Integrand size = 18, antiderivative size = 555 \[ \int \frac {\log ^2\left (x+\sqrt {1+x}\right )}{(1+x)^2} \, dx=\log (1+x)+\frac {2 \log \left (x+\sqrt {1+x}\right )}{\sqrt {1+x}}-6 \log \left (\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\frac {\log ^2\left (x+\sqrt {1+x}\right )}{1+x}-\left (1+\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+x}\right )+6 \log \left (\frac {1}{2} \left (-1+\sqrt {5}\right )\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+x}\right )+\left (3+\sqrt {5}\right ) \log \left (x+\sqrt {1+x}\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+x}\right )-\frac {1}{2} \left (3+\sqrt {5}\right ) \log ^2\left (1-\sqrt {5}+2 \sqrt {1+x}\right )-\left (1-\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+x}\right )+\left (3-\sqrt {5}\right ) \log \left (x+\sqrt {1+x}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+x}\right )-\left (3-\sqrt {5}\right ) \log \left (-\frac {1-\sqrt {5}+2 \sqrt {1+x}}{2 \sqrt {5}}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+x}\right )-\frac {1}{2} \left (3-\sqrt {5}\right ) \log ^2\left (1+\sqrt {5}+2 \sqrt {1+x}\right )-\left (3+\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+x}\right ) \log \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{2 \sqrt {5}}\right )+6 \log \left (\sqrt {1+x}\right ) \log \left (1+\frac {2 \sqrt {1+x}}{1+\sqrt {5}}\right )+6 \operatorname {PolyLog}\left (2,-\frac {2 \sqrt {1+x}}{1+\sqrt {5}}\right )-\left (3+\sqrt {5}\right ) \operatorname {PolyLog}\left (2,-\frac {1-\sqrt {5}+2 \sqrt {1+x}}{2 \sqrt {5}}\right )-\left (3-\sqrt {5}\right ) \operatorname {PolyLog}\left (2,\frac {1+\sqrt {5}+2 \sqrt {1+x}}{2 \sqrt {5}}\right )-6 \operatorname {PolyLog}\left (2,1+\frac {2 \sqrt {1+x}}{1-\sqrt {5}}\right ) \]
ln(1+x)-3*ln(1+x)*ln(x+(1+x)^(1/2))-ln(x+(1+x)^(1/2))^2/(1+x)+6*ln(1/2*5^( 1/2)-1/2)*ln(1-5^(1/2)+2*(1+x)^(1/2))+3*ln(1+x)*ln(1+2*(1+x)^(1/2)/(5^(1/2 )+1))+6*polylog(2,-2*(1+x)^(1/2)/(5^(1/2)+1))-6*polylog(2,1+2*(1+x)^(1/2)/ (-5^(1/2)+1))-ln(1+5^(1/2)+2*(1+x)^(1/2))*(-5^(1/2)+1)+ln(x+(1+x)^(1/2))*l n(1+5^(1/2)+2*(1+x)^(1/2))*(3-5^(1/2))-ln(1/10*(-1+5^(1/2)-2*(1+x)^(1/2))* 5^(1/2))*ln(1+5^(1/2)+2*(1+x)^(1/2))*(3-5^(1/2))-1/2*ln(1+5^(1/2)+2*(1+x)^ (1/2))^2*(3-5^(1/2))-polylog(2,1/10*(1+5^(1/2)+2*(1+x)^(1/2))*5^(1/2))*(3- 5^(1/2))-ln(1-5^(1/2)+2*(1+x)^(1/2))*(5^(1/2)+1)+ln(x+(1+x)^(1/2))*ln(1-5^ (1/2)+2*(1+x)^(1/2))*(3+5^(1/2))-1/2*ln(1-5^(1/2)+2*(1+x)^(1/2))^2*(3+5^(1 /2))-ln(1-5^(1/2)+2*(1+x)^(1/2))*ln(1/10*(1+5^(1/2)+2*(1+x)^(1/2))*5^(1/2) )*(3+5^(1/2))-polylog(2,1/10*(-1+5^(1/2)-2*(1+x)^(1/2))*5^(1/2))*(3+5^(1/2 ))+2*ln(x+(1+x)^(1/2))/(1+x)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(1280\) vs. \(2(555)=1110\).
