3.33.28 $\int \frac{-2x-e^{5}x-x^2+e^x(-6e^5{x}^3-6x^4)+e^{2x}(-6e^5{x}^3-6x^4)+(e^5+x+e^x(-18e^5{x}^2-18x^3)+e^{2x}(-18e^5{x}^2-18x^3))\log (e^5+x)+(e^x(-18e^{5}x-18x^2)+e^{2x}(-18e^{5}x-18x^2)){\log }^2(e^5+x)+(e^x(-6e^5-6x)+e^{2x}(-6e^5-6x)){\log }^3(e^5+x)}{e^5{x}^3+x^4+(3e^5{x}^2+3x^3)\log (e^5+x)+(3e^{5}x+3x^2){\log }^2(e^5+x)+(e^5+x){\log }^3(e^5+x)}dx$

Optimal. Leaf size=22 $$-3{(1+e^x)}^2+\frac{x}{{(x+\log (e^5+x))}^2}$$

________________________________________________________________________________________

Rubi [F]  time = 2.15, 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 = {} $$\begin{array}{c}\int \frac{-2x-e^{5}x-x^2+e^x(-6e^5{x}^3-6x^4)+e^{2x}(-6e^5{x}^3-6x^4)+(e^5+x+e^x(-18e^5{x}^2-18x^3)+e^{2x}(-18e^5{x}^2-18x^3))\log (e^5+x)+(e^x(-18e^{5}x-18x^2)+e^{2x}(-18e^{5}x-18x^2)){\log }^2(e^5+x)+(e^x(-6e^5-6x)+e^{2x}(-6e^5-6x)){\log }^3(e^5+x)}{e^5{x}^3+x^4+(3e^5{x}^2+3x^3)\log (e^5+x)+(3e^{5}x+3x^2){\log }^2(e^5+x)+(e^5+x){\log }^3(e^5+x)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(-2*x - E^5*x - x^2 + E^x*(-6*E^5*x^3 - 6*x^4) + E^(2*x)*(-6*E^5*x^3 - 6*x^4) + (E^5 + x + E^x*(-18*E^5*x^
2 - 18*x^3) + E^(2*x)*(-18*E^5*x^2 - 18*x^3))*Log[E^5 + x] + (E^x*(-18*E^5*x - 18*x^2) + E^(2*x)*(-18*E^5*x -
18*x^2))*Log[E^5 + x]^2 + (E^x*(-6*E^5 - 6*x) + E^(2*x)*(-6*E^5 - 6*x))*Log[E^5 + x]^3)/(E^5*x^3 + x^4 + (3*E^
5*x^2 + 3*x^3)*Log[E^5 + x] + (3*E^5*x + 3*x^2)*Log[E^5 + x]^2 + (E^5 + x)*Log[E^5 + x]^3),x]

[Out]

