Integrand size = 18, antiderivative size = 981 \[ \int \frac {\log \left (x+\sqrt {1+x}\right )}{1+x^2} \, dx =\text {Too large to display} \]
-1/2*I*ln((1-I)^(1/2)-(1+x)^(1/2))*ln((1-5^(1/2)+2*(1+x)^(1/2))/(1+2*(1-I) ^(1/2)-5^(1/2)))+1/2*I*ln((1+I)^(1/2)+(1+x)^(1/2))*ln((1-5^(1/2)+2*(1+x)^( 1/2))/(1-2*(1+I)^(1/2)-5^(1/2)))-1/2*I*ln((1-I)^(1/2)-(1+x)^(1/2))*ln((1+5 ^(1/2)+2*(1+x)^(1/2))/(1+2*(1-I)^(1/2)+5^(1/2)))-1/2*I*ln(x+(1+x)^(1/2))*l n((1+I)^(1/2)+(1+x)^(1/2))+1/2*I*ln(x+(1+x)^(1/2))*ln((1-I)^(1/2)+(1+x)^(1 /2))-1/2*I*polylog(2,-2*((1-I)^(1/2)+(1+x)^(1/2))/(1-2*(1-I)^(1/2)+5^(1/2) ))+1/2*I*ln((1-I)^(1/2)-(1+x)^(1/2))*ln(x+(1+x)^(1/2))+1/2*I*ln((1+I)^(1/2 )-(1+x)^(1/2))*ln((1+5^(1/2)+2*(1+x)^(1/2))/(1+2*(1+I)^(1/2)+5^(1/2)))-1/2 *I*ln((1+I)^(1/2)-(1+x)^(1/2))*ln(x+(1+x)^(1/2))-1/2*I*ln((1-I)^(1/2)+(1+x )^(1/2))*ln((1-5^(1/2)+2*(1+x)^(1/2))/(1-2*(1-I)^(1/2)-5^(1/2)))+1/2*I*ln( (1+I)^(1/2)-(1+x)^(1/2))*ln((1-5^(1/2)+2*(1+x)^(1/2))/(1+2*(1+I)^(1/2)-5^( 1/2)))+1/2*I*ln((1+I)^(1/2)+(1+x)^(1/2))*ln((1+5^(1/2)+2*(1+x)^(1/2))/(1-2 *(1+I)^(1/2)+5^(1/2)))-1/2*I*polylog(2,-2*((1-I)^(1/2)+(1+x)^(1/2))/(1-2*( 1-I)^(1/2)-5^(1/2)))+1/2*I*polylog(2,2*((1+I)^(1/2)-(1+x)^(1/2))/(1+2*(1+I )^(1/2)+5^(1/2)))+1/2*I*polylog(2,-2*((1+I)^(1/2)+(1+x)^(1/2))/(1-2*(1+I)^ (1/2)-5^(1/2)))+1/2*I*polylog(2,-2*((1+I)^(1/2)+(1+x)^(1/2))/(1-2*(1+I)^(1 /2)+5^(1/2)))-1/2*I*ln((1-I)^(1/2)+(1+x)^(1/2))*ln((1+5^(1/2)+2*(1+x)^(1/2 ))/(1-2*(1-I)^(1/2)+5^(1/2)))-1/2*I*polylog(2,2*((1-I)^(1/2)-(1+x)^(1/2))/ (1+2*(1-I)^(1/2)-5^(1/2)))-1/2*I*polylog(2,2*((1-I)^(1/2)-(1+x)^(1/2))/(1+ 2*(1-I)^(1/2)+5^(1/2)))+1/2*I*polylog(2,2*((1+I)^(1/2)-(1+x)^(1/2))/(1+...
