3.94.47 \(\int \frac {-12 x^3+e^{2 x} (24 x^2-24 x^3)+e^{1-x} x (e^{4 x} (-1+x)-x^2-3 x^3+e^{2 x} (2 x+6 x^2-8 x^3))}{3 e^{4 x} x-6 e^{2 x} x^2+3 x^3+e^{1-x} x (e^{4 x} x-2 e^{2 x} x^2+x^3)} \, dx\)

Optimal. Leaf size=31 \[ \frac {4 x^2}{e^{2 x}-x}-\log \left (3+e^{1-x} x\right ) \]

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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 {-12 x^3+e^{2 x} \left (24 x^2-24 x^3\right )+e^{1-x} x \left (e^{4 x} (-1+x)-x^2-3 x^3+e^{2 x} \left (2 x+6 x^2-8 x^3\right )\right )}{3 e^{4 x} x-6 e^{2 x} x^2+3 x^3+e^{1-x} x \left (e^{4 x} x-2 e^{2 x} x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-12*x^3 + E^(2*x)*(24*x^2 - 24*x^3) + E^(1 - x)*x*(E^(4*x)*(-1 + x) - x^2 - 3*x^3 + E^(2*x)*(2*x + 6*x^2
- 8*x^3)))/(3*E^(4*x)*x - 6*E^(2*x)*x^2 + 3*x^3 + E^(1 - x)*x*(E^(4*x)*x - 2*E^(2*x)*x^2 + x^3)),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{1+4 x} (-1+x)-24 e^{3 x} (-1+x) x-12 e^x x^2-e x^2 (1+3 x)-2 e^{1+2 x} x \left (-1-3 x+4 x^2\right )}{\left (e^{2 x}-x\right )^2 \left (3 e^x+e x\right )} \, dx\\ &=\int \left (-\frac {8 (-1+x) x}{e^{2 x}-x}-\frac {4 x^2 (-1+2 x)}{\left (e^{2 x}-x\right )^2}+\frac {e (-1+x)}{3 e^x+e x}\right ) \, dx\\ &=-\left (4 \int \frac {x^2 (-1+2 x)}{\left (e^{2 x}-x\right )^2} \, dx\right )-8 \int \frac {(-1+x) x}{e^{2 x}-x} \, dx+e \int \frac {-1+x}{3 e^x+e x} \, dx\\ &=-\left (4 \int \left (-\frac {x^2}{\left (e^{2 x}-x\right )^2}+\frac {2 x^3}{\left (e^{2 x}-x\right )^2}\right ) \, dx\right )-8 \int \left (-\frac {x}{e^{2 x}-x}+\frac {x^2}{e^{2 x}-x}\right ) \, dx+e \int \left (-\frac {1}{3 e^x+e x}+\frac {x}{3 e^x+e x}\right ) \, dx\\ &=4 \int \frac {x^2}{\left (e^{2 x}-x\right )^2} \, dx+8 \int \frac {x}{e^{2 x}-x} \, dx-8 \int \frac {x^2}{e^{2 x}-x} \, dx-8 \int \frac {x^3}{\left (e^{2 x}-x\right )^2} \, dx-e \int \frac {1}{3 e^x+e x} \, dx+e \int \frac {x}{3 e^x+e x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 30, normalized size = 0.97 \begin {gather*} x+\frac {4 x^2}{e^{2 x}-x}-\log \left (3 e^x+e x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-12*x^3 + E^(2*x)*(24*x^2 - 24*x^3) + E^(1 - x)*x*(E^(4*x)*(-1 + x) - x^2 - 3*x^3 + E^(2*x)*(2*x +
6*x^2 - 8*x^3)))/(3*E^(4*x)*x - 6*E^(2*x)*x^2 + 3*x^3 + E^(1 - x)*x*(E^(4*x)*x - 2*E^(2*x)*x^2 + x^3)),x]

[Out]

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

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fricas [B]  time = 0.69, size = 64, normalized size = 2.06 \begin {gather*} \frac {4 \, x e^{\left (-2 \, x + 2 \, \log \relax (x) + 2\right )} - {\left (x e^{2} - e^{\left (-2 \, x + 2 \, \log \relax (x) + 2\right )}\right )} \log \left (e^{\left (-x + \log \relax (x) + 1\right )} + 3\right )}{x e^{2} - e^{\left (-2 \, x + 2 \, \log \relax (x) + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-1)*exp(2*x)^2+(-8*x^3+6*x^2+2*x)*exp(2*x)-3*x^3-x^2)*exp(1+log(x)-x)+(-24*x^3+24*x^2)*exp(2*x)-
12*x^3)/((x*exp(2*x)^2-2*exp(2*x)*x^2+x^3)*exp(1+log(x)-x)+3*x*exp(2*x)^2-6*exp(2*x)*x^2+3*x^3),x, algorithm="
fricas")

