3.72.69 \(\int \frac {-2 e^{x+\frac {1}{\log (64)}} x+e^{\frac {1}{\log (64)}} (1+x)}{-x-2 e^{x+\frac {1}{\log (64)}} x+e^{\frac {1}{\log (64)}} x^2+e^{\frac {1}{\log (64)}} x \log (x)} \, dx\)

Optimal. Leaf size=22 \[ \log \left (2 e^x+e^{-\frac {1}{\log (64)}}-x-\log (x)\right ) \]

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Rubi [F]  time = 0.85, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-2 e^{x+\frac {1}{\log (64)}} x+e^{\frac {1}{\log (64)}} (1+x)}{-x-2 e^{x+\frac {1}{\log (64)}} x+e^{\frac {1}{\log (64)}} x^2+e^{\frac {1}{\log (64)}} x \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2*E^(x + Log[64]^(-1))*x + E^Log[64]^(-1)*(1 + x))/(-x - 2*E^(x + Log[64]^(-1))*x + E^Log[64]^(-1)*x^2 +
E^Log[64]^(-1)*x*Log[x]),x]

[Out]

-(E^Log[64]^(-1)*Defer[Int][(1 + 2*E^(x + Log[64]^(-1)) - E^Log[64]^(-1)*x - E^Log[64]^(-1)*Log[x])^(-1), x])
+ 2*E^Log[64]^(-1)*Defer[Int][E^x/(1 + 2*E^(x + Log[64]^(-1)) - E^Log[64]^(-1)*x - E^Log[64]^(-1)*Log[x]), x]
+ E^Log[64]^(-1)*Defer[Int][1/(x*(-1 - 2*E^(x + Log[64]^(-1)) + E^Log[64]^(-1)*x + E^Log[64]^(-1)*Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {1}{\log (64)}} \left (-1-x+2 e^x x\right )}{x+2 e^{x+\frac {1}{\log (64)}} x-e^{\frac {1}{\log (64)}} x^2-e^{\frac {1}{\log (64)}} x \log (x)} \, dx\\ &=e^{\frac {1}{\log (64)}} \int \frac {-1-x+2 e^x x}{x+2 e^{x+\frac {1}{\log (64)}} x-e^{\frac {1}{\log (64)}} x^2-e^{\frac {1}{\log (64)}} x \log (x)} \, dx\\ &=e^{\frac {1}{\log (64)}} \int \left (-\frac {1}{1+2 e^{x+\frac {1}{\log (64)}}-e^{\frac {1}{\log (64)}} x-e^{\frac {1}{\log (64)}} \log (x)}+\frac {2 e^x}{1+2 e^{x+\frac {1}{\log (64)}}-e^{\frac {1}{\log (64)}} x-e^{\frac {1}{\log (64)}} \log (x)}+\frac {1}{x \left (-1-2 e^{x+\frac {1}{\log (64)}}+e^{\frac {1}{\log (64)}} x+e^{\frac {1}{\log (64)}} \log (x)\right )}\right ) \, dx\\ &=-\left (e^{\frac {1}{\log (64)}} \int \frac {1}{1+2 e^{x+\frac {1}{\log (64)}}-e^{\frac {1}{\log (64)}} x-e^{\frac {1}{\log (64)}} \log (x)} \, dx\right )+e^{\frac {1}{\log (64)}} \int \frac {1}{x \left (-1-2 e^{x+\frac {1}{\log (64)}}+e^{\frac {1}{\log (64)}} x+e^{\frac {1}{\log (64)}} \log (x)\right )} \, dx+\left (2 e^{\frac {1}{\log (64)}}\right ) \int \frac {e^x}{1+2 e^{x+\frac {1}{\log (64)}}-e^{\frac {1}{\log (64)}} x-e^{\frac {1}{\log (64)}} \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.45, size = 32, normalized size = 1.45 \begin {gather*} \log \left (1+2 e^{x+\frac {1}{\log (64)}}-e^{\frac {1}{\log (64)}} x-e^{\frac {1}{\log (64)}} \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*E^(x + Log[64]^(-1))*x + E^Log[64]^(-1)*(1 + x))/(-x - 2*E^(x + Log[64]^(-1))*x + E^Log[64]^(-1)
*x^2 + E^Log[64]^(-1)*x*Log[x]),x]

[Out]

Log[1 + 2*E^(x + Log[64]^(-1)) - E^Log[64]^(-1)*x - E^Log[64]^(-1)*Log[x]]

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fricas [A]  time = 0.47, size = 38, normalized size = 1.73 \begin {gather*} \log \left (x e^{\left (\frac {1}{6 \, \log \relax (2)}\right )} + e^{\left (\frac {1}{6 \, \log \relax (2)}\right )} \log \relax (x) - 2 \, e^{\left (\frac {6 \, x \log \relax (2) + 1}{6 \, \log \relax (2)}\right )} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*exp(1/6/log(2))*exp(x)+(x+1)*exp(1/6/log(2)))/(x*exp(1/6/log(2))*log(x)-2*x*exp(1/6/log(2))*ex
p(x)+x^2*exp(1/6/log(2))-x),x, algorithm="fricas")

