3.72.96 \(\int \frac {e^x (3 x-2 x^2-x^3+e^5 (x-x^2))+e^x (-3+e^5 (-1+x)+2 x+x^2) \log (3+e^5+x)+\frac {(e^x (-x-4 x^2-x^3+e^5 (-x-x^2))+e^x (-3+e^5 (-1+x)+2 x+x^2) \log (3+e^5+x)) (x^2-2 x \log (3+e^5+x)+\log ^2(3+e^5+x))}{e}}{-3 x^3-e^5 x^3-x^4+(3 x^2+e^5 x^2+x^3) \log (3+e^5+x)} \, dx\)

Optimal. Leaf size=26 \[ \frac {e^x \left (1+\frac {\left (x-\log \left (3+e^5+x\right )\right )^2}{e}\right )}{x} \]

________________________________________________________________________________________

Rubi [F]  time = 2.90, 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^x \left (3 x-2 x^2-x^3+e^5 \left (x-x^2\right )\right )+e^x \left (-3+e^5 (-1+x)+2 x+x^2\right ) \log \left (3+e^5+x\right )+\frac {\left (e^x \left (-x-4 x^2-x^3+e^5 \left (-x-x^2\right )\right )+e^x \left (-3+e^5 (-1+x)+2 x+x^2\right ) \log \left (3+e^5+x\right )\right ) \left (x^2-2 x \log \left (3+e^5+x\right )+\log ^2\left (3+e^5+x\right )\right )}{e}}{-3 x^3-e^5 x^3-x^4+\left (3 x^2+e^5 x^2+x^3\right ) \log \left (3+e^5+x\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x*(3*x - 2*x^2 - x^3 + E^5*(x - x^2)) + E^x*(-3 + E^5*(-1 + x) + 2*x + x^2)*Log[3 + E^5 + x] + ((E^x*(-
x - 4*x^2 - x^3 + E^5*(-x - x^2)) + E^x*(-3 + E^5*(-1 + x) + 2*x + x^2)*Log[3 + E^5 + x])*(x^2 - 2*x*Log[3 + E
^5 + x] + Log[3 + E^5 + x]^2))/E)/(-3*x^3 - E^5*x^3 - x^4 + (3*x^2 + E^5*x^2 + x^3)*Log[3 + E^5 + x]),x]

[Out]

