3.101.99 \(\int \frac {-4+4 x+7 x^2-8 x^3-2 x^4-e^{2 x} x^4+4 x^5-x^6+e^x (-4 x^2+2 x^3+4 x^4-2 x^5)+(4-4 x-7 x^2+8 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x (4 x^2-2 x^3-4 x^4+2 x^5)) \log (x)+(6-4 x-5 x^2+6 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x (3 x^2-3 x^3-4 x^4+2 x^5)) \log ^2(x)}{(4-4 x-7 x^2+8 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x (4 x^2-2 x^3-4 x^4+2 x^5)) \log ^2(x)} \, dx\)

Optimal. Leaf size=27 \[ x+\frac {x}{2-x+x^2 \left (-2+e^x+x\right )}+\frac {x}{\log (x)} \]

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Rubi [F]  time = 1.71, 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 {-4+4 x+7 x^2-8 x^3-2 x^4-e^{2 x} x^4+4 x^5-x^6+e^x \left (-4 x^2+2 x^3+4 x^4-2 x^5\right )+\left (4-4 x-7 x^2+8 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (4 x^2-2 x^3-4 x^4+2 x^5\right )\right ) \log (x)+\left (6-4 x-5 x^2+6 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (3 x^2-3 x^3-4 x^4+2 x^5\right )\right ) \log ^2(x)}{\left (4-4 x-7 x^2+8 x^3+2 x^4+e^{2 x} x^4-4 x^5+x^6+e^x \left (4 x^2-2 x^3-4 x^4+2 x^5\right )\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-4 + 4*x + 7*x^2 - 8*x^3 - 2*x^4 - E^(2*x)*x^4 + 4*x^5 - x^6 + E^x*(-4*x^2 + 2*x^3 + 4*x^4 - 2*x^5) + (4
- 4*x - 7*x^2 + 8*x^3 + 2*x^4 + E^(2*x)*x^4 - 4*x^5 + x^6 + E^x*(4*x^2 - 2*x^3 - 4*x^4 + 2*x^5))*Log[x] + (6 -
4*x - 5*x^2 + 6*x^3 + 2*x^4 + E^(2*x)*x^4 - 4*x^5 + x^6 + E^x*(3*x^2 - 3*x^3 - 4*x^4 + 2*x^5))*Log[x]^2)/((4
- 4*x - 7*x^2 + 8*x^3 + 2*x^4 + E^(2*x)*x^4 - 4*x^5 + x^6 + E^x*(4*x^2 - 2*x^3 - 4*x^4 + 2*x^5))*Log[x]^2),x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.48, size = 32, normalized size = 1.19 \begin {gather*} x+\frac {x}{2-x-2 x^2+e^x x^2+x^3}+\frac {x}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4 + 4*x + 7*x^2 - 8*x^3 - 2*x^4 - E^(2*x)*x^4 + 4*x^5 - x^6 + E^x*(-4*x^2 + 2*x^3 + 4*x^4 - 2*x^5)
+ (4 - 4*x - 7*x^2 + 8*x^3 + 2*x^4 + E^(2*x)*x^4 - 4*x^5 + x^6 + E^x*(4*x^2 - 2*x^3 - 4*x^4 + 2*x^5))*Log[x]
+ (6 - 4*x - 5*x^2 + 6*x^3 + 2*x^4 + E^(2*x)*x^4 - 4*x^5 + x^6 + E^x*(3*x^2 - 3*x^3 - 4*x^4 + 2*x^5))*Log[x]^2
)/((4 - 4*x - 7*x^2 + 8*x^3 + 2*x^4 + E^(2*x)*x^4 - 4*x^5 + x^6 + E^x*(4*x^2 - 2*x^3 - 4*x^4 + 2*x^5))*Log[x]^
2),x]

[Out]

