\(\int \frac {-7+7 x-3 x^2+e^{-4-2 e^x-2 x} (-4+3 x+e^x (-4+3 x))+e^{-2-e^x-x} (-4+7 x-3 x^2+e^x (4 x-3 x^2))}{-4+3 x} \, dx\) [567]

Optimal result

Integrand size = 81, antiderivative size = 32 \[ \int \frac {-7+7 x-3 x^2+e^{-4-2 e^x-2 x} \left (-4+3 x+e^x (-4+3 x)\right )+e^{-2-e^x-x} \left (-4+7 x-3 x^2+e^x \left (4 x-3 x^2\right )\right )}{-4+3 x} \, dx=x-\frac {1}{2} \left (-e^{-2-e^x-x}+x\right )^2-\log (-4+3 x) \]

[Out]

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

Rubi [F]

\[ \int \frac {-7+7 x-3 x^2+e^{-4-2 e^x-2 x} \left (-4+3 x+e^x (-4+3 x)\right )+e^{-2-e^x-x} \left (-4+7 x-3 x^2+e^x \left (4 x-3 x^2\right )\right )}{-4+3 x} \, dx=\int \frac {-7+7 x-3 x^2+e^{-4-2 e^x-2 x} \left (-4+3 x+e^x (-4+3 x)\right )+e^{-2-e^x-x} \left (-4+7 x-3 x^2+e^x \left (4 x-3 x^2\right )\right )}{-4+3 x} \, dx \]

[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{align*} \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+\text {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 \text {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 \operatorname {ExpIntegralEi}\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+\text {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 \operatorname {ExpIntegralEi}\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-\text {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 \operatorname {ExpIntegralEi}\left (-2 e^x\right )}{e^4}-\frac {\operatorname {ExpIntegralEi}\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{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.41 \[ \int \frac {-7+7 x-3 x^2+e^{-4-2 e^x-2 x} \left (-4+3 x+e^x (-4+3 x)\right )+e^{-2-e^x-x} \left (-4+7 x-3 x^2+e^x \left (4 x-3 x^2\right )\right )}{-4+3 x} \, dx=-\frac {1}{2} e^{-2 \left (2+e^x+x\right )}+x+e^{-2-e^x-x} x-\frac {x^2}{2}-\log (4-3 x) \]

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

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.25

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

[In]

int((((-4+3*x)*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
)/(-4+3*x),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(-4+3*x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.22 \[ \int \frac {-7+7 x-3 x^2+e^{-4-2 e^x-2 x} \left (-4+3 x+e^x (-4+3 x)\right )+e^{-2-e^x-x} \left (-4+7 x-3 x^2+e^x \left (4 x-3 x^2\right )\right )}{-4+3 x} \, dx=-\frac {1}{2} \, x^{2} + x e^{\left (-x - e^{x} - 2\right )} + x - \frac {1}{2} \, e^{\left (-2 \, x - 2 \, e^{x} - 4\right )} - \log \left (3 \, x - 4\right ) \]

[In]

integrate((((-4+3*x)*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)/(-4+3*x),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)

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.22 \[ \int \frac {-7+7 x-3 x^2+e^{-4-2 e^x-2 x} \left (-4+3 x+e^x (-4+3 x)\right )+e^{-2-e^x-x} \left (-4+7 x-3 x^2+e^x \left (4 x-3 x^2\right )\right )}{-4+3 x} \, dx=- \frac {x^{2}}{2} + x e^{- x - e^{x} - 2} + x - \frac {e^{- 2 x - 2 e^{x} - 4}}{2} - \log {\left (3 x - 4 \right )} \]

[In]

integrate((((-4+3*x)*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)/(-4+3*x),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \[ \int \frac {-7+7 x-3 x^2+e^{-4-2 e^x-2 x} \left (-4+3 x+e^x (-4+3 x)\right )+e^{-2-e^x-x} \left (-4+7 x-3 x^2+e^x \left (4 x-3 x^2\right )\right )}{-4+3 x} \, dx=-\frac {1}{2} \, x^{2} + \frac {1}{2} \, {\left (2 \, x e^{\left (x - e^{x} + 2\right )} - e^{\left (-2 \, e^{x}\right )}\right )} e^{\left (-2 \, x - 4\right )} + x - \log \left (3 \, x - 4\right ) \]

[In]

integrate((((-4+3*x)*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)/(-4+3*x),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)

Giac [F]

\[ \int \frac {-7+7 x-3 x^2+e^{-4-2 e^x-2 x} \left (-4+3 x+e^x (-4+3 x)\right )+e^{-2-e^x-x} \left (-4+7 x-3 x^2+e^x \left (4 x-3 x^2\right )\right )}{-4+3 x} \, dx=\int { -\frac {3 \, x^{2} + {\left (3 \, x^{2} + {\left (3 \, x^{2} - 4 \, x\right )} e^{x} - 7 \, x + 4\right )} e^{\left (-x - e^{x} - 2\right )} - {\left ({\left (3 \, x - 4\right )} e^{x} + 3 \, x - 4\right )} e^{\left (-2 \, x - 2 \, e^{x} - 4\right )} - 7 \, x + 7}{3 \, x - 4} \,d x } \]

[In]

integrate((((-4+3*x)*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)/(-4+3*x),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)

Mupad [B] (verification not implemented)

Time = 7.99 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {-7+7 x-3 x^2+e^{-4-2 e^x-2 x} \left (-4+3 x+e^x (-4+3 x)\right )+e^{-2-e^x-x} \left (-4+7 x-3 x^2+e^x \left (4 x-3 x^2\right )\right )}{-4+3 x} \, dx=x-\ln \left (x-\frac {4}{3}\right )-\frac {{\mathrm {e}}^{-2\,x-2\,{\mathrm {e}}^x-4}}{2}+x\,{\mathrm {e}}^{-x-{\mathrm {e}}^x-2}-\frac {x^2}{2} \]

[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