E^x/x + E^(-1 + x)*x - 2*E^(-1 + x)*Log[3 + E^5 + x] + (2*ExpIntegralEi[x]*Log[3 + E^5 + x])/(E*(3 + E^5)) - (
2*E^(-4 - E^5)*ExpIntegralEi[3 + E^5 + x]*Log[3 + E^5 + x])/(3 + E^5) - (2*Defer[Int][ExpIntegralEi[x]/(3 + E^
5 + x), x])/(E*(3 + E^5)) + (2*E^(-4 - E^5)*Defer[Int][ExpIntegralEi[3 + E^5 + x]/(3 + E^5 + x), x])/(3 + E^5)
- Defer[Int][(E^(-1 + x)*Log[3 + E^5 + x]^2)/x^2, x] + Defer[Int][(E^(-1 + x)*Log[3 + E^5 + x]^2)/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x \left (3 x-2 x^2-x^3+e^5 \left (x-x^2\right )\right )+e^x \left (-3+e^5 (-1+x)+2 x+x^2\right ) \log \left (3+e^5+x\right )+\frac {\left (e^x \left (-x-4 x^2-x^3+e^5 \left (-x-x^2\right )\right )+e^x \left (-3+e^5 (-1+x)+2 x+x^2\right ) \log \left (3+e^5+x\right )\right ) \left (x^2-2 x \log \left (3+e^5+x\right )+\log ^2\left (3+e^5+x\right )\right )}{e}}{\left (-3-e^5\right ) x^3-x^4+\left (3 x^2+e^5 x^2+x^3\right ) \log \left (3+e^5+x\right )} \, dx\\ &=\int \frac {e^{-1+x} \left (e^6 (-1+x)+e^5 x^2 (1+x)+e \left (-3+2 x+x^2\right )+x^2 \left (1+4 x+x^2\right )-2 x \left (-1+\left (3+e^5\right ) x+x^2\right ) \log \left (3+e^5+x\right )+\left (-3+e^5 (-1+x)+2 x+x^2\right ) \log ^2\left (3+e^5+x\right )\right )}{x^2 \left (3+e^5+x\right )} \, dx\\ &=\int \left (\frac {e^{-1+x} \left (-e \left (3+e^5\right )+e \left (2+e^5\right ) x+\left (1+e+e^5\right ) x^2+\left (4+e^5\right ) x^3+x^4\right )}{x^2 \left (3+e^5+x\right )}+\frac {2 e^{-1+x} \left (1-\left (3+e^5\right ) x-x^2\right ) \log \left (3+e^5+x\right )}{x \left (3+e^5+x\right )}+\frac {e^{-1+x} (-1+x) \log ^2\left (3+e^5+x\right )}{x^2}\right ) \, dx\\ &=2 \int \frac {e^{-1+x} \left (1-\left (3+e^5\right ) x-x^2\right ) \log \left (3+e^5+x\right )}{x \left (3+e^5+x\right )} \, dx+\int \frac {e^{-1+x} \left (-e \left (3+e^5\right )+e \left (2+e^5\right ) x+\left (1+e+e^5\right ) x^2+\left (4+e^5\right ) x^3+x^4\right )}{x^2 \left (3+e^5+x\right )} \, dx+\int \frac {e^{-1+x} (-1+x) \log ^2\left (3+e^5+x\right )}{x^2} \, dx\\ &=-2 e^{-1+x} \log \left (3+e^5+x\right )+\frac {2 \text {Ei}(x) \log \left (3+e^5+x\right )}{e \left (3+e^5\right )}-\frac {2 e^{-4-e^5} \text {Ei}\left (3+e^5+x\right ) \log \left (3+e^5+x\right )}{3+e^5}-2 \int \frac {-e^x \left (3+e^5\right )+\text {Ei}(x)-e^{-3-e^5} \text {Ei}\left (3+e^5+x\right )}{e \left (3+e^5\right ) \left (3+e^5+x\right )} \, dx+\int \left (e^{-1+x}-\frac {e^x}{x^2}+\frac {e^x}{x}+e^{-1+x} x-\frac {2 e^{-1+x}}{3+e^5+x}\right ) \, dx+\int \left (-\frac {e^{-1+x} \log ^2\left (3+e^5+x\right )}{x^2}+\frac {e^{-1+x} \log ^2\left (3+e^5+x\right )}{x}\right ) \, dx\\ &=-2 e^{-1+x} \log \left (3+e^5+x\right )+\frac {2 \text {Ei}(x) \log \left (3+e^5+x\right )}{e \left (3+e^5\right )}-\frac {2 e^{-4-e^5} \text {Ei}\left (3+e^5+x\right ) \log \left (3+e^5+x\right )}{3+e^5}-2 \int \frac {e^{-1+x}}{3+e^5+x} \, dx-\frac {2 \int \frac {-e^x \left (3+e^5\right )+\text {Ei}(x)-e^{-3-e^5} \text {Ei}\left (3+e^5+x\right )}{3+e^5+x} \, dx}{e \left (3+e^5\right )}+\int e^{-1+x} \, dx-\int \frac {e^x}{x^2} \, dx+\int \frac {e^x}{x} \, dx+\int e^{-1+x} x \, dx-\int \frac {e^{-1+x} \log ^2\left (3+e^5+x\right )}{x^2} \, dx+\int \frac {e^{-1+x} \log ^2\left (3+e^5+x\right )}{x} \, dx\\ &=e^{-1+x}+\frac {e^x}{x}+e^{-1+x} x+\text {Ei}(x)-2 e^{-4-e^5} \text {Ei}\left (3+e^5+x\right )-2 e^{-1+x} \log \left (3+e^5+x\right )+\frac {2 \text {Ei}(x) \log \left (3+e^5+x\right )}{e \left (3+e^5\right )}-\frac {2 e^{-4-e^5} \text {Ei}\left (3+e^5+x\right ) \log \left (3+e^5+x\right )}{3+e^5}-\frac {2 \int \left (-\frac {e^x \left (3+e^5\right )}{3+e^5+x}+\frac {e^{-3-e^5} \left (e^{3+e^5} \text {Ei}(x)-\text {Ei}\left (3+e^5+x\right )\right )}{3+e^5+x}\right ) \, dx}{e \left (3+e^5\right )}-\int e^{-1+x} \, dx-\int \frac {e^x}{x} \, dx-\int \frac {e^{-1+x} \log ^2\left (3+e^5+x\right )}{x^2} \, dx+\int \frac {e^{-1+x} \log ^2\left (3+e^5+x\right )}{x} \, dx\\ &=\frac {e^x}{x}+e^{-1+x} x-2 e^{-4-e^5} \text {Ei}\left (3+e^5+x\right )-2 e^{-1+x} \log \left (3+e^5+x\right )+\frac {2 \text {Ei}(x) \log \left (3+e^5+x\right )}{e \left (3+e^5\right )}-\frac {2 e^{-4-e^5} \text {Ei}\left (3+e^5+x\right ) \log \left (3+e^5+x\right )}{3+e^5}+\frac {2 \int \frac {e^x}{3+e^5+x} \, dx}{e}-\frac {\left (2 e^{-4-e^5}\right ) \int \frac {e^{3+e^5} \text {Ei}(x)-\text {Ei}\left (3+e^5+x\right )}{3+e^5+x} \, dx}{3+e^5}-\int \frac {e^{-1+x} \log ^2\left (3+e^5+x\right )}{x^2} \, dx+\int \frac {e^{-1+x} \log ^2\left (3+e^5+x\right )}{x} \, dx\\ &=\frac {e^x}{x}+e^{-1+x} x-2 e^{-1+x} \log \left (3+e^5+x\right )+\frac {2 \text {Ei}(x) \log \left (3+e^5+x\right )}{e \left (3+e^5\right )}-\frac {2 e^{-4-e^5} \text {Ei}\left (3+e^5+x\right ) \log \left (3+e^5+x\right )}{3+e^5}-\frac {\left (2 e^{-4-e^5}\right ) \int \left (\frac {e^{3+e^5} \text {Ei}(x)}{3+e^5+x}-\frac {\text {Ei}\left (3+e^5+x\right )}{3+e^5+x}\right ) \, dx}{3+e^5}-\int \frac {e^{-1+x} \log ^2\left (3+e^5+x\right )}{x^2} \, dx+\int \frac {e^{-1+x} \log ^2\left (3+e^5+x\right )}{x} \, dx\\ &=\frac {e^x}{x}+e^{-1+x} x-2 e^{-1+x} \log \left (3+e^5+x\right )+\frac {2 \text {Ei}(x) \log \left (3+e^5+x\right )}{e \left (3+e^5\right )}-\frac {2 e^{-4-e^5} \text {Ei}\left (3+e^5+x\right ) \log \left (3+e^5+x\right )}{3+e^5}-\frac {2 \int \frac {\text {Ei}(x)}{3+e^5+x} \, dx}{e \left (3+e^5\right )}+\frac {\left (2 e^{-4-e^5}\right ) \int \frac {\text {Ei}\left (3+e^5+x\right )}{3+e^5+x} \, dx}{3+e^5}-\int \frac {e^{-1+x} \log ^2\left (3+e^5+x\right )}{x^2} \, dx+\int \frac {e^{-1+x} \log ^2\left (3+e^5+x\right )}{x} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 33, normalized size = 1.27 \begin {gather*} \frac {e^{-1+x} \left (e+x^2-2 x \log \left (3+e^5+x\right )+\log ^2\left (3+e^5+x\right )\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(3*x - 2*x^2 - x^3 + E^5*(x - x^2)) + E^x*(-3 + E^5*(-1 + x) + 2*x + x^2)*Log[3 + E^5 + x] + ((
E^x*(-x - 4*x^2 - x^3 + E^5*(-x - x^2)) + E^x*(-3 + E^5*(-1 + x) + 2*x + x^2)*Log[3 + E^5 + x])*(x^2 - 2*x*Log
[3 + E^5 + x] + Log[3 + E^5 + x]^2))/E)/(-3*x^3 - E^5*x^3 - x^4 + (3*x^2 + E^5*x^2 + x^3)*Log[3 + E^5 + x]),x]

