3.83.19 \(\int \frac {9-12 e^x+e^{x^2 \log ^2(2)} (-3-6 x^2 \log ^2(2)+e^x (3+6 x \log ^2(2)))}{-4 e^x+e^{x^2 \log ^2(2)} (e^x-x)+3 x} \, dx\)

Optimal. Leaf size=25 \[ 3 \log \left (x+\left (-4+e^{x^2 \log ^2(2)}\right ) \left (-e^x+x\right )\right ) \]

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Rubi [F]  time = 4.31, 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 {9-12 e^x+e^{x^2 \log ^2(2)} \left (-3-6 x^2 \log ^2(2)+e^x \left (3+6 x \log ^2(2)\right )\right )}{-4 e^x+e^{x^2 \log ^2(2)} \left (e^x-x\right )+3 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(9 - 12*E^x + E^(x^2*Log[2]^2)*(-3 - 6*x^2*Log[2]^2 + E^x*(3 + 6*x*Log[2]^2)))/(-4*E^x + E^(x^2*Log[2]^2)*
(E^x - x) + 3*x),x]

[Out]

3*x + 3*x^2*Log[2]^2 - 3*Defer[Int][(E^x - x)^(-1), x] + 3*Defer[Int][x/(E^x - x), x] - 3*Defer[Int][E^x/((E^x
- x)*(-4*E^x + E^(x + x^2*Log[2]^2) + 3*x - E^(x^2*Log[2]^2)*x)), x] + 3*Defer[Int][(E^x*x)/((E^x - x)*(-4*E^
x + E^(x + x^2*Log[2]^2) + 3*x - E^(x^2*Log[2]^2)*x)), x] + 24*Log[2]^2*Defer[Int][(E^(2*x)*x)/((E^x - x)*(-4*
E^x + E^(x + x^2*Log[2]^2) + 3*x - E^(x^2*Log[2]^2)*x)), x] - 42*Log[2]^2*Defer[Int][(E^x*x^2)/((E^x - x)*(-4*
E^x + E^(x + x^2*Log[2]^2) + 3*x - E^(x^2*Log[2]^2)*x)), x] + 18*Log[2]^2*Defer[Int][x^3/((E^x - x)*(-4*E^x +
E^(x + x^2*Log[2]^2) + 3*x - E^(x^2*Log[2]^2)*x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {3 \left (-1+e^x+2 e^x x \log ^2(2)-2 x^2 \log ^2(2)\right )}{e^x-x}+\frac {3 \left (-e^x+e^x x+8 e^{2 x} x \log ^2(2)-14 e^x x^2 \log ^2(2)+6 x^3 \log ^2(2)\right )}{\left (e^x-x\right ) \left (-4 e^x+e^{x+x^2 \log ^2(2)}+3 x-e^{x^2 \log ^2(2)} x\right )}\right ) \, dx\\ &=3 \int \frac {-1+e^x+2 e^x x \log ^2(2)-2 x^2 \log ^2(2)}{e^x-x} \, dx+3 \int \frac {-e^x+e^x x+8 e^{2 x} x \log ^2(2)-14 e^x x^2 \log ^2(2)+6 x^3 \log ^2(2)}{\left (e^x-x\right ) \left (-4 e^x+e^{x+x^2 \log ^2(2)}+3 x-e^{x^2 \log ^2(2)} x\right )} \, dx\\ &=3 \int \left (1+\frac {-1+x}{e^x-x}+2 x \log ^2(2)\right ) \, dx+3 \int \left (-\frac {e^x}{\left (e^x-x\right ) \left (-4 e^x+e^{x+x^2 \log ^2(2)}+3 x-e^{x^2 \log ^2(2)} x\right )}+\frac {e^x x}{\left (e^x-x\right ) \left (-4 e^x+e^{x+x^2 \log ^2(2)}+3 x-e^{x^2 \log ^2(2)} x\right )}+\frac {8 e^{2 x} x \log ^2(2)}{\left (e^x-x\right ) \left (-4 e^x+e^{x+x^2 \log ^2(2)}+3 x-e^{x^2 \log ^2(2)} x\right )}-\frac {14 e^x x^2 \log ^2(2)}{\left (e^x-x\right ) \left (-4 e^x+e^{x+x^2 \log ^2(2)}+3 x-e^{x^2 \log ^2(2)} x\right )}+\frac {6 x^3 \log ^2(2)}{\left (e^x-x\right ) \left (-4 e^x+e^{x+x^2 \log ^2(2)}+3 x-e^{x^2 \log ^2(2)} x\right )}\right ) \, dx\\ &=3 x+3 x^2 \log ^2(2)+3 \int \frac {-1+x}{e^x-x} \, dx-3 \int \frac {e^x}{\left (e^x-x\right ) \left (-4 e^x+e^{x+x^2 \log ^2(2)}+3 x-e^{x^2 \log ^2(2)} x\right )} \, dx+3 \int \frac {e^x x}{\left (e^x-x\right ) \left (-4 e^x+e^{x+x^2 \log ^2(2)}+3 x-e^{x^2 \log ^2(2)} x\right )} \, dx+\left (18 \log ^2(2)\right ) \int \frac {x^3}{\left (e^x-x\right ) \left (-4 e^x+e^{x+x^2 \log ^2(2)}+3 x-e^{x^2 \log ^2(2)} x\right )} \, dx+\left (24 \log ^2(2)\right ) \int \frac {e^{2 x} x}{\left (e^x-x\right ) \left (-4 e^x+e^{x+x^2 \log ^2(2)}+3 x-e^{x^2 \log ^2(2)} x\right )} \, dx-\left (42 \log ^2(2)\right ) \int \frac {e^x x^2}{\left (e^x-x\right ) \left (-4 e^x+e^{x+x^2 \log ^2(2)}+3 x-e^{x^2 \log ^2(2)} x\right )} \, dx\\ &=3 x+3 x^2 \log ^2(2)-3 \int \frac {e^x}{\left (e^x-x\right ) \left (-4 e^x+e^{x+x^2 \log ^2(2)}+3 x-e^{x^2 \log ^2(2)} x\right )} \, dx+3 \int \frac {e^x x}{\left (e^x-x\right ) \left (-4 e^x+e^{x+x^2 \log ^2(2)}+3 x-e^{x^2 \log ^2(2)} x\right )} \, dx+3 \int \left (-\frac {1}{e^x-x}+\frac {x}{e^x-x}\right ) \, dx+\left (18 \log ^2(2)\right ) \int \frac {x^3}{\left (e^x-x\right ) \left (-4 e^x+e^{x+x^2 \log ^2(2)}+3 x-e^{x^2 \log ^2(2)} x\right )} \, dx+\left (24 \log ^2(2)\right ) \int \frac {e^{2 x} x}{\left (e^x-x\right ) \left (-4 e^x+e^{x+x^2 \log ^2(2)}+3 x-e^{x^2 \log ^2(2)} x\right )} \, dx-\left (42 \log ^2(2)\right ) \int \frac {e^x x^2}{\left (e^x-x\right ) \left (-4 e^x+e^{x+x^2 \log ^2(2)}+3 x-e^{x^2 \log ^2(2)} x\right )} \, dx\\ &=3 x+3 x^2 \log ^2(2)-3 \int \frac {1}{e^x-x} \, dx+3 \int \frac {x}{e^x-x} \, dx-3 \int \frac {e^x}{\left (e^x-x\right ) \left (-4 e^x+e^{x+x^2 \log ^2(2)}+3 x-e^{x^2 \log ^2(2)} x\right )} \, dx+3 \int \frac {e^x x}{\left (e^x-x\right ) \left (-4 e^x+e^{x+x^2 \log ^2(2)}+3 x-e^{x^2 \log ^2(2)} x\right )} \, dx+\left (18 \log ^2(2)\right ) \int \frac {x^3}{\left (e^x-x\right ) \left (-4 e^x+e^{x+x^2 \log ^2(2)}+3 x-e^{x^2 \log ^2(2)} x\right )} \, dx+\left (24 \log ^2(2)\right ) \int \frac {e^{2 x} x}{\left (e^x-x\right ) \left (-4 e^x+e^{x+x^2 \log ^2(2)}+3 x-e^{x^2 \log ^2(2)} x\right )} \, dx-\left (42 \log ^2(2)\right ) \int \frac {e^x x^2}{\left (e^x-x\right ) \left (-4 e^x+e^{x+x^2 \log ^2(2)}+3 x-e^{x^2 \log ^2(2)} x\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.97, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {9-12 e^x+e^{x^2 \log ^2(2)} \left (-3-6 x^2 \log ^2(2)+e^x \left (3+6 x \log ^2(2)\right )\right )}{-4 e^x+e^{x^2 \log ^2(2)} \left (e^x-x\right )+3 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(9 - 12*E^x + E^(x^2*Log[2]^2)*(-3 - 6*x^2*Log[2]^2 + E^x*(3 + 6*x*Log[2]^2)))/(-4*E^x + E^(x^2*Log[
2]^2)*(E^x - x) + 3*x),x]