-6*E^x - 3*E^(2*x) + E^5*Defer[Int][(x + Log[E^5 + x])^(-3), x] - (2 + E^5)*Defer[Int][(x + Log[E^5 + x])^(-3)
, x] - 2*Defer[Int][x/(x + Log[E^5 + x])^3, x] - E^10*Defer[Int][1/((E^5 + x)*(x + Log[E^5 + x])^3), x] + E^5*
(2 + E^5)*Defer[Int][1/((E^5 + x)*(x + Log[E^5 + x])^3), x] + Defer[Int][(x + Log[E^5 + x])^(-2), x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{(-2-e^5)x-x^2+e^x(-6e^5{x}^3-6x^4)+e^{2x}(-6e^5{x}^3-6x^4)+(e^5+x+e^x(-18e^5{x}^2-18x^3)+e^{2x}(-18e^5{x}^2-18x^3))\log (e^5+x)+(e^x(-18e^{5}x-18x^2)+e^{2x}(-18e^{5}x-18x^2)){\log }^2(e^5+x)+(e^x(-6e^5-6x)+e^{2x}(-6e^5-6x)){\log }^3(e^5+x)}{e^5{x}^3+x^4+(3e^5{x}^2+3x^3)\log (e^5+x)+(3e^{5}x+3x^2){\log }^2(e^5+x)+(e^5+x){\log }^3(e^5+x)}dx\\ & =\int \frac{-x(2+e^5+x+6e^{5+x}{x}^2+6e^{5+2x}{x}^2+6e^x{x}^3+6e^{2x}{x}^3)-(e^5+x)(-1+18e^x{x}^2+18e^{2x}{x}^2)\log (e^5+x)-18e^x(1+e^x)x(e^5+x){\log }^2(e^5+x)-6e^x(1+e^x)(e^5+x){\log }^3(e^5+x)}{(e^5+x){(x+\log (e^5+x))}^3}dx\\ & =\int (-6e^x-6e^{2x}-\frac{2(1+\frac{e^5}{2})x}{(e^5+x){(x+\log (e^5+x))}^3}-\frac{x^2}{(e^5+x){(x+\log (e^5+x))}^3}+\frac{\log (e^5+x)}{{(x+\log (e^5+x))}^3})dx\\ & =-(6\int e^{x}dx)-6\int e^{2x}dx-(2+e^5)\int \frac{x}{(e^5+x){(x+\log (e^5+x))}^3}dx-\int \frac{x^2}{(e^5+x){(x+\log (e^5+x))}^3}dx+\int \frac{\log (e^5+x)}{{(x+\log (e^5+x))}^3}dx\\ & =-6e^x-3e^{2x}-(2+e^5)\int (\frac{1}{{(x+\log (e^5+x))}^3}-\frac{e^5}{(e^5+x){(x+\log (e^5+x))}^3})dx-\int (-\frac{e^5}{{(x+\log (e^5+x))}^3}+\frac{x}{{(x+\log (e^5+x))}^3}+\frac{e^{10}}{(e^5+x){(x+\log (e^5+x))}^3})dx+\int (-\frac{x}{{(x+\log (e^5+x))}^3}+\frac{1}{{(x+\log (e^5+x))}^2})dx\\ & =-6e^x-3e^{2x}+e^5\int \frac{1}{{(x+\log (e^5+x))}^3}dx-e^{10}\int \frac{1}{(e^5+x){(x+\log (e^5+x))}^3}dx-(2+e^5)\int \frac{1}{{(x+\log (e^5+x))}^3}dx+\left(e^5(2+e^5)\right)\int \frac{1}{(e^5+x){(x+\log (e^5+x))}^3}dx-2\int \frac{x}{{(x+\log (e^5+x))}^3}dx+\int \frac{1}{{(x+\log (e^5+x))}^2}dx\end{array}\end{array}$$

________________________________________________________________________________________

Mathematica [A]  time = 0.13, size = 25, normalized size = 1.14 $$\begin{array}{c}-6e^x-3e^{2x}+\frac{x}{{(x+\log (e^5+x))}^2}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-2*x - E^5*x - x^2 + E^x*(-6*E^5*x^3 - 6*x^4) + E^(2*x)*(-6*E^5*x^3 - 6*x^4) + (E^5 + x + E^x*(-18*
E^5*x^2 - 18*x^3) + E^(2*x)*(-18*E^5*x^2 - 18*x^3))*Log[E^5 + x] + (E^x*(-18*E^5*x - 18*x^2) + E^(2*x)*(-18*E^
5*x - 18*x^2))*Log[E^5 + x]^2 + (E^x*(-6*E^5 - 6*x) + E^(2*x)*(-6*E^5 - 6*x))*Log[E^5 + x]^3)/(E^5*x^3 + x^4 +
(3*E^5*x^2 + 3*x^3)*Log[E^5 + x] + (3*E^5*x + 3*x^2)*Log[E^5 + x]^2 + (E^5 + x)*Log[E^5 + x]^3),x]

