3.17.31 $\int \frac{512x^5+768x^6+256x^7+e^x(-512x^3+512x^4+1152x^5+384x^6)+(-1024x^4-1280x^5-384x^6+e^x(-1024x^3-1280x^4-384x^5))\log (\frac{e^x+x}{x})}{-e^x{x}^3-x^4+(3e^x{x}^2+3x^3)\log (\frac{e^x+x}{x})+(-3e^{x}x-3x^2){\log }^2(\frac{e^x+x}{x})+(e^x+x){\log }^3(\frac{e^x+x}{x})}dx$

Optimal. Leaf size=29 $$-3+e-\frac{64x^4(2+x{)}^2}{{(x-\log \left(\frac{e^x+x}{x}\right))}^2}$$

________________________________________________________________________________________

Rubi [F]  time = 5.42, 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 = {} $$\begin{array}{c}\int \frac{512x^5+768x^6+256x^7+e^x(-512x^3+512x^4+1152x^5+384x^6)+(-1024x^4-1280x^5-384x^6+e^x(-1024x^3-1280x^4-384x^5))\log \left(\frac{e^x+x}{x}\right)}{-e^x{x}^3-x^4+(3e^x{x}^2+3x^3)\log \left(\frac{e^x+x}{x}\right)+(-3e^{x}x-3x^2){\log }^2\left(\frac{e^x+x}{x}\right)+(e^x+x){\log }^3\left(\frac{e^x+x}{x}\right)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(512*x^5 + 768*x^6 + 256*x^7 + E^x*(-512*x^3 + 512*x^4 + 1152*x^5 + 384*x^6) + (-1024*x^4 - 1280*x^5 - 384
*x^6 + E^x*(-1024*x^3 - 1280*x^4 - 384*x^5))*Log[(E^x + x)/x])/(-(E^x*x^3) - x^4 + (3*E^x*x^2 + 3*x^3)*Log[(E^
x + x)/x] + (-3*E^x*x - 3*x^2)*Log[(E^x + x)/x]^2 + (E^x + x)*Log[(E^x + x)/x]^3),x]

[Out]