[Out]

Integrate[(9 - 12*E^x + E^(x^2*Log[2]^2)*(-3 - 6*x^2*Log[2]^2 + E^x*(3 + 6*x*Log[2]^2)))/(-4*E^x + E^(x^2*Log[
2]^2)*(E^x - x) + 3*x), x]

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fricas [A]  time = 0.90, size = 46, normalized size = 1.84 \begin {gather*} 3 \, \log \left (-x + e^{x}\right ) + 3 \, \log \left (\frac {{\left (x - e^{x}\right )} e^{\left (x^{2} \log \relax (2)^{2}\right )} - 3 \, x + 4 \, e^{x}}{x - e^{x}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((6*x*log(2)^2+3)*exp(x)-6*x^2*log(2)^2-3)*exp(x^2*log(2)^2)-12*exp(x)+9)/((exp(x)-x)*exp(x^2*log(2
)^2)-4*exp(x)+3*x),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.20, size = 35, normalized size = 1.40 \begin {gather*} 3 \, \log \left (x e^{\left (x^{2} \log \relax (2)^{2}\right )} - 3 \, x - e^{\left (x^{2} \log \relax (2)^{2} + x\right )} + 4 \, e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((6*x*log(2)^2+3)*exp(x)-6*x^2*log(2)^2-3)*exp(x^2*log(2)^2)-12*exp(x)+9)/((exp(x)-x)*exp(x^2*log(2
)^2)-4*exp(x)+3*x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.12, size = 36, normalized size = 1.44

method	result	size
norman	\(3 \ln \left ({\mathrm e}^{x^{2} \ln \relax (2)^{2}} x -{\mathrm e}^{x^{2} \ln \relax (2)^{2}} {\mathrm e}^{x}-3 x +4 \,{\mathrm e}^{x}\right )\)	\(36\)
risch	\(3 \ln \left ({\mathrm e}^{x}-x \right )+3 \ln \left ({\mathrm e}^{x^{2} \ln \relax (2)^{2}}-\frac {3 x -4 \,{\mathrm e}^{x}}{x -{\mathrm e}^{x}}\right )\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((6*x*ln(2)^2+3)*exp(x)-6*x^2*ln(2)^2-3)*exp(x^2*ln(2)^2)-12*exp(x)+9)/((exp(x)-x)*exp(x^2*ln(2)^2)-4*exp
(x)+3*x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.50, size = 46, normalized size = 1.84 \begin {gather*} 3 \, \log \left (-x + e^{x}\right ) + 3 \, \log \left (\frac {{\left (x - e^{x}\right )} e^{\left (x^{2} \log \relax (2)^{2}\right )} - 3 \, x + 4 \, e^{x}}{x - e^{x}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((6*x*log(2)^2+3)*exp(x)-6*x^2*log(2)^2-3)*exp(x^2*log(2)^2)-12*exp(x)+9)/((exp(x)-x)*exp(x^2*log(2
)^2)-4*exp(x)+3*x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.58, size = 34, normalized size = 1.36 \begin {gather*} 3\,\ln \left (3\,x+{\mathrm {e}}^{{\ln \relax (2)}^2\,x^2+x}-4\,{\mathrm {e}}^x-x\,{\mathrm {e}}^{x^2\,{\ln \relax (2)}^2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*exp(x) + exp(x^2*log(2)^2)*(6*x^2*log(2)^2 - exp(x)*(6*x*log(2)^2 + 3) + 3) - 9)/(4*exp(x) - 3*x + exp
(x^2*log(2)^2)*(x - exp(x))),x)

[Out]

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

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sympy [A]  time = 0.58, size = 34, normalized size = 1.36 \begin {gather*} 3 \log {\left (- x + e^{x} \right )} + 3 \log {\left (\frac {- 3 x + 4 e^{x}}{x - e^{x}} + e^{x^{2} \log {\relax (2 )}^{2}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((6*x*ln(2)**2+3)*exp(x)-6*x**2*ln(2)**2-3)*exp(x**2*ln(2)**2)-12*exp(x)+9)/((exp(x)-x)*exp(x**2*ln
(2)**2)-4*exp(x)+3*x),x)

[Out]

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

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