3.41.57 $\int \frac{-8x^4-16x^5-8x^6+(16x^4+40x^5+24x^6+e^{2+x}(-4x^3-4x^4)+e^3(16x^3+40x^4+24x^5))\log (e^3+x)+e^x(e^5(12x^2+20x^3+4x^4)+e^2(12x^3+20x^4+4x^5)){\log }^2(e^3+x)+e^{2x}(e^7(2x+2x^2)+e^4(2x^2+2x^3)){\log }^3(e^3+x)}{(e^3+x){\log }^3(e^3+x)}dx$

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

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Rubi [F]  time = 9.84, 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 = {} $$\begin{array}{c}\int \frac{-8x^4-16x^5-8x^6+(16x^4+40x^5+24x^6+e^{2+x}(-4x^3-4x^4)+e^3(16x^3+40x^4+24x^5))\log (e^3+x)+e^x(e^5(12x^2+20x^3+4x^4)+e^2(12x^3+20x^4+4x^5)){\log }^2(e^3+x)+e^{2x}(e^7(2x+2x^2)+e^4(2x^2+2x^3)){\log }^3(e^3+x)}{(e^3+x){\log }^3(e^3+x)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(-8*x^4 - 16*x^5 - 8*x^6 + (16*x^4 + 40*x^5 + 24*x^6 + E^(2 + x)*(-4*x^3 - 4*x^4) + E^3*(16*x^3 + 40*x^4 +
24*x^5))*Log[E^3 + x] + E^x*(E^5*(12*x^2 + 20*x^3 + 4*x^4) + E^2*(12*x^3 + 20*x^4 + 4*x^5))*Log[E^3 + x]^2 +
E^(2*x)*(E^7*(2*x + 2*x^2) + E^4*(2*x^2 + 2*x^3))*Log[E^3 + x]^3)/((E^3 + x)*Log[E^3 + x]^3),x]

[Out]

