3.94.66 \(\int \frac {16 e^4+e^{4+2 x} x^2+e^{4+x} (-4+4 x)}{-32+16 x+e^x (-20 x+8 x^2)+e^{2 x} (-3 x^2+x^3)} \, dx\)

Optimal. Leaf size=23 \[ e^4 \log \left (2-x+\frac {x}{4 e^{-x}+x}\right ) \]

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Rubi [F]  time = 1.45, 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 {16 e^4+e^{4+2 x} x^2+e^{4+x} (-4+4 x)}{-32+16 x+e^x \left (-20 x+8 x^2\right )+e^{2 x} \left (-3 x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(16*E^4 + E^(4 + 2*x)*x^2 + E^(4 + x)*(-4 + 4*x))/(-32 + 16*x + E^x*(-20*x + 8*x^2) + E^(2*x)*(-3*x^2 + x^
3)),x]

[Out]

E^4*Log[3 - x] + 4*E^4*Defer[Int][(4 + E^x*x)^(-1), x] + 4*E^4*Defer[Int][1/(x*(4 + E^x*x)), x] + 4*E^4*Defer[
Int][(-8 + 4*x - 3*E^x*x + E^x*x^2)^(-1), x] - 4*E^4*Defer[Int][1/((-3 + x)*(-8 + 4*x - 3*E^x*x + E^x*x^2)), x
] + 8*E^4*Defer[Int][1/(x*(-8 + 4*x - 3*E^x*x + E^x*x^2)), x] - 4*E^4*Defer[Int][x/(-8 + 4*x - 3*E^x*x + E^x*x
^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^4 \left (-16+4 e^x-4 e^x x-e^{2 x} x^2\right )}{32-16 x-e^x \left (-20 x+8 x^2\right )-e^{2 x} \left (-3 x^2+x^3\right )} \, dx\\ &=e^4 \int \frac {-16+4 e^x-4 e^x x-e^{2 x} x^2}{32-16 x-e^x \left (-20 x+8 x^2\right )-e^{2 x} \left (-3 x^2+x^3\right )} \, dx\\ &=e^4 \int \left (\frac {1}{-3+x}+\frac {4 (1+x)}{x \left (4+e^x x\right )}-\frac {4 \left (6+2 x-4 x^2+x^3\right )}{(-3+x) x \left (-8+4 x-3 e^x x+e^x x^2\right )}\right ) \, dx\\ &=e^4 \log (3-x)+\left (4 e^4\right ) \int \frac {1+x}{x \left (4+e^x x\right )} \, dx-\left (4 e^4\right ) \int \frac {6+2 x-4 x^2+x^3}{(-3+x) x \left (-8+4 x-3 e^x x+e^x x^2\right )} \, dx\\ &=e^4 \log (3-x)+\left (4 e^4\right ) \int \left (\frac {1}{4+e^x x}+\frac {1}{x \left (4+e^x x\right )}\right ) \, dx-\left (4 e^4\right ) \int \left (-\frac {1}{-8+4 x-3 e^x x+e^x x^2}+\frac {1}{(-3+x) \left (-8+4 x-3 e^x x+e^x x^2\right )}-\frac {2}{x \left (-8+4 x-3 e^x x+e^x x^2\right )}+\frac {x}{-8+4 x-3 e^x x+e^x x^2}\right ) \, dx\\ &=e^4 \log (3-x)+\left (4 e^4\right ) \int \frac {1}{4+e^x x} \, dx+\left (4 e^4\right ) \int \frac {1}{x \left (4+e^x x\right )} \, dx+\left (4 e^4\right ) \int \frac {1}{-8+4 x-3 e^x x+e^x x^2} \, dx-\left (4 e^4\right ) \int \frac {1}{(-3+x) \left (-8+4 x-3 e^x x+e^x x^2\right )} \, dx-\left (4 e^4\right ) \int \frac {x}{-8+4 x-3 e^x x+e^x x^2} \, dx+\left (8 e^4\right ) \int \frac {1}{x \left (-8+4 x-3 e^x x+e^x x^2\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.38, size = 53, normalized size = 2.30 \begin {gather*} e^4 \left (\log (3-x)+\log (x)-\log ((3-x) x)-\log \left (4+e^x x\right )+\log \left (8-4 x+3 e^x x-e^x x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(16*E^4 + E^(4 + 2*x)*x^2 + E^(4 + x)*(-4 + 4*x))/(-32 + 16*x + E^x*(-20*x + 8*x^2) + E^(2*x)*(-3*x^
2 + x^3)),x]

