3.11.20 \(\int \frac {e^{\frac {x}{1+x}} (x^2+(-5-5 x-6 x^2) \log (2))+e^{\frac {2 x}{1+x}} (-2-4 x-2 x^2-243 x^4-486 x^5-243 x^6+(2+4 x+2 x^2-1620 x^3-2997 x^4-1134 x^5+243 x^6) \log (2))}{(x^2+2 x^3+x^4) \log (2)+e^{\frac {x}{1+x}} (-4 x-8 x^2-4 x^3+162 x^5+324 x^6+162 x^7) \log (2)+e^{\frac {2 x}{1+x}} (4+8 x+4 x^2-324 x^4-648 x^5-324 x^6+6561 x^8+13122 x^9+6561 x^{10}) \log (2)} \, dx\) [1020]

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

[Out]

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

________________________________________________________________________________________

Rubi [F]
time = 24.58, 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 {gather*} \int \frac {e^{\frac {x}{1+x}} \left (x^2+\left (-5-5 x-6 x^2\right ) \log (2)\right )+e^{\frac {2 x}{1+x}} \left (-2-4 x-2 x^2-243 x^4-486 x^5-243 x^6+\left (2+4 x+2 x^2-1620 x^3-2997 x^4-1134 x^5+243 x^6\right ) \log (2)\right )}{\left (x^2+2 x^3+x^4\right ) \log (2)+e^{\frac {x}{1+x}} \left (-4 x-8 x^2-4 x^3+162 x^5+324 x^6+162 x^7\right ) \log (2)+e^{\frac {2 x}{1+x}} \left (4+8 x+4 x^2-324 x^4-648 x^5-324 x^6+6561 x^8+13122 x^9+6561 x^{10}\right ) \log (2)} \, dx \end {gather*}

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

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Mathematica [F]
time = 3.17, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{\frac {x}{1+x}} \left (x^2+\left (-5-5 x-6 x^2\right ) \log (2)\right )+e^{\frac {2 x}{1+x}} \left (-2-4 x-2 x^2-243 x^4-486 x^5-243 x^6+\left (2+4 x+2 x^2-1620 x^3-2997 x^4-1134 x^5+243 x^6\right ) \log (2)\right )}{\left (x^2+2 x^3+x^4\right ) \log (2)+e^{\frac {x}{1+x}} \left (-4 x-8 x^2-4 x^3+162 x^5+324 x^6+162 x^7\right ) \log (2)+e^{\frac {2 x}{1+x}} \left (4+8 x+4 x^2-324 x^4-648 x^5-324 x^6+6561 x^8+13122 x^9+6561 x^{10}\right ) \log (2)} \, dx \end {gather*}

Verification is not applicable to the result.

[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]

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]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(83\) vs. \(2(33)=66\).
time = 1.38, size = 84, normalized size = 2.55

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

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 = 0.59, size = 45, normalized size = 1.36 \begin {gather*} -\frac {x {\left (\log \left (2\right ) - 1\right )} e - 5 \, e \log \left (2\right )}{81 \, x^{4} e \log \left (2\right ) + x e^{\left (\frac {1}{x + 1}\right )} \log \left (2\right ) - 2 \, e \log \left (2\right )} \end {gather*}

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
/(1+x))^2+((-6*x^2-5*x-5)*log(2)+x^2)*exp(x/(1+x)))/((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/(1+x))^2+(162*x^7+324*x^6+162*x^5-4*x^3-8*x^2-4*x)*log(2)*exp(x/(1+x))+(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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Fricas [A]
time = 0.37, size = 45, normalized size = 1.36 \begin {gather*} -\frac {{\left ({\left (x - 5\right )} \log \left (2\right ) - x\right )} e^{\left (\frac {x}{x + 1}\right )}}{{\left (81 \, x^{4} - 2\right )} e^{\left (\frac {x}{x + 1}\right )} \log \left (2\right ) + x \log \left (2\right )} \end {gather*}

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
/(1+x))^2+((-6*x^2-5*x-5)*log(2)+x^2)*exp(x/(1+x)))/((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/(1+x))^2+(162*x^7+324*x^6+162*x^5-4*x^3-8*x^2-4*x)*log(2)*exp(x/(1+x))+(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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (22) = 44\).
time = 0.95, size = 85, normalized size = 2.58 \begin {gather*} - \frac {x \left (-1 + \log {\left (2 \right )}\right ) - 5 \log {\left (2 \right )}}{81 x^{4} \log {\left (2 \right )} - 2 \log {\left (2 \right )}} + \frac {- x^{2} + x^{2} \log {\left (2 \right )} - 5 x \log {\left (2 \right )}}{81 x^{5} \log {\left (2 \right )} - 2 x \log {\left (2 \right )} + \left (6561 x^{8} \log {\left (2 \right )} - 324 x^{4} \log {\left (2 \right )} + 4 \log {\left (2 \right )}\right ) e^{\frac {x}{x + 1}}} \end {gather*}