512*Defer[Int][x^3/(x - Log[(E^x + x)/x])^3, x] + 512*Defer[Int][x^4/(x - Log[(E^x + x)/x])^3, x] + 128*Defer[
Int][x^5/(x - Log[(E^x + x)/x])^3, x] - 512*Defer[Int][x^4/((E^x + x)*(x - Log[(E^x + x)/x])^3), x] + 384*Defe
r[Int][x^6/((E^x + x)*(x - Log[(E^x + x)/x])^3), x] + 128*Defer[Int][x^7/((E^x + x)*(x - Log[(E^x + x)/x])^3),
x] - 1024*Defer[Int][x^3/(x - Log[(E^x + x)/x])^2, x] - 1280*Defer[Int][x^4/(x - Log[(E^x + x)/x])^2, x] - 38
4*Defer[Int][x^5/(x - Log[(E^x + x)/x])^2, x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{128x^3(2+x)(-2x^2(1+x)-e^x(-2+3x+3x^2)+(e^x+x)(4+3x)\log \left(\frac{e^x+x}{x}\right))}{(e^x+x){(x-\log \left(\frac{e^x+x}{x}\right))}^3}dx\\ & =128\int \frac{x^3(2+x)(-2x^2(1+x)-e^x(-2+3x+3x^2)+(e^x+x)(4+3x)\log \left(\frac{e^x+x}{x}\right))}{(e^x+x){(x-\log \left(\frac{e^x+x}{x}\right))}^3}dx\\ & =128\int (\frac{(-1+x)x^4(2+x{)}^2}{(e^x+x){(x-\log \left(\frac{e^x+x}{x}\right))}^3}-\frac{x^3(2+x)(-2+3x+3x^2-4\log \left(\frac{e^x+x}{x}\right)-3x\log \left(\frac{e^x+x}{x}\right))}{{(x-\log \left(\frac{e^x+x}{x}\right))}^3})dx\\ & =128\int \frac{(-1+x)x^4(2+x{)}^2}{(e^x+x){(x-\log \left(\frac{e^x+x}{x}\right))}^3}dx-128\int \frac{x^3(2+x)(-2+3x+3x^2-4\log \left(\frac{e^x+x}{x}\right)-3x\log \left(\frac{e^x+x}{x}\right))}{{(x-\log \left(\frac{e^x+x}{x}\right))}^3}dx\\ & =128\int (-\frac{4x^4}{(e^x+x){(x-\log \left(\frac{e^x+x}{x}\right))}^3}+\frac{3x^6}{(e^x+x){(x-\log \left(\frac{e^x+x}{x}\right))}^3}+\frac{x^7}{(e^x+x){(x-\log \left(\frac{e^x+x}{x}\right))}^3})dx-128\int (-\frac{x^3(2+x{)}^2}{{(x-\log \left(\frac{e^x+x}{x}\right))}^3}+\frac{x^3(8+10x+3x^2)}{{(x-\log \left(\frac{e^x+x}{x}\right))}^2})dx\\ & =128\int \frac{x^3(2+x{)}^2}{{(x-\log \left(\frac{e^x+x}{x}\right))}^3}dx+128\int \frac{x^7}{(e^x+x){(x-\log \left(\frac{e^x+x}{x}\right))}^3}dx-128\int \frac{x^3(8+10x+3x^2)}{{(x-\log \left(\frac{e^x+x}{x}\right))}^2}dx+384\int \frac{x^6}{(e^x+x){(x-\log \left(\frac{e^x+x}{x}\right))}^3}dx-512\int \frac{x^4}{(e^x+x){(x-\log \left(\frac{e^x+x}{x}\right))}^3}dx\\ & =128\int (\frac{4x^3}{{(x-\log \left(\frac{e^x+x}{x}\right))}^3}+\frac{4x^4}{{(x-\log \left(\frac{e^x+x}{x}\right))}^3}+\frac{x^5}{{(x-\log \left(\frac{e^x+x}{x}\right))}^3})dx-128\int (\frac{8x^3}{{(x-\log \left(\frac{e^x+x}{x}\right))}^2}+\frac{10x^4}{{(x-\log \left(\frac{e^x+x}{x}\right))}^2}+\frac{3x^5}{{(x-\log \left(\frac{e^x+x}{x}\right))}^2})dx+128\int \frac{x^7}{(e^x+x){(x-\log \left(\frac{e^x+x}{x}\right))}^3}dx+384\int \frac{x^6}{(e^x+x){(x-\log \left(\frac{e^x+x}{x}\right))}^3}dx-512\int \frac{x^4}{(e^x+x){(x-\log \left(\frac{e^x+x}{x}\right))}^3}dx\\ & =128\int \frac{x^5}{{(x-\log \left(\frac{e^x+x}{x}\right))}^3}dx+128\int \frac{x^7}{(e^x+x){(x-\log \left(\frac{e^x+x}{x}\right))}^3}dx+384\int \frac{x^6}{(e^x+x){(x-\log \left(\frac{e^x+x}{x}\right))}^3}dx-384\int \frac{x^5}{{(x-\log \left(\frac{e^x+x}{x}\right))}^2}dx+512\int \frac{x^3}{{(x-\log \left(\frac{e^x+x}{x}\right))}^3}dx+512\int \frac{x^4}{{(x-\log \left(\frac{e^x+x}{x}\right))}^3}dx-512\int \frac{x^4}{(e^x+x){(x-\log \left(\frac{e^x+x}{x}\right))}^3}dx-1024\int \frac{x^3}{{(x-\log \left(\frac{e^x+x}{x}\right))}^2}dx-1280\int \frac{x^4}{{(x-\log \left(\frac{e^x+x}{x}\right))}^2}dx\end{array}\end{array}$$

________________________________________________________________________________________

Mathematica [A]  time = 1.19, size = 26, normalized size = 0.90 $$\begin{array}{c}-\frac{64x^4(2+x{)}^2}{{(-x+\log \left(\frac{e^x+x}{x}\right))}^2}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(512*x^5 + 768*x^6 + 256*x^7 + E^x*(-512*x^3 + 512*x^4 + 1152*x^5 + 384*x^6) + (-1024*x^4 - 1280*x^5
- 384*x^6 + E^x*(-1024*x^3 - 1280*x^4 - 384*x^5))*Log[(E^x + x)/x])/(-(E^x*x^3) - x^4 + (3*E^x*x^2 + 3*x^3)*L
og[(E^x + x)/x] + (-3*E^x*x - 3*x^2)*Log[(E^x + x)/x]^2 + (E^x + x)*Log[(E^x + x)/x]^3),x]

