3.6.67 \(\int \frac {-7+7 x-3 x^2+e^{-4-2 e^x-2 x} (-4+3 x+e^x (-4+3 x))+e^{-2-e^x-x} (-4+7 x-3 x^2+e^x (4 x-3 x^2))}{-4+3 x} \, dx\)

Optimal. Leaf size=32 \[ x-\frac {1}{2} \left (-e^{-2-e^x-x}+x\right )^2-\log (-4+3 x) \]

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Rubi [F]  time = 1.42, 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 {-7+7 x-3 x^2+e^{-4-2 e^x-2 x} \left (-4+3 x+e^x (-4+3 x)\right )+e^{-2-e^x-x} \left (-4+7 x-3 x^2+e^x \left (4 x-3 x^2\right )\right )}{-4+3 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (e^{-2 \left (2+e^x+x\right )}-e^{-4-2 e^x-x} \left (-1-e^{2+e^x}+e^{2+e^x} x\right )-\frac {e^{-2-e^x} \left (7 e^{2+e^x}-4 x-7 e^{2+e^x} x+3 x^2+3 e^{2+e^x} x^2\right )}{-4+3 x}\right ) \, dx\\ &=\int e^{-2 \left (2+e^x+x\right )} \, dx-\int e^{-4-2 e^x-x} \left (-1-e^{2+e^x}+e^{2+e^x} x\right ) \, dx-\int \frac {e^{-2-e^x} \left (7 e^{2+e^x}-4 x-7 e^{2+e^x} x+3 x^2+3 e^{2+e^x} x^2\right )}{-4+3 x} \, dx\\ &=\int e^{-2 \left (2+e^x+x\right )} \, dx-\int e^{-4-2 e^x-x} \left (-1+e^{2+e^x} (-1+x)\right ) \, dx-\int \left (e^{-2-e^x} x+\frac {7-7 x+3 x^2}{-4+3 x}\right ) \, dx\\ &=\int e^{-2 \left (2+e^x+x\right )} \, dx-\int \left (-e^{-4-2 e^x-x}+e^{-2-e^x-x} (-1+x)\right ) \, dx-\int e^{-2-e^x} x \, dx-\int \frac {7-7 x+3 x^2}{-4+3 x} \, dx\\ &=\int e^{-4-2 e^x-x} \, dx+\int e^{-2 \left (2+e^x+x\right )} \, dx-\int e^{-2-e^x-x} (-1+x) \, dx-\int e^{-2-e^x} x \, dx-\int \left (-1+x+\frac {3}{-4+3 x}\right ) \, dx\\ &=x-\frac {x^2}{2}-\log (4-3 x)+\int e^{-2 \left (2+e^x+x\right )} \, dx-\int e^{-2-e^x} x \, dx-\int \left (-e^{-2-e^x-x}+e^{-2-e^x-x} x\right ) \, dx+\operatorname {Subst}\left (\int \frac {e^{-4-2 x}}{x^2} \, dx,x,e^x\right )\\ &=-e^{-4-2 e^x-x}+x-\frac {x^2}{2}-\log (4-3 x)-2 \operatorname {Subst}\left (\int \frac {e^{-4-2 x}}{x} \, dx,x,e^x\right )+\int e^{-2-e^x-x} \, dx+\int e^{-2 \left (2+e^x+x\right )} \, dx-\int e^{-2-e^x} x \, dx-\int e^{-2-e^x-x} x \, dx\\ &=-e^{-4-2 e^x-x}+x-\frac {x^2}{2}-\frac {2 \text {Ei}\left (-2 e^x\right )}{e^4}-\log (4-3 x)+\int e^{-2 \left (2+e^x+x\right )} \, dx-\int e^{-2-e^x} x \, dx-\int e^{-2-e^x-x} x \, dx+\operatorname {Subst}\left (\int \frac {e^{-2-x}}{x^2} \, dx,x,e^x\right )\\ &=-e^{-4-2 e^x-x}-e^{-2-e^x-x}+x-\frac {x^2}{2}-\frac {2 \text {Ei}\left (-2 e^x\right )}{e^4}-\log (4-3 x)+\int e^{-2 \left (2+e^x+x\right )} \, dx-\int e^{-2-e^x} x \, dx-\int e^{-2-e^x-x} x \, dx-\operatorname {Subst}\left (\int \frac {e^{-2-x}}{x} \, dx,x,e^x\right )\\ &=-e^{-4-2 e^x-x}-e^{-2-e^x-x}+x-\frac {x^2}{2}-\frac {2 \text {Ei}\left (-2 e^x\right )}{e^4}-\frac {\text {Ei}\left (-e^x\right )}{e^2}-\log (4-3 x)+\int e^{-2 \left (2+e^x+x\right )} \, dx-\int e^{-2-e^x} x \, dx-\int e^{-2-e^x-x} x \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.32, size = 47, normalized size = 1.47 \begin {gather*} -\frac {1}{2} e^{-4-2 e^x-2 x}+x+e^{-2-e^x-x} x-\frac {x^2}{2}-\log (4-3 x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.83, size = 39, normalized size = 1.22 \begin {gather*} -\frac {1}{2} \, x^{2} + x e^{\left (-x - e^{x} - 2\right )} + x - \frac {1}{2} \, e^{\left (-2 \, x - 2 \, e^{x} - 4\right )} - \log \left (3 \, x - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {3 \, x^{2} + {\left (3 \, x^{2} + {\left (3 \, x^{2} - 4 \, x\right )} e^{x} - 7 \, x + 4\right )} e^{\left (-x - e^{x} - 2\right )} - {\left ({\left (3 \, x - 4\right )} e^{x} + 3 \, x - 4\right )} e^{\left (-2 \, x - 2 \, e^{x} - 4\right )} - 7 \, x + 7}{3 \, x - 4}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(-(3*x^2 + (3*x^2 + (3*x^2 - 4*x)*e^x - 7*x + 4)*e^(-x - e^x - 2) - ((3*x - 4)*e^x + 3*x - 4)*e^(-2*x
- 2*e^x - 4) - 7*x + 7)/(3*x - 4), x)

