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

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

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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 = {} $$\begin{array}{c}\int \frac{e^x(-16+4x)+x^{\frac{1}{x}}(x^2-x^3-x^2\log (x))}{4e^{x}x+2x^5+x^{4+\frac{1}{x}}}dx\end{array}$$

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{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{4e^x(-4+x)-x^{2+\frac{1}{x}}(-1+x+\log (x))}{4e^{x}x+2x^5+x^{4+\frac{1}{x}}}dx\\ & =\int (\frac{1-x-\log (x)}{x^2}+\frac{2(-2e^x-6e^{x}x+2e^x{x}^2-x^4+x^5+2e^x\log (x)+x^4\log (x))}{x^2(4e^x+2x^4+x^{3+\frac{1}{x}})})dx\\ & =2\int \frac{-2e^x-6e^{x}x+2e^x{x}^2-x^4+x^5+2e^x\log (x)+x^4\log (x)}{x^2(4e^x+2x^4+x^{3+\frac{1}{x}})}dx+\int \frac{1-x-\log (x)}{x^2}dx\\ & =2\int (\frac{2e^x}{4e^x+2x^4+x^{3+\frac{1}{x}}}-\frac{2e^x}{x^2(4e^x+2x^4+x^{3+\frac{1}{x}})}-\frac{6e^x}{x(4e^x+2x^4+x^{3+\frac{1}{x}})}-\frac{x^2}{4e^x+2x^4+x^{3+\frac{1}{x}}}+\frac{x^3}{4e^x+2x^4+x^{3+\frac{1}{x}}}+\frac{2e^x\log (x)}{x^2(4e^x+2x^4+x^{3+\frac{1}{x}})}+\frac{x^2\log (x)}{4e^x+2x^4+x^{3+\frac{1}{x}}})dx+\int (\frac{1-x}{x^2}-\frac{\log (x)}{x^2})dx\\ & =-(2\int \frac{x^2}{4e^x+2x^4+x^{3+\frac{1}{x}}}dx)+2\int \frac{x^3}{4e^x+2x^4+x^{3+\frac{1}{x}}}dx+2\int \frac{x^2\log (x)}{4e^x+2x^4+x^{3+\frac{1}{x}}}dx+4\int \frac{e^x}{4e^x+2x^4+x^{3+\frac{1}{x}}}dx-4\int \frac{e^x}{x^2(4e^x+2x^4+x^{3+\frac{1}{x}})}dx+4\int \frac{e^x\log (x)}{x^2(4e^x+2x^4+x^{3+\frac{1}{x}})}dx-12\int \frac{e^x}{x(4e^x+2x^4+x^{3+\frac{1}{x}})}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}{4e^x+2x^4+x^{3+\frac{1}{x}}}dx+2\int \frac{x^3}{4e^x+2x^4+x^{3+\frac{1}{x}}}dx-2\int \frac{\int \frac{x^2}{4e^x+2x^4+x^{3+\frac{1}{x}}}dx}{x}dx+4\int \frac{e^x}{4e^x+2x^4+x^{3+\frac{1}{x}}}dx-4\int \frac{e^x}{x^2(4e^x+2x^4+x^{3+\frac{1}{x}})}dx-4\int \frac{\int \frac{e^x}{4e^x{x}^2+2x^6+x^{5+\frac{1}{x}}}dx}{x}dx-12\int \frac{e^x}{x(4e^x+2x^4+x^{3+\frac{1}{x}})}dx+(2\log (x))\int \frac{x^2}{4e^x+2x^4+x^{3+\frac{1}{x}}}dx+(4\log (x))\int \frac{e^x}{x^2(4e^x+2x^4+x^{3+\frac{1}{x}})}dx+\int (\frac{1}{x^2}-\frac{1}{x})dx\\ & =-\log (x)+\frac{\log (x)}{x}-2\int \frac{x^2}{4e^x+2x^4+x^{3+\frac{1}{x}}}dx+2\int \frac{x^3}{4e^x+2x^4+x^{3+\frac{1}{x}}}dx-2\int \frac{\int \frac{x^2}{4e^x+2x^4+x^{3+\frac{1}{x}}}dx}{x}dx+4\int \frac{e^x}{4e^x+2x^4+x^{3+\frac{1}{x}}}dx-4\int \frac{e^x}{x^2(4e^x+2x^4+x^{3+\frac{1}{x}})}dx-4\int \frac{\int \frac{e^x}{4e^x{x}^2+2x^6+x^{5+\frac{1}{x}}}dx}{x}dx-12\int \frac{e^x}{x(4e^x+2x^4+x^{3+\frac{1}{x}})}dx+(2\log (x))\int \frac{x^2}{4e^x+2x^4+x^{3+\frac{1}{x}}}dx+(4\log (x))\int \frac{e^x}{x^2(4e^x+2x^4+x^{3+\frac{1}{x}})}dx\end{array}\end{array}$$

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

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 $$\begin{array}{c}-\log (x)+\log \left(\frac{x^3{x}^{\left(\frac{1}{x}\right)}+2x^4+4e^x}{x^3}\right)\end{array}$$

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 $$\begin{array}{c}\frac{x\log (x^3{x}^{\left(\frac{1}{x}\right)}+2x^4+4e^x)-3x\log (x)-\log (x)}{x}+\frac{\log (x)}{x}-\log (x)\end{array}$$

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	$-\ln (x)+\ln (x^{\frac{1}{x}}+\frac{2x^4+4{\mathrm{e}}^x}{x^3})$	$26$

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 $$\begin{array}{c}-\log (x)+\log \left(\frac{x^3{x}^{\left(\frac{1}{x}\right)}+2x^4+4e^x}{x^3}\right)\end{array}$$

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 $$\begin{array}{c}\ln \left(\frac{4{\mathrm{e}}^x+2x^4+x^{1/x}{x}^3}{x^3}\right)-\ln (x)\end{array}$$

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 $$\begin{array}{c}-\log (x)+\log (e^{\frac{\log (x)}{x}}+\frac{2x^4+4e^x}{x^3})\end{array}$$

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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