3.73.72 \(\int \frac {e^x (-16+4 x)+x^{\frac {1}{x}} (x^2-x^3-x^2 \log (x))}{4 e^x x+2 x^5+x^{4+\frac {1}{x}}} \, dx\)

Optimal. Leaf size=24 \[ 5+\log \left (\frac {\frac {4 e^x}{x^3}+2 x+x^{\frac {1}{x}}}{x}\right ) \]

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Rubi [F]  time = 1.91, 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 {e^x (-16+4 x)+x^{\frac {1}{x}} \left (x^2-x^3-x^2 \log (x)\right )}{4 e^x x+2 x^5+x^{4+\frac {1}{x}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x*(-16 + 4*x) + x^x^(-1)*(x^2 - x^3 - x^2*Log[x]))/(4*E^x*x + 2*x^5 + x^(4 + x^(-1))),x]

[Out]

-Log[x] + Log[x]/x + 4*Defer[Int][E^x/(4*E^x + 2*x^4 + x^(3 + x^(-1))), x] - 4*Defer[Int][E^x/(x^2*(4*E^x + 2*
x^4 + x^(3 + x^(-1)))), x] + 4*Log[x]*Defer[Int][E^x/(x^2*(4*E^x + 2*x^4 + x^(3 + x^(-1)))), x] - 12*Defer[Int
][E^x/(x*(4*E^x + 2*x^4 + x^(3 + x^(-1)))), x] - 2*Defer[Int][x^2/(4*E^x + 2*x^4 + x^(3 + x^(-1))), x] + 2*Log
[x]*Defer[Int][x^2/(4*E^x + 2*x^4 + x^(3 + x^(-1))), x] + 2*Defer[Int][x^3/(4*E^x + 2*x^4 + x^(3 + x^(-1))), x
] - 2*Defer[Int][Defer[Int][x^2/(4*E^x + 2*x^4 + x^(3 + x^(-1))), x]/x, x] - 4*Defer[Int][Defer[Int][E^x/(4*E^
x*x^2 + 2*x^6 + x^(5 + x^(-1))), x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 e^x (-4+x)-x^{2+\frac {1}{x}} (-1+x+\log (x))}{4 e^x x+2 x^5+x^{4+\frac {1}{x}}} \, dx\\ &=\int \left (\frac {1-x-\log (x)}{x^2}+\frac {2 \left (-2 e^x-6 e^x x+2 e^x x^2-x^4+x^5+2 e^x \log (x)+x^4 \log (x)\right )}{x^2 \left (4 e^x+2 x^4+x^{3+\frac {1}{x}}\right )}\right ) \, dx\\ &=2 \int \frac {-2 e^x-6 e^x x+2 e^x x^2-x^4+x^5+2 e^x \log (x)+x^4 \log (x)}{x^2 \left (4 e^x+2 x^4+x^{3+\frac {1}{x}}\right )} \, dx+\int \frac {1-x-\log (x)}{x^2} \, dx\\ &=2 \int \left (\frac {2 e^x}{4 e^x+2 x^4+x^{3+\frac {1}{x}}}-\frac {2 e^x}{x^2 \left (4 e^x+2 x^4+x^{3+\frac {1}{x}}\right )}-\frac {6 e^x}{x \left (4 e^x+2 x^4+x^{3+\frac {1}{x}}\right )}-\frac {x^2}{4 e^x+2 x^4+x^{3+\frac {1}{x}}}+\frac {x^3}{4 e^x+2 x^4+x^{3+\frac {1}{x}}}+\frac {2 e^x \log (x)}{x^2 \left (4 e^x+2 x^4+x^{3+\frac {1}{x}}\right )}+\frac {x^2 \log (x)}{4 e^x+2 x^4+x^{3+\frac {1}{x}}}\right ) \, dx+\int \left (\frac {1-x}{x^2}-\frac {\log (x)}{x^2}\right ) \, dx\\ &=-\left (2 \int \frac {x^2}{4 e^x+2 x^4+x^{3+\frac {1}{x}}} \, dx\right )+2 \int \frac {x^3}{4 e^x+2 x^4+x^{3+\frac {1}{x}}} \, dx+2 \int \frac {x^2 \log (x)}{4 e^x+2 x^4+x^{3+\frac {1}{x}}} \, dx+4 \int \frac {e^x}{4 e^x+2 x^4+x^{3+\frac {1}{x}}} \, dx-4 \int \frac {e^x}{x^2 \left (4 e^x+2 x^4+x^{3+\frac {1}{x}}\right )} \, dx+4 \int \frac {e^x \log (x)}{x^2 \left (4 e^x+2 x^4+x^{3+\frac {1}{x}}\right )} \, dx-12 \int \frac {e^x}{x \left (4 e^x+2 x^4+x^{3+\frac {1}{x}}\right )} \, dx+\int \frac {1-x}{x^2} \, dx-\int \frac {\log (x)}{x^2} \, dx\\ &=\frac {1}{x}+\frac {\log (x)}{x}-2 \int \frac {x^2}{4 e^x+2 x^4+x^{3+\frac {1}{x}}} \, dx+2 \int \frac {x^3}{4 e^x+2 x^4+x^{3+\frac {1}{x}}} \, dx-2 \int \frac {\int \frac {x^2}{4 e^x+2 x^4+x^{3+\frac {1}{x}}} \, dx}{x} \, dx+4 \int \frac {e^x}{4 e^x+2 x^4+x^{3+\frac {1}{x}}} \, dx-4 \int \frac {e^x}{x^2 \left (4 e^x+2 x^4+x^{3+\frac {1}{x}}\right )} \, dx-4 \int \frac {\int \frac {e^x}{4 e^x x^2+2 x^6+x^{5+\frac {1}{x}}} \, dx}{x} \, dx-12 \int \frac {e^x}{x \left (4 e^x+2 x^4+x^{3+\frac {1}{x}}\right )} \, dx+(2 \log (x)) \int \frac {x^2}{4 e^x+2 x^4+x^{3+\frac {1}{x}}} \, dx+(4 \log (x)) \int \frac {e^x}{x^2 \left (4 e^x+2 x^4+x^{3+\frac {1}{x}}\right )} \, dx+\int \left (\frac {1}{x^2}-\frac {1}{x}\right ) \, dx\\ &=-\log (x)+\frac {\log (x)}{x}-2 \int \frac {x^2}{4 e^x+2 x^4+x^{3+\frac {1}{x}}} \, dx+2 \int \frac {x^3}{4 e^x+2 x^4+x^{3+\frac {1}{x}}} \, dx-2 \int \frac {\int \frac {x^2}{4 e^x+2 x^4+x^{3+\frac {1}{x}}} \, dx}{x} \, dx+4 \int \frac {e^x}{4 e^x+2 x^4+x^{3+\frac {1}{x}}} \, dx-4 \int \frac {e^x}{x^2 \left (4 e^x+2 x^4+x^{3+\frac {1}{x}}\right )} \, dx-4 \int \frac {\int \frac {e^x}{4 e^x x^2+2 x^6+x^{5+\frac {1}{x}}} \, dx}{x} \, dx-12 \int \frac {e^x}{x \left (4 e^x+2 x^4+x^{3+\frac {1}{x}}\right )} \, dx+(2 \log (x)) \int \frac {x^2}{4 e^x+2 x^4+x^{3+\frac {1}{x}}} \, dx+(4 \log (x)) \int \frac {e^x}{x^2 \left (4 e^x+2 x^4+x^{3+\frac {1}{x}}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.42, size = 24, normalized size = 1.00 \begin {gather*} -4 \log (x)+\log \left (4 e^x+2 x^4+x^{3+\frac {1}{x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(-16 + 4*x) + x^x^(-1)*(x^2 - x^3 - x^2*Log[x]))/(4*E^x*x + 2*x^5 + x^(4 + x^(-1))),x]

