3.11.20 $\int \frac{e^{\frac{x}{1+x}}(x^2+(-5-5x-6x^2)\log (2))+e^{\frac{2x}{1+x}}(-2-4x-2x^2-243x^4-486x^5-243x^6+(2+4x+2x^2-1620x^3-2997x^4-1134x^5+243x^6)\log (2))}{(x^2+2x^3+x^4)\log (2)+e^{\frac{x}{1+x}}(-4x-8x^2-4x^3+162x^5+324x^6+162x^7)\log (2)+e^{\frac{2x}{1+x}}(4+8x+4x^2-324x^4-648x^5-324x^6+6561x^8+13122x^9+6561x^{10})\log (2)}dx$

Optimal. Leaf size=33 $$\frac{5-x+\frac{x}{\log (2)}}{-2+e^{-\frac{x}{1+x}}x+81x^4}$$

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Rubi [F]  time = 39.31, 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 = {} $$\begin{array}{c}\int \frac{e^{\frac{x}{1+x}}(x^2+(-5-5x-6x^2)\log (2))+e^{\frac{2x}{1+x}}(-2-4x-2x^2-243x^4-486x^5-243x^6+(2+4x+2x^2-1620x^3-2997x^4-1134x^5+243x^6)\log (2))}{(x^2+2x^3+x^4)\log (2)+e^{\frac{x}{1+x}}(-4x-8x^2-4x^3+162x^5+324x^6+162x^7)\log (2)+e^{\frac{2x}{1+x}}(4+8x+4x^2-324x^4-648x^5-324x^6+6561x^8+13122x^9+6561x^{10})\log (2)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(E^(x/(1 + x))*(x^2 + (-5 - 5*x - 6*x^2)*Log[2]) + E^((2*x)/(1 + x))*(-2 - 4*x - 2*x^2 - 243*x^4 - 486*x^5
- 243*x^6 + (2 + 4*x + 2*x^2 - 1620*x^3 - 2997*x^4 - 1134*x^5 + 243*x^6)*Log[2]))/((x^2 + 2*x^3 + x^4)*Log[2]
+ E^(x/(1 + x))*(-4*x - 8*x^2 - 4*x^3 + 162*x^5 + 324*x^6 + 162*x^7)*Log[2] + E^((2*x)/(1 + x))*(4 + 8*x + 4*
x^2 - 324*x^4 - 648*x^5 - 324*x^6 + 6561*x^8 + 13122*x^9 + 6561*x^10)*Log[2]),x]

[Out]

((1 + 14*Log[2])*Defer[Int][E^(x/(1 + x))/(x + E^(x/(1 + x))*(-2 + 81*x^4))^2, x])/Log[2] - (5*I)*2^(1/4)*Defe
r[Int][E^(x/(1 + x))/((I*2^(1/4) - 3*x)*(x + E^(x/(1 + x))*(-2 + 81*x^4))^2), x] + ((24964 + 16670*Log[2] + 63
99*Log[16] - 6723*Log[1024])*Defer[Int][E^(x/(1 + x))/((I*2^(1/4) - 3*x)*(x + E^(x/(1 + x))*(-2 + 81*x^4))^2),
x])/(37446*Sqrt[2]*Log[2]) - 5*2^(1/4)*Defer[Int][E^(x/(1 + x))/((2^(1/4) - 3*x)*(x + E^(x/(1 + x))*(-2 + 81*
x^4))^2), x] - ((24964 + 16670*Log[2] + 6399*Log[16] - 6723*Log[1024])*Defer[Int][E^(x/(1 + x))/((2^(1/4) - 3*