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maple [A]  time = 0.27, size = 40, normalized size = 1.25

method	result	size
risch	\(x +{\mathrm e}^{-{\mathrm e}^{x}-x -2} x -\frac {x^{2}}{2}-\frac {{\mathrm e}^{-2 \,{\mathrm e}^{x}-2 x -4}}{2}-\ln \left (3 x -4\right )\)	\(40\)
norman	\(x +{\mathrm e}^{-{\mathrm e}^{x}-x -2} x -\frac {x^{2}}{2}-\frac {{\mathrm e}^{-2 \,{\mathrm e}^{x}-2 x -4}}{2}-\ln \left (3 x -4\right )\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.80, size = 42, normalized size = 1.31 \begin {gather*} -\frac {1}{2} \, x^{2} + \frac {1}{2} \, {\left (2 \, x e^{\left (x - e^{x} + 2\right )} - e^{\left (-2 \, e^{x}\right )}\right )} e^{\left (-2 \, x - 4\right )} + x - \log \left (3 \, x - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.55, size = 37, normalized size = 1.16 \begin {gather*} x-\ln \left (x-\frac {4}{3}\right )-\frac {{\mathrm {e}}^{-2\,x-2\,{\mathrm {e}}^x-4}}{2}+x\,{\mathrm {e}}^{-x-{\mathrm {e}}^x-2}-\frac {x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((7*x + exp(- x - exp(x) - 2)*(7*x + exp(x)*(4*x - 3*x^2) - 3*x^2 - 4) - 3*x^2 + exp(- 2*x - 2*exp(x) - 4)*
(3*x + exp(x)*(3*x - 4) - 4) - 7)/(3*x - 4),x)

[Out]

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

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sympy [A]  time = 0.25, size = 39, normalized size = 1.22 \begin {gather*} - \frac {x^{2}}{2} + x e^{- x - e^{x} - 2} + x - \frac {e^{- 2 x - 2 e^{x} - 4}}{2} - \log {\left (3 x - 4 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x-4)*exp(x)+3*x-4)*exp(-exp(x)-x-2)**2+((-3*x**2+4*x)*exp(x)-3*x**2+7*x-4)*exp(-exp(x)-x-2)-3*x
**2+7*x-7)/(3*x-4),x)

[Out]

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

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