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/(1+x))**2+((-6*x**2-5*x-5)*ln(2)+x**2)*exp(x/(1+x)))/((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/(1+x))**2+(162*x**7+324*x**6+162*x**5-4*x**3-8*x**2-4*x)*ln(2)*exp(x/(1
+x))+(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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (32) = 64\).
time = 3.43, size = 356, normalized size = 10.79 \begin {gather*} -\frac {\frac {1707 \, x e^{\left (\frac {x}{x + 1}\right )} \log \left (2\right )}{x + 1} - \frac {2679 \, x^{2} e^{\left (\frac {x}{x + 1}\right )} \log \left (2\right )}{{\left (x + 1\right )}^{2}} + \frac {1865 \, x^{3} e^{\left (\frac {x}{x + 1}\right )} \log \left (2\right )}{{\left (x + 1\right )}^{3}} - 407 \, e^{\left (\frac {x}{x + 1}\right )} \log \left (2\right ) - \frac {87 \, x e^{\left (\frac {x}{x + 1}\right )}}{x + 1} + \frac {249 \, x^{2} e^{\left (\frac {x}{x + 1}\right )}}{{\left (x + 1\right )}^{2}} - \frac {245 \, x^{3} e^{\left (\frac {x}{x + 1}\right )}}{{\left (x + 1\right )}^{3}} + \frac {6 \, x \log \left (2\right )}{x + 1} - \frac {18 \, x^{2} \log \left (2\right )}{{\left (x + 1\right )}^{2}} + \frac {18 \, x^{3} \log \left (2\right )}{{\left (x + 1\right )}^{3}} - \frac {6 \, x^{4} \log \left (2\right )}{{\left (x + 1\right )}^{4}} - \frac {x}{x + 1} + \frac {3 \, x^{2}}{{\left (x + 1\right )}^{2}} - \frac {3 \, x^{3}}{{\left (x + 1\right )}^{3}} + \frac {x^{4}}{{\left (x + 1\right )}^{4}} + 2 \, e^{\left (\frac {x}{x + 1}\right )}}{79 \, {\left (\frac {8 \, x e^{\left (\frac {x}{x + 1}\right )} \log \left (2\right )}{x + 1} - \frac {12 \, x^{2} e^{\left (\frac {x}{x + 1}\right )} \log \left (2\right )}{{\left (x + 1\right )}^{2}} + \frac {8 \, x^{3} e^{\left (\frac {x}{x + 1}\right )} \log \left (2\right )}{{\left (x + 1\right )}^{3}} + \frac {79 \, x^{4} e^{\left (\frac {x}{x + 1}\right )} \log \left (2\right )}{{\left (x + 1\right )}^{4}} - 2 \, e^{\left (\frac {x}{x + 1}\right )} \log \left (2\right ) + \frac {x \log \left (2\right )}{x + 1} - \frac {3 \, x^{2} \log \left (2\right )}{{\left (x + 1\right )}^{2}} + \frac {3 \, x^{3} \log \left (2\right )}{{\left (x + 1\right )}^{3}} - \frac {x^{4} \log \left (2\right )}{{\left (x + 1\right )}^{4}}\right )}} \end {gather*}

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
/(1+x))^2+((-6*x^2-5*x-5)*log(2)+x^2)*exp(x/(1+x)))/((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/(1+x))^2+(162*x^7+324*x^6+162*x^5-4*x^3-8*x^2-4*x)*log(2)*exp(x/(1+x))+(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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {{\mathrm {e}}^{\frac {x}{x+1}}\,\left (\ln \left (2\right )\,\left (6\,x^2+5\,x+5\right )-x^2\right )+{\mathrm {e}}^{\frac {2\,x}{x+1}}\,\left (4\,x-\ln \left (2\right )\,\left (243\,x^6-1134\,x^5-2997\,x^4-1620\,x^3+2\,x^2+4\,x+2\right )+2\,x^2+243\,x^4+486\,x^5+243\,x^6+2\right )}{\ln \left (2\right )\,\left (x^4+2\,x^3+x^2\right )-{\mathrm {e}}^{\frac {x}{x+1}}\,\ln \left (2\right )\,\left (-162\,x^7-324\,x^6-162\,x^5+4\,x^3+8\,x^2+4\,x\right )+{\mathrm {e}}^{\frac {2\,x}{x+1}}\,\ln \left (2\right )\,\left (6561\,x^{10}+13122\,x^9+6561\,x^8-324\,x^6-648\,x^5-324\,x^4+4\,x^2+8\,x+4\right )} \,d x \end {gather*}

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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