3.71.70 \(\int \frac {-8 e^x+8 x+4 x^2-2 x^3+x^4+(-4 x^2+x^4+e^x (4-x^2)) \log (\frac {4-x^2}{x})}{4 x^2-x^4+e^x (-4+x^2)} \, dx\)

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

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Rubi [F]  time = 0.94, 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 {-8 e^x+8 x+4 x^2-2 x^3+x^4+\left (-4 x^2+x^4+e^x \left (4-x^2\right )\right ) \log \left (\frac {4-x^2}{x}\right )}{4 x^2-x^4+e^x \left (-4+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-8*E^x + 8*x + 4*x^2 - 2*x^3 + x^4 + (-4*x^2 + x^4 + E^x*(4 - x^2))*Log[(4 - x^2)/x])/(4*x^2 - x^4 + E^x*
(-4 + x^2)),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 e^x-8 x-4 x^2+2 x^3-x^4-\left (-4 x^2+x^4+e^x \left (4-x^2\right )\right ) \log \left (\frac {4-x^2}{x}\right )}{\left (4-x^2\right ) \left (e^x-x^2\right )} \, dx\\ &=\int \left (-\frac {(-2+x) x}{-e^x+x^2}+\frac {-8+4 \log \left (\frac {4}{x}-x\right )-x^2 \log \left (\frac {4}{x}-x\right )}{-4+x^2}\right ) \, dx\\ &=-\int \frac {(-2+x) x}{-e^x+x^2} \, dx+\int \frac {-8+4 \log \left (\frac {4}{x}-x\right )-x^2 \log \left (\frac {4}{x}-x\right )}{-4+x^2} \, dx\\ &=-\int \left (-\frac {2 x}{-e^x+x^2}+\frac {x^2}{-e^x+x^2}\right ) \, dx+\int \left (-\frac {8}{-4+x^2}-\log \left (\frac {4-x^2}{x}\right )\right ) \, dx\\ &=2 \int \frac {x}{-e^x+x^2} \, dx-8 \int \frac {1}{-4+x^2} \, dx-\int \frac {x^2}{-e^x+x^2} \, dx-\int \log \left (\frac {4-x^2}{x}\right ) \, dx\\ &=4 \tanh ^{-1}\left (\frac {x}{2}\right )-x \log \left (\frac {4-x^2}{x}\right )+2 \int \frac {x}{-e^x+x^2} \, dx+\int \frac {-4-x^2}{4-x^2} \, dx-\int \frac {x^2}{-e^x+x^2} \, dx\\ &=x+4 \tanh ^{-1}\left (\frac {x}{2}\right )-x \log \left (\frac {4-x^2}{x}\right )+2 \int \frac {x}{-e^x+x^2} \, dx-8 \int \frac {1}{4-x^2} \, dx-\int \frac {x^2}{-e^x+x^2} \, dx\\ &=x-x \log \left (\frac {4-x^2}{x}\right )+2 \int \frac {x}{-e^x+x^2} \, dx-\int \frac {x^2}{-e^x+x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.20, size = 24, normalized size = 1.00 \begin {gather*} -x \log \left (\frac {4}{x}-x\right )+\log \left (e^x-x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8*E^x + 8*x + 4*x^2 - 2*x^3 + x^4 + (-4*x^2 + x^4 + E^x*(4 - x^2))*Log[(4 - x^2)/x])/(4*x^2 - x^4
+ E^x*(-4 + x^2)),x]

[Out]

-(x*Log[4/x - x]) + Log[E^x - x^2]

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fricas [A]  time = 0.92, size = 24, normalized size = 1.00 \begin {gather*} -x \log \left (-\frac {x^{2} - 4}{x}\right ) + \log \left (-x^{2} + e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2+4)*exp(x)+x^4-4*x^2)*log((-x^2+4)/x)-8*exp(x)+x^4-2*x^3+4*x^2+8*x)/((x^2-4)*exp(x)-x^4+4*x^2
),x, algorithm="fricas")

[Out]

-x*log(-(x^2 - 4)/x) + log(-x^2 + e^x)

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giac [A]  time = 0.16, size = 24, normalized size = 1.00 \begin {gather*} -x \log \left (-\frac {x^{2} - 4}{x}\right ) + \log \left (x^{2} - e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2+4)*exp(x)+x^4-4*x^2)*log((-x^2+4)/x)-8*exp(x)+x^4-2*x^3+4*x^2+8*x)/((x^2-4)*exp(x)-x^4+4*x^2
),x, algorithm="giac")

[Out]

-x*log(-(x^2 - 4)/x) + log(x^2 - e^x)

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maple [A]  time = 0.13, size = 26, normalized size = 1.08

method	result	size
norman	\(-x \ln \left (\frac {-x^{2}+4}{x}\right )+\ln \left (-{\mathrm e}^{x}+x^{2}\right )\)	\(26\)
risch	\(-x \ln \left (x^{2}-4\right )+x \ln \relax (x )+\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (x^{2}-4\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}-4\right )}{x}\right )}{2}-\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}-4\right )}{x}\right )^{2}}{2}+i \pi x \mathrm {csgn}\left (\frac {i \left (x^{2}-4\right )}{x}\right )^{2}-\frac {i \pi x \,\mathrm {csgn}\left (i \left (x^{2}-4\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}-4\right )}{x}\right )^{2}}{2}-\frac {i \pi x \mathrm {csgn}\left (\frac {i \left (x^{2}-4\right )}{x}\right )^{3}}{2}-i \pi x +\ln \left ({\mathrm e}^{x}-x^{2}\right )\)	\(154\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x^2+4)*exp(x)+x^4-4*x^2)*ln((-x^2+4)/x)-8*exp(x)+x^4-2*x^3+4*x^2+8*x)/((x^2-4)*exp(x)-x^4+4*x^2),x,met
hod=_RETURNVERBOSE)

[Out]

-x*ln((-x^2+4)/x)+ln(-exp(x)+x^2)

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maxima [A]  time = 0.42, size = 30, normalized size = 1.25 \begin {gather*} -x \log \left (x + 2\right ) + x \log \relax (x) - x \log \left (-x + 2\right ) + \log \left (-x^{2} + e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2+4)*exp(x)+x^4-4*x^2)*log((-x^2+4)/x)-8*exp(x)+x^4-2*x^3+4*x^2+8*x)/((x^2-4)*exp(x)-x^4+4*x^2
),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 4.39, size = 24, normalized size = 1.00 \begin {gather*} \ln \left ({\mathrm {e}}^x-x^2\right )-x\,\ln \left (-\frac {x^2-4}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x - 8*exp(x) - log(-(x^2 - 4)/x)*(exp(x)*(x^2 - 4) + 4*x^2 - x^4) + 4*x^2 - 2*x^3 + x^4)/(exp(x)*(x^2 -
4) + 4*x^2 - x^4),x)

[Out]

log(exp(x) - x^2) - x*log(-(x^2 - 4)/x)

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sympy [A]  time = 0.39, size = 17, normalized size = 0.71 \begin {gather*} - x \log {\left (\frac {4 - x^{2}}{x} \right )} + \log {\left (- x^{2} + e^{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x**2+4)*exp(x)+x**4-4*x**2)*ln((-x**2+4)/x)-8*exp(x)+x**4-2*x**3+4*x**2+8*x)/((x**2-4)*exp(x)-x*
*4+4*x**2),x)

[Out]

-x*log((4 - x**2)/x) + log(-x**2 + exp(x))

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