3.87.42 \(\int \frac {-2 e^{-6-x}+2 x+(-x-e^{-6-x} x) \log (196 x^2)}{(e^{-6-x} x-x^2) \log (196 x^2)} \, dx\)

Optimal. Leaf size=21 \[ \log \left (\frac {e^{-6-x}-x}{\log \left (196 x^2\right )}\right ) \]

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Rubi [F]  time = 0.86, 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^{-6-x}+2 x+\left (-x-e^{-6-x} x\right ) \log \left (196 x^2\right )}{\left (e^{-6-x} x-x^2\right ) \log \left (196 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2*E^(-6 - x) + 2*x + (-x - E^(-6 - x)*x)*Log[196*x^2])/((E^(-6 - x)*x - x^2)*Log[196*x^2]),x]

[Out]

Log[x] - Log[Log[196*x^2]] + Defer[Int][(-1 + E^(6 + x)*x)^(-1), x] + Defer[Int][1/(x*(-1 + E^(6 + x)*x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1+x}{x \left (-1+e^{6+x} x\right )}+\frac {-2+\log \left (196 x^2\right )}{x \log \left (196 x^2\right )}\right ) \, dx\\ &=\int \frac {1+x}{x \left (-1+e^{6+x} x\right )} \, dx+\int \frac {-2+\log \left (196 x^2\right )}{x \log \left (196 x^2\right )} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {-2+x}{x} \, dx,x,\log \left (196 x^2\right )\right )+\int \left (\frac {1}{-1+e^{6+x} x}+\frac {1}{x \left (-1+e^{6+x} x\right )}\right ) \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (1-\frac {2}{x}\right ) \, dx,x,\log \left (196 x^2\right )\right )+\int \frac {1}{-1+e^{6+x} x} \, dx+\int \frac {1}{x \left (-1+e^{6+x} x\right )} \, dx\\ &=\log (x)-\log \left (\log \left (196 x^2\right )\right )+\int \frac {1}{-1+e^{6+x} x} \, dx+\int \frac {1}{x \left (-1+e^{6+x} x\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 24, normalized size = 1.14 \begin {gather*} -x+\log \left (1-e^{6+x} x\right )-\log \left (\log \left (196 x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*E^(-6 - x) + 2*x + (-x - E^(-6 - x)*x)*Log[196*x^2])/((E^(-6 - x)*x - x^2)*Log[196*x^2]),x]

[Out]

-x + Log[1 - E^(6 + x)*x] - Log[Log[196*x^2]]

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fricas [A]  time = 1.16, size = 21, normalized size = 1.00 \begin {gather*} \log \left (-x + e^{\left (-x - 6\right )}\right ) - \log \left (\log \left (196 \, x^{2}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x*exp(-x-6)-x)*log(196*x^2)-2*exp(-x-6)+2*x)/(x*exp(-x-6)-x^2)/log(196*x^2),x, algorithm="fricas"
)

[Out]

log(-x + e^(-x - 6)) - log(log(196*x^2))

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giac [A]  time = 0.21, size = 21, normalized size = 1.00 \begin {gather*} \log \left (-x e^{6} + e^{\left (-x\right )}\right ) - \log \left (\log \left (196 \, x^{2}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x*exp(-x-6)-x)*log(196*x^2)-2*exp(-x-6)+2*x)/(x*exp(-x-6)-x^2)/log(196*x^2),x, algorithm="giac")

[Out]

log(-x*e^6 + e^(-x)) - log(log(196*x^2))

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maple [A]  time = 0.14, size = 22, normalized size = 1.05

method	result	size
default	\(-\ln \left (\ln \left (196 x^{2}\right )\right )+\ln \left (x -{\mathrm e}^{-x -6}\right )\)	\(22\)
norman	\(-\ln \left (\ln \left (196 x^{2}\right )\right )+\ln \left (x -{\mathrm e}^{-x -6}\right )\)	\(22\)
risch	\(6+\ln \left ({\mathrm e}^{-x -6}-x \right )-\ln \left (\ln \relax (x )-\frac {i \left (\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (2)+4 i \ln \relax (7)\right )}{4}\right )\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x*exp(-x-6)-x)*ln(196*x^2)-2*exp(-x-6)+2*x)/(x*exp(-x-6)-x^2)/ln(196*x^2),x,method=_RETURNVERBOSE)

[Out]

-ln(ln(196*x^2))+ln(x-exp(-x-6))

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maxima [A]  time = 0.51, size = 31, normalized size = 1.48 \begin {gather*} -x + \log \relax (x) + \log \left (\frac {{\left (x e^{\left (x + 6\right )} - 1\right )} e^{\left (-6\right )}}{x}\right ) - \log \left (\log \relax (7) + \log \relax (2) + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x*exp(-x-6)-x)*log(196*x^2)-2*exp(-x-6)+2*x)/(x*exp(-x-6)-x^2)/log(196*x^2),x, algorithm="maxima"
)

[Out]

-x + log(x) + log((x*e^(x + 6) - 1)*e^(-6)/x) - log(log(7) + log(2) + log(x))

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mupad [B]  time = 5.39, size = 21, normalized size = 1.00 \begin {gather*} \ln \left (x-{\mathrm {e}}^{-x-6}\right )-\ln \left (\ln \left (196\,x^2\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*exp(- x - 6) - 2*x + log(196*x^2)*(x + x*exp(- x - 6)))/(log(196*x^2)*(x*exp(- x - 6) - x^2)),x)

[Out]

log(x - exp(- x - 6)) - log(log(196*x^2))

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sympy [A]  time = 0.31, size = 17, normalized size = 0.81 \begin {gather*} \log {\left (- x + e^{- x - 6} \right )} - \log {\left (\log {\left (196 x^{2} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x*exp(-x-6)-x)*ln(196*x**2)-2*exp(-x-6)+2*x)/(x*exp(-x-6)-x**2)/ln(196*x**2),x)

[Out]

log(-x + exp(-x - 6)) - log(log(196*x**2))

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