E^(4 + 2*x)*x^2 - 80*E^12*ExpIntegralEi[2*Log[E^3 + x]] - 96*E^6*(1 - E^3)^2*ExpIntegralEi[2*Log[E^3 + x]] + 6
4*E^9*(2 - E^3)*ExpIntegralEi[2*Log[E^3 + x]] + 16*E^6*(6 - 20*E^3 + 15*E^6)*ExpIntegralEi[2*Log[E^3 + x]] + 3
60*E^9*ExpIntegralEi[3*Log[E^3 + x]] + 144*E^3*(1 - E^3)^2*ExpIntegralEi[3*Log[E^3 + x]] - 216*E^6*(2 - E^3)*E
xpIntegralEi[3*Log[E^3 + x]] - 144*E^3*(1 - 5*E^3 + 5*E^6)*ExpIntegralEi[3*Log[E^3 + x]] - 640*E^6*ExpIntegral
Ei[4*Log[E^3 + x]] - 64*(1 - E^3)^2*ExpIntegralEi[4*Log[E^3 + x]] + 256*E^3*(2 - E^3)*ExpIntegralEi[4*Log[E^3
+ x]] + 64*(1 - 10*E^3 + 15*E^6)*ExpIntegralEi[4*Log[E^3 + x]] + 500*E^3*ExpIntegralEi[5*Log[E^3 + x]] + 200*(
1 - 3*E^3)*ExpIntegralEi[5*Log[E^3 + x]] - 100*(2 - E^3)*ExpIntegralEi[5*Log[E^3 + x]] + (4*E^12*(1 - E^3)^2)/
Log[E^3 + x]^2 - (4*E^9*(1 - E^3)^2*(E^3 + x))/Log[E^3 + x]^2 + (4*E^6*(1 - E^3)^2*x*(E^3 + x))/Log[E^3 + x]^2
- (4*E^3*(1 - E^3)^2*x^2*(E^3 + x))/Log[E^3 + x]^2 + (4*(1 - E^3)^2*x^3*(E^3 + x))/Log[E^3 + x]^2 + (4*(2 - E
^3)*x^4*(E^3 + x))/Log[E^3 + x]^2 + (4*x^5*(E^3 + x))/Log[E^3 + x]^2 + (8*E^9*(2 - 5*E^3 + 3*E^6)*(E^3 + x))/L
og[E^3 + x] + (16*(1 - E^3)^2*x^3*(E^3 + x))/Log[E^3 + x] + (16*E^3*(2 - E^3)*x^3*(E^3 + x))/Log[E^3 + x] + (2
0*E^3*x^4*(E^3 + x))/Log[E^3 + x] + (20*(2 - E^3)*x^4*(E^3 + x))/Log[E^3 + x] + (24*x^5*(E^3 + x))/Log[E^3 + x
] - (8*E^6*(6 - 20*E^3 + 15*E^6)*(E^3 + x)^2)/Log[E^3 + x] + (48*E^3*(1 - 5*E^3 + 5*E^6)*(E^3 + x)^3)/Log[E^3
+ x] - (16*(1 - 10*E^3 + 15*E^6)*(E^3 + x)^4)/Log[E^3 + x] - (40*(1 - 3*E^3)*(E^3 + x)^5)/Log[E^3 + x] - (24*(
E^3 + x)^6)/Log[E^3 + x] + 4*E^15*LogIntegral[E^3 + x] + 16*E^9*(1 - E^3)^2*LogIntegral[E^3 + x] - 4*E^12*(2 -
E^3)*LogIntegral[E^3 + x] - 8*E^9*(2 - 5*E^3 + 3*E^6)*LogIntegral[E^3 + x] - 4*E^6*(1 - E^3)*Defer[Int][E^(2
+ x)/Log[E^3 + x]^2, x] - 8*(1 - E^3)*Defer[Int][E^(8 + x)/Log[E^3 + x]^2, x] + 4*Defer[Int][E^(11 + x)/Log[E^
3 + x]^2, x] + 4*(1 - E^3)*Defer[Int][E^(11 + x)/((E^3 + x)*Log[E^3 + x]^2), x] + 12*(1 - E^3)*Defer[Int][(E^(
5 + x)*(E^3 + x))/Log[E^3 + x]^2, x] - 12*Defer[Int][(E^(8 + x)*(E^3 + x))/Log[E^3 + x]^2, x] - 4*(1 - E^3)*De
fer[Int][(E^(2 + x)*(E^3 + x)^2)/Log[E^3 + x]^2, x] + 12*Defer[Int][(E^(5 + x)*(E^3 + x)^2)/Log[E^3 + x]^2, x]
- 4*Defer[Int][(E^(2 + x)*(E^3 + x)^3)/Log[E^3 + x]^2, x] + 4*(3 - 5*E^3 + E^6)*Defer[Int][E^(8 + x)/Log[E^3
+ x], x] - 4*E^3*(6 - 15*E^3 + 4*E^6)*Defer[Int][(E^(2 + x)*(E^3 + x))/Log[E^3 + x], x] + 12*(1 - 5*E^3 + 2*E^
6)*Defer[Int][(E^(2 + x)*(E^3 + x)^2)/Log[E^3 + x], x] + 4*(5 - 4*E^3)*Defer[Int][(E^(2 + x)*(E^3 + x)^3)/Log[