\[ \int \frac {\log \left (x+\sqrt {1+x}\right )}{1+x^2} \, dx=\int \frac {\log \left (x+\sqrt {1+x}\right )}{1+x^2} \, dx \]
Time = 1.37 (sec) , antiderivative size = 983, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3010, 7292, 3008, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\log \left (x+\sqrt {x+1}\right )}{x^2+1} \, dx\) |
\(\Big \downarrow \) 3010 |
\(\displaystyle 2 \int \frac {\sqrt {x+1} \log \left (x+\sqrt {x+1}\right )}{x^2+1}d\sqrt {x+1}\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle 2 \int \frac {\sqrt {x+1} \log \left (x+\sqrt {x+1}\right )}{(x+1)^2-2 (x+1)+2}d\sqrt {x+1}\) |
\(\Big \downarrow \) 3008 |
\(\displaystyle 2 \int \left (\frac {i \sqrt {x+1} \log \left (x+\sqrt {x+1}\right )}{(2+2 i)-2 (x+1)}+\frac {i \sqrt {x+1} \log \left (x+\sqrt {x+1}\right )}{2 (x+1)-(2-2 i)}\right )d\sqrt {x+1}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\frac {1}{4} i \log \left (\sqrt {1-i}-\sqrt {x+1}\right ) \log \left (x+\sqrt {x+1}\right )-\frac {1}{4} i \log \left (\sqrt {1+i}-\sqrt {x+1}\right ) \log \left (x+\sqrt {x+1}\right )+\frac {1}{4} i \log \left (\sqrt {x+1}+\sqrt {1-i}\right ) \log \left (x+\sqrt {x+1}\right )-\frac {1}{4} i \log \left (\sqrt {x+1}+\sqrt {1+i}\right ) \log \left (x+\sqrt {x+1}\right )-\frac {1}{4} i \log \left (\sqrt {x+1}+\sqrt {1-i}\right ) \log \left (\frac {2 \sqrt {x+1}-\sqrt {5}+1}{1-2 \sqrt {1-i}-\sqrt {5}}\right )-\frac {1}{4} i \log \left (\sqrt {1-i}-\sqrt {x+1}\right ) \log \left (\frac {2 \sqrt {x+1}-\sqrt {5}+1}{1+2 \sqrt {1-i}-\sqrt {5}}\right )+\frac {1}{4} i \log \left (\sqrt {x+1}+\sqrt {1+i}\right ) \log \left (\frac {2 \sqrt {x+1}-\sqrt {5}+1}{1-2 \sqrt {1+i}-\sqrt {5}}\right )+\frac {1}{4} i \log \left (\sqrt {1+i}-\sqrt {x+1}\right ) \log \left (\frac {2 \sqrt {x+1}-\sqrt {5}+1}{1+2 \sqrt {1+i}-\sqrt {5}}\right )-\frac {1}{4} i \log \left (\sqrt {x+1}+\sqrt {1-i}\right ) \log \left (\frac {2 \sqrt {x+1}+\sqrt {5}+1}{1-2 \sqrt {1-i}+\sqrt {5}}\right )-\frac {1}{4} i \log \left (\sqrt {1-i}-\sqrt {x+1}\right ) \log \left (\frac {2 \sqrt {x+1}+\sqrt {5}+1}{1+2 \sqrt {1-i}+\sqrt {5}}\right )+\frac {1}{4} i \log \left (\sqrt {x+1}+\sqrt {1+i}\right ) \log \left (\frac {2 \sqrt {x+1}+\sqrt {5}+1}{1-2 \sqrt {1+i}+\sqrt {5}}\right )+\frac {1}{4} i \log \left (\sqrt {1+i}-\sqrt {x+1}\right ) \log \left (\frac {2 \sqrt {x+1}+\sqrt {5}+1}{1+2 \sqrt {1+i}+\sqrt {5}}\right )-\frac {1}{4} i \operatorname {PolyLog}\left (2,\frac {2 \left (\sqrt {1-i}-\sqrt {x+1}\right )}{1+2 \sqrt {1-i}-\sqrt {5}}\right )-\frac {1}{4} i \operatorname {PolyLog}\left (2,\frac {2 \left (\sqrt {1-i}-\sqrt {x+1}\right )}{1+2 \sqrt {1-i}+\sqrt {5}}\right )+\frac {1}{4} i \operatorname {PolyLog}\left (2,\frac {2 \left (\sqrt {1+i}-\sqrt {x+1}\right )}{1+2 \sqrt {1+i}-\sqrt {5}}\right )+\frac {1}{4} i \operatorname {PolyLog}\left (2,\frac {2 \left (\sqrt {1+i}-\sqrt {x+1}\right )}{1+2 \sqrt {1+i}+\sqrt {5}}\right )-\frac {1}{4} i \operatorname {PolyLog}\left (2,-\frac {2 \left (\sqrt {x+1}+\sqrt {1-i}\right )}{1-2 \sqrt {1-i}-\sqrt {5}}\right )-\frac {1}{4} i \operatorname {PolyLog}\left (2,-\frac {2 \left (\sqrt {x+1}+\sqrt {1-i}\right )}{1-2 \sqrt {1-i}+\sqrt {5}}\right )+\frac {1}{4} i \operatorname {PolyLog}\left (2,-\frac {2 \left (\sqrt {x+1}+\sqrt {1+i}\right )}{1-2 \sqrt {1+i}-\sqrt {5}}\right )+\frac {1}{4} i \operatorname {PolyLog}\left (2,-\frac {2 \left (\sqrt {x+1}+\sqrt {1+i}\right )}{1-2 \sqrt {1+i}+\sqrt {5}}\right )\right )\) |
2*((I/4)*Log[Sqrt[1 - I] - Sqrt[1 + x]]*Log[x + Sqrt[1 + x]] - (I/4)*Log[S qrt[1 + I] - Sqrt[1 + x]]*Log[x + Sqrt[1 + x]] + (I/4)*Log[Sqrt[1 - I] + S qrt[1 + x]]*Log[x + Sqrt[1 + x]] - (I/4)*Log[Sqrt[1 + I] + Sqrt[1 + x]]*Lo g[x + Sqrt[1 + x]] - (I/4)*Log[Sqrt[1 - I] + Sqrt[1 + x]]*Log[(1 - Sqrt[5] + 2*Sqrt[1 + x])/(1 - 2*Sqrt[1 - I] - Sqrt[5])] - (I/4)*Log[Sqrt[1 - I] - Sqrt[1 + x]]*Log[(1 - Sqrt[5] + 2*Sqrt[1 + x])/(1 + 2*Sqrt[1 - I] - Sqrt[ 5])] + (I/4)*Log[Sqrt[1 + I] + Sqrt[1 + x]]*Log[(1 - Sqrt[5] + 2*Sqrt[1 + x])/(1 - 2*Sqrt[1 + I] - Sqrt[5])] + (I/4)*Log[Sqrt[1 + I] - Sqrt[1 + x]]* Log[(1 - Sqrt[5] + 2*Sqrt[1 + x])/(1 + 2*Sqrt[1 + I] - Sqrt[5])] - (I/4)*L og[Sqrt[1 - I] + Sqrt[1 + x]]*Log[(1 + Sqrt[5] + 2*Sqrt[1 + x])/(1 - 2*Sqr t[1 - I] + Sqrt[5])] - (I/4)*Log[Sqrt[1 - I] - Sqrt[1 + x]]*Log[(1 + Sqrt[ 5] + 2*Sqrt[1 + x])/(1 + 2*Sqrt[1 - I] + Sqrt[5])] + (I/4)*Log[Sqrt[1 + I] + Sqrt[1 + x]]*Log[(1 + Sqrt[5] + 2*Sqrt[1 + x])/(1 - 2*Sqrt[1 + I] + Sqr t[5])] + (I/4)*Log[Sqrt[1 + I] - Sqrt[1 + x]]*Log[(1 + Sqrt[5] + 2*Sqrt[1 + x])/(1 + 2*Sqrt[1 + I] + Sqrt[5])] - (I/4)*PolyLog[2, (2*(Sqrt[1 - I] - Sqrt[1 + x]))/(1 + 2*Sqrt[1 - I] - Sqrt[5])] - (I/4)*PolyLog[2, (2*(Sqrt[1 - I] - Sqrt[1 + x]))/(1 + 2*Sqrt[1 - I] + Sqrt[5])] + (I/4)*PolyLog[2, (2 *(Sqrt[1 + I] - Sqrt[1 + x]))/(1 + 2*Sqrt[1 + I] - Sqrt[5])] + (I/4)*PolyL og[2, (2*(Sqrt[1 + I] - Sqrt[1 + x]))/(1 + 2*Sqrt[1 + I] + Sqrt[5])] - (I/ 4)*PolyLog[2, (-2*(Sqrt[1 - I] + Sqrt[1 + x]))/(1 - 2*Sqrt[1 - I] - Sqr...