x)*(x + E^(x/(1 + x))*(-2 + 81*x^4))^2), x])/(37446*Sqrt[2]*Log[2]) + (3*(1 - Log[2])*Defer[Int][(E^(x/(1 + x)
)*x)/(x + E^(x/(1 + x))*(-2 + 81*x^4))^2, x])/Log[2] + ((79 - 484*Log[2] + Log[1024])*Defer[Int][E^(x/(1 + x))
/((1 + x)^2*(x + E^(x/(1 + x))*(-2 + 81*x^4))^2), x])/(79*Log[2]) - ((12482 - 41079*Log[2] + 158*Log[16] - 324
*Log[1024])*Defer[Int][E^(x/(1 + x))/((1 + x)*(x + E^(x/(1 + x))*(-2 + 81*x^4))^2), x])/(6241*Log[2]) - (5*I)*
2^(1/4)*Defer[Int][E^(x/(1 + x))/((I*2^(1/4) + 3*x)*(x + E^(x/(1 + x))*(-2 + 81*x^4))^2), x] - ((24964 + 16670
*Log[2] + 6399*Log[16] - 6723*Log[1024])*Defer[Int][E^(x/(1 + x))/((I*2^(1/4) + 3*x)*(x + E^(x/(1 + x))*(-2 +
81*x^4))^2), x])/(37446*Sqrt[2]*Log[2]) - 5*2^(1/4)*Defer[Int][E^(x/(1 + x))/((2^(1/4) + 3*x)*(x + E^(x/(1 + x
))*(-2 + 81*x^4))^2), x] + ((24964 + 16670*Log[2] + 6399*Log[16] - 6723*Log[1024])*Defer[Int][E^(x/(1 + x))/((
2^(1/4) + 3*x)*(x + E^(x/(1 + x))*(-2 + 81*x^4))^2), x])/(37446*Sqrt[2]*Log[2]) - ((3 - Log[8])*Defer[Int][E^(
x/(1 + x))/(x + E^(x/(1 + x))*(-2 + 81*x^4)), x])/Log[2] + (I*2^(1/4)*(1 + 566*Log[2] - 81*Log[128])*Defer[Int
][E^(x/(1 + x))/((I*2^(1/4) - 3*x)*(x + E^(x/(1 + x))*(-2 + 81*x^4))), x])/Log[2] + (3*Log[1024]*Defer[Int][E^
(x/(1 + x))/((I*2^(1/4) - 3*x)*(x + E^(x/(1 + x))*(-2 + 81*x^4))), x])/(2*Log[2]) + (2^(1/4)*(1 + 566*Log[2] -
81*Log[128])*Defer[Int][E^(x/(1 + x))/((2^(1/4) - 3*x)*(x + E^(x/(1 + x))*(-2 + 81*x^4))), x])/Log[2] + (3*Lo
g[1024]*Defer[Int][E^(x/(1 + x))/((2^(1/4) - 3*x)*(x + E^(x/(1 + x))*(-2 + 81*x^4))), x])/(2*Log[2]) + (I*2^(1
/4)*(1 + 566*Log[2] - 81*Log[128])*Defer[Int][E^(x/(1 + x))/((I*2^(1/4) + 3*x)*(x + E^(x/(1 + x))*(-2 + 81*x^4
))), x])/Log[2] - (3*Log[1024]*Defer[Int][E^(x/(1 + x))/((I*2^(1/4) + 3*x)*(x + E^(x/(1 + x))*(-2 + 81*x^4))),
x])/(2*Log[2]) + (2^(1/4)*(1 + 566*Log[2] - 81*Log[128])*Defer[Int][E^(x/(1 + x))/((2^(1/4) + 3*x)*(x + E^(x/
(1 + x))*(-2 + 81*x^4))), x])/Log[2] - (3*Log[1024]*Defer[Int][E^(x/(1 + x))/((2^(1/4) + 3*x)*(x + E^(x/(1 + x
))*(-2 + 81*x^4))), x])/(2*Log[2])