[Out]

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

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giac [A]  time = 0.75, size = 52, normalized size = 1.68 \begin {gather*} -\frac {7 \, x^{2} + x e^{\left (2 \, x\right )} + x \log \left (x e + 3 \, e^{x}\right ) - e^{\left (2 \, x\right )} \log \left (x e + 3 \, e^{x}\right )}{x - e^{\left (2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-1)*exp(2*x)^2+(-8*x^3+6*x^2+2*x)*exp(2*x)-3*x^3-x^2)*exp(1+log(x)-x)+(-24*x^3+24*x^2)*exp(2*x)-
12*x^3)/((x*exp(2*x)^2-2*exp(2*x)*x^2+x^3)*exp(1+log(x)-x)+3*x*exp(2*x)^2-6*exp(2*x)*x^2+3*x^3),x, algorithm="
giac")

[Out]

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

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maple [A]  time = 0.26, size = 29, normalized size = 0.94

method	result	size
risch	\(x -\frac {4 x^{2}}{x -{\mathrm e}^{2 x}}-\ln \left (\frac {x \,{\mathrm e}}{3}+{\mathrm e}^{x}\right )\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x-1)*exp(2*x)^2+(-8*x^3+6*x^2+2*x)*exp(2*x)-3*x^3-x^2)*exp(1+ln(x)-x)+(-24*x^3+24*x^2)*exp(2*x)-12*x^3)
/((x*exp(2*x)^2-2*exp(2*x)*x^2+x^3)*exp(1+ln(x)-x)+3*x*exp(2*x)^2-6*exp(2*x)*x^2+3*x^3),x,method=_RETURNVERBOS
E)

[Out]

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

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maxima [A]  time = 0.57, size = 36, normalized size = 1.16 \begin {gather*} -\frac {3 \, x^{2} + x e^{\left (2 \, x\right )}}{x - e^{\left (2 \, x\right )}} - \log \left (\frac {1}{3} \, x e + e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-1)*exp(2*x)^2+(-8*x^3+6*x^2+2*x)*exp(2*x)-3*x^3-x^2)*exp(1+log(x)-x)+(-24*x^3+24*x^2)*exp(2*x)-
12*x^3)/((x*exp(2*x)^2-2*exp(2*x)*x^2+x^3)*exp(1+log(x)-x)+3*x*exp(2*x)^2-6*exp(2*x)*x^2+3*x^3),x, algorithm="
maxima")

[Out]

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

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mupad [B]  time = 0.35, size = 50, normalized size = 1.61 \begin {gather*} -\frac {x\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{2\,x}\,\ln \left ({\mathrm {e}}^x+\frac {x\,\mathrm {e}}{3}\right )+x\,\ln \left ({\mathrm {e}}^x+\frac {x\,\mathrm {e}}{3}\right )+3\,x^2}{x-{\mathrm {e}}^{2\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(log(x) - x + 1)*(exp(2*x)*(2*x + 6*x^2 - 8*x^3) + exp(4*x)*(x - 1) - x^2 - 3*x^3) + exp(2*x)*(24*x^2
- 24*x^3) - 12*x^3)/(3*x*exp(4*x) - 6*x^2*exp(2*x) + exp(log(x) - x + 1)*(x*exp(4*x) - 2*x^2*exp(2*x) + x^3) +
3*x^3),x)

[Out]

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

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sympy [A]  time = 0.34, size = 37, normalized size = 1.19 \begin {gather*} - 4 x - \frac {4 x}{x e^{- 2 x} - 1} - \log {\relax (x )} - \log {\left (\frac {1}{\sqrt {e^{2 x}}} + \frac {3}{e x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-1)*exp(2*x)**2+(-8*x**3+6*x**2+2*x)*exp(2*x)-3*x**3-x**2)*exp(1+ln(x)-x)+(-24*x**3+24*x**2)*exp
(2*x)-12*x**3)/((x*exp(2*x)**2-2*exp(2*x)*x**2+x**3)*exp(1+ln(x)-x)+3*x*exp(2*x)**2-6*exp(2*x)*x**2+3*x**3),x)

[Out]

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

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