E^3 + x], x] + 4*Defer[Int][(E^(2 + x)*(E^3 + x)^4)/Log[E^3 + x], x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int 2x(e^{4+2x}(1+x)-\frac{4x^3(1+x{)}^2}{(e^3+x){\log }^3(e^3+x)}+\frac{2x^2(1+x)(-e^{2+x}+2x(2+3x)+e^3(4+6x))}{(e^3+x){\log }^2(e^3+x)}+\frac{2e^{2+x}x(3+5x+x^2)}{\log (e^3+x)})dx\\ & =2\int x(e^{4+2x}(1+x)-\frac{4x^3(1+x{)}^2}{(e^3+x){\log }^3(e^3+x)}+\frac{2x^2(1+x)(-e^{2+x}+2x(2+3x)+e^3(4+6x))}{(e^3+x){\log }^2(e^3+x)}+\frac{2e^{2+x}x(3+5x+x^2)}{\log (e^3+x)})dx\\ & =2\int (e^{4+2x}x(1+x)+\frac{4x^3(1+x)(-x-x^2+2e^3\log (e^3+x)+2(1+\frac{3e^3}{2})x\log (e^3+x)+3x^2\log (e^3+x))}{(e^3+x){\log }^3(e^3+x)}+\frac{2e^{2+x}{x}^2(-x-x^2+3e^3\log (e^3+x)+3(1+\frac{5e^3}{3})x\log (e^3+x)+5(1+\frac{e^3}{5})x^2\log (e^3+x)+x^3\log (e^3+x))}{(e^3+x){\log }^2(e^3+x)})dx\\ & =2\int e^{4+2x}x(1+x)dx+4\int \frac{e^{2+x}{x}^2(-x-x^2+3e^3\log (e^3+x)+3(1+\frac{5e^3}{3})x\log (e^3+x)+5(1+\frac{e^3}{5})x^2\log (e^3+x)+x^3\log (e^3+x))}{(e^3+x){\log }^2(e^3+x)}dx+8\int \frac{x^3(1+x)(-x-x^2+2e^3\log (e^3+x)+2(1+\frac{3e^3}{2})x\log (e^3+x)+3x^2\log (e^3+x))}{(e^3+x){\log }^3(e^3+x)}dx\\ & =2\int (e^{4+2x}x+e^{4+2x}{x}^2)dx+4\int \frac{e^{2+x}{x}^2(-x(1+x)+(e^3+x)(3+5x+x^2)\log (e^3+x))}{(e^3+x){\log }^2(e^3+x)}dx+8\int \frac{x^3(1+x)(-x(1+x)+(e^3+x)(2+3x)\log (e^3+x))}{(e^3+x){\log }^3(e^3+x)}dx\\ & =2\int e^{4+2x}xdx+2\int e^{4+2x}{x}^{2}dx+4\int (-\frac{e^{2+x}{x}^3(1+x)}{(e^3+x){\log }^2(e^3+x)}+\frac{e^{2+x}{x}^2(3+5x+x^2)}{\log (e^3+x)})dx+8\int (-\frac{x^4(1+x{)}^2}{(e^3+x){\log }^3(e^3+x)}+\frac{x^3(1+x)(2+3x)}{{\log }^2(e^3+x)})dx\\ & =e^{4+2x}x+e^{4+2x}{x}^2-2\int e^{4+2x}xdx-4\int \frac{e^{2+x}{x}^3(1+x)}{(e^3+x){\log }^2(e^3+x)}dx+4\int \frac{e^{2+x}{x}^2(3+5x+x^2)}{\log (e^3+x)}dx-8\int \frac{x^4(1+x{)}^2}{(e^3+x){\log }^3(e^3+x)}dx+8\int \frac{x^3(1+x)(2+3x)}{{\log }^2(e^3+x)}dx-\int e^{4+2x}dx\\ & =-\frac{1}{2}{e}^{4+2x}+e^{4+2x}{x}^2-4\int (\frac{e^{2+x}(e^6-e^9)}{{\log }^2(e^3+x)}+\frac{e^{5+x}(-1+e^3)x}{{\log }^2(e^3+x)}-\frac{e^{2+x}(-1+e^3)x^2}{{\log }^2(e^3+x)}+\frac{e^{2+x}{x}^3}{{\log }^2(e^3+x)}+\frac{e^{11+x}(-1+e^3)}{(e^3+x){\log }^2(e^3+x)})dx+4\int (\frac{e^{8+x}(3-5e^3+e^6)}{\log (e^3+x)}+\frac{e^{2+x}(-6e^3+15e^6-4e^9)(e^3+x)}{\log (e^3+x)}+\frac{3e^{2+x}(1-5e^3+2e^6){(e^3+x)}^2}{\log (e^3+x)}+\frac{e^{2+x}(5-4e^3){(e^3+x)}^3}{\log (e^3+x)}+\frac{e^{2+x}{(e^3+x)}^4}{\log (e^3+x)})dx-8\int (-\frac{e^9{(-1+e^3)}^2}{{\log }^3(e^3+x)}+\frac{e^6{(-1+e^3)}^{2}x}{{\log }^3(e^3+x)}-\frac{e^3{(-1+e^3)}^2{x}^2}{{\log }^3(e^3+x)}+\frac{{(-1+e^3)}^2{x}^3}{{\log }^3(e^3+x)}-\frac{(-2+e^3)x^4}{{\log }^3(e^3+x)}+\frac{x^5}{{\log }^3(e^3+x)}+\frac{e^{12}{(-1+e^3)}^2}{(e^3+x){\log }^3(e^3+x)})dx+8\int (-\frac{e^9(2-5e^3+3e^6)}{{\log }^2(e^3+x)}+\frac{e^6(6-20e^3+15e^6)(e^3+x)}{{\log }^2(e^3+x)}-\frac{6(e^3-5e^6+5e^9){(e^3+x)}^2}{{\log }^2(e^3+x)}+\frac{2(1-10e^3+15e^6){(e^3+x)}^3}{{\log }^2(e^3+x)}-\frac{5(-1+3e^3){(e^3+x)}^4}{{\log }^2(e^3+x)}+\frac{3{(e^3+x)}^5}{{\log }^2(e^3+x)})dx+\int e^{4+2x}dx\\ & =\text{Rest of rules removed due to large latex content}\end{array}\end{array}$$