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{e^{\frac{x}{1+x}}(x^2(1-6\log (2))-5\log (2)-5x\log (2)+e^{\frac{x}{1+x}}(-2+243x^6(-1+\log (2))-1620x^3\log (2)-81x^4(3+37\log (2))+x^2(-2+\log (4))+\log (4)+x(-4+\log (16))-162x^5(3+\log (128))))}{(1+x{)}^2{(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2\log (2)}dx\\ & =\frac{\int \frac{e^{\frac{x}{1+x}}(x^2(1-6\log (2))-5\log (2)-5x\log (2)+e^{\frac{x}{1+x}}(-2+243x^6(-1+\log (2))-1620x^3\log (2)-81x^4(3+37\log (2))+x^2(-2+\log (4))+\log (4)+x(-4+\log (16))-162x^5(3+\log (128))))}{(1+x{)}^2{(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{\log (2)}\\ & =\frac{\int (\frac{e^{\frac{x}{1+x}}(-2x^3(1-\log (2))-243x^7(1-\log (2))-1215x^4\log (2)-567x^6(1+\frac{8\log (2)}{7})-243x^5(1+\frac{32\log (2)}{3})-2x(1+\log (16))-2x^2(1+\log (16))-\log (1024))}{(1+x{)}^2(2-81x^4){(2e^{\frac{x}{1+x}}-x-81e^{\frac{x}{1+x}}{x}^4)}^2}+\frac{e^{\frac{x}{1+x}}(-2(1-\log (2))-4x(1-\log (2))-2x^2(1-\log (2))-243x^6(1-\log (2))-1620x^3\log (2)-243x^4(1+\frac{37\log (2)}{3})-486x^5(1+\log \left(4\sqrt[3]{2}\right)))}{(1+x{)}^2(2-81x^4)(2e^{\frac{x}{1+x}}-x-81e^{\frac{x}{1+x}}{x}^4)})dx}{\log (2)}\\ & =\frac{\int \frac{e^{\frac{x}{1+x}}(-2x^3(1-\log (2))-243x^7(1-\log (2))-1215x^4\log (2)-567x^6(1+\frac{8\log (2)}{7})-243x^5(1+\frac{32\log (2)}{3})-2x(1+\log (16))-2x^2(1+\log (16))-\log (1024))}{(1+x{)}^2(2-81x^4){(2e^{\frac{x}{1+x}}-x-81e^{\frac{x}{1+x}}{x}^4)}^2}dx}{\log (2)}+\frac{\int \frac{e^{\frac{x}{1+x}}(-2(1-\log (2))-4x(1-\log (2))-2x^2(1-\log (2))-243x^6(1-\log (2))-1620x^3\log (2)-243x^4(1+\frac{37\log (2)}{3})-486x^5(1+\log \left(4\sqrt[3]{2}\right)))}{(1+x{)}^2(2-81x^4)(2e^{\frac{x}{1+x}}-x-81e^{\frac{x}{1+x}}{x}^4)}dx}{\log (2)}\\ & =\frac{\int (-\frac{3e^{\frac{x}{1+x}}x(-1+\log (2))}{{(-2e^{\frac{x}{1+x}}+x+81e^{\frac{x}{1+x}}{x}^4)}^2}+\frac{e^{\frac{x}{1+x}}(1+14\log (2))}{{(-2e^{\frac{x}{1+x}}+x+81e^{\frac{x}{1+x}}{x}^4)}^2}+\frac{e^{\frac{x}{1+x}}(-180434\log (2)+316\log (16)-2x(24964+16670\log (2)+6399\log (16)-6723\log (1024))-7047\log (1024))}{6241(2-81x^4){(2e^{\frac{x}{1+x}}-x-81e^{\frac{x}{1+x}}{x}^4)}^2}-\frac{e^{\frac{x}{1+x}}(-79+484\log (2)-\log (1024))}{79(1+x{)}^2{(-2e^{\frac{x}{1+x}}+x+81e^{\frac{x}{1+x}}{x}^4)}^2}+\frac{e^{\frac{x}{1+x}}(-12482+41079\log (2)-158\log (16)+324\log (1024))}{6241(1+x){(-2e^{\frac{x}{1+x}}+x+81e^{\frac{x}{1+x}}{x}^4)}^2})dx}{\log (2)}+\frac{\int \frac{e^{\frac{x}{1+x}}(-2-243x^5(1-\log (2))+1136\log (2)+x^4(-243-243\log (2)-486\log \left(4\sqrt[3]{2}\right))+x^2(1134\log (2)-486\log \left(4\sqrt[3]{2}\right))-486\log \left(4\sqrt[3]{2}\right)+x^3(-2754\log (2)+486\log \left(4\sqrt[3]{2}\right))+x(-2-1132\log (2)+486\log \left(4\sqrt[3]{2}\right)))}{(1+x)(2-81x^4)(2e^{\frac{x}{1+x}}-x-81e^{\frac{x}{1+x}}{x}^4)}dx}{\log (2)}\\ & =\frac{\int \frac{e^{\frac{x}{1+x}}(-180434\log (2)+316\log (16)-2x(24964+16670\log (2)+6399\log (16)-6723\log (1024))-7047\log (1024))}{(2-81x^4){(2e^{\frac{x}{1+x}}-x-81e^{\frac{x}{1+x}}{x}^4)}^2}dx}{6241\log (2)}+\frac{\int \frac{e^{\frac{x}{1+x}}(-2-243x^4(1-\log (2))-4534\log (2)+x(3402\log (2)-1458\log \left(4\sqrt[3]{2}\right))+x^3(-486\log (2)-486\log \left(4\sqrt[3]{2}\right))+1944\log \left(4\sqrt[3]{2}\right)+x^2(-2268\log (2)+972\log \left(4\sqrt[3]{2}\right)))}{(2-81x^4)(2e^{\frac{x}{1+x}}-x-81e^{\frac{x}{1+x}}{x}^4)}dx}{\log (2)}-\frac{(3(-1+\log (2)))\int \frac{e^{\frac{x}{1+x}}x}{{(-2e^{\frac{x}{1+x}}+x+81e^{\frac{x}{1+x}}{x}^4)}^2}dx}{\log (2)}+\frac{(1+14\log (2))\int \frac{e^{\frac{x}{1+x}}}{{(-2e^{\frac{x}{1+x}}+x+81e^{\frac{x}{1+x}}{x}^4)}^2}dx}{\log (2)}-\frac{(12482-41079\log (2)+158\log (16)-324\log (1024))\int \frac{e^{\frac{x}{1+x}}}{(1+x){(-2e^{\frac{x}{1+x}}+x+81e^{\frac{x}{1+x}}{x}^4)}^2}dx}{6241\log (2)}+\frac{(79-484\log (2)+\log (1024))\int \frac{e^{\frac{x}{1+x}}}{(1+x{)}^2{(-2e^{\frac{x}{1+x}}+x+81e^{\frac{x}{1+x}}{x}^4)}^2}dx}{79\log (2)}\\ & =\frac{\int \frac{e^{\frac{x}{1+x}}(-180434\log (2)+316\log (16)-2x(24964+16670\log (2)+6399\log (16)-6723\log (1024))-7047\log (1024))}{(2-81x^4){(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{6241\log (2)}+\frac{\int \frac{e^{\frac{x}{1+x}}(243x^4(1-\log (2))+2(1+2267\log (2)-324\log (128))+162x^3\log (1024))}{(2-81x^4)(x+e^{\frac{x}{1+x}}(-2+81x^4))}dx}{\log (2)}-\frac{(3(-1+\log (2)))\int \frac{e^{\frac{x}{1+x}}x}{{(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{\log (2)}+\frac{(1+14\log (2))\int \frac{e^{\frac{x}{1+x}}}{{(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{\log (2)}-\frac{(12482-41079\log (2)+158\log (16)-324\log (1024))\int \frac{e^{\frac{x}{1+x}}}{(1+x){(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{6241\log (2)}+\frac{(79-484\log (2)+\log (1024))\int \frac{e^{\frac{x}{1+x}}}{(1+x{)}^2{(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{79\log (2)}\\ & =\frac{\int (\frac{2e^{\frac{x}{1+x}}x(24964+16670\log (2)+6399\log (16)-6723\log (1024))}{(-2+81x^4){(-2e^{\frac{x}{1+x}}+x+81e^{\frac{x}{1+x}}{x}^4)}^2}+\frac{180434e^{\frac{x}{1+x}}\log (2)(1+\frac{-316\log (16)+7047\log (1024)}{180434\log (2)})}{(-2+81x^4){(-2e^{\frac{x}{1+x}}+x+81e^{\frac{x}{1+x}}{x}^4)}^2})dx}{6241\log (2)}+\frac{\int (\frac{e^{\frac{x}{1+x}}(-3+\log (8))}{-2e^{\frac{x}{1+x}}+x+81e^{\frac{x}{1+x}}{x}^4}-\frac{2e^{\frac{x}{1+x}}(4+2264\log (2)-324\log (128)+81x^3\log (1024))}{(-2+81x^4)(-2e^{\frac{x}{1+x}}+x+81e^{\frac{x}{1+x}}{x}^4)})dx}{\log (2)}-\frac{(3(-1+\log (2)))\int \frac{e^{\frac{x}{1+x}}x}{{(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{\log (2)}+\frac{(1+14\log (2))\int \frac{e^{\frac{x}{1+x}}}{{(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{\log (2)}-\frac{(12482-41079\log (2)+158\log (16)-324\log (1024))\int \frac{e^{\frac{x}{1+x}}}{(1+x){(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{6241\log (2)}+\frac{(79-484\log (2)+\log (1024))\int \frac{e^{\frac{x}{1+x}}}{(1+x{)}^2{(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{79\log (2)}\\ & =-\frac{2\int \frac{e^{\frac{x}{1+x}}(4+2264\log (2)-324\log (128)+81x^3\log (1024))}{(-2+81x^4)(-2e^{\frac{x}{1+x}}+x+81e^{\frac{x}{1+x}}{x}^4)}dx}{\log (2)}-\frac{(3(-1+\log (2)))\int \frac{e^{\frac{x}{1+x}}x}{{(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{\log (2)}+\frac{(1+14\log (2))\int \frac{e^{\frac{x}{1+x}}}{{(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{\log (2)}+\frac{(180434+\frac{34603\log (4)}{\log (2)})\int \frac{e^{\frac{x}{1+x}}}{(-2+81x^4){(-2e^{\frac{x}{1+x}}+x+81e^{\frac{x}{1+x}}{x}^4)}^2}dx}{6241}-\frac{(3-\log (8))\int \frac{e^{\frac{x}{1+x}}}{-2e^{\frac{x}{1+x}}+x+81e^{\frac{x}{1+x}}{x}^4}dx}{\log (2)}+\frac{(2(24964+16670\log (2)+6399\log (16)-6723\log (1024)))\int \frac{e^{\frac{x}{1+x}}x}{(-2+81x^4){(-2e^{\frac{x}{1+x}}+x+81e^{\frac{x}{1+x}}{x}^4)}^2}dx}{6241\log (2)}-\frac{(12482-41079\log (2)+158\log (16)-324\log (1024))\int \frac{e^{\frac{x}{1+x}}}{(1+x){(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{6241\log (2)}+\frac{(79-484\log (2)+\log (1024))\int \frac{e^{\frac{x}{1+x}}}{(1+x{)}^2{(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{79\log (2)}\\ & =-\frac{2\int (\frac{4e^{\frac{x}{1+x}}(1+566\log (2)-81\log (128))}{(-2+81x^4)(-2e^{\frac{x}{1+x}}+x+81e^{\frac{x}{1+x}}{x}^4)}+\frac{81e^{\frac{x}{1+x}}{x}^3\log (1024)}{(-2+81x^4)(-2e^{\frac{x}{1+x}}+x+81e^{\frac{x}{1+x}}{x}^4)})dx}{\log (2)}-\frac{(3(-1+\log (2)))\int \frac{e^{\frac{x}{1+x}}x}{{(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{\log (2)}+\frac{(1+14\log (2))\int \frac{e^{\frac{x}{1+x}}}{{(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{\log (2)}+\frac{(180434+\frac{34603\log (4)}{\log (2)})\int \frac{e^{\frac{x}{1+x}}}{(-2+81x^4){(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{6241}-\frac{(3-\log (8))\int \frac{e^{\frac{x}{1+x}}}{x+e^{\frac{x}{1+x}}(-2+81x^4)}dx}{\log (2)}+\frac{(2(24964+16670\log (2)+6399\log (16)-6723\log (1024)))\int \frac{e^{\frac{x}{1+x}}x}{(-2+81x^4){(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{6241\log (2)}-\frac{(12482-41079\log (2)+158\log (16)-324\log (1024))\int \frac{e^{\frac{x}{1+x}}}{(1+x){(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{6241\log (2)}+\frac{(79-484\log (2)+\log (1024))\int \frac{e^{\frac{x}{1+x}}}{(1+x{)}^2{(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{79\log (2)}\\ & =-\frac{(3(-1+\log (2)))\int \frac{e^{\frac{x}{1+x}}x}{{(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{\log (2)}+\frac{(1+14\log (2))\int \frac{e^{\frac{x}{1+x}}}{{(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{\log (2)}+\frac{(180434+\frac{34603\log (4)}{\log (2)})\int (-\frac{e^{\frac{x}{1+x}}}{2\sqrt{2}(\sqrt{2}-9x^2){(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}-\frac{e^{\frac{x}{1+x}}}{2\sqrt{2}(\sqrt{2}+9x^2){(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2})dx}{6241}-\frac{(3-\log (8))\int \frac{e^{\frac{x}{1+x}}}{x+e^{\frac{x}{1+x}}(-2+81x^4)}dx}{\log (2)}-\frac{(8(1+566\log (2)-81\log (128)))\int \frac{e^{\frac{x}{1+x}}}{(-2+81x^4)(-2e^{\frac{x}{1+x}}+x+81e^{\frac{x}{1+x}}{x}^4)}dx}{\log (2)}+\frac{(2(24964+16670\log (2)+6399\log (16)-6723\log (1024)))\int (\frac{9e^{\frac{x}{1+x}}x}{2\sqrt{2}(-9\sqrt{2}+81x^2){(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}-\frac{9e^{\frac{x}{1+x}}x}{2\sqrt{2}(9\sqrt{2}+81x^2){(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2})dx}{6241\log (2)}-\frac{(12482-41079\log (2)+158\log (16)-324\log (1024))\int \frac{e^{\frac{x}{1+x}}}{(1+x){(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{6241\log (2)}-\frac{(162\log (1024))\int \frac{e^{\frac{x}{1+x}}{x}^3}{(-2+81x^4)(-2e^{\frac{x}{1+x}}+x+81e^{\frac{x}{1+x}}{x}^4)}dx}{\log (2)}+\frac{(79-484\log (2)+\log (1024))\int \frac{e^{\frac{x}{1+x}}}{(1+x{)}^2{(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{79\log (2)}\\ & =-\frac{(3(-1+\log (2)))\int \frac{e^{\frac{x}{1+x}}x}{{(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{\log (2)}+\frac{(1+14\log (2))\int \frac{e^{\frac{x}{1+x}}}{{(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{\log (2)}-\frac{(180434+\frac{34603\log (4)}{\log (2)})\int \frac{e^{\frac{x}{1+x}}}{(\sqrt{2}-9x^2){(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{12482\sqrt{2}}-\frac{(180434+\frac{34603\log (4)}{\log (2)})\int \frac{e^{\frac{x}{1+x}}}{(\sqrt{2}+9x^2){(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{12482\sqrt{2}}-\frac{(3-\log (8))\int \frac{e^{\frac{x}{1+x}}}{x+e^{\frac{x}{1+x}}(-2+81x^4)}dx}{\log (2)}-\frac{(8(1+566\log (2)-81\log (128)))\int \frac{e^{\frac{x}{1+x}}}{(-2+81x^4)(x+e^{\frac{x}{1+x}}(-2+81x^4))}dx}{\log (2)}+\frac{(9(24964+16670\log (2)+6399\log (16)-6723\log (1024)))\int \frac{e^{\frac{x}{1+x}}x}{(-9\sqrt{2}+81x^2){(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{6241\sqrt{2}\log (2)}-\frac{(9(24964+16670\log (2)+6399\log (16)-6723\log (1024)))\int \frac{e^{\frac{x}{1+x}}x}{(9\sqrt{2}+81x^2){(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{6241\sqrt{2}\log (2)}-\frac{(12482-41079\log (2)+158\log (16)-324\log (1024))\int \frac{e^{\frac{x}{1+x}}}{(1+x){(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{6241\log (2)}-\frac{(162\log (1024))\int \frac{e^{\frac{x}{1+x}}{x}^3}{(-2+81x^4)(x+e^{\frac{x}{1+x}}(-2+81x^4))}dx}{\log (2)}+\frac{(79-484\log (2)+\log (1024))\int \frac{e^{\frac{x}{1+x}}}{(1+x{)}^2{(x+e^{\frac{x}{1+x}}(-2+81x^4))}^2}dx}{79\log (2)}\\ & =\text{Rest of rules removed due to large latex content}\end{array}\end{array}$$