[Out]

-4*Log[x] + Log[4*E^x + 2*x^4 + x^(3 + x^(-1))]

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fricas [A]  time = 0.77, size = 29, normalized size = 1.21 \begin {gather*} -\log \relax (x) + \log \left (\frac {x^{3} x^{\left (\frac {1}{x}\right )} + 2 \, x^{4} + 4 \, e^{x}}{x^{3}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2*log(x)-x^3+x^2)*exp(log(x)/x)+(4*x-16)*exp(x))/(x^4*exp(log(x)/x)+4*exp(x)*x+2*x^5),x, algori
thm="fricas")

[Out]

-log(x) + log((x^3*x^(1/x) + 2*x^4 + 4*e^x)/x^3)

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giac [B]  time = 0.29, size = 47, normalized size = 1.96 \begin {gather*} \frac {x \log \left (x^{3} x^{\left (\frac {1}{x}\right )} + 2 \, x^{4} + 4 \, e^{x}\right ) - 3 \, x \log \relax (x) - \log \relax (x)}{x} + \frac {\log \relax (x)}{x} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2*log(x)-x^3+x^2)*exp(log(x)/x)+(4*x-16)*exp(x))/(x^4*exp(log(x)/x)+4*exp(x)*x+2*x^5),x, algori
thm="giac")

[Out]

(x*log(x^3*x^(1/x) + 2*x^4 + 4*e^x) - 3*x*log(x) - log(x))/x + log(x)/x - log(x)

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maple [A]  time = 0.03, size = 26, normalized size = 1.08

method	result	size
risch	\(-\ln \relax (x )+\ln \left (x^{\frac {1}{x}}+\frac {2 x^{4}+4 \,{\mathrm e}^{x}}{x^{3}}\right )\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^2*ln(x)-x^3+x^2)*exp(ln(x)/x)+(4*x-16)*exp(x))/(x^4*exp(ln(x)/x)+4*exp(x)*x+2*x^5),x,method=_RETURNVE
RBOSE)

[Out]

-ln(x)+ln(x^(1/x)+2*(x^4+2*exp(x))/x^3)

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maxima [A]  time = 0.40, size = 29, normalized size = 1.21 \begin {gather*} -\log \relax (x) + \log \left (\frac {x^{3} x^{\left (\frac {1}{x}\right )} + 2 \, x^{4} + 4 \, e^{x}}{x^{3}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2*log(x)-x^3+x^2)*exp(log(x)/x)+(4*x-16)*exp(x))/(x^4*exp(log(x)/x)+4*exp(x)*x+2*x^5),x, algori
thm="maxima")

[Out]

-log(x) + log((x^3*x^(1/x) + 2*x^4 + 4*e^x)/x^3)

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mupad [B]  time = 4.51, size = 29, normalized size = 1.21 \begin {gather*} \ln \left (\frac {4\,{\mathrm {e}}^x+2\,x^4+x^{1/x}\,x^3}{x^3}\right )-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(log(x)/x)*(x^2*log(x) - x^2 + x^3) - exp(x)*(4*x - 16))/(4*x*exp(x) + x^4*exp(log(x)/x) + 2*x^5),x)

[Out]

log((4*exp(x) + 2*x^4 + x^(1/x)*x^3)/x^3) - log(x)

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sympy [A]  time = 0.52, size = 24, normalized size = 1.00 \begin {gather*} - \log {\relax (x )} + \log {\left (e^{\frac {\log {\relax (x )}}{x}} + \frac {2 x^{4} + 4 e^{x}}{x^{3}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**2*ln(x)-x**3+x**2)*exp(ln(x)/x)+(4*x-16)*exp(x))/(x**4*exp(ln(x)/x)+4*exp(x)*x+2*x**5),x)

[Out]

-log(x) + log(exp(log(x)/x) + (2*x**4 + 4*exp(x))/x**3)

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