3.1.30.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With [{u = ExpandIntegrand[(a + b*Log[c*RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u ]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalFuncti onQ[RGx, x] && IGtQ[n, 0]
Int[((a_.) + Log[u_]*(b_.))*(RFx_), x_Symbol] :> With[{lst = SubstForFracti onalPowerOfLinear[RFx*(a + b*Log[u]), x]}, Simp[lst[[2]]*lst[[4]] Subst[I nt[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x] /; !FalseQ[lst]] /; FreeQ[{ a, b}, x] && RationalFunctionQ[RFx, x]
Time = 0.14 (sec) , antiderivative size = 722, normalized size of antiderivative = 0.74
method | result | size |
derivativedivides | \(\frac {i \left (\ln \left (\sqrt {1+x}-\sqrt {1-i}\right ) \ln \left (x +\sqrt {1+x}\right )-\ln \left (\sqrt {1+x}-\sqrt {1-i}\right ) \ln \left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1-i}-\sqrt {5}}\right )-\ln \left (\sqrt {1+x}-\sqrt {1-i}\right ) \ln \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1-i}+\sqrt {5}}\right )-\operatorname {dilog}\left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1-i}-\sqrt {5}}\right )-\operatorname {dilog}\left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1-i}+\sqrt {5}}\right )\right )}{2}+\frac {i \left (\ln \left (\sqrt {1-i}+\sqrt {1+x}\right ) \ln \left (x +\sqrt {1+x}\right )-\ln \left (\sqrt {1-i}+\sqrt {1+x}\right ) \ln \left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1-i}-\sqrt {5}}\right )-\ln \left (\sqrt {1-i}+\sqrt {1+x}\right ) \ln \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1-i}+\sqrt {5}}\right )-\operatorname {dilog}\left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1-i}-\sqrt {5}}\right )-\operatorname {dilog}\left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1-i}+\sqrt {5}}\right )\right )}{2}-\frac {i \left (\ln \left (\sqrt {1+x}-\sqrt {1+i}\right ) \ln \left (x +\sqrt {1+x}\right )-\ln \left (\sqrt {1+x}-\sqrt {1+i}\right ) \ln \left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1+i}-\sqrt {5}}\right )-\ln \left (\sqrt {1+x}-\sqrt {1+i}\right ) \ln \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1+i}+\sqrt {5}}\right )-\operatorname {dilog}\left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1+i}-\sqrt {5}}\right )-\operatorname {dilog}\left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1+i}+\sqrt {5}}\right )\right )}{2}-\frac {i \left (\ln \left (\sqrt {1+i}+\sqrt {1+x}\right ) \ln \left (x +\sqrt {1+x}\right )-\ln \left (\sqrt {1+i}+\sqrt {1+x}\right ) \ln \left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1+i}-\sqrt {5}}\right )-\ln \left (\sqrt {1+i}+\sqrt {1+x}\right ) \ln \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1+i}+\sqrt {5}}\right )-\operatorname {dilog}\left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1+i}-\sqrt {5}}\right )-\operatorname {dilog}\left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1+i}+\sqrt {5}}\right )\right )}{2}\) | \(722\) |
