3.88.88 \(\int \frac {-6+2 e^2-2 x}{45+5 e^4+e^2 (-30-2 x)+6 x+x^2} \, dx\)

Optimal. Leaf size=26 \[ 9 e^2-\log \left (4+\left (1+\frac {x}{3-e^2}\right )^2\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 25, normalized size of antiderivative = 0.96, number of steps used = 1, number of rules used = 1, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {1587} \begin {gather*} -\log \left (x^2+6 x-2 e^2 (x+15)+5 \left (9+e^4\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-6 + 2*E^2 - 2*x)/(45 + 5*E^4 + E^2*(-30 - 2*x) + 6*x + x^2),x]

[Out]

-Log[5*(9 + E^4) + 6*x + x^2 - 2*E^2*(15 + x)]

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\log \left (5 \left (9+e^4\right )+6 x+x^2-2 e^2 (15+x)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 24, normalized size = 0.92 \begin {gather*} -\log \left (45+5 e^4+6 x+x^2-2 e^2 (15+x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6 + 2*E^2 - 2*x)/(45 + 5*E^4 + E^2*(-30 - 2*x) + 6*x + x^2),x]

[Out]

-Log[45 + 5*E^4 + 6*x + x^2 - 2*E^2*(15 + x)]

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fricas [A]  time = 0.63, size = 22, normalized size = 0.85 \begin {gather*} -\log \left (x^{2} - 2 \, {\left (x + 15\right )} e^{2} + 6 \, x + 5 \, e^{4} + 45\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(2)-2*x-6)/(5*exp(2)^2+(-2*x-30)*exp(2)+x^2+6*x+45),x, algorithm="fricas")

[Out]

-log(x^2 - 2*(x + 15)*e^2 + 6*x + 5*e^4 + 45)

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giac [A]  time = 0.19, size = 24, normalized size = 0.92 \begin {gather*} -\log \left (x^{2} - 2 \, x e^{2} + 6 \, x + 5 \, e^{4} - 30 \, e^{2} + 45\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(2)-2*x-6)/(5*exp(2)^2+(-2*x-30)*exp(2)+x^2+6*x+45),x, algorithm="giac")

[Out]

-log(x^2 - 2*x*e^2 + 6*x + 5*e^4 - 30*e^2 + 45)

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maple [A]  time = 0.92, size = 25, normalized size = 0.96

method	result	size
risch	\(-\ln \left (x^{2}+\left (-2 \,{\mathrm e}^{2}+6\right ) x +5 \,{\mathrm e}^{4}-30 \,{\mathrm e}^{2}+45\right )\)	\(25\)
default	\(-\ln \left (5 \,{\mathrm e}^{4}-2 \,{\mathrm e}^{2} x +x^{2}-30 \,{\mathrm e}^{2}+6 x +45\right )\)	\(27\)
norman	\(-\ln \left (5 \,{\mathrm e}^{4}-2 \,{\mathrm e}^{2} x +x^{2}-30 \,{\mathrm e}^{2}+6 x +45\right )\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*exp(2)-2*x-6)/(5*exp(2)^2+(-2*x-30)*exp(2)+x^2+6*x+45),x,method=_RETURNVERBOSE)

[Out]

-ln(x^2+(-2*exp(2)+6)*x+5*exp(4)-30*exp(2)+45)

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maxima [A]  time = 0.35, size = 22, normalized size = 0.85 \begin {gather*} -\log \left (x^{2} - 2 \, {\left (x + 15\right )} e^{2} + 6 \, x + 5 \, e^{4} + 45\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(2)-2*x-6)/(5*exp(2)^2+(-2*x-30)*exp(2)+x^2+6*x+45),x, algorithm="maxima")

[Out]

-log(x^2 - 2*(x + 15)*e^2 + 6*x + 5*e^4 + 45)

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mupad [B]  time = 0.14, size = 24, normalized size = 0.92 \begin {gather*} -\ln \left (x^2+\left (6-2\,{\mathrm {e}}^2\right )\,x+5\,{\left ({\mathrm {e}}^2-3\right )}^2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - 2*exp(2) + 6)/(6*x + 5*exp(4) + x^2 - exp(2)*(2*x + 30) + 45),x)

[Out]

-log(5*(exp(2) - 3)^2 + x^2 - x*(2*exp(2) - 6))

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sympy [A]  time = 0.18, size = 26, normalized size = 1.00 \begin {gather*} - \log {\left (x^{2} + x \left (6 - 2 e^{2}\right ) - 30 e^{2} + 45 + 5 e^{4} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(2)-2*x-6)/(5*exp(2)**2+(-2*x-30)*exp(2)+x**2+6*x+45),x)

[Out]

-log(x**2 + x*(6 - 2*exp(2)) - 30*exp(2) + 45 + 5*exp(4))

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