[Out]

(E^(-1 + x)*(E + x^2 - 2*x*Log[3 + E^5 + x] + Log[3 + E^5 + x]^2))/x

________________________________________________________________________________________

fricas [A]  time = 1.14, size = 41, normalized size = 1.58 \begin {gather*} -\frac {{\left (2 \, x e^{x} \log \left (x + e^{5} + 3\right ) - e^{x} \log \left (x + e^{5} + 3\right )^{2} - {\left (x^{2} + e\right )} e^{x}\right )} e^{\left (-1\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x-1)*exp(5)+x^2+2*x-3)*exp(x)*log(exp(5)+3+x)+((-x^2-x)*exp(5)-x^3-4*x^2-x)*exp(x))*exp(log(log(
exp(5)+3+x)^2-2*x*log(exp(5)+3+x)+x^2)-1)+((x-1)*exp(5)+x^2+2*x-3)*exp(x)*log(exp(5)+3+x)+((-x^2+x)*exp(5)-x^3
-2*x^2+3*x)*exp(x))/((x^2*exp(5)+x^3+3*x^2)*log(exp(5)+3+x)-x^3*exp(5)-x^4-3*x^3),x, algorithm="fricas")

[Out]

-(2*x*e^x*log(x + e^5 + 3) - e^x*log(x + e^5 + 3)^2 - (x^2 + e)*e^x)*e^(-1)/x

________________________________________________________________________________________

giac [A]  time = 0.33, size = 39, normalized size = 1.50 \begin {gather*} \frac {{\left (x^{2} e^{x} - 2 \, x e^{x} \log \left (x + e^{5} + 3\right ) + e^{x} \log \left (x + e^{5} + 3\right )^{2} + e^{\left (x + 1\right )}\right )} e^{\left (-1\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x-1)*exp(5)+x^2+2*x-3)*exp(x)*log(exp(5)+3+x)+((-x^2-x)*exp(5)-x^3-4*x^2-x)*exp(x))*exp(log(log(
exp(5)+3+x)^2-2*x*log(exp(5)+3+x)+x^2)-1)+((x-1)*exp(5)+x^2+2*x-3)*exp(x)*log(exp(5)+3+x)+((-x^2+x)*exp(5)-x^3
-2*x^2+3*x)*exp(x))/((x^2*exp(5)+x^3+3*x^2)*log(exp(5)+3+x)-x^3*exp(5)-x^4-3*x^3),x, algorithm="giac")

[Out]

(x^2*e^x - 2*x*e^x*log(x + e^5 + 3) + e^x*log(x + e^5 + 3)^2 + e^(x + 1))*e^(-1)/x

________________________________________________________________________________________

maple [F]  time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (\left (x -1\right ) {\mathrm e}^{5}+x^{2}+2 x -3\right ) {\mathrm e}^{x} \ln \left ({\mathrm e}^{5}+3+x \right )+\left (\left (-x^{2}-x \right ) {\mathrm e}^{5}-x^{3}-4 x^{2}-x \right ) {\mathrm e}^{x}\right ) {\mathrm e}^{\ln \left (\ln \left ({\mathrm e}^{5}+3+x \right )^{2}-2 x \ln \left ({\mathrm e}^{5}+3+x \right )+x^{2}\right )-1}+\left (\left (x -1\right ) {\mathrm e}^{5}+x^{2}+2 x -3\right ) {\mathrm e}^{x} \ln \left ({\mathrm e}^{5}+3+x \right )+\left (\left (-x^{2}+x \right ) {\mathrm e}^{5}-x^{3}-2 x^{2}+3 x \right ) {\mathrm e}^{x}}{\left (x^{2} {\mathrm e}^{5}+x^{3}+3 x^{2}\right ) \ln \left ({\mathrm e}^{5}+3+x \right )-x^{3} {\mathrm e}^{5}-x^{4}-3 x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((x-1)*exp(5)+x^2+2*x-3)*exp(x)*ln(exp(5)+3+x)+((-x^2-x)*exp(5)-x^3-4*x^2-x)*exp(x))*exp(ln(ln(exp(5)+3+
x)^2-2*x*ln(exp(5)+3+x)+x^2)-1)+((x-1)*exp(5)+x^2+2*x-3)*exp(x)*ln(exp(5)+3+x)+((-x^2+x)*exp(5)-x^3-2*x^2+3*x)
*exp(x))/((x^2*exp(5)+x^3+3*x^2)*ln(exp(5)+3+x)-x^3*exp(5)-x^4-3*x^3),x)

[Out]

int(((((x-1)*exp(5)+x^2+2*x-3)*exp(x)*ln(exp(5)+3+x)+((-x^2-x)*exp(5)-x^3-4*x^2-x)*exp(x))*exp(ln(ln(exp(5)+3+
x)^2-2*x*ln(exp(5)+3+x)+x^2)-1)+((x-1)*exp(5)+x^2+2*x-3)*exp(x)*ln(exp(5)+3+x)+((-x^2+x)*exp(5)-x^3-2*x^2+3*x)
*exp(x))/((x^2*exp(5)+x^3+3*x^2)*ln(exp(5)+3+x)-x^3*exp(5)-x^4-3*x^3),x)

