3.71.87 \(\int \frac {5+e^{-2+x}-2 x}{3+e^{-2+x}+5 x-x^2+2 \log ^2(5)-\log ^4(5)} \, dx\)

Optimal. Leaf size=24 \[ \log \left (-4-e^{-2+x}-5 x+x^2+\left (-1+\log ^2(5)\right )^2\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.09, antiderivative size = 37, normalized size of antiderivative = 1.54, number of steps used = 1, number of rules used = 1, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {6684} \begin {gather*} \log \left (-e^2 x^2+5 e^2 x+e^x+e^2 \left (3-\log ^2(5)\right ) \left (1+\log ^2(5)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.33, size = 28, normalized size = 1.17 \begin {gather*} \log \left (3+e^{-2+x}+5 x-x^2+2 \log ^2(5)-\log ^4(5)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.62, size = 27, normalized size = 1.12 \begin {gather*} \log \left (-\log \relax (5)^{4} - x^{2} + 2 \, \log \relax (5)^{2} + 5 \, x + e^{\left (x - 2\right )} + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.19, size = 36, normalized size = 1.50 \begin {gather*} \log \left (-e^{2} \log \relax (5)^{4} - x^{2} e^{2} + 2 \, e^{2} \log \relax (5)^{2} + 5 \, x e^{2} + 3 \, e^{2} + e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.04, size = 26, normalized size = 1.08

method	result	size
norman	\(\ln \left (\ln \relax (5)^{4}-2 \ln \relax (5)^{2}+x^{2}-{\mathrm e}^{x -2}-5 x -3\right )\)	\(26\)
derivativedivides	\(\ln \left ({\mathrm e}^{x -2}-\ln \relax (5)^{4}+2 \ln \relax (5)^{2}-x^{2}+5 x +3\right )\)	\(28\)
default	\(\ln \left ({\mathrm e}^{x -2}-\ln \relax (5)^{4}+2 \ln \relax (5)^{2}-x^{2}+5 x +3\right )\)	\(28\)
risch	\(2+\ln \left ({\mathrm e}^{x -2}-\ln \relax (5)^{4}+2 \ln \relax (5)^{2}-x^{2}+5 x +3\right )\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.37, size = 25, normalized size = 1.04 \begin {gather*} \log \left (\log \relax (5)^{4} + x^{2} - 2 \, \log \relax (5)^{2} - 5 \, x - e^{\left (x - 2\right )} - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.13, size = 25, normalized size = 1.04 \begin {gather*} \ln \left ({\ln \relax (5)}^4-{\mathrm {e}}^{x-2}-2\,{\ln \relax (5)}^2-5\,x+x^2-3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.16, size = 26, normalized size = 1.08 \begin {gather*} \log {\left (- x^{2} + 5 x + e^{x - 2} - \log {\relax (5 )}^{4} + 3 + 2 \log {\relax (5 )}^{2} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________