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Mathematica [B]  time = 0.28, size = 180, normalized size = 5.45 $$\begin{array}{c}\frac{3944312x+19785288\log (2)-1524858x\log (2)+72868\log (4)-595423x\log (4)-36434\log (16)+51192x\log (16)-9104\log (128)-204768x\log (128)+\frac{3944312e^{\frac{1}{1+x}}x(243x^7(-1+\log (2))-10\log (2)-1215x^4\log (2)-81x^5(3+32\log (2))+x^3(-2+\log (4))-x(2+\log (256))-x^2(2+\log (256))-81x^6(7+\log (256)))}{(2+2x+2x^2+243x^4+567x^5+243x^6)(e^{\frac{1}{1+x}}x+e(-2+81x^4))}}{3944312(-2+81x^4)\log (2)}\end{array}$$

Warning: Unable to verify antiderivative.

[In]

Integrate[(E^(x/(1 + x))*(x^2 + (-5 - 5*x - 6*x^2)*Log[2]) + E^((2*x)/(1 + x))*(-2 - 4*x - 2*x^2 - 243*x^4 - 4
86*x^5 - 243*x^6 + (2 + 4*x + 2*x^2 - 1620*x^3 - 2997*x^4 - 1134*x^5 + 243*x^6)*Log[2]))/((x^2 + 2*x^3 + x^4)*
Log[2] + E^(x/(1 + x))*(-4*x - 8*x^2 - 4*x^3 + 162*x^5 + 324*x^6 + 162*x^7)*Log[2] + E^((2*x)/(1 + x))*(4 + 8*
x + 4*x^2 - 324*x^4 - 648*x^5 - 324*x^6 + 6561*x^8 + 13122*x^9 + 6561*x^10)*Log[2]),x]

[Out]

(3944312*x + 19785288*Log[2] - 1524858*x*Log[2] + 72868*Log[4] - 595423*x*Log[4] - 36434*Log[16] + 51192*x*Log
[16] - 9104*Log[128] - 204768*x*Log[128] + (3944312*E^(1 + x)^(-1)*x*(243*x^7*(-1 + Log[2]) - 10*Log[2] - 1215
*x^4*Log[2] - 81*x^5*(3 + 32*Log[2]) + x^3*(-2 + Log[4]) - x*(2 + Log[256]) - x^2*(2 + Log[256]) - 81*x^6*(7 +
Log[256])))/((2 + 2*x + 2*x^2 + 243*x^4 + 567*x^5 + 243*x^6)*(E^(1 + x)^(-1)*x + E*(-2 + 81*x^4))))/(3944312*
(-2 + 81*x^4)*Log[2])

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fricas [A]  time = 0.69, size = 45, normalized size = 1.36 $$\begin{array}{c}-\frac{((x-5)\log (2)-x)e^{\left(\frac{x}{x+1}\right)}}{(81x^4-2)e^{\left(\frac{x}{x+1}\right)}\log (2)+x\log (2)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((243*x^6-1134*x^5-2997*x^4-1620*x^3+2*x^2+4*x+2)*log(2)-243*x^6-486*x^5-243*x^4-2*x^2-4*x-2)*exp(x
/(x+1))^2+((-6*x^2-5*x-5)*log(2)+x^2)*exp(x/(x+1)))/((6561*x^10+13122*x^9+6561*x^8-324*x^6-648*x^5-324*x^4+4*x
^2+8*x+4)*log(2)*exp(x/(x+1))^2+(162*x^7+324*x^6+162*x^5-4*x^3-8*x^2-4*x)*log(2)*exp(x/(x+1))+(x^4+2*x^3+x^2)*
log(2)),x, algorithm="fricas")

[Out]

-((x - 5)*log(2) - x)*e^(x/(x + 1))/((81*x^4 - 2)*e^(x/(x + 1))*log(2) + x*log(2))

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giac [B]  time = 19.53, size = 356, normalized size = 10.79 $$\begin{array}{c}-\frac{\frac{1707x{e}^{\left(\frac{x}{x+1}\right)}\log (2)}{x+1}-\frac{2679x^2{e}^{\left(\frac{x}{x+1}\right)}\log (2)}{{(x+1)}^2}+\frac{1865x^3{e}^{\left(\frac{x}{x+1}\right)}\log (2)}{{(x+1)}^3}-407e^{\left(\frac{x}{x+1}\right)}\log (2)-\frac{87x{e}^{\left(\frac{x}{x+1}\right)}}{x+1}+\frac{249x^2{e}^{\left(\frac{x}{x+1}\right)}}{{(x+1)}^2}-\frac{245x^3{e}^{\left(\frac{x}{x+1}\right)}}{{(x+1)}^3}+\frac{6x\log (2)}{x+1}-\frac{18x^2\log (2)}{{(x+1)}^2}+\frac{18x^3\log (2)}{{(x+1)}^3}-\frac{6x^4\log (2)}{{(x+1)}^4}-\frac{x}{x+1}+\frac{3x^2}{{(x+1)}^2}-\frac{3x^3}{{(x+1)}^3}+\frac{x^4}{{(x+1)}^4}+2e^{\left(\frac{x}{x+1}\right)}}{79(\frac{8x{e}^{\left(\frac{x}{x+1}\right)}\log (2)}{x+1}-\frac{12x^2{e}^{\left(\frac{x}{x+1}\right)}\log (2)}{{(x+1)}^2}+\frac{8x^3{e}^{\left(\frac{x}{x+1}\right)}\log (2)}{{(x+1)}^3}+\frac{79x^4{e}^{\left(\frac{x}{x+1}\right)}\log (2)}{{(x+1)}^4}-2e^{\left(\frac{x}{x+1}\right)}\log (2)+\frac{x\log (2)}{x+1}-\frac{3x^2\log (2)}{{(x+1)}^2}+\frac{3x^3\log (2)}{{(x+1)}^3}-\frac{x^4\log (2)}{{(x+1)}^4})}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((243*x^6-1134*x^5-2997*x^4-1620*x^3+2*x^2+4*x+2)*log(2)-243*x^6-486*x^5-243*x^4-2*x^2-4*x-2)*exp(x
/(x+1))^2+((-6*x^2-5*x-5)*log(2)+x^2)*exp(x/(x+1)))/((6561*x^10+13122*x^9+6561*x^8-324*x^6-648*x^5-324*x^4+4*x
^2+8*x+4)*log(2)*exp(x/(x+1))^2+(162*x^7+324*x^6+162*x^5-4*x^3-8*x^2-4*x)*log(2)*exp(x/(x+1))+(x^4+2*x^3+x^2)*
log(2)),x, algorithm="giac")