[Out]

(-64*x^4*(2 + x)^2)/(-x + Log[(E^x + x)/x])^2

________________________________________________________________________________________

fricas [A]  time = 0.70, size = 45, normalized size = 1.55 $$\begin{array}{c}-\frac{64(x^6+4x^5+4x^4)}{x^2-2x\log \left(\frac{x+e^x}{x}\right)+\log {\left(\frac{x+e^x}{x}\right)}^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-384*x^5-1280*x^4-1024*x^3)*exp(x)-384*x^6-1280*x^5-1024*x^4)*log(1/x*(exp(x)+x))+(384*x^6+1152*x
^5+512*x^4-512*x^3)*exp(x)+256*x^7+768*x^6+512*x^5)/((exp(x)+x)*log(1/x*(exp(x)+x))^3+(-3*exp(x)*x-3*x^2)*log(
1/x*(exp(x)+x))^2+(3*exp(x)*x^2+3*x^3)*log(1/x*(exp(x)+x))-exp(x)*x^3-x^4),x, algorithm="fricas")

[Out]

-64*(x^6 + 4*x^5 + 4*x^4)/(x^2 - 2*x*log((x + e^x)/x) + log((x + e^x)/x)^2)

________________________________________________________________________________________

giac [A]  time = 0.80, size = 45, normalized size = 1.55 $$\begin{array}{c}-\frac{64(x^6+4x^5+4x^4)}{x^2-2x\log \left(\frac{x+e^x}{x}\right)+\log {\left(\frac{x+e^x}{x}\right)}^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-384*x^5-1280*x^4-1024*x^3)*exp(x)-384*x^6-1280*x^5-1024*x^4)*log(1/x*(exp(x)+x))+(384*x^6+1152*x
^5+512*x^4-512*x^3)*exp(x)+256*x^7+768*x^6+512*x^5)/((exp(x)+x)*log(1/x*(exp(x)+x))^3+(-3*exp(x)*x-3*x^2)*log(
1/x*(exp(x)+x))^2+(3*exp(x)*x^2+3*x^3)*log(1/x*(exp(x)+x))-exp(x)*x^3-x^4),x, algorithm="giac")

[Out]

-64*(x^6 + 4*x^5 + 4*x^4)/(x^2 - 2*x*log((x + e^x)/x) + log((x + e^x)/x)^2)

________________________________________________________________________________________

maple [C]  time = 0.48, size = 127, normalized size = 4.38

method	result	size
risch	$-\frac{256(x^2+4x+4)x^4}{{(i\pi \mathrm{csgn}\left(\frac{i}{x}\right)\mathrm{csgn}\left(i({\mathrm{e}}^x+x)\right)\mathrm{csgn}\left(\frac{i({\mathrm{e}}^x+x)}{x}\right)-i\pi \mathrm{csgn}\left(\frac{i}{x}\right)\mathrm{csgn}{\left(\frac{i({\mathrm{e}}^x+x)}{x}\right)}^2-i\pi \mathrm{csgn}\left(i({\mathrm{e}}^x+x)\right)\mathrm{csgn}{\left(\frac{i({\mathrm{e}}^x+x)}{x}\right)}^2+i\pi \mathrm{csgn}{\left(\frac{i({\mathrm{e}}^x+x)}{x}\right)}^3+2x+2\ln (x)-2\ln ({\mathrm{e}}^x+x))}^2}$	$127$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-384*x^5-1280*x^4-1024*x^3)*exp(x)-384*x^6-1280*x^5-1024*x^4)*ln(1/x*(exp(x)+x))+(384*x^6+1152*x^5+512*
x^4-512*x^3)*exp(x)+256*x^7+768*x^6+512*x^5)/((exp(x)+x)*ln(1/x*(exp(x)+x))^3+(-3*exp(x)*x-3*x^2)*ln(1/x*(exp(
x)+x))^2+(3*exp(x)*x^2+3*x^3)*ln(1/x*(exp(x)+x))-exp(x)*x^3-x^4),x,method=_RETURNVERBOSE)

