3.90.61 \(\int \frac {9+18 x^2-72 x^3+90 x^4-36 x^5+e^x (9-9 x+18 x^2-54 x^3+36 x^4)}{x+e^x x-x^2} \, dx\)

Optimal. Leaf size=26 \[ 9 \log \left (\frac {e^{\left (x-x^2\right )^2} x}{1+e^x-x}\right ) \]

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Rubi [F]  time = 0.26, 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 {9+18 x^2-72 x^3+90 x^4-36 x^5+e^x \left (9-9 x+18 x^2-54 x^3+36 x^4\right )}{x+e^x x-x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(9 + 18*x^2 - 72*x^3 + 90*x^4 - 36*x^5 + E^x*(9 - 9*x + 18*x^2 - 54*x^3 + 36*x^4))/(x + E^x*x - x^2),x]

[Out]

-9*x + 9*x^2 - 18*x^3 + 9*x^4 + 9*Log[x] + 18*Defer[Int][(1 + E^x - x)^(-1), x] - 9*Defer[Int][x/(1 + E^x - x)
, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {9 (-2+x)}{1+e^x-x}+\frac {9 \left (1-x+2 x^2-6 x^3+4 x^4\right )}{x}\right ) \, dx\\ &=-\left (9 \int \frac {-2+x}{1+e^x-x} \, dx\right )+9 \int \frac {1-x+2 x^2-6 x^3+4 x^4}{x} \, dx\\ &=-\left (9 \int \left (-\frac {2}{1+e^x-x}+\frac {x}{1+e^x-x}\right ) \, dx\right )+9 \int \left (-1+\frac {1}{x}+2 x-6 x^2+4 x^3\right ) \, dx\\ &=-9 x+9 x^2-18 x^3+9 x^4+9 \log (x)-9 \int \frac {x}{1+e^x-x} \, dx+18 \int \frac {1}{1+e^x-x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 31, normalized size = 1.19 \begin {gather*} -9 \left (-x^2+2 x^3-x^4+\log \left (1+e^x-x\right )-\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(9 + 18*x^2 - 72*x^3 + 90*x^4 - 36*x^5 + E^x*(9 - 9*x + 18*x^2 - 54*x^3 + 36*x^4))/(x + E^x*x - x^2)
,x]

[Out]

-9*(-x^2 + 2*x^3 - x^4 + Log[1 + E^x - x] - Log[x])

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fricas [A]  time = 0.56, size = 30, normalized size = 1.15 \begin {gather*} 9 \, x^{4} - 18 \, x^{3} + 9 \, x^{2} + 9 \, \log \relax (x) - 9 \, \log \left (-x + e^{x} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((36*x^4-54*x^3+18*x^2-9*x+9)*exp(x)-36*x^5+90*x^4-72*x^3+18*x^2+9)/(exp(x)*x-x^2+x),x, algorithm="f
ricas")

[Out]

9*x^4 - 18*x^3 + 9*x^2 + 9*log(x) - 9*log(-x + e^x + 1)

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giac [A]  time = 0.14, size = 30, normalized size = 1.15 \begin {gather*} 9 \, x^{4} - 18 \, x^{3} + 9 \, x^{2} - 9 \, \log \left (x - e^{x} - 1\right ) + 9 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((36*x^4-54*x^3+18*x^2-9*x+9)*exp(x)-36*x^5+90*x^4-72*x^3+18*x^2+9)/(exp(x)*x-x^2+x),x, algorithm="g
iac")

[Out]

9*x^4 - 18*x^3 + 9*x^2 - 9*log(x - e^x - 1) + 9*log(x)

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maple [A]  time = 0.04, size = 31, normalized size = 1.19

method	result	size
norman	\(9 x^{4}-18 x^{3}+9 x^{2}+9 \ln \relax (x )-9 \ln \left (x -{\mathrm e}^{x}-1\right )\)	\(31\)
risch	\(9 x^{4}-18 x^{3}+9 x^{2}+9 \ln \relax (x )-9 \ln \left (1+{\mathrm e}^{x}-x \right )\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((36*x^4-54*x^3+18*x^2-9*x+9)*exp(x)-36*x^5+90*x^4-72*x^3+18*x^2+9)/(exp(x)*x-x^2+x),x,method=_RETURNVERBO
SE)

[Out]

9*x^4-18*x^3+9*x^2+9*ln(x)-9*ln(x-exp(x)-1)

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maxima [A]  time = 0.41, size = 30, normalized size = 1.15 \begin {gather*} 9 \, x^{4} - 18 \, x^{3} + 9 \, x^{2} + 9 \, \log \relax (x) - 9 \, \log \left (-x + e^{x} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((36*x^4-54*x^3+18*x^2-9*x+9)*exp(x)-36*x^5+90*x^4-72*x^3+18*x^2+9)/(exp(x)*x-x^2+x),x, algorithm="m
axima")

[Out]

9*x^4 - 18*x^3 + 9*x^2 + 9*log(x) - 9*log(-x + e^x + 1)

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mupad [B]  time = 0.13, size = 30, normalized size = 1.15 \begin {gather*} 9\,\ln \relax (x)-9\,\ln \left (x-{\mathrm {e}}^x-1\right )+9\,x^2-18\,x^3+9\,x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(18*x^2 - 9*x - 54*x^3 + 36*x^4 + 9) + 18*x^2 - 72*x^3 + 90*x^4 - 36*x^5 + 9)/(x + x*exp(x) - x^2)
,x)

[Out]

9*log(x) - 9*log(x - exp(x) - 1) + 9*x^2 - 18*x^3 + 9*x^4

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sympy [A]  time = 0.14, size = 29, normalized size = 1.12 \begin {gather*} 9 x^{4} - 18 x^{3} + 9 x^{2} + 9 \log {\relax (x )} - 9 \log {\left (- x + e^{x} + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((36*x**4-54*x**3+18*x**2-9*x+9)*exp(x)-36*x**5+90*x**4-72*x**3+18*x**2+9)/(exp(x)*x-x**2+x),x)

[Out]

9*x**4 - 18*x**3 + 9*x**2 + 9*log(x) - 9*log(-x + exp(x) + 1)

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