________________________________________________________________________________________

maxima [A]  time = 0.42, size = 41, normalized size = 1.58 \begin {gather*} -\frac {{\left (2 \, x e^{x} \log \left (x + e^{5} + 3\right ) - e^{x} \log \left (x + e^{5} + 3\right )^{2} - {\left (x^{2} + e\right )} e^{x}\right )} e^{\left (-1\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x-1)*exp(5)+x^2+2*x-3)*exp(x)*log(exp(5)+3+x)+((-x^2-x)*exp(5)-x^3-4*x^2-x)*exp(x))*exp(log(log(
exp(5)+3+x)^2-2*x*log(exp(5)+3+x)+x^2)-1)+((x-1)*exp(5)+x^2+2*x-3)*exp(x)*log(exp(5)+3+x)+((-x^2+x)*exp(5)-x^3
-2*x^2+3*x)*exp(x))/((x^2*exp(5)+x^3+3*x^2)*log(exp(5)+3+x)-x^3*exp(5)-x^4-3*x^3),x, algorithm="maxima")

[Out]

-(2*x*e^x*log(x + e^5 + 3) - e^x*log(x + e^5 + 3)^2 - (x^2 + e)*e^x)*e^(-1)/x

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} -\int \frac {{\mathrm {e}}^x\,\left (3\,x+{\mathrm {e}}^5\,\left (x-x^2\right )-2\,x^2-x^3\right )-{\mathrm {e}}^{\ln \left (x^2-2\,x\,\ln \left (x+{\mathrm {e}}^5+3\right )+{\ln \left (x+{\mathrm {e}}^5+3\right )}^2\right )-1}\,\left ({\mathrm {e}}^x\,\left (x+{\mathrm {e}}^5\,\left (x^2+x\right )+4\,x^2+x^3\right )-{\mathrm {e}}^x\,\ln \left (x+{\mathrm {e}}^5+3\right )\,\left (2\,x+{\mathrm {e}}^5\,\left (x-1\right )+x^2-3\right )\right )+{\mathrm {e}}^x\,\ln \left (x+{\mathrm {e}}^5+3\right )\,\left (2\,x+{\mathrm {e}}^5\,\left (x-1\right )+x^2-3\right )}{x^3\,{\mathrm {e}}^5-\ln \left (x+{\mathrm {e}}^5+3\right )\,\left (x^2\,{\mathrm {e}}^5+3\,x^2+x^3\right )+3\,x^3+x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x)*(3*x + exp(5)*(x - x^2) - 2*x^2 - x^3) - exp(log(log(x + exp(5) + 3)^2 - 2*x*log(x + exp(5) + 3)
+ x^2) - 1)*(exp(x)*(x + exp(5)*(x + x^2) + 4*x^2 + x^3) - exp(x)*log(x + exp(5) + 3)*(2*x + exp(5)*(x - 1) +
x^2 - 3)) + exp(x)*log(x + exp(5) + 3)*(2*x + exp(5)*(x - 1) + x^2 - 3))/(x^3*exp(5) - log(x + exp(5) + 3)*(x^
2*exp(5) + 3*x^2 + x^3) + 3*x^3 + x^4),x)

[Out]

-int((exp(x)*(3*x + exp(5)*(x - x^2) - 2*x^2 - x^3) - exp(log(log(x + exp(5) + 3)^2 - 2*x*log(x + exp(5) + 3)
+ x^2) - 1)*(exp(x)*(x + exp(5)*(x + x^2) + 4*x^2 + x^3) - exp(x)*log(x + exp(5) + 3)*(2*x + exp(5)*(x - 1) +
x^2 - 3)) + exp(x)*log(x + exp(5) + 3)*(2*x + exp(5)*(x - 1) + x^2 - 3))/(x^3*exp(5) - log(x + exp(5) + 3)*(x^
2*exp(5) + 3*x^2 + x^3) + 3*x^3 + x^4), x)

________________________________________________________________________________________

sympy [A]  time = 0.60, size = 36, normalized size = 1.38 \begin {gather*} \frac {\left (x^{2} - 2 x \log {\left (x + 3 + e^{5} \right )} + \log {\left (x + 3 + e^{5} \right )}^{2} + e\right ) e^{x}}{e x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x-1)*exp(5)+x**2+2*x-3)*exp(x)*ln(exp(5)+3+x)+((-x**2-x)*exp(5)-x**3-4*x**2-x)*exp(x))*exp(ln(ln
(exp(5)+3+x)**2-2*x*ln(exp(5)+3+x)+x**2)-1)+((x-1)*exp(5)+x**2+2*x-3)*exp(x)*ln(exp(5)+3+x)+((-x**2+x)*exp(5)-
x**3-2*x**2+3*x)*exp(x))/((x**2*exp(5)+x**3+3*x**2)*ln(exp(5)+3+x)-x**3*exp(5)-x**4-3*x**3),x)

[Out]

(x**2 - 2*x*log(x + 3 + exp(5)) + log(x + 3 + exp(5))**2 + E)*exp(-1)*exp(x)/x

________________________________________________________________________________________