[Out]

log(x*e^(1/6/log(2)) + e^(1/6/log(2))*log(x) - 2*e^(1/6*(6*x*log(2) + 1)/log(2)) - 1)

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giac [A]  time = 0.19, size = 35, normalized size = 1.59 \begin {gather*} \log \left (-x e^{\left (\frac {1}{6 \, \log \relax (2)}\right )} - e^{\left (\frac {1}{6 \, \log \relax (2)}\right )} \log \relax (x) + 2 \, e^{\left (x + \frac {1}{6 \, \log \relax (2)}\right )} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*exp(1/6/log(2))*exp(x)+(x+1)*exp(1/6/log(2)))/(x*exp(1/6/log(2))*log(x)-2*x*exp(1/6/log(2))*ex
p(x)+x^2*exp(1/6/log(2))-x),x, algorithm="giac")

[Out]

log(-x*e^(1/6/log(2)) - e^(1/6/log(2))*log(x) + 2*e^(x + 1/6/log(2)) + 1)

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maple [A]  time = 0.10, size = 34, normalized size = 1.55

method	result	size
norman	\(\ln \left ({\mathrm e}^{\frac {1}{6 \ln \relax (2)}} \ln \relax (x )-2 \,{\mathrm e}^{\frac {1}{6 \ln \relax (2)}} {\mathrm e}^{x}+x \,{\mathrm e}^{\frac {1}{6 \ln \relax (2)}}-1\right )\)	\(34\)
risch	\(\ln \left (\ln \relax (x )-\left ({\mathrm e}^{\frac {5}{6 \ln \relax (2)}}-{\mathrm e}^{\frac {1}{\ln \relax (2)}} x +2 \,{\mathrm e}^{\frac {x \ln \relax (2)+1}{\ln \relax (2)}}\right ) {\mathrm e}^{-\frac {1}{\ln \relax (2)}}\right )\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x*exp(1/6/ln(2))*exp(x)+(x+1)*exp(1/6/ln(2)))/(x*exp(1/6/ln(2))*ln(x)-2*x*exp(1/6/ln(2))*exp(x)+x^2*ex
p(1/6/ln(2))-x),x,method=_RETURNVERBOSE)

[Out]

ln(exp(1/6/ln(2))*ln(x)-2*exp(1/6/ln(2))*exp(x)+x*exp(1/6/ln(2))-1)

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maxima [B]  time = 0.49, size = 42, normalized size = 1.91 \begin {gather*} \log \left (-\frac {1}{2} \, {\left (x e^{\left (\frac {1}{6 \, \log \relax (2)}\right )} + e^{\left (\frac {1}{6 \, \log \relax (2)}\right )} \log \relax (x) - 2 \, e^{\left (x + \frac {1}{6 \, \log \relax (2)}\right )} - 1\right )} e^{\left (-\frac {1}{6 \, \log \relax (2)}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*exp(1/6/log(2))*exp(x)+(x+1)*exp(1/6/log(2)))/(x*exp(1/6/log(2))*log(x)-2*x*exp(1/6/log(2))*ex
p(x)+x^2*exp(1/6/log(2))-x),x, algorithm="maxima")

[Out]

log(-1/2*(x*e^(1/6/log(2)) + e^(1/6/log(2))*log(x) - 2*e^(x + 1/6/log(2)) - 1)*e^(-1/6/log(2)))

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mupad [B]  time = 4.59, size = 18, normalized size = 0.82 \begin {gather*} \ln \left (x-{\mathrm {e}}^{-\frac {1}{6\,\ln \relax (2)}}-2\,{\mathrm {e}}^x+\ln \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(1/(6*log(2)))*(x + 1) - 2*x*exp(1/(6*log(2)))*exp(x))/(x - x^2*exp(1/(6*log(2))) + 2*x*exp(1/(6*log(
2)))*exp(x) - x*exp(1/(6*log(2)))*log(x)),x)

[Out]

log(x - exp(-1/(6*log(2))) - 2*exp(x) + log(x))

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sympy [A]  time = 0.40, size = 37, normalized size = 1.68 \begin {gather*} \log {\left (\frac {- x e^{\frac {1}{6 \log {\relax (2 )}}} - e^{\frac {1}{6 \log {\relax (2 )}}} \log {\relax (x )} + 1}{2 e^{\frac {1}{6 \log {\relax (2 )}}}} + e^{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*exp(1/6/ln(2))*exp(x)+(x+1)*exp(1/6/ln(2)))/(x*exp(1/6/ln(2))*ln(x)-2*x*exp(1/6/ln(2))*exp(x)+
x**2*exp(1/6/ln(2))-x),x)

[Out]

log((-x*exp(1/(6*log(2))) - exp(1/(6*log(2)))*log(x) + 1)*exp(-1/(6*log(2)))/2 + exp(x))

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