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Mathematica [A]  time = 0.16, size = 33, normalized size = 1.27 $$\begin{array}{c}\frac{x^2{(2x(1+x)+e^{2+x}\log (e^3+x))}^2}{{\log }^2(e^3+x)}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-8*x^4 - 16*x^5 - 8*x^6 + (16*x^4 + 40*x^5 + 24*x^6 + E^(2 + x)*(-4*x^3 - 4*x^4) + E^3*(16*x^3 + 40
*x^4 + 24*x^5))*Log[E^3 + x] + E^x*(E^5*(12*x^2 + 20*x^3 + 4*x^4) + E^2*(12*x^3 + 20*x^4 + 4*x^5))*Log[E^3 + x
]^2 + E^(2*x)*(E^7*(2*x + 2*x^2) + E^4*(2*x^2 + 2*x^3))*Log[E^3 + x]^3)/((E^3 + x)*Log[E^3 + x]^3),x]

[Out]

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

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fricas [B]  time = 0.62, size = 59, normalized size = 2.27 $$\begin{array}{c}\frac{4x^6+8x^5+x^2{e}^{(2x+4)}\log {(x+e^3)}^2+4x^4+4(x^4+x^3)e^{(x+2)}\log (x+e^3)}{\log {(x+e^3)}^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2+2*x)*exp(1)^4*exp(3)+(2*x^3+2*x^2)*exp(1)^4)*exp(x)^2*log(exp(3)+x)^3+((4*x^4+20*x^3+12*x^2
)*exp(1)^2*exp(3)+(4*x^5+20*x^4+12*x^3)*exp(1)^2)*exp(x)*log(exp(3)+x)^2+((-4*x^4-4*x^3)*exp(1)^2*exp(x)+(24*x
^5+40*x^4+16*x^3)*exp(3)+24*x^6+40*x^5+16*x^4)*log(exp(3)+x)-8*x^6-16*x^5-8*x^4)/(exp(3)+x)/log(exp(3)+x)^3,x,
algorithm="fricas")

[Out]

(4*x^6 + 8*x^5 + x^2*e^(2*x + 4)*log(x + e^3)^2 + 4*x^4 + 4*(x^4 + x^3)*e^(x + 2)*log(x + e^3))/log(x + e^3)^2

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giac [B]  time = 0.35, size = 88, normalized size = 3.38 $$\begin{array}{c}\frac{4x^6\log (x+e^3)+4x^4{e}^{(x+2)}\log {(x+e^3)}^2+8x^5\log (x+e^3)+4x^3{e}^{(x+2)}\log {(x+e^3)}^2+x^2{e}^{(2x+4)}\log {(x+e^3)}^3+4x^4\log (x+e^3)}{\log {(x+e^3)}^3}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2+2*x)*exp(1)^4*exp(3)+(2*x^3+2*x^2)*exp(1)^4)*exp(x)^2*log(exp(3)+x)^3+((4*x^4+20*x^3+12*x^2
)*exp(1)^2*exp(3)+(4*x^5+20*x^4+12*x^3)*exp(1)^2)*exp(x)*log(exp(3)+x)^2+((-4*x^4-4*x^3)*exp(1)^2*exp(x)+(24*x
^5+40*x^4+16*x^3)*exp(3)+24*x^6+40*x^5+16*x^4)*log(exp(3)+x)-8*x^6-16*x^5-8*x^4)/(exp(3)+x)/log(exp(3)+x)^3,x,
algorithm="giac")

[Out]

(4*x^6*log(x + e^3) + 4*x^4*e^(x + 2)*log(x + e^3)^2 + 8*x^5*log(x + e^3) + 4*x^3*e^(x + 2)*log(x + e^3)^2 + x
^2*e^(2*x + 4)*log(x + e^3)^3 + 4*x^4*log(x + e^3))/log(x + e^3)^3

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maple [B]  time = 0.12, size = 55, normalized size = 2.12