[Out]

E^4*(Log[3 - x] + Log[x] - Log[(3 - x)*x] - Log[4 + E^x*x] + Log[8 - 4*x + 3*E^x*x - E^x*x^2])

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fricas [B]  time = 1.07, size = 62, normalized size = 2.70 \begin {gather*} e^{4} \log \left (x - 3\right ) + e^{4} \log \left (\frac {4 \, {\left (x - 2\right )} e^{4} + {\left (x^{2} - 3 \, x\right )} e^{\left (x + 4\right )}}{x^{2} - 3 \, x}\right ) - e^{4} \log \left (\frac {x e^{\left (x + 4\right )} + 4 \, e^{4}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*exp(4)*exp(x)^2+(4*x-4)*exp(4)*exp(x)+16*exp(4))/((x^3-3*x^2)*exp(x)^2+(8*x^2-20*x)*exp(x)+16*x
-32),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.61, size = 32, normalized size = 1.39 \begin {gather*} e^{4} \log \left (x^{2} e^{x} - 3 \, x e^{x} + 4 \, x - 8\right ) - e^{4} \log \left (x e^{x} + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*exp(4)*exp(x)^2+(4*x-4)*exp(4)*exp(x)+16*exp(4))/((x^3-3*x^2)*exp(x)^2+(8*x^2-20*x)*exp(x)+16*x
-32),x, algorithm="giac")

[Out]

e^4*log(x^2*e^x - 3*x*e^x + 4*x - 8) - e^4*log(x*e^x + 4)

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maple [A]  time = 0.12, size = 33, normalized size = 1.43

method	result	size
norman	\({\mathrm e}^{4} \ln \left ({\mathrm e}^{x} x^{2}-3 \,{\mathrm e}^{x} x +4 x -8\right )-{\mathrm e}^{4} \ln \left ({\mathrm e}^{x} x +4\right )\)	\(33\)
risch	\({\mathrm e}^{4} \ln \left (x -3\right )+{\mathrm e}^{4} \ln \left ({\mathrm e}^{x}+\frac {4 x -8}{x \left (x -3\right )}\right )-{\mathrm e}^{4} \ln \left ({\mathrm e}^{x}+\frac {4}{x}\right )\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*exp(4)*exp(x)^2+(4*x-4)*exp(4)*exp(x)+16*exp(4))/((x^3-3*x^2)*exp(x)^2+(8*x^2-20*x)*exp(x)+16*x-32),x
,method=_RETURNVERBOSE)

[Out]

exp(4)*ln(exp(x)*x^2-3*exp(x)*x+4*x-8)-exp(4)*ln(exp(x)*x+4)

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maxima [B]  time = 0.40, size = 52, normalized size = 2.26 \begin {gather*} e^{4} \log \left (x - 3\right ) + e^{4} \log \left (\frac {{\left (x^{2} - 3 \, x\right )} e^{x} + 4 \, x - 8}{x^{2} - 3 \, x}\right ) - e^{4} \log \left (\frac {x e^{x} + 4}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*exp(4)*exp(x)^2+(4*x-4)*exp(4)*exp(x)+16*exp(4))/((x^3-3*x^2)*exp(x)^2+(8*x^2-20*x)*exp(x)+16*x
-32),x, algorithm="maxima")

[Out]

e^4*log(x - 3) + e^4*log(((x^2 - 3*x)*e^x + 4*x - 8)/(x^2 - 3*x)) - e^4*log((x*e^x + 4)/x)

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mupad [B]  time = 0.27, size = 30, normalized size = 1.30 \begin {gather*} {\mathrm {e}}^4\,\left (\ln \left (4\,x+x^2\,{\mathrm {e}}^x-3\,x\,{\mathrm {e}}^x-8\right )-\ln \left (x\,{\mathrm {e}}^x+4\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(16*exp(4) + exp(4)*exp(x)*(4*x - 4) + x^2*exp(2*x)*exp(4))/(exp(2*x)*(3*x^2 - x^3) - 16*x + exp(x)*(20*x
- 8*x^2) + 32),x)

[Out]

exp(4)*(log(4*x + x^2*exp(x) - 3*x*exp(x) - 8) - log(x*exp(x) + 4))

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2*exp(4)*exp(x)**2+(4*x-4)*exp(4)*exp(x)+16*exp(4))/((x**3-3*x**2)*exp(x)**2+(8*x**2-20*x)*exp(x
)+16*x-32),x)

[Out]

Exception raised: PolynomialError

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