Time = 6.90 (sec) , antiderivative size = 1280, normalized size of antiderivative = 2.31 \[ \int \frac {\log ^2\left (x+\sqrt {1+x}\right )}{(1+x)^2} \, dx =\text {Too large to display} \]
(2*Log[1 + x])/(-1 + Sqrt[5]) - (2*Log[1 + x])/(1 + Sqrt[5]) - (4*Log[-1 + Sqrt[5] - 2*Sqrt[1 + x]])/(-1 + Sqrt[5]) + (Log[100]*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]])/Sqrt[5] - 6*Log[(2*Sqrt[1 + x])/(-1 + Sqrt[5])]*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]] + 3*Log[1 + x]*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]] - 3*Log[-1 + Sqrt[5] - 2*Sqrt[1 + x]]*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]] - Sqrt[5]*Log[-1 + Sqrt[5] - 2*Sqrt[1 + x]]*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]] + (3*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]]^2)/2 + (Sqrt[5]*Log[1/2 - Sqr t[5]/2 + Sqrt[1 + x]]^2)/2 + (Log[8]*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]])/( 2*Sqrt[5]) - 3*Log[-1 + Sqrt[5] - 2*Sqrt[1 + x]]*Log[(1 + Sqrt[5])/2 + Sqr t[1 + x]] - Sqrt[5]*Log[-1 + Sqrt[5] - 2*Sqrt[1 + x]]*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]] + (3*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]]^2)/2 - Log[(1 + Sqr t[5])/2 + Sqrt[1 + x]]^2/Sqrt[5] + (2*Log[x + Sqrt[1 + x]])/Sqrt[1 + x] - 3*Log[1 + x]*Log[x + Sqrt[1 + x]] + 3*Log[-1 + Sqrt[5] - 2*Sqrt[1 + x]]*Lo g[x + Sqrt[1 + x]] + Sqrt[5]*Log[-1 + Sqrt[5] - 2*Sqrt[1 + x]]*Log[x + Sqr t[1 + x]] - Log[x + Sqrt[1 + x]]^2/(1 + x) + (4*Log[1 + Sqrt[5] + 2*Sqrt[1 + x]])/(1 + Sqrt[5]) - 3*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]]*Log[1 + Sqrt[ 5] + 2*Sqrt[1 + x]] + Sqrt[5]*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]]*Log[1 + S qrt[5] + 2*Sqrt[1 + x]] - 3*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]]*Log[1 + Sqr t[5] + 2*Sqrt[1 + x]] + (7*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]]*Log[1 + Sqrt [5] + 2*Sqrt[1 + x]])/(2*Sqrt[5]) + 3*Log[x + Sqrt[1 + x]]*Log[1 + Sqrt...
Time = 0.95 (sec) , antiderivative size = 580, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {7267, 3005, 25, 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.