[Out]

-6*E^x - 3*E^(2*x) + x/(x + Log[E^5 + x])^2

________________________________________________________________________________________

fricas [B]  time = 0.71, size = 80, normalized size = 3.64 $$\begin{array}{c}-\frac{3x^2{e}^{\left(2x\right)}+6x^2{e}^x+3(e^{\left(2x\right)}+2e^x)\log {(x+e^5)}^2+6(x{e}^{\left(2x\right)}+2x{e}^x)\log (x+e^5)-x}{x^2+2x\log (x+e^5)+\log {(x+e^5)}^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*exp(5)-6*x)*exp(x)^2+(-6*exp(5)-6*x)*exp(x))*log(exp(5)+x)^3+((-18*x*exp(5)-18*x^2)*exp(x)^2+(
-18*x*exp(5)-18*x^2)*exp(x))*log(exp(5)+x)^2+((-18*x^2*exp(5)-18*x^3)*exp(x)^2+(-18*x^2*exp(5)-18*x^3)*exp(x)+
exp(5)+x)*log(exp(5)+x)+(-6*x^3*exp(5)-6*x^4)*exp(x)^2+(-6*x^3*exp(5)-6*x^4)*exp(x)-x*exp(5)-x^2-2*x)/((exp(5)
+x)*log(exp(5)+x)^3+(3*x*exp(5)+3*x^2)*log(exp(5)+x)^2+(3*x^2*exp(5)+3*x^3)*log(exp(5)+x)+x^3*exp(5)+x^4),x, a
lgorithm="fricas")

[Out]

-(3*x^2*e^(2*x) + 6*x^2*e^x + 3*(e^(2*x) + 2*e^x)*log(x + e^5)^2 + 6*(x*e^(2*x) + 2*x*e^x)*log(x + e^5) - x)/(
x^2 + 2*x*log(x + e^5) + log(x + e^5)^2)

________________________________________________________________________________________

giac [B]  time = 1.11, size = 89, normalized size = 4.05 $$\begin{array}{c}-\frac{3x^2{e}^{\left(2x\right)}+6x^2{e}^x+6x{e}^{\left(2x\right)}\log (x+e^5)+12x{e}^x\log (x+e^5)+3e^{\left(2x\right)}\log {(x+e^5)}^2+6e^x\log {(x+e^5)}^2-x}{x^2+2x\log (x+e^5)+\log {(x+e^5)}^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*exp(5)-6*x)*exp(x)^2+(-6*exp(5)-6*x)*exp(x))*log(exp(5)+x)^3+((-18*x*exp(5)-18*x^2)*exp(x)^2+(
-18*x*exp(5)-18*x^2)*exp(x))*log(exp(5)+x)^2+((-18*x^2*exp(5)-18*x^3)*exp(x)^2+(-18*x^2*exp(5)-18*x^3)*exp(x)+
exp(5)+x)*log(exp(5)+x)+(-6*x^3*exp(5)-6*x^4)*exp(x)^2+(-6*x^3*exp(5)-6*x^4)*exp(x)-x*exp(5)-x^2-2*x)/((exp(5)
+x)*log(exp(5)+x)^3+(3*x*exp(5)+3*x^2)*log(exp(5)+x)^2+(3*x^2*exp(5)+3*x^3)*log(exp(5)+x)+x^3*exp(5)+x^4),x, a
lgorithm="giac")

[Out]

-(3*x^2*e^(2*x) + 6*x^2*e^x + 6*x*e^(2*x)*log(x + e^5) + 12*x*e^x*log(x + e^5) + 3*e^(2*x)*log(x + e^5)^2 + 6*
e^x*log(x + e^5)^2 - x)/(x^2 + 2*x*log(x + e^5) + log(x + e^5)^2)