[Out]

-1/79*(1707*x*e^(x/(x + 1))*log(2)/(x + 1) - 2679*x^2*e^(x/(x + 1))*log(2)/(x + 1)^2 + 1865*x^3*e^(x/(x + 1))*
log(2)/(x + 1)^3 - 407*e^(x/(x + 1))*log(2) - 87*x*e^(x/(x + 1))/(x + 1) + 249*x^2*e^(x/(x + 1))/(x + 1)^2 - 2
45*x^3*e^(x/(x + 1))/(x + 1)^3 + 6*x*log(2)/(x + 1) - 18*x^2*log(2)/(x + 1)^2 + 18*x^3*log(2)/(x + 1)^3 - 6*x^
4*log(2)/(x + 1)^4 - x/(x + 1) + 3*x^2/(x + 1)^2 - 3*x^3/(x + 1)^3 + x^4/(x + 1)^4 + 2*e^(x/(x + 1)))/(8*x*e^(
x/(x + 1))*log(2)/(x + 1) - 12*x^2*e^(x/(x + 1))*log(2)/(x + 1)^2 + 8*x^3*e^(x/(x + 1))*log(2)/(x + 1)^3 + 79*
x^4*e^(x/(x + 1))*log(2)/(x + 1)^4 - 2*e^(x/(x + 1))*log(2) + x*log(2)/(x + 1) - 3*x^2*log(2)/(x + 1)^2 + 3*x^
3*log(2)/(x + 1)^3 - x^4*log(2)/(x + 1)^4)

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maple [B]  time = 1.18, size = 84, normalized size = 2.55

method	result	size
risch	$\frac{81(-\frac{\ln (2)}{81}+\frac{1}{81})x+5\ln (2)}{\ln (2)(81x^4-2)}+\frac{(x\ln (2)-5\ln (2)-x)x}{\ln (2)(81x^4-2)(81{\mathrm{e}}^{\frac{x}{x+1}}{x}^4-2{\mathrm{e}}^{\frac{x}{x+1}}+x)}$	$84$
norman	$\frac{5{\mathrm{e}}^{\frac{x}{x+1}}+\frac{(1+4\ln (2))x{\mathrm{e}}^{\frac{x}{x+1}}}{\ln (2)}-\frac{(\ln (2)-1)x^2{\mathrm{e}}^{\frac{x}{x+1}}}{\ln (2)}}{81{\mathrm{e}}^{\frac{x}{x+1}}{x}^5+81{\mathrm{e}}^{\frac{x}{x+1}}{x}^4+x^2-2x{\mathrm{e}}^{\frac{x}{x+1}}+x-2{\mathrm{e}}^{\frac{x}{x+1}}}$	$108$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((243*x^6-1134*x^5-2997*x^4-1620*x^3+2*x^2+4*x+2)*ln(2)-243*x^6-486*x^5-243*x^4-2*x^2-4*x-2)*exp(x/(x+1))
^2+((-6*x^2-5*x-5)*ln(2)+x^2)*exp(x/(x+1)))/((6561*x^10+13122*x^9+6561*x^8-324*x^6-648*x^5-324*x^4+4*x^2+8*x+4
)*ln(2)*exp(x/(x+1))^2+(162*x^7+324*x^6+162*x^5-4*x^3-8*x^2-4*x)*ln(2)*exp(x/(x+1))+(x^4+2*x^3+x^2)*ln(2)),x,m
ethod=_RETURNVERBOSE)

[Out]

81*((-1/81*ln(2)+1/81)*x+5/81*ln(2))/ln(2)/(81*x^4-2)+(x*ln(2)-5*ln(2)-x)*x/ln(2)/(81*x^4-2)/(81*exp(x/(x+1))*
x^4-2*exp(x/(x+1))+x)