\(\displaystyle \int \frac {\log ^2\left (x+\sqrt {x+1}\right )}{(x+1)^2} \, dx\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle 2 \int \frac {\log ^2\left (x+\sqrt {x+1}\right )}{(x+1)^{3/2}}d\sqrt {x+1}\) |
\(\Big \downarrow \) 3005 |
\(\displaystyle 2 \left (\int -\frac {\left (2 \sqrt {x+1}+1\right ) \log \left (x+\sqrt {x+1}\right )}{(x+1) \left (-x-\sqrt {x+1}\right )}d\sqrt {x+1}-\frac {\log ^2\left (x+\sqrt {x+1}\right )}{2 (x+1)}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle 2 \left (-\int \frac {\left (2 \sqrt {x+1}+1\right ) \log \left (x+\sqrt {x+1}\right )}{(x+1) \left (-x-\sqrt {x+1}\right )}d\sqrt {x+1}-\frac {\log ^2\left (x+\sqrt {x+1}\right )}{2 (x+1)}\right )\) |
\(\Big \downarrow \) 3008 |
\(\displaystyle 2 \left (-\int \left (\frac {3 \log \left (x+\sqrt {x+1}\right )}{\sqrt {x+1}}+\frac {\log \left (x+\sqrt {x+1}\right )}{x+1}+\frac {\left (-3 \sqrt {x+1}-4\right ) \log \left (x+\sqrt {x+1}\right )}{x+\sqrt {x+1}}\right )d\sqrt {x+1}-\frac {\log ^2\left (x+\sqrt {x+1}\right )}{2 (x+1)}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (3 \operatorname {PolyLog}\left (2,-\frac {2 \sqrt {x+1}}{1+\sqrt {5}}\right )-\frac {1}{2} \left (3+\sqrt {5}\right ) \operatorname {PolyLog}\left (2,-\frac {2 \sqrt {x+1}-\sqrt {5}+1}{2 \sqrt {5}}\right )-\frac {1}{2} \left (3-\sqrt {5}\right ) \operatorname {PolyLog}\left (2,\frac {2 \sqrt {x+1}+\sqrt {5}+1}{2 \sqrt {5}}\right )-3 \operatorname {PolyLog}\left (2,\frac {2 \sqrt {x+1}}{1-\sqrt {5}}+1\right )-\frac {\log ^2\left (x+\sqrt {x+1}\right )}{2 (x+1)}-\frac {1}{4} \left (3+\sqrt {5}\right ) \log ^2\left (2 \sqrt {x+1}-\sqrt {5}+1\right )-\frac {1}{4} \left (3-\sqrt {5}\right ) \log ^2\left (2 \sqrt {x+1}+\sqrt {5}+1\right )-3 \log \left (\sqrt {x+1}\right ) \log \left (x+\sqrt {x+1}\right )+\frac {1}{2} \left (3+\sqrt {5}\right ) \log \left (2 \sqrt {x+1}-\sqrt {5}+1\right ) \log \left (x+\sqrt {x+1}\right )+\frac {1}{2} \left (3-\sqrt {5}\right ) \log \left (2 \sqrt {x+1}+\sqrt {5}+1\right ) \log \left (x+\sqrt {x+1}\right )+\frac {\log \left (x+\sqrt {x+1}\right )}{\sqrt {x+1}}+\log \left (\sqrt {x+1}\right )+3 \log \left (\frac {1}{2} \left (\sqrt {5}-1\right )\right ) \log \left (2 \sqrt {x+1}-\sqrt {5}+1\right )-\frac {1}{2} \left (1+\sqrt {5}\right ) \log \left (2 \sqrt {x+1}-\sqrt {5}+1\right )-\frac {1}{2} \left (3-\sqrt {5}\right ) \log \left (-\frac {2 \sqrt {x+1}-\sqrt {5}+1}{2 \sqrt {5}}\right ) \log \left (2 \sqrt {x+1}+\sqrt {5}+1\right )-\frac {1}{2} \left (1-\sqrt {5}\right ) \log \left (2 \sqrt {x+1}+\sqrt {5}+1\right )-\frac {1}{2} \left (3+\sqrt {5}\right ) \log \left (2 \sqrt {x+1}-\sqrt {5}+1\right ) \log \left (\frac {2 \sqrt {x+1}+\sqrt {5}+1}{2 \sqrt {5}}\right )+3 \log \left (\sqrt {x+1}\right ) \log \left (\frac {2 \sqrt {x+1}}{1+\sqrt {5}}+1\right )\right )\) |