________________________________________________________________________________________

maple [A]  time = 0.36, size = 23, normalized size = 1.05

method	result	size
risch	$-3{\mathrm{e}}^{2x}-6{\mathrm{e}}^x+\frac{x}{{(\ln ({\mathrm{e}}^5+x)+x)}^2}$	$23$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-6*exp(5)-6*x)*exp(x)^2+(-6*exp(5)-6*x)*exp(x))*ln(exp(5)+x)^3+((-18*x*exp(5)-18*x^2)*exp(x)^2+(-18*x*e
xp(5)-18*x^2)*exp(x))*ln(exp(5)+x)^2+((-18*x^2*exp(5)-18*x^3)*exp(x)^2+(-18*x^2*exp(5)-18*x^3)*exp(x)+exp(5)+x
)*ln(exp(5)+x)+(-6*x^3*exp(5)-6*x^4)*exp(x)^2+(-6*x^3*exp(5)-6*x^4)*exp(x)-x*exp(5)-x^2-2*x)/((exp(5)+x)*ln(ex
p(5)+x)^3+(3*x*exp(5)+3*x^2)*ln(exp(5)+x)^2+(3*x^2*exp(5)+3*x^3)*ln(exp(5)+x)+x^3*exp(5)+x^4),x,method=_RETURN
VERBOSE)

[Out]

-3*exp(2*x)-6*exp(x)+x/(ln(exp(5)+x)+x)^2

________________________________________________________________________________________

maxima [B]  time = 0.81, size = 80, normalized size = 3.64 $$\begin{array}{c}-\frac{3x^2{e}^{\left(2x\right)}+6x^2{e}^x+3(e^{\left(2x\right)}+2e^x)\log {(x+e^5)}^2+6(x{e}^{\left(2x\right)}+2x{e}^x)\log (x+e^5)-x}{x^2+2x\log (x+e^5)+\log {(x+e^5)}^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*exp(5)-6*x)*exp(x)^2+(-6*exp(5)-6*x)*exp(x))*log(exp(5)+x)^3+((-18*x*exp(5)-18*x^2)*exp(x)^2+(
-18*x*exp(5)-18*x^2)*exp(x))*log(exp(5)+x)^2+((-18*x^2*exp(5)-18*x^3)*exp(x)^2+(-18*x^2*exp(5)-18*x^3)*exp(x)+
exp(5)+x)*log(exp(5)+x)+(-6*x^3*exp(5)-6*x^4)*exp(x)^2+(-6*x^3*exp(5)-6*x^4)*exp(x)-x*exp(5)-x^2-2*x)/((exp(5)
+x)*log(exp(5)+x)^3+(3*x*exp(5)+3*x^2)*log(exp(5)+x)^2+(3*x^2*exp(5)+3*x^3)*log(exp(5)+x)+x^3*exp(5)+x^4),x, a
lgorithm="maxima")

[Out]

-(3*x^2*e^(2*x) + 6*x^2*e^x + 3*(e^(2*x) + 2*e^x)*log(x + e^5)^2 + 6*(x*e^(2*x) + 2*x*e^x)*log(x + e^5) - x)/(
x^2 + 2*x*log(x + e^5) + log(x + e^5)^2)

