3.67.37 $\int \frac{e^x(-20+12x+7x^2+x^4)+e^3(-x^3+3x^4+2x^5)+e^x(-2+x+x^2)\log (2+x)+(e^3(20x^2+8x^3+x^4+x^5)+e^3(2x^2+x^3)\log (2+x))\log (10-x+x^2+\log (2+x))}{20x^2+8x^3+x^4+x^5+(2x^2+x^3)\log (2+x)}dx$

Optimal. Leaf size=26 $$\frac{e^x}{x}+e^{3}x\log (10-x+x^2+\log (2+x))$$

________________________________________________________________________________________

Rubi [F]  time = 2.33, 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 = {} $$\begin{array}{c}\int \frac{e^x(-20+12x+7x^2+x^4)+e^3(-x^3+3x^4+2x^5)+e^x(-2+x+x^2)\log (2+x)+(e^3(20x^2+8x^3+x^4+x^5)+e^3(2x^2+x^3)\log (2+x))\log (10-x+x^2+\log (2+x))}{20x^2+8x^3+x^4+x^5+(2x^2+x^3)\log (2+x)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(E^x*(-20 + 12*x + 7*x^2 + x^4) + E^3*(-x^3 + 3*x^4 + 2*x^5) + E^x*(-2 + x + x^2)*Log[2 + x] + (E^3*(20*x^
2 + 8*x^3 + x^4 + x^5) + E^3*(2*x^2 + x^3)*Log[2 + x])*Log[10 - x + x^2 + Log[2 + x]])/(20*x^2 + 8*x^3 + x^4 +
x^5 + (2*x^2 + x^3)*Log[2 + x]),x]

[Out]

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

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{e^x(-20+12x+7x^2+x^4)+e^3(-x^3+3x^4+2x^5)+e^x(-2+x+x^2)\log (2+x)+(e^3(20x^2+8x^3+x^4+x^5)+e^3(2x^2+x^3)\log (2+x))\log (10-x+x^2+\log (2+x))}{x^2(2+x)(10-x+x^2+\log (2+x))}dx\\ & =\int (\frac{e^x(-1+x)}{x^2}+\frac{e^3(-x+3x^2+2x^3+20\log (10-x+x^2+\log (2+x))+8x\log (10-x+x^2+\log (2+x))+x^2\log (10-x+x^2+\log (2+x))+x^3\log (10-x+x^2+\log (2+x))+2\log (2+x)\log (10-x+x^2+\log (2+x))+x\log (2+x)\log (10-x+x^2+\log (2+x)))}{(2+x)(10-x+x^2+\log (2+x))})dx\\ & =e^3\int \frac{-x+3x^2+2x^3+20\log (10-x+x^2+\log (2+x))+8x\log (10-x+x^2+\log (2+x))+x^2\log (10-x+x^2+\log (2+x))+x^3\log (10-x+x^2+\log (2+x))+2\log (2+x)\log (10-x+x^2+\log (2+x))+x\log (2+x)\log (10-x+x^2+\log (2+x))}{(2+x)(10-x+x^2+\log (2+x))}dx+\int \frac{e^x(-1+x)}{x^2}dx\\ & =\frac{e^x}{x}+e^3\int (\frac{x(-1+3x+2x^2)}{(2+x)(10-x+x^2+\log (2+x))}+\log (10-x+x^2+\log (2+x)))dx\\ & =\frac{e^x}{x}+e^3\int \frac{x(-1+3x+2x^2)}{(2+x)(10-x+x^2+\log (2+x))}dx+e^3\int \log (10-x+x^2+\log (2+x))dx\\ & =\frac{e^x}{x}+e^3\int (\frac{1}{10-x+x^2+\log (2+x)}-\frac{x}{10-x+x^2+\log (2+x)}+\frac{2x^2}{10-x+x^2+\log (2+x)}-\frac{2}{(2+x)(10-x+x^2+\log (2+x))})dx+e^3\int \log (10-x+x^2+\log (2+x))dx\\ & =\frac{e^x}{x}+e^3\int \frac{1}{10-x+x^2+\log (2+x)}dx-e^3\int \frac{x}{10-x+x^2+\log (2+x)}dx+e^3\int \log (10-x+x^2+\log (2+x))dx+\left(2e^3\right)\int \frac{x^2}{10-x+x^2+\log (2+x)}dx-\left(2e^3\right)\int \frac{1}{(2+x)(10-x+x^2+\log (2+x))}dx\end{array}\end{array}$$

________________________________________________________________________________________

Mathematica [A]  time = 0.13, size = 26, normalized size = 1.00 $$\begin{array}{c}\frac{e^x}{x}+e^{3}x\log (10-x+x^2+\log (2+x))\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(-20 + 12*x + 7*x^2 + x^4) + E^3*(-x^3 + 3*x^4 + 2*x^5) + E^x*(-2 + x + x^2)*Log[2 + x] + (E^3*
(20*x^2 + 8*x^3 + x^4 + x^5) + E^3*(2*x^2 + x^3)*Log[2 + x])*Log[10 - x + x^2 + Log[2 + x]])/(20*x^2 + 8*x^3 +
x^4 + x^5 + (2*x^2 + x^3)*Log[2 + x]),x]

[Out]

E^x/x + E^3*x*Log[10 - x + x^2 + Log[2 + x]]

