E^E^5*(-3 - 3*x - 6*x^3 - 6*x^4 + 2*x^5) + E^(2*E^5)*(6*x + 3*x^2 + 3*x^4 + 3
*x^5 - x^6))/(-4*x^2 - x^3 + x^4 + E^E^5*(3*x + 8*x^3 + 2*x^4 - 2*x^5) + E^(2*E^5)*(-3*x^2 - 4*x^4 - x^5 + x^6
)),x]

[Out]

-x - Log[x] - Log[1 - E^E^5*x] + Log[-3*E^E^5 + 4*x + x^2 - 4*E^E^5*x^2 - x^3 - E^E^5*x^3 + E^E^5*x^4]

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fricas [A]  time = 0.58, size = 50, normalized size = 1.56 \begin {gather*} -x + \log \left (-x^{3} + x^{2} + {\left (x^{4} - x^{3} - 4 \, x^{2} - 3\right )} e^{\left (e^{5}\right )} + 4 \, x\right ) - \log \left (x^{2} e^{\left (e^{5}\right )} - x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^6+3*x^5+3*x^4+3*x^2+6*x)*exp(exp(5))^2+(2*x^5-6*x^4-6*x^3-3*x-3)*exp(exp(5))-x^4+3*x^3+3*x^2)/(
(x^6-x^5-4*x^4-3*x^2)*exp(exp(5))^2+(-2*x^5+2*x^4+8*x^3+3*x)*exp(exp(5))+x^4-x^3-4*x^2),x, algorithm="fricas")

[Out]

-x + log(-x^3 + x^2 + (x^4 - x^3 - 4*x^2 - 3)*e^(e^5) + 4*x) - log(x^2*e^(e^5) - x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^6+3*x^5+3*x^4+3*x^2+6*x)*exp(exp(5))^2+(2*x^5-6*x^4-6*x^3-3*x-3)*exp(exp(5))-x^4+3*x^3+3*x^2)/(
(x^6-x^5-4*x^4-3*x^2)*exp(exp(5))^2+(-2*x^5+2*x^4+8*x^3+3*x)*exp(exp(5))+x^4-x^3-4*x^2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.31, size = 56, normalized size = 1.75

method	result	size
risch	\(-x -\ln \left (-x^{2} {\mathrm e}^{{\mathrm e}^{5}}+x \right )+\ln \left (-{\mathrm e}^{{\mathrm e}^{5}} x^{4}+\left ({\mathrm e}^{{\mathrm e}^{5}}+1\right ) x^{3}+\left (-1+4 \,{\mathrm e}^{{\mathrm e}^{5}}\right ) x^{2}-4 x +3 \,{\mathrm e}^{{\mathrm e}^{5}}\right )\)	\(56\)
norman	\(-x -\ln \relax (x )-\ln \left (x \,{\mathrm e}^{{\mathrm e}^{5}}-1\right )+\ln \left ({\mathrm e}^{{\mathrm e}^{5}} x^{4}-{\mathrm e}^{{\mathrm e}^{5}} x^{3}-4 x^{2} {\mathrm e}^{{\mathrm e}^{5}}-x^{3}+x^{2}-3 \,{\mathrm e}^{{\mathrm e}^{5}}+4 x \right )\)	\(60\)
default	\(-x -\ln \relax (x )-\left (\munderset {\textit {\_R} =\RootOf \left ({\mathrm e}^{2 \,{\mathrm e}^{5}} \textit {\_Z}^{5}-\left (2 \,{\mathrm e}^{{\mathrm e}^{5}}+{\mathrm e}^{2 \,{\mathrm e}^{5}}\right ) \textit {\_Z}^{4}-\left (-2 \,{\mathrm e}^{{\mathrm e}^{5}}+4 \,{\mathrm e}^{2 \,{\mathrm e}^{5}}-1\right ) \textit {\_Z}^{3}-\left (-8 \,{\mathrm e}^{{\mathrm e}^{5}}+1\right ) \textit {\_Z}^{2}-\left (3 \,{\mathrm e}^{2 \,{\mathrm e}^{5}}+4\right ) \textit {\_Z} +3 \,{\mathrm e}^{{\mathrm e}^{5}}\right )}{\sum }\frac {\left (-4+3 \,{\mathrm e}^{2 \,{\mathrm e}^{5}} \textit {\_R}^{4}+2 \left (-3 \,{\mathrm e}^{{\mathrm e}^{5}}-{\mathrm e}^{2 \,{\mathrm e}^{5}}\right ) \textit {\_R}^{3}+\left (4 \,{\mathrm e}^{{\mathrm e}^{5}}-4 \,{\mathrm e}^{2 \,{\mathrm e}^{5}}+3\right ) \textit {\_R}^{2}+2 \left (-1+4 \,{\mathrm e}^{{\mathrm e}^{5}}\right ) \textit {\_R} +3 \,{\mathrm e}^{2 \,{\mathrm e}^{5}}\right ) \ln \left (x -\textit {\_R} \right )}{-5 \,{\mathrm e}^{2 \,{\mathrm e}^{5}} \textit {\_R}^{4}+8 \,{\mathrm e}^{{\mathrm e}^{5}} \textit {\_R}^{3}+4 \,{\mathrm e}^{2 \,{\mathrm e}^{5}} \textit {\_R}^{3}-6 \textit {\_R}^{2} {\mathrm e}^{{\mathrm e}^{5}}+12 \,{\mathrm e}^{2 \,{\mathrm e}^{5}} \textit {\_R}^{2}-16 \textit {\_R} \,{\mathrm e}^{{\mathrm e}^{5}}-3 \textit {\_R}^{2}+3 \,{\mathrm e}^{2 \,{\mathrm e}^{5}}+2 \textit {\_R} +4}\right )\)	\(232\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^6+3*x^5+3*x^4+3*x^2+6*x)*exp(exp(5))^2+(2*x^5-6*x^4-6*x^3-3*x-3)*exp(exp(5))-x^4+3*x^3+3*x^2)/((x^6-x
^5-4*x^4-3*x^2)*exp(exp(5))^2+(-2*x^5+2*x^4+8*x^3+3*x)*exp(exp(5))+x^4-x^3-4*x^2),x,method=_RETURNVERBOSE)