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

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fricas [B]  time = 1.00, size = 75, normalized size = 2.78 \begin {gather*} \frac {x^{4} + x^{3} e^{x} - 2 \, x^{3} - x^{2} + {\left (x^{4} + x^{3} e^{x} - 2 \, x^{3} - x^{2} + 3 \, x\right )} \log \relax (x) + 2 \, x}{{\left (x^{3} + x^{2} e^{x} - 2 \, x^{2} - x + 2\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)^2*x^4+(2*x^5-4*x^4-3*x^3+3*x^2)*exp(x)+x^6-4*x^5+2*x^4+6*x^3-5*x^2-4*x+6)*log(x)^2+(exp(x)^
2*x^4+(2*x^5-4*x^4-2*x^3+4*x^2)*exp(x)+x^6-4*x^5+2*x^4+8*x^3-7*x^2-4*x+4)*log(x)-exp(x)^2*x^4+(-2*x^5+4*x^4+2*
x^3-4*x^2)*exp(x)-x^6+4*x^5-2*x^4-8*x^3+7*x^2+4*x-4)/(exp(x)^2*x^4+(2*x^5-4*x^4-2*x^3+4*x^2)*exp(x)+x^6-4*x^5+
2*x^4+8*x^3-7*x^2-4*x+4)/log(x)^2,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.30, size = 90, normalized size = 3.33 \begin {gather*} \frac {x^{4} \log \relax (x) + x^{3} e^{x} \log \relax (x) + x^{4} + x^{3} e^{x} - 2 \, x^{3} \log \relax (x) - 2 \, x^{3} - x^{2} \log \relax (x) - x^{2} + 3 \, x \log \relax (x) + 2 \, x}{x^{3} \log \relax (x) + x^{2} e^{x} \log \relax (x) - 2 \, x^{2} \log \relax (x) - x \log \relax (x) + 2 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)^2*x^4+(2*x^5-4*x^4-3*x^3+3*x^2)*exp(x)+x^6-4*x^5+2*x^4+6*x^3-5*x^2-4*x+6)*log(x)^2+(exp(x)^
2*x^4+(2*x^5-4*x^4-2*x^3+4*x^2)*exp(x)+x^6-4*x^5+2*x^4+8*x^3-7*x^2-4*x+4)*log(x)-exp(x)^2*x^4+(-2*x^5+4*x^4+2*
x^3-4*x^2)*exp(x)-x^6+4*x^5-2*x^4-8*x^3+7*x^2+4*x-4)/(exp(x)^2*x^4+(2*x^5-4*x^4-2*x^3+4*x^2)*exp(x)+x^6-4*x^5+
2*x^4+8*x^3-7*x^2-4*x+4)/log(x)^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 50, normalized size = 1.85