[Out]

-256*(x^2+4*x+4)*x^4/(I*Pi*csgn(I/x)*csgn(I*(exp(x)+x))*csgn(I/x*(exp(x)+x))-I*Pi*csgn(I/x)*csgn(I/x*(exp(x)+x
))^2-I*Pi*csgn(I*(exp(x)+x))*csgn(I/x*(exp(x)+x))^2+I*Pi*csgn(I/x*(exp(x)+x))^3+2*x+2*ln(x)-2*ln(exp(x)+x))^2

________________________________________________________________________________________

maxima [A]  time = 0.68, size = 49, normalized size = 1.69 $$\begin{array}{c}-\frac{64(x^6+4x^5+4x^4)}{x^2-2(x+\log (x))\log (x+e^x)+\log {(x+e^x)}^2+2x\log (x)+\log (x{)}^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-384*x^5-1280*x^4-1024*x^3)*exp(x)-384*x^6-1280*x^5-1024*x^4)*log(1/x*(exp(x)+x))+(384*x^6+1152*x
^5+512*x^4-512*x^3)*exp(x)+256*x^7+768*x^6+512*x^5)/((exp(x)+x)*log(1/x*(exp(x)+x))^3+(-3*exp(x)*x-3*x^2)*log(
1/x*(exp(x)+x))^2+(3*exp(x)*x^2+3*x^3)*log(1/x*(exp(x)+x))-exp(x)*x^3-x^4),x, algorithm="maxima")

[Out]

-64*(x^6 + 4*x^5 + 4*x^4)/(x^2 - 2*(x + log(x))*log(x + e^x) + log(x + e^x)^2 + 2*x*log(x) + log(x)^2)

________________________________________________________________________________________

mupad [B]  time = 1.64, size = 540, normalized size = 18.62 $$\begin{array}{c}\frac{\frac{64x(x+{\mathrm{e}}^x)(24x^5{\mathrm{e}}^x+48x^6{\mathrm{e}}^x+43x^7{\mathrm{e}}^x+19x^8{\mathrm{e}}^x+3x^9{\mathrm{e}}^x-8x^3{\mathrm{e}}^{2x}+22x^4{\mathrm{e}}^{2x}+47x^5{\mathrm{e}}^{2x}+18x^6{\mathrm{e}}^{2x}+16x^7+30x^8+12x^9)}{{({\mathrm{e}}^x+x^2)}^3}-\frac{64x\ln \left(\frac{x+{\mathrm{e}}^x}{x}\right)(x+{\mathrm{e}}^x)(40x^4{\mathrm{e}}^x+68x^5{\mathrm{e}}^x+49x^6{\mathrm{e}}^x+19x^7{\mathrm{e}}^x+3x^8{\mathrm{e}}^x+32x^3{\mathrm{e}}^{2x}+50x^4{\mathrm{e}}^{2x}+18x^5{\mathrm{e}}^{2x}+24x^6+40x^7+15x^8)}{{({\mathrm{e}}^x+x^2)}^3}}{x-\ln \left(\frac{x+{\mathrm{e}}^x}{x}\right)}+\frac{\frac{64x^4(x+2)(3x^2{\mathrm{e}}^x-2{\mathrm{e}}^x+3x{\mathrm{e}}^x+2x^2+2x^3)}{{\mathrm{e}}^x+x^2}-\frac{64x^4\ln \left(\frac{x+{\mathrm{e}}^x}{x}\right)(x+{\mathrm{e}}^x)(3x^2+10x+8)}{{\mathrm{e}}^x+x^2}}{x^2-2x\ln \left(\frac{x+{\mathrm{e}}^x}{x}\right)+{\ln \left(\frac{x+{\mathrm{e}}^x}{x}\right)}^2}-2048x^4-3200x^5-1152x^6-\frac{64(-3x^{11}+41x^{10}+13x^9-188x^8-28x^7+144x^6)}{({\mathrm{e}}^x+x^2)(2x-x^2)}-\frac{64(-3x^{15}+8x^{14}+13x^{13}-46x^{12}+4x^{11}+56x^{10}-32x^9)}{(2x-x^2)({\mathrm{e}}^{3x}+3x^4{\mathrm{e}}^x+3x^2{\mathrm{e}}^{2x}+x^6)}+\frac{64(-6x^{13}+31x^{12}+12x^{11}-151x^{10}+50x^9+144x^8-80x^7)}{(2x-x^2)({\mathrm{e}}^{2x}+2x^2{\mathrm{e}}^x+x^4)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x)*(512*x^4 - 512*x^3 + 1152*x^5 + 384*x^6) + 512*x^5 + 768*x^6 + 256*x^7 - log((x + exp(x))/x)*(exp
(x)*(1024*x^3 + 1280*x^4 + 384*x^5) + 1024*x^4 + 1280*x^5 + 384*x^6))/(x^3*exp(x) + log((x + exp(x))/x)^2*(3*x
*exp(x) + 3*x^2) - log((x + exp(x))/x)*(3*x^2*exp(x) + 3*x^3) - log((x + exp(x))/x)^3*(x + exp(x)) + x^4),x)