[Out]

-x-ln(-x^2*exp(exp(5))+x)+ln(-exp(exp(5))*x^4+(exp(exp(5))+1)*x^3+(-1+4*exp(exp(5)))*x^2-4*x+3*exp(exp(5)))

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maxima [A]  time = 0.36, size = 57, normalized size = 1.78 \begin {gather*} -x + \log \left (x^{4} e^{\left (e^{5}\right )} - x^{3} {\left (e^{\left (e^{5}\right )} + 1\right )} - x^{2} {\left (4 \, e^{\left (e^{5}\right )} - 1\right )} + 4 \, x - 3 \, e^{\left (e^{5}\right )}\right ) - \log \left (x e^{\left (e^{5}\right )} - 1\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^6+3*x^5+3*x^4+3*x^2+6*x)*exp(exp(5))^2+(2*x^5-6*x^4-6*x^3-3*x-3)*exp(exp(5))-x^4+3*x^3+3*x^2)/(
(x^6-x^5-4*x^4-3*x^2)*exp(exp(5))^2+(-2*x^5+2*x^4+8*x^3+3*x)*exp(exp(5))+x^4-x^3-4*x^2),x, algorithm="maxima")

[Out]

-x + log(x^4*e^(e^5) - x^3*(e^(e^5) + 1) - x^2*(4*e^(e^5) - 1) + 4*x - 3*e^(e^5)) - log(x*e^(e^5) - 1) - log(x
)

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mupad [B]  time = 7.83, size = 62, normalized size = 1.94 \begin {gather*} \ln \left (4\,x\,{\mathrm {e}}^{-{\mathrm {e}}^5}+x^2\,{\mathrm {e}}^{-{\mathrm {e}}^5}-x^3\,{\mathrm {e}}^{-{\mathrm {e}}^5}-4\,x^2-x^3+x^4-3\right )-\ln \left (x^2-x\,{\mathrm {e}}^{-{\mathrm {e}}^5}\right )-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*exp(5))*(6*x + 3*x^2 + 3*x^4 + 3*x^5 - x^6) - exp(exp(5))*(3*x + 6*x^3 + 6*x^4 - 2*x^5 + 3) + 3*x^
2 + 3*x^3 - x^4)/(exp(2*exp(5))*(3*x^2 + 4*x^4 + x^5 - x^6) - exp(exp(5))*(3*x + 8*x^3 + 2*x^4 - 2*x^5) + 4*x^
2 + x^3 - x^4),x)

[Out]

log(4*x*exp(-exp(5)) + x^2*exp(-exp(5)) - x^3*exp(-exp(5)) - 4*x^2 - x^3 + x^4 - 3) - log(x^2 - x*exp(-exp(5))
) - x

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sympy [B]  time = 5.03, size = 61, normalized size = 1.91 \begin {gather*} - x - \log {\left (x^{2} - \frac {x}{e^{e^{5}}} \right )} + \log {\left (x^{4} + \frac {x^{3} \left (- e^{e^{5}} - 1\right )}{e^{e^{5}}} + \frac {x^{2} \left (1 - 4 e^{e^{5}}\right )}{e^{e^{5}}} + \frac {4 x}{e^{e^{5}}} - 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**6+3*x**5+3*x**4+3*x**2+6*x)*exp(exp(5))**2+(2*x**5-6*x**4-6*x**3-3*x-3)*exp(exp(5))-x**4+3*x**
3+3*x**2)/((x**6-x**5-4*x**4-3*x**2)*exp(exp(5))**2+(-2*x**5+2*x**4+8*x**3+3*x)*exp(exp(5))+x**4-x**3-4*x**2),
x)

[Out]

-x - log(x**2 - x*exp(-exp(5))) + log(x**4 + x**3*(-exp(exp(5)) - 1)*exp(-exp(5)) + x**2*(1 - 4*exp(exp(5)))*e
xp(-exp(5)) + 4*x*exp(-exp(5)) - 3)

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