method	result	size
risch	\(\frac {x \left (x^{3}+{\mathrm e}^{x} x^{2}-2 x^{2}-x +3\right )}{{\mathrm e}^{x} x^{2}+x^{3}-2 x^{2}-x +2}+\frac {x}{\ln \relax (x )}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp(x)^2*x^4+(2*x^5-4*x^4-3*x^3+3*x^2)*exp(x)+x^6-4*x^5+2*x^4+6*x^3-5*x^2-4*x+6)*ln(x)^2+(exp(x)^2*x^4+(
2*x^5-4*x^4-2*x^3+4*x^2)*exp(x)+x^6-4*x^5+2*x^4+8*x^3-7*x^2-4*x+4)*ln(x)-exp(x)^2*x^4+(-2*x^5+4*x^4+2*x^3-4*x^
2)*exp(x)-x^6+4*x^5-2*x^4-8*x^3+7*x^2+4*x-4)/(exp(x)^2*x^4+(2*x^5-4*x^4-2*x^3+4*x^2)*exp(x)+x^6-4*x^5+2*x^4+8*
x^3-7*x^2-4*x+4)/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.42, size = 78, normalized size = 2.89 \begin {gather*} \frac {x^{4} - 2 \, x^{3} - x^{2} + {\left (x^{3} \log \relax (x) + x^{3}\right )} e^{x} + {\left (x^{4} - 2 \, x^{3} - x^{2} + 3 \, x\right )} \log \relax (x) + 2 \, x}{x^{2} e^{x} \log \relax (x) + {\left (x^{3} - 2 \, x^{2} - x + 2\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)^2*x^4+(2*x^5-4*x^4-3*x^3+3*x^2)*exp(x)+x^6-4*x^5+2*x^4+6*x^3-5*x^2-4*x+6)*log(x)^2+(exp(x)^
2*x^4+(2*x^5-4*x^4-2*x^3+4*x^2)*exp(x)+x^6-4*x^5+2*x^4+8*x^3-7*x^2-4*x+4)*log(x)-exp(x)^2*x^4+(-2*x^5+4*x^4+2*
x^3-4*x^2)*exp(x)-x^6+4*x^5-2*x^4-8*x^3+7*x^2+4*x-4)/(exp(x)^2*x^4+(2*x^5-4*x^4-2*x^3+4*x^2)*exp(x)+x^6-4*x^5+
2*x^4+8*x^3-7*x^2-4*x+4)/log(x)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 6.82, size = 76, normalized size = 2.81 \begin {gather*} \frac {x\,\left (3\,\ln \relax (x)-x+x^2\,{\mathrm {e}}^x-2\,x^2\,\ln \relax (x)+x^3\,\ln \relax (x)-x\,\ln \relax (x)-2\,x^2+x^3+x^2\,{\mathrm {e}}^x\,\ln \relax (x)+2\right )}{\ln \relax (x)\,\left (x^2\,{\mathrm {e}}^x-x-2\,x^2+x^3+2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^4*exp(2*x) - log(x)*(x^4*exp(2*x) - 4*x + exp(x)*(4*x^2 - 2*x^3 - 4*x^4 + 2*x^5) - 7*x^2 + 8*x^3 + 2*x
^4 - 4*x^5 + x^6 + 4) - 4*x + exp(x)*(4*x^2 - 2*x^3 - 4*x^4 + 2*x^5) - log(x)^2*(x^4*exp(2*x) - 4*x + exp(x)*(
3*x^2 - 3*x^3 - 4*x^4 + 2*x^5) - 5*x^2 + 6*x^3 + 2*x^4 - 4*x^5 + x^6 + 6) - 7*x^2 + 8*x^3 + 2*x^4 - 4*x^5 + x^
6 + 4)/(log(x)^2*(x^4*exp(2*x) - 4*x + exp(x)*(4*x^2 - 2*x^3 - 4*x^4 + 2*x^5) - 7*x^2 + 8*x^3 + 2*x^4 - 4*x^5
+ x^6 + 4)),x)

[Out]

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

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sympy [A]  time = 0.46, size = 26, normalized size = 0.96 \begin {gather*} x + \frac {x}{\log {\relax (x )}} + \frac {x}{x^{3} + x^{2} e^{x} - 2 x^{2} - x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)**2*x**4+(2*x**5-4*x**4-3*x**3+3*x**2)*exp(x)+x**6-4*x**5+2*x**4+6*x**3-5*x**2-4*x+6)*ln(x)*
*2+(exp(x)**2*x**4+(2*x**5-4*x**4-2*x**3+4*x**2)*exp(x)+x**6-4*x**5+2*x**4+8*x**3-7*x**2-4*x+4)*ln(x)-exp(x)**
2*x**4+(-2*x**5+4*x**4+2*x**3-4*x**2)*exp(x)-x**6+4*x**5-2*x**4-8*x**3+7*x**2+4*x-4)/(exp(x)**2*x**4+(2*x**5-4
*x**4-2*x**3+4*x**2)*exp(x)+x**6-4*x**5+2*x**4+8*x**3-7*x**2-4*x+4)/ln(x)**2,x)

[Out]

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

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