\(\int -\frac {2 x}{-6-e^5+x^2+2 \log (4)} \, dx\) [7885]

Optimal result

Integrand size = 19, antiderivative size = 19 \[ \int -\frac {2 x}{-6-e^5+x^2+2 \log (4)} \, dx=1-\log \left (6+e^5-x^2-2 \log (4)\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {12, 266} \[ \int -\frac {2 x}{-6-e^5+x^2+2 \log (4)} \, dx=-\log \left (-x^2+e^5+6-2 \log (4)\right ) \]

[In]

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

[Out]

-Log[6 + E^5 - x^2 - 2*Log[4]]

Rule 12

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

Rule 266

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

Rubi steps \begin{align*} \text {integral}& = -\left (2 \int \frac {x}{-6-e^5+x^2+2 \log (4)} \, dx\right ) \\ & = -\log \left (6+e^5-x^2-2 \log (4)\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int -\frac {2 x}{-6-e^5+x^2+2 \log (4)} \, dx=-\log \left (6+e^5-x^2-2 \log (4)\right ) \]

[In]

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

[Out]

-Log[6 + E^5 - x^2 - 2*Log[4]]

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89

method	result	size
derivativedivides	\(-\ln \left (4 \ln \left (2\right )-{\mathrm e}^{5}+x^{2}-6\right )\)	\(17\)
default	\(-\ln \left (6-x^{2}-4 \ln \left (2\right )+{\mathrm e}^{5}\right )\)	\(17\)
norman	\(-\ln \left (6-x^{2}-4 \ln \left (2\right )+{\mathrm e}^{5}\right )\)	\(17\)
risch	\(-\ln \left (4 \ln \left (2\right )-{\mathrm e}^{5}+x^{2}-6\right )\)	\(17\)
parallelrisch	\(-\ln \left (4 \ln \left (2\right )-{\mathrm e}^{5}+x^{2}-6\right )\)	\(17\)
meijerg	\(\frac {\left ({\mathrm e}^{5}-4 \ln \left (2\right )+6\right ) \ln \left (1-\frac {x^{2}}{{\mathrm e}^{5}-4 \ln \left (2\right )+6}\right )}{-{\mathrm e}^{5}+4 \ln \left (2\right )-6}\)	\(40\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int -\frac {2 x}{-6-e^5+x^2+2 \log (4)} \, dx=-\log \left (x^{2} - e^{5} + 4 \, \log \left (2\right ) - 6\right ) \]

[In]

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

[Out]

-log(x^2 - e^5 + 4*log(2) - 6)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int -\frac {2 x}{-6-e^5+x^2+2 \log (4)} \, dx=- \log {\left (x^{2} - e^{5} - 6 + 4 \log {\left (2 \right )} \right )} \]

[In]

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

[Out]

-log(x**2 - exp(5) - 6 + 4*log(2))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int -\frac {2 x}{-6-e^5+x^2+2 \log (4)} \, dx=-\log \left (x^{2} - e^{5} + 4 \, \log \left (2\right ) - 6\right ) \]

[In]

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

[Out]

-log(x^2 - e^5 + 4*log(2) - 6)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int -\frac {2 x}{-6-e^5+x^2+2 \log (4)} \, dx=-\log \left ({\left | x^{2} - e^{5} + 4 \, \log \left (2\right ) - 6 \right |}\right ) \]

[In]

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

[Out]

-log(abs(x^2 - e^5 + 4*log(2) - 6))

Mupad [B] (verification not implemented)

Time = 13.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int -\frac {2 x}{-6-e^5+x^2+2 \log (4)} \, dx=-\ln \left (x^2-{\mathrm {e}}^5+\ln \left (16\right )-6\right ) \]

[In]

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

[Out]

-log(log(16) - exp(5) + x^2 - 6)