[Out]

((64*x*(x + exp(x))*(24*x^5*exp(x) + 48*x^6*exp(x) + 43*x^7*exp(x) + 19*x^8*exp(x) + 3*x^9*exp(x) - 8*x^3*exp(
2*x) + 22*x^4*exp(2*x) + 47*x^5*exp(2*x) + 18*x^6*exp(2*x) + 16*x^7 + 30*x^8 + 12*x^9))/(exp(x) + x^2)^3 - (64
*x*log((x + exp(x))/x)*(x + exp(x))*(40*x^4*exp(x) + 68*x^5*exp(x) + 49*x^6*exp(x) + 19*x^7*exp(x) + 3*x^8*exp
(x) + 32*x^3*exp(2*x) + 50*x^4*exp(2*x) + 18*x^5*exp(2*x) + 24*x^6 + 40*x^7 + 15*x^8))/(exp(x) + x^2)^3)/(x -
log((x + exp(x))/x)) + ((64*x^4*(x + 2)*(3*x^2*exp(x) - 2*exp(x) + 3*x*exp(x) + 2*x^2 + 2*x^3))/(exp(x) + x^2)
- (64*x^4*log((x + exp(x))/x)*(x + exp(x))*(10*x + 3*x^2 + 8))/(exp(x) + x^2))/(log((x + exp(x))/x)^2 - 2*x*l
og((x + exp(x))/x) + x^2) - 2048*x^4 - 3200*x^5 - 1152*x^6 - (64*(144*x^6 - 28*x^7 - 188*x^8 + 13*x^9 + 41*x^1
0 - 3*x^11))/((exp(x) + x^2)*(2*x - x^2)) - (64*(56*x^10 - 32*x^9 + 4*x^11 - 46*x^12 + 13*x^13 + 8*x^14 - 3*x^
15))/((2*x - x^2)*(exp(3*x) + 3*x^4*exp(x) + 3*x^2*exp(2*x) + x^6)) + (64*(144*x^8 - 80*x^7 + 50*x^9 - 151*x^1
0 + 12*x^11 + 31*x^12 - 6*x^13))/((2*x - x^2)*(exp(2*x) + 2*x^2*exp(x) + x^4))

________________________________________________________________________________________

sympy [A]  time = 0.27, size = 41, normalized size = 1.41 $$\begin{array}{c}\frac{-64x^6-256x^5-256x^4}{x^2-2x\log \left(\frac{x+e^x}{x}\right)+\log {\left(\frac{x+e^x}{x}\right)}^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-384*x**5-1280*x**4-1024*x**3)*exp(x)-384*x**6-1280*x**5-1024*x**4)*ln(1/x*(exp(x)+x))+(384*x**6+
1152*x**5+512*x**4-512*x**3)*exp(x)+256*x**7+768*x**6+512*x**5)/((exp(x)+x)*ln(1/x*(exp(x)+x))**3+(-3*exp(x)*x
-3*x**2)*ln(1/x*(exp(x)+x))**2+(3*exp(x)*x**2+3*x**3)*ln(1/x*(exp(x)+x))-exp(x)*x**3-x**4),x)

[Out]

(-64*x**6 - 256*x**5 - 256*x**4)/(x**2 - 2*x*log((x + exp(x))/x) + log((x + exp(x))/x)**2)

________________________________________________________________________________________