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maxima [A]  time = 1.96, size = 45, normalized size = 1.36 $$\begin{array}{c}-\frac{x(\log (2)-1)e-5e\log (2)}{81x^{4}e\log (2)+x{e}^{\left(\frac{1}{x+1}\right)}\log (2)-2e\log (2)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((243*x^6-1134*x^5-2997*x^4-1620*x^3+2*x^2+4*x+2)*log(2)-243*x^6-486*x^5-243*x^4-2*x^2-4*x-2)*exp(x
/(x+1))^2+((-6*x^2-5*x-5)*log(2)+x^2)*exp(x/(x+1)))/((6561*x^10+13122*x^9+6561*x^8-324*x^6-648*x^5-324*x^4+4*x
^2+8*x+4)*log(2)*exp(x/(x+1))^2+(162*x^7+324*x^6+162*x^5-4*x^3-8*x^2-4*x)*log(2)*exp(x/(x+1))+(x^4+2*x^3+x^2)*
log(2)),x, algorithm="maxima")

[Out]

-(x*(log(2) - 1)*e - 5*e*log(2))/(81*x^4*e*log(2) + x*e^(1/(x + 1))*log(2) - 2*e*log(2))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 $$\begin{array}{c}\int -\frac{{\mathrm{e}}^{\frac{x}{x+1}}(\ln (2)(6x^2+5x+5)-x^2)+{\mathrm{e}}^{\frac{2x}{x+1}}(4x-\ln (2)(243x^6-1134x^5-2997x^4-1620x^3+2x^2+4x+2)+2x^2+243x^4+486x^5+243x^6+2)}{\ln (2)(x^4+2x^3+x^2)-{\mathrm{e}}^{\frac{x}{x+1}}\ln (2)(-162x^7-324x^6-162x^5+4x^3+8x^2+4x)+{\mathrm{e}}^{\frac{2x}{x+1}}\ln (2)(6561x^{10}+13122x^9+6561x^8-324x^6-648x^5-324x^4+4x^2+8x+4)}dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x/(x + 1))*(log(2)*(5*x + 6*x^2 + 5) - x^2) + exp((2*x)/(x + 1))*(4*x - log(2)*(4*x + 2*x^2 - 1620*x
^3 - 2997*x^4 - 1134*x^5 + 243*x^6 + 2) + 2*x^2 + 243*x^4 + 486*x^5 + 243*x^6 + 2))/(log(2)*(x^2 + 2*x^3 + x^4
) - exp(x/(x + 1))*log(2)*(4*x + 8*x^2 + 4*x^3 - 162*x^5 - 324*x^6 - 162*x^7) + exp((2*x)/(x + 1))*log(2)*(8*x
+ 4*x^2 - 324*x^4 - 648*x^5 - 324*x^6 + 6561*x^8 + 13122*x^9 + 6561*x^10 + 4)),x)

[Out]

int(-(exp(x/(x + 1))*(log(2)*(5*x + 6*x^2 + 5) - x^2) + exp((2*x)/(x + 1))*(4*x - log(2)*(4*x + 2*x^2 - 1620*x
^3 - 2997*x^4 - 1134*x^5 + 243*x^6 + 2) + 2*x^2 + 243*x^4 + 486*x^5 + 243*x^6 + 2))/(log(2)*(x^2 + 2*x^3 + x^4
) - exp(x/(x + 1))*log(2)*(4*x + 8*x^2 + 4*x^3 - 162*x^5 - 324*x^6 - 162*x^7) + exp((2*x)/(x + 1))*log(2)*(8*x
+ 4*x^2 - 324*x^4 - 648*x^5 - 324*x^6 + 6561*x^8 + 13122*x^9 + 6561*x^10 + 4)), x)

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sympy [B]  time = 1.62, size = 85, normalized size = 2.58 $$\begin{array}{c}-\frac{x(-1+\log (2))-5\log (2)}{81x^4\log (2)-2\log (2)}+\frac{-x^2+x^2\log (2)-5x\log (2)}{81x^5\log (2)-2x\log (2)+(6561x^8\log (2)-324x^4\log (2)+4\log (2))e^{\frac{x}{x+1}}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((243*x**6-1134*x**5-2997*x**4-1620*x**3+2*x**2+4*x+2)*ln(2)-243*x**6-486*x**5-243*x**4-2*x**2-4*x-
2)*exp(x/(x+1))**2+((-6*x**2-5*x-5)*ln(2)+x**2)*exp(x/(x+1)))/((6561*x**10+13122*x**9+6561*x**8-324*x**6-648*x
**5-324*x**4+4*x**2+8*x+4)*ln(2)*exp(x/(x+1))**2+(162*x**7+324*x**6+162*x**5-4*x**3-8*x**2-4*x)*ln(2)*exp(x/(x
+1))+(x**4+2*x**3+x**2)*ln(2)),x)

[Out]

-(x*(-1 + log(2)) - 5*log(2))/(81*x**4*log(2) - 2*log(2)) + (-x**2 + x**2*log(2) - 5*x*log(2))/(81*x**5*log(2)
- 2*x*log(2) + (6561*x**8*log(2) - 324*x**4*log(2) + 4*log(2))*exp(x/(x + 1)))

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