________________________________________________________________________________________

fricas [A]  time = 0.55, size = 26, normalized size = 1.00 $$\begin{array}{c}\frac{x^2{e}^3\log (x^2-x+\log (x+2)+10)+e^x}{x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^3+2*x^2)*exp(3)*log(2+x)+(x^5+x^4+8*x^3+20*x^2)*exp(3))*log(log(2+x)+x^2-x+10)+(x^2+x-2)*exp(x)
*log(2+x)+(x^4+7*x^2+12*x-20)*exp(x)+(2*x^5+3*x^4-x^3)*exp(3))/((x^3+2*x^2)*log(2+x)+x^5+x^4+8*x^3+20*x^2),x,
algorithm="fricas")

[Out]

(x^2*e^3*log(x^2 - x + log(x + 2) + 10) + e^x)/x

________________________________________________________________________________________

giac [A]  time = 0.20, size = 26, normalized size = 1.00 $$\begin{array}{c}\frac{x^2{e}^3\log (x^2-x+\log (x+2)+10)+e^x}{x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^3+2*x^2)*exp(3)*log(2+x)+(x^5+x^4+8*x^3+20*x^2)*exp(3))*log(log(2+x)+x^2-x+10)+(x^2+x-2)*exp(x)
*log(2+x)+(x^4+7*x^2+12*x-20)*exp(x)+(2*x^5+3*x^4-x^3)*exp(3))/((x^3+2*x^2)*log(2+x)+x^5+x^4+8*x^3+20*x^2),x,
algorithm="giac")

[Out]

(x^2*e^3*log(x^2 - x + log(x + 2) + 10) + e^x)/x

________________________________________________________________________________________

maple [A]  time = 0.09, size = 25, normalized size = 0.96

method	result	size
risch	$\frac{{\mathrm{e}}^x}{x}+{\mathrm{e}}^3\ln (\ln (2+x)+x^2-x+10)x$	$25$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^3+2*x^2)*exp(3)*ln(2+x)+(x^5+x^4+8*x^3+20*x^2)*exp(3))*ln(ln(2+x)+x^2-x+10)+(x^2+x-2)*exp(x)*ln(2+x)+
(x^4+7*x^2+12*x-20)*exp(x)+(2*x^5+3*x^4-x^3)*exp(3))/((x^3+2*x^2)*ln(2+x)+x^5+x^4+8*x^3+20*x^2),x,method=_RETU
RNVERBOSE)

[Out]

exp(x)/x+exp(3)*ln(ln(2+x)+x^2-x+10)*x

________________________________________________________________________________________

maxima [A]  time = 0.42, size = 26, normalized size = 1.00 $$\begin{array}{c}\frac{x^2{e}^3\log (x^2-x+\log (x+2)+10)+e^x}{x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^3+2*x^2)*exp(3)*log(2+x)+(x^5+x^4+8*x^3+20*x^2)*exp(3))*log(log(2+x)+x^2-x+10)+(x^2+x-2)*exp(x)
*log(2+x)+(x^4+7*x^2+12*x-20)*exp(x)+(2*x^5+3*x^4-x^3)*exp(3))/((x^3+2*x^2)*log(2+x)+x^5+x^4+8*x^3+20*x^2),x,
algorithm="maxima")

[Out]

(x^2*e^3*log(x^2 - x + log(x + 2) + 10) + e^x)/x

________________________________________________________________________________________

mupad [B]  time = 0.36, size = 24, normalized size = 0.92 $$\begin{array}{c}\frac{{\mathrm{e}}^x}{x}+x{\mathrm{e}}^3\ln (\ln (x+2)-x+x^2+10)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(log(x + 2) - x + x^2 + 10)*(exp(3)*(20*x^2 + 8*x^3 + x^4 + x^5) + log(x + 2)*exp(3)*(2*x^2 + x^3)) +
exp(x)*(12*x + 7*x^2 + x^4 - 20) + exp(3)*(3*x^4 - x^3 + 2*x^5) + log(x + 2)*exp(x)*(x + x^2 - 2))/(log(x + 2)
*(2*x^2 + x^3) + 20*x^2 + 8*x^3 + x^4 + x^5),x)

[Out]

exp(x)/x + x*exp(3)*log(log(x + 2) - x + x^2 + 10)

________________________________________________________________________________________

sympy [A]  time = 1.27, size = 42, normalized size = 1.62 $$\begin{array}{c}(x{e}^3+e^3)\log (x^2-x+\log (x+2)+10)-e^3\log (x^2-x+\log (x+2)+10)+\frac{e^x}{x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**3+2*x**2)*exp(3)*ln(2+x)+(x**5+x**4+8*x**3+20*x**2)*exp(3))*ln(ln(2+x)+x**2-x+10)+(x**2+x-2)*e
xp(x)*ln(2+x)+(x**4+7*x**2+12*x-20)*exp(x)+(2*x**5+3*x**4-x**3)*exp(3))/((x**3+2*x**2)*ln(2+x)+x**5+x**4+8*x**
3+20*x**2),x)

[Out]

(x*exp(3) + exp(3))*log(x**2 - x + log(x + 2) + 10) - exp(3)*log(x**2 - x + log(x + 2) + 10) + exp(x)/x

________________________________________________________________________________________