3.16.19 \(\int \frac {-7 x+e^x (-x+x^2)-2 \log (x)+2 \log ^2(x)}{-7 x^2-e^x x^2+x \log ^2(x)} \, dx\)

Optimal. Leaf size=19 \[ \log \left (\frac {x}{7+e^x-\frac {\log ^2(x)}{x}}\right ) \]

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Rubi [F]  time = 1.43, 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 {-7 x+e^x \left (-x+x^2\right )-2 \log (x)+2 \log ^2(x)}{-7 x^2-e^x x^2+x \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-7*x + E^x*(-x + x^2) - 2*Log[x] + 2*Log[x]^2)/(-7*x^2 - E^x*x^2 + x*Log[x]^2),x]

[Out]

-x + Log[x] + 7*Defer[Int][x/(7*x + E^x*x - Log[x]^2), x] + 2*Defer[Int][Log[x]/(x*(7*x + E^x*x - Log[x]^2)),
x] - Defer[Int][Log[x]^2/(x*(7*x + E^x*x - Log[x]^2)), x] + Defer[Int][Log[x]^2/(-7*x - E^x*x + Log[x]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1-x}{x}+\frac {7 x^2+2 \log (x)-\log ^2(x)-x \log ^2(x)}{x \left (7 x+e^x x-\log ^2(x)\right )}\right ) \, dx\\ &=\int \frac {1-x}{x} \, dx+\int \frac {7 x^2+2 \log (x)-\log ^2(x)-x \log ^2(x)}{x \left (7 x+e^x x-\log ^2(x)\right )} \, dx\\ &=\int \left (-1+\frac {1}{x}\right ) \, dx+\int \left (\frac {7 x}{7 x+e^x x-\log ^2(x)}+\frac {2 \log (x)}{x \left (7 x+e^x x-\log ^2(x)\right )}-\frac {\log ^2(x)}{x \left (7 x+e^x x-\log ^2(x)\right )}+\frac {\log ^2(x)}{-7 x-e^x x+\log ^2(x)}\right ) \, dx\\ &=-x+\log (x)+2 \int \frac {\log (x)}{x \left (7 x+e^x x-\log ^2(x)\right )} \, dx+7 \int \frac {x}{7 x+e^x x-\log ^2(x)} \, dx-\int \frac {\log ^2(x)}{x \left (7 x+e^x x-\log ^2(x)\right )} \, dx+\int \frac {\log ^2(x)}{-7 x-e^x x+\log ^2(x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.33, size = 23, normalized size = 1.21 \begin {gather*} 2 \log (x)-\log \left (7 x+e^x x-\log ^2(x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-7*x + E^x*(-x + x^2) - 2*Log[x] + 2*Log[x]^2)/(-7*x^2 - E^x*x^2 + x*Log[x]^2),x]

[Out]

2*Log[x] - Log[7*x + E^x*x - Log[x]^2]

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fricas [A]  time = 0.59, size = 21, normalized size = 1.11 \begin {gather*} -\log \left (-x e^{x} + \log \relax (x)^{2} - 7 \, x\right ) + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(x)^2-2*log(x)+(x^2-x)*exp(x)-7*x)/(x*log(x)^2-exp(x)*x^2-7*x^2),x, algorithm="fricas")

[Out]

-log(-x*e^x + log(x)^2 - 7*x) + 2*log(x)

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giac [A]  time = 0.33, size = 21, normalized size = 1.11 \begin {gather*} -\log \left (-x e^{x} + \log \relax (x)^{2} - 7 \, x\right ) + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(x)^2-2*log(x)+(x^2-x)*exp(x)-7*x)/(x*log(x)^2-exp(x)*x^2-7*x^2),x, algorithm="giac")

[Out]

-log(-x*e^x + log(x)^2 - 7*x) + 2*log(x)

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maple [A]  time = 0.03, size = 22, normalized size = 1.16

method	result	size
risch	\(2 \ln \relax (x )-\ln \left (\ln \relax (x )^{2}-{\mathrm e}^{x} x -7 x \right )\)	\(22\)
norman	\(2 \ln \relax (x )-\ln \left ({\mathrm e}^{x} x -\ln \relax (x )^{2}+7 x \right )\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*ln(x)^2-2*ln(x)+(x^2-x)*exp(x)-7*x)/(x*ln(x)^2-exp(x)*x^2-7*x^2),x,method=_RETURNVERBOSE)

[Out]

2*ln(x)-ln(ln(x)^2-exp(x)*x-7*x)

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maxima [A]  time = 0.71, size = 24, normalized size = 1.26 \begin {gather*} \log \relax (x) - \log \left (\frac {x e^{x} - \log \relax (x)^{2} + 7 \, x}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(x)^2-2*log(x)+(x^2-x)*exp(x)-7*x)/(x*log(x)^2-exp(x)*x^2-7*x^2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.15, size = 22, normalized size = 1.16 \begin {gather*} 2\,\ln \relax (x)-\ln \left (-{\ln \relax (x)}^2+7\,x+x\,{\mathrm {e}}^x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((7*x + 2*log(x) - 2*log(x)^2 + exp(x)*(x - x^2))/(x^2*exp(x) - x*log(x)^2 + 7*x^2),x)

[Out]

2*log(x) - log(7*x - log(x)^2 + x*exp(x))

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sympy [A]  time = 0.35, size = 17, normalized size = 0.89 \begin {gather*} \log {\relax (x )} - \log {\left (e^{x} + \frac {7 x - \log {\relax (x )}^{2}}{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*ln(x)**2-2*ln(x)+(x**2-x)*exp(x)-7*x)/(x*ln(x)**2-exp(x)*x**2-7*x**2),x)

[Out]

log(x) - log(exp(x) + (7*x - log(x)**2)/x)

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