________________________________________________________________________________________

mupad [B]  time = 2.70, size = 181, normalized size = 8.23 $$\begin{array}{c}\frac{\frac{x(x+{\mathrm{e}}^5+2)}{2(x+{\mathrm{e}}^5+1)}-\frac{\ln (x+{\mathrm{e}}^5)(x+{\mathrm{e}}^5)}{2(x+{\mathrm{e}}^5+1)}}{x^2+2x\ln (x+{\mathrm{e}}^5)+{\ln (x+{\mathrm{e}}^5)}^2}-6{\mathrm{e}}^x-3{\mathrm{e}}^{2x}+\frac{\frac{(x+{\mathrm{e}}^5)(x+2{\mathrm{e}}^5+{\mathrm{e}}^{10}+2x{\mathrm{e}}^5+x^2+1)}{2{(x+{\mathrm{e}}^5+1)}^3}-\frac{\ln (x+{\mathrm{e}}^5)(x+{\mathrm{e}}^5)}{2{(x+{\mathrm{e}}^5+1)}^3}}{x+\ln (x+{\mathrm{e}}^5)}+\frac{x+{\mathrm{e}}^5}{2x^3+(6{\mathrm{e}}^5+6)x^2+(12{\mathrm{e}}^5+6{\mathrm{e}}^{10}+6)x+6{\mathrm{e}}^5+6{\mathrm{e}}^{10}+2{\mathrm{e}}^{15}+2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x + x*exp(5) + log(x + exp(5))^2*(exp(x)*(18*x*exp(5) + 18*x^2) + exp(2*x)*(18*x*exp(5) + 18*x^2)) + e
xp(x)*(6*x^3*exp(5) + 6*x^4) - log(x + exp(5))*(x + exp(5) - exp(x)*(18*x^2*exp(5) + 18*x^3) - exp(2*x)*(18*x^
2*exp(5) + 18*x^3)) + log(x + exp(5))^3*(exp(x)*(6*x + 6*exp(5)) + exp(2*x)*(6*x + 6*exp(5))) + exp(2*x)*(6*x^
3*exp(5) + 6*x^4) + x^2)/(log(x + exp(5))^2*(3*x*exp(5) + 3*x^2) + log(x + exp(5))*(3*x^2*exp(5) + 3*x^3) + x^
3*exp(5) + log(x + exp(5))^3*(x + exp(5)) + x^4),x)

[Out]

((x*(x + exp(5) + 2))/(2*(x + exp(5) + 1)) - (log(x + exp(5))*(x + exp(5)))/(2*(x + exp(5) + 1)))/(log(x + exp
(5))^2 + 2*x*log(x + exp(5)) + x^2) - 6*exp(x) - 3*exp(2*x) + (((x + exp(5))*(x + 2*exp(5) + exp(10) + 2*x*exp
(5) + x^2 + 1))/(2*(x + exp(5) + 1)^3) - (log(x + exp(5))*(x + exp(5)))/(2*(x + exp(5) + 1)^3))/(x + log(x + e
xp(5))) + (x + exp(5))/(6*exp(5) + 6*exp(10) + 2*exp(15) + x^2*(6*exp(5) + 6) + x*(12*exp(5) + 6*exp(10) + 6)
+ 2*x^3 + 2)

________________________________________________________________________________________

sympy [A]  time = 0.59, size = 34, normalized size = 1.55 $$\begin{array}{c}\frac{x}{x^2+2x\log (x+e^5)+\log {(x+e^5)}^2}-3e^{2x}-6e^x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*exp(5)-6*x)*exp(x)**2+(-6*exp(5)-6*x)*exp(x))*ln(exp(5)+x)**3+((-18*x*exp(5)-18*x**2)*exp(x)**
2+(-18*x*exp(5)-18*x**2)*exp(x))*ln(exp(5)+x)**2+((-18*x**2*exp(5)-18*x**3)*exp(x)**2+(-18*x**2*exp(5)-18*x**3
)*exp(x)+exp(5)+x)*ln(exp(5)+x)+(-6*x**3*exp(5)-6*x**4)*exp(x)**2+(-6*x**3*exp(5)-6*x**4)*exp(x)-x*exp(5)-x**2
-2*x)/((exp(5)+x)*ln(exp(5)+x)**3+(3*x*exp(5)+3*x**2)*ln(exp(5)+x)**2+(3*x**2*exp(5)+3*x**3)*ln(exp(5)+x)+x**3
*exp(5)+x**4),x)

[Out]

x/(x**2 + 2*x*log(x + exp(5)) + log(x + exp(5))**2) - 3*exp(2*x) - 6*exp(x)

________________________________________________________________________________________