default | \(\frac {i \left (\ln \left (\sqrt {1+x}-\sqrt {1-i}\right ) \ln \left (x +\sqrt {1+x}\right )-\ln \left (\sqrt {1+x}-\sqrt {1-i}\right ) \ln \left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1-i}-\sqrt {5}}\right )-\ln \left (\sqrt {1+x}-\sqrt {1-i}\right ) \ln \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1-i}+\sqrt {5}}\right )-\operatorname {dilog}\left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1-i}-\sqrt {5}}\right )-\operatorname {dilog}\left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1-i}+\sqrt {5}}\right )\right )}{2}+\frac {i \left (\ln \left (\sqrt {1-i}+\sqrt {1+x}\right ) \ln \left (x +\sqrt {1+x}\right )-\ln \left (\sqrt {1-i}+\sqrt {1+x}\right ) \ln \left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1-i}-\sqrt {5}}\right )-\ln \left (\sqrt {1-i}+\sqrt {1+x}\right ) \ln \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1-i}+\sqrt {5}}\right )-\operatorname {dilog}\left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1-i}-\sqrt {5}}\right )-\operatorname {dilog}\left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1-i}+\sqrt {5}}\right )\right )}{2}-\frac {i \left (\ln \left (\sqrt {1+x}-\sqrt {1+i}\right ) \ln \left (x +\sqrt {1+x}\right )-\ln \left (\sqrt {1+x}-\sqrt {1+i}\right ) \ln \left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1+i}-\sqrt {5}}\right )-\ln \left (\sqrt {1+x}-\sqrt {1+i}\right ) \ln \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1+i}+\sqrt {5}}\right )-\operatorname {dilog}\left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1+i}-\sqrt {5}}\right )-\operatorname {dilog}\left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1+i}+\sqrt {5}}\right )\right )}{2}-\frac {i \left (\ln \left (\sqrt {1+i}+\sqrt {1+x}\right ) \ln \left (x +\sqrt {1+x}\right )-\ln \left (\sqrt {1+i}+\sqrt {1+x}\right ) \ln \left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1+i}-\sqrt {5}}\right )-\ln \left (\sqrt {1+i}+\sqrt {1+x}\right ) \ln \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1+i}+\sqrt {5}}\right )-\operatorname {dilog}\left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1+i}-\sqrt {5}}\right )-\operatorname {dilog}\left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1+i}+\sqrt {5}}\right )\right )}{2}\) | \(722\) |
parts | \(\frac {i \left (\ln \left (\sqrt {1+x}-\sqrt {1-i}\right ) \ln \left (x +\sqrt {1+x}\right )-\ln \left (\sqrt {1+x}-\sqrt {1-i}\right ) \ln \left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1-i}-\sqrt {5}}\right )-\ln \left (\sqrt {1+x}-\sqrt {1-i}\right ) \ln \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1-i}+\sqrt {5}}\right )-\operatorname {dilog}\left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1-i}-\sqrt {5}}\right )-\operatorname {dilog}\left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1-i}+\sqrt {5}}\right )\right )}{2}+\frac {i \left (\ln \left (\sqrt {1-i}+\sqrt {1+x}\right ) \ln \left (x +\sqrt {1+x}\right )-\ln \left (\sqrt {1-i}+\sqrt {1+x}\right ) \ln \left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1-i}-\sqrt {5}}\right )-\ln \left (\sqrt {1-i}+\sqrt {1+x}\right ) \ln \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1-i}+\sqrt {5}}\right )-\operatorname {dilog}\left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1-i}-\sqrt {5}}\right )-\operatorname {dilog}\left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1-i}+\sqrt {5}}\right )\right )}{2}-\frac {i \left (\ln \left (\sqrt {1+x}-\sqrt {1+i}\right ) \ln \left (x +\sqrt {1+x}\right )-\ln \left (\sqrt {1+x}-\sqrt {1+i}\right ) \ln \left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1+i}-\sqrt {5}}\right )-\ln \left (\sqrt {1+x}-\sqrt {1+i}\right ) \ln \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1+i}+\sqrt {5}}\right )-\operatorname {dilog}\left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1+i}-\sqrt {5}}\right )-\operatorname {dilog}\left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1+i}+\sqrt {5}}\right )\right )}{2}-\frac {i \left (\ln \left (\sqrt {1+i}+\sqrt {1+x}\right ) \ln \left (x +\sqrt {1+x}\right )-\ln \left (\sqrt {1+i}+\sqrt {1+x}\right ) \ln \left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1+i}-\sqrt {5}}\right )-\ln \left (\sqrt {1+i}+\sqrt {1+x}\right ) \ln \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1+i}+\sqrt {5}}\right )-\operatorname {dilog}\left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1+i}-\sqrt {5}}\right )-\operatorname {dilog}\left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1+i}+\sqrt {5}}\right )\right )}{2}\) | \(722\) |
1/2*I*(ln((1+x)^(1/2)-(1-I)^(1/2))*ln(x+(1+x)^(1/2))-ln((1+x)^(1/2)-(1-I)^ (1/2))*ln((1-5^(1/2)+2*(1+x)^(1/2))/(1+2*(1-I)^(1/2)-5^(1/2)))-ln((1+x)^(1 /2)-(1-I)^(1/2))*ln((1+5^(1/2)+2*(1+x)^(1/2))/(1+2*(1-I)^(1/2)+5^(1/2)))-d ilog((1-5^(1/2)+2*(1+x)^(1/2))/(1+2*(1-I)^(1/2)-5^(1/2)))-dilog((1+5^(1/2) +2*(1+x)^(1/2))/(1+2*(1-I)^(1/2)+5^(1/2))))+1/2*I*(ln((1-I)^(1/2)+(1+x)^(1 /2))*ln(x+(1+x)^(1/2))-ln((1-I)^(1/2)+(1+x)^(1/2))*ln((1-5^(1/2)+2*(1+x)^( 1/2))/(1-2*(1-I)^(1/2)-5^(1/2)))-ln((1-I)^(1/2)+(1+x)^(1/2))*ln((1+5^(1/2) +2*(1+x)^(1/2))/(1-2*(1-I)^(1/2)+5^(1/2)))-dilog((1-5^(1/2)+2*(1+x)^(1/2)) /(1-2*(1-I)^(1/2)-5^(1/2)))-dilog((1+5^(1/2)+2*(1+x)^(1/2))/(1-2*(1-I)^(1/ 2)+5^(1/2))))-1/2*I*(ln((1+x)^(1/2)-(1+I)^(1/2))*ln(x+(1+x)^(1/2))-ln((1+x )^(1/2)-(1+I)^(1/2))*ln((1-5^(1/2)+2*(1+x)^(1/2))/(1+2*(1+I)^(1/2)-5^(1/2) ))-ln((1+x)^(1/2)-(1+I)^(1/2))*ln((1+5^(1/2)+2*(1+x)^(1/2))/(1+2*(1+I)^(1/ 2)+5^(1/2)))-dilog((1-5^(1/2)+2*(1+x)^(1/2))/(1+2*(1+I)^(1/2)-5^(1/2)))-di log((1+5^(1/2)+2*(1+x)^(1/2))/(1+2*(1+I)^(1/2)+5^(1/2))))-1/2*I*(ln((1+I)^ (1/2)+(1+x)^(1/2))*ln(x+(1+x)^(1/2))-ln((1+I)^(1/2)+(1+x)^(1/2))*ln((1-5^( 1/2)+2*(1+x)^(1/2))/(1-2*(1+I)^(1/2)-5^(1/2)))-ln((1+I)^(1/2)+(1+x)^(1/2)) *ln((1+5^(1/2)+2*(1+x)^(1/2))/(1-2*(1+I)^(1/2)+5^(1/2)))-dilog((1-5^(1/2)+ 2*(1+x)^(1/2))/(1-2*(1+I)^(1/2)-5^(1/2)))-dilog((1+5^(1/2)+2*(1+x)^(1/2))/ (1-2*(1+I)^(1/2)+5^(1/2))))
\[ \int \frac {\log \left (x+\sqrt {1+x}\right )}{1+x^2} \, dx=\int { \frac {\log \left (x + \sqrt {x + 1}\right )}{x^{2} + 1} \,d x } \]
\[ \int \frac {\log \left (x+\sqrt {1+x}\right )}{1+x^2} \, dx=\int \frac {\log {\left (x + \sqrt {x + 1} \right )}}{x^{2} + 1}\, dx \]
\[ \int \frac {\log \left (x+\sqrt {1+x}\right )}{1+x^2} \, dx=\int { \frac {\log \left (x + \sqrt {x + 1}\right )}{x^{2} + 1} \,d x } \]
\[ \int \frac {\log \left (x+\sqrt {1+x}\right )}{1+x^2} \, dx=\int { \frac {\log \left (x + \sqrt {x + 1}\right )}{x^{2} + 1} \,d x } \]
Timed out. \[ \int \frac {\log \left (x+\sqrt {1+x}\right )}{1+x^2} \, dx=\int \frac {\ln \left (x+\sqrt {x+1}\right )}{x^2+1} \,d x \]
\[ \int \frac {\log \left (x+\sqrt {1+x}\right )}{1+x^2} \, dx=\int \frac {\mathrm {log}\left (\sqrt {x +1}+x \right )}{x^{2}+1}d x \]