2*(Log[Sqrt[1 + x]] + Log[x + Sqrt[1 + x]]/Sqrt[1 + x] - 3*Log[Sqrt[1 + x] ]*Log[x + Sqrt[1 + x]] - Log[x + Sqrt[1 + x]]^2/(2*(1 + x)) - ((1 + Sqrt[5 ])*Log[1 - Sqrt[5] + 2*Sqrt[1 + x]])/2 + 3*Log[(-1 + Sqrt[5])/2]*Log[1 - S qrt[5] + 2*Sqrt[1 + x]] + ((3 + Sqrt[5])*Log[x + Sqrt[1 + x]]*Log[1 - Sqrt [5] + 2*Sqrt[1 + x]])/2 - ((3 + Sqrt[5])*Log[1 - Sqrt[5] + 2*Sqrt[1 + x]]^ 2)/4 - ((1 - Sqrt[5])*Log[1 + Sqrt[5] + 2*Sqrt[1 + x]])/2 + ((3 - Sqrt[5]) *Log[x + Sqrt[1 + x]]*Log[1 + Sqrt[5] + 2*Sqrt[1 + x]])/2 - ((3 - Sqrt[5]) *Log[-1/2*(1 - Sqrt[5] + 2*Sqrt[1 + x])/Sqrt[5]]*Log[1 + Sqrt[5] + 2*Sqrt[ 1 + x]])/2 - ((3 - Sqrt[5])*Log[1 + Sqrt[5] + 2*Sqrt[1 + x]]^2)/4 - ((3 + Sqrt[5])*Log[1 - Sqrt[5] + 2*Sqrt[1 + x]]*Log[(1 + Sqrt[5] + 2*Sqrt[1 + x] )/(2*Sqrt[5])])/2 + 3*Log[Sqrt[1 + x]]*Log[1 + (2*Sqrt[1 + x])/(1 + Sqrt[5 ])] + 3*PolyLog[2, (-2*Sqrt[1 + x])/(1 + Sqrt[5])] - ((3 + Sqrt[5])*PolyLo g[2, -1/2*(1 - Sqrt[5] + 2*Sqrt[1 + x])/Sqrt[5]])/2 - ((3 - Sqrt[5])*PolyL og[2, (1 + Sqrt[5] + 2*Sqrt[1 + x])/(2*Sqrt[5])])/2 - 3*PolyLog[2, 1 + (2* Sqrt[1 + x])/(1 - Sqrt[5])])
3.1.31.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_. ), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((a + b*Log[c*RFx^p])^n/(e*(m + 1))) , x] - Simp[b*n*(p/(e*(m + 1))) Int[SimplifyIntegrand[(d + e*x)^(m + 1)*( a + b*Log[c*RFx^p])^(n - 1)*(D[RFx, x]/RFx), x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]
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[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
\[\int \frac {\ln \left (x +\sqrt {1+x}\right )^{2}}{\left (1+x \right )^{2}}d x\]
\[ \int \frac {\log ^2\left (x+\sqrt {1+x}\right )}{(1+x)^2} \, dx=\int { \frac {\log \left (x + \sqrt {x + 1}\right )^{2}}{{\left (x + 1\right )}^{2}} \,d x } \]
\[ \int \frac {\log ^2\left (x+\sqrt {1+x}\right )}{(1+x)^2} \, dx=\int \frac {\log {\left (x + \sqrt {x + 1} \right )}^{2}}{\left (x + 1\right )^{2}}\, dx \]
\[ \int \frac {\log ^2\left (x+\sqrt {1+x}\right )}{(1+x)^2} \, dx=\int { \frac {\log \left (x + \sqrt {x + 1}\right )^{2}}{{\left (x + 1\right )}^{2}} \,d x } \]
-log(x + sqrt(x + 1))^2/(x + 1) + integrate((2*x + sqrt(x + 1) + 2)*log(x + sqrt(x + 1))/(x^3 + 2*x^2 + (x^2 + 2*x + 1)*sqrt(x + 1) + x), x)
\[ \int \frac {\log ^2\left (x+\sqrt {1+x}\right )}{(1+x)^2} \, dx=\int { \frac {\log \left (x + \sqrt {x + 1}\right )^{2}}{{\left (x + 1\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\log ^2\left (x+\sqrt {1+x}\right )}{(1+x)^2} \, dx=\int \frac {{\ln \left (x+\sqrt {x+1}\right )}^2}{{\left (x+1\right )}^2} \,d x \]
\[ \int \frac {\log ^2\left (x+\sqrt {1+x}\right )}{(1+x)^2} \, dx=\int \frac {\mathrm {log}\left (\sqrt {x +1}+x \right )^{2}}{x^{2}+2 x +1}d x \]