3.24.12 \(\int \frac {e^x (2-x)+x}{(e^x x+x^2) \log (\frac {20 e^x+20 x}{x^2})} \, dx\)

Optimal. Leaf size=16 \[ -1-\log \left (\log \left (\frac {20 \left (e^x+x\right )}{x^2}\right )\right ) \]

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Rubi [A]  time = 0.12, antiderivative size = 14, normalized size of antiderivative = 0.88, number of steps used = 1, number of rules used = 1, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {6684} \begin {gather*} -\log \left (\log \left (\frac {20 \left (x+e^x\right )}{x^2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^x*(2 - x) + x)/((E^x*x + x^2)*Log[(20*E^x + 20*x)/x^2]),x]

[Out]

-Log[Log[(20*(E^x + x))/x^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 (\log \left (\frac {20 \left (e^x+x\right )}{x^2}\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 14, normalized size = 0.88 \begin {gather*} -\log \left (\log \left (\frac {20 \left (e^x+x\right )}{x^2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(2 - x) + x)/((E^x*x + x^2)*Log[(20*E^x + 20*x)/x^2]),x]

[Out]

-Log[Log[(20*(E^x + x))/x^2]]

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fricas [A]  time = 0.91, size = 13, normalized size = 0.81 \begin {gather*} -\log \left (\log \left (\frac {20 \, {\left (x + e^{x}\right )}}{x^{2}}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2-x)*exp(x)+x)/(exp(x)*x+x^2)/log((20*exp(x)+20*x)/x^2),x, algorithm="fricas")

[Out]

-log(log(20*(x + e^x)/x^2))

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giac [A]  time = 0.22, size = 13, normalized size = 0.81 \begin {gather*} -\log \left (\log \left (\frac {20 \, {\left (x + e^{x}\right )}}{x^{2}}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2-x)*exp(x)+x)/(exp(x)*x+x^2)/log((20*exp(x)+20*x)/x^2),x, algorithm="giac")

[Out]

-log(log(20*(x + e^x)/x^2))

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maple [A]  time = 0.13, size = 17, normalized size = 1.06

method	result	size
norman	\(-\ln \left (\ln \left (\frac {20 \,{\mathrm e}^{x}+20 x}{x^{2}}\right )\right )\)	\(17\)
risch	\(-\ln \left (\ln \left ({\mathrm e}^{x}+x \right )-\frac {i \left (\pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x^{2}}\right )^{3}-\pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x^{2}}\right )^{2} \mathrm {csgn}\left (\frac {i}{x^{2}}\right )-\pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x^{2}}\right )^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right )+\pi \,\mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right )-\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 i \ln \relax (5)+4 i \ln \relax (2)-4 i \ln \relax (x )\right )}{2}\right )\)	\(165\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2-x)*exp(x)+x)/(exp(x)*x+x^2)/ln((20*exp(x)+20*x)/x^2),x,method=_RETURNVERBOSE)

[Out]

-ln(ln((20*exp(x)+20*x)/x^2))

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maxima [A]  time = 0.81, size = 19, normalized size = 1.19 \begin {gather*} -\log \left (\log \relax (5) + 2 \, \log \relax (2) + \log \left (x + e^{x}\right ) - 2 \, \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2-x)*exp(x)+x)/(exp(x)*x+x^2)/log((20*exp(x)+20*x)/x^2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.81, size = 16, normalized size = 1.00 \begin {gather*} -\ln \left (\ln \left (\frac {20\,x+20\,{\mathrm {e}}^x}{x^2}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - exp(x)*(x - 2))/(log((20*x + 20*exp(x))/x^2)*(x*exp(x) + x^2)),x)

[Out]

-log(log((20*x + 20*exp(x))/x^2))

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sympy [A]  time = 0.26, size = 15, normalized size = 0.94 \begin {gather*} - \log {\left (\log {\left (\frac {20 x + 20 e^{x}}{x^{2}} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2-x)*exp(x)+x)/(exp(x)*x+x**2)/ln((20*exp(x)+20*x)/x**2),x)

[Out]

-log(log((20*x + 20*exp(x))/x**2))

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