3.87.26 \(\int \frac {e^8 (-3-2 x)-3 x-x^2+e^x (3+x+e^8 (4+x))}{-3 x-x^2+e^x (3+x)} \, dx\)

Optimal. Leaf size=19 \[ 2+x+e^8 \log \left ((3+x) \left (-e^x+x\right )\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 0.48, 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 {e^8 (-3-2 x)-3 x-x^2+e^x \left (3+x+e^8 (4+x)\right )}{-3 x-x^2+e^x (3+x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.12, size = 22, normalized size = 1.16 \begin {gather*} x+e^8 \log \left (e^x-x\right )+e^8 \log (3+x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x + E^8*Log[E^x - x] + E^8*Log[3 + x]

________________________________________________________________________________________

fricas [A]  time = 0.60, size = 19, normalized size = 1.00 \begin {gather*} e^{8} \log \left (x + 3\right ) + e^{8} \log \left (-x + e^{x}\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.14, size = 19, normalized size = 1.00 \begin {gather*} e^{8} \log \left (x + 3\right ) + e^{8} \log \left (-x + e^{x}\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.06, size = 20, normalized size = 1.05

method	result	size
norman	\(x +{\mathrm e}^{8} \ln \left (3+x \right )+{\mathrm e}^{8} \ln \left (x -{\mathrm e}^{x}\right )\)	\(20\)
risch	\(x +{\mathrm e}^{8} \ln \left (3+x \right )+{\mathrm e}^{8} \ln \left ({\mathrm e}^{x}-x \right )\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x+exp(8)*ln(3+x)+exp(8)*ln(x-exp(x))

________________________________________________________________________________________

maxima [A]  time = 0.39, size = 19, normalized size = 1.00 \begin {gather*} e^{8} \log \left (x + 3\right ) + e^{8} \log \left (-x + e^{x}\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.10, size = 19, normalized size = 1.00 \begin {gather*} x+\ln \left (x+3\right )\,{\mathrm {e}}^8+{\mathrm {e}}^8\,\ln \left (x-{\mathrm {e}}^x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + log(x + 3)*exp(8) + exp(8)*log(x - exp(x))

________________________________________________________________________________________

sympy [A]  time = 0.16, size = 19, normalized size = 1.00 \begin {gather*} x + e^{8} \log {\left (- x + e^{x} \right )} + e^{8} \log {\left (x + 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + exp(8)*log(-x + exp(x)) + exp(8)*log(x + 3)

________________________________________________________________________________________