method	result	size
risch	$x^2{\mathrm{e}}^{2x+4}+\frac{4x^3(\ln ({\mathrm{e}}^3+x)x{\mathrm{e}}^{2+x}+\ln ({\mathrm{e}}^3+x){\mathrm{e}}^{2+x}+x^3+2x^2+x)}{\ln {({\mathrm{e}}^3+x)}^2}$	$55$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^2+2*x)*exp(1)^4*exp(3)+(2*x^3+2*x^2)*exp(1)^4)*exp(x)^2*ln(exp(3)+x)^3+((4*x^4+20*x^3+12*x^2)*exp(1
)^2*exp(3)+(4*x^5+20*x^4+12*x^3)*exp(1)^2)*exp(x)*ln(exp(3)+x)^2+((-4*x^4-4*x^3)*exp(1)^2*exp(x)+(24*x^5+40*x^
4+16*x^3)*exp(3)+24*x^6+40*x^5+16*x^4)*ln(exp(3)+x)-8*x^6-16*x^5-8*x^4)/(exp(3)+x)/ln(exp(3)+x)^3,x,method=_RE
TURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.41, size = 63, normalized size = 2.42 $$\begin{array}{c}\frac{4x^6+8x^5+x^2{e}^{(2x+4)}\log {(x+e^3)}^2+4x^4+4(x^4{e}^2+x^3{e}^2)e^x\log (x+e^3)}{\log {(x+e^3)}^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2+2*x)*exp(1)^4*exp(3)+(2*x^3+2*x^2)*exp(1)^4)*exp(x)^2*log(exp(3)+x)^3+((4*x^4+20*x^3+12*x^2
)*exp(1)^2*exp(3)+(4*x^5+20*x^4+12*x^3)*exp(1)^2)*exp(x)*log(exp(3)+x)^2+((-4*x^4-4*x^3)*exp(1)^2*exp(x)+(24*x
^5+40*x^4+16*x^3)*exp(3)+24*x^6+40*x^5+16*x^4)*log(exp(3)+x)-8*x^6-16*x^5-8*x^4)/(exp(3)+x)/log(exp(3)+x)^3,x,
algorithm="maxima")

[Out]

(4*x^6 + 8*x^5 + x^2*e^(2*x + 4)*log(x + e^3)^2 + 4*x^4 + 4*(x^4*e^2 + x^3*e^2)*e^x*log(x + e^3))/log(x + e^3)
^2

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mupad [B]  time = 4.06, size = 79, normalized size = 3.04 $$\begin{array}{c}\frac{4x^4}{{\ln (x+{\mathrm{e}}^3)}^2}+\frac{8x^5}{{\ln (x+{\mathrm{e}}^3)}^2}+\frac{4x^6}{{\ln (x+{\mathrm{e}}^3)}^2}+x^2{\mathrm{e}}^{2x+4}+\frac{4x^3{\mathrm{e}}^{x+2}}{\ln (x+{\mathrm{e}}^3)}+\frac{4x^4{\mathrm{e}}^{x+2}}{\ln (x+{\mathrm{e}}^3)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x + exp(3))*(exp(3)*(16*x^3 + 40*x^4 + 24*x^5) + 16*x^4 + 40*x^5 + 24*x^6 - exp(2)*exp(x)*(4*x^3 + 4*
x^4)) - 8*x^4 - 16*x^5 - 8*x^6 + exp(x)*log(x + exp(3))^2*(exp(2)*(12*x^3 + 20*x^4 + 4*x^5) + exp(5)*(12*x^2 +
20*x^3 + 4*x^4)) + exp(2*x)*log(x + exp(3))^3*(exp(7)*(2*x + 2*x^2) + exp(4)*(2*x^2 + 2*x^3)))/(log(x + exp(3
))^3*(x + exp(3))),x)

[Out]

(4*x^4)/log(x + exp(3))^2 + (8*x^5)/log(x + exp(3))^2 + (4*x^6)/log(x + exp(3))^2 + x^2*exp(2*x + 4) + (4*x^3*
exp(x + 2))/log(x + exp(3)) + (4*x^4*exp(x + 2))/log(x + exp(3))

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sympy [B]  time = 0.47, size = 68, normalized size = 2.62 $$\begin{array}{c}\frac{x^2{e}^4{e}^{2x}\log (x+e^3)+(4x^4{e}^2+4x^3{e}^2)e^x}{\log (x+e^3)}+\frac{4x^6+8x^5+4x^4}{\log {(x+e^3)}^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**2+2*x)*exp(1)**4*exp(3)+(2*x**3+2*x**2)*exp(1)**4)*exp(x)**2*ln(exp(3)+x)**3+((4*x**4+20*x**
3+12*x**2)*exp(1)**2*exp(3)+(4*x**5+20*x**4+12*x**3)*exp(1)**2)*exp(x)*ln(exp(3)+x)**2+((-4*x**4-4*x**3)*exp(1
)**2*exp(x)+(24*x**5+40*x**4+16*x**3)*exp(3)+24*x**6+40*x**5+16*x**4)*ln(exp(3)+x)-8*x**6-16*x**5-8*x**4)/(exp
(3)+x)/ln(exp(3)+x)**3,x)

[Out]

(x**2*exp(4)*exp(2*x)*log(x + exp(3)) + (4*x**4*exp(2) + 4*x**3*exp(2))*exp(x))/log(x + exp(3)) + (4*x**6 + 8*
x**5 + 4*x**4)/log(x + exp(3))**2

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