\(\int \frac {\log ^2(x+\sqrt {1+x})}{(1+x)^2} \, dx\) [31]

Optimal result

Integrand size = 18, antiderivative size = 555 \[ \int \frac {\log ^2\left (x+\sqrt {1+x}\right )}{(1+x)^2} \, dx=\log (1+x)+\frac {2 \log \left (x+\sqrt {1+x}\right )}{\sqrt {1+x}}-6 \log \left (\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\frac {\log ^2\left (x+\sqrt {1+x}\right )}{1+x}-\left (1+\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+x}\right )+6 \log \left (\frac {1}{2} \left (-1+\sqrt {5}\right )\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+x}\right )+\left (3+\sqrt {5}\right ) \log \left (x+\sqrt {1+x}\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+x}\right )-\frac {1}{2} \left (3+\sqrt {5}\right ) \log ^2\left (1-\sqrt {5}+2 \sqrt {1+x}\right )-\left (1-\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+x}\right )+\left (3-\sqrt {5}\right ) \log \left (x+\sqrt {1+x}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+x}\right )-\left (3-\sqrt {5}\right ) \log \left (-\frac {1-\sqrt {5}+2 \sqrt {1+x}}{2 \sqrt {5}}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+x}\right )-\frac {1}{2} \left (3-\sqrt {5}\right ) \log ^2\left (1+\sqrt {5}+2 \sqrt {1+x}\right )-\left (3+\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+x}\right ) \log \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{2 \sqrt {5}}\right )+6 \log \left (\sqrt {1+x}\right ) \log \left (1+\frac {2 \sqrt {1+x}}{1+\sqrt {5}}\right )+6 \operatorname {PolyLog}\left (2,-\frac {2 \sqrt {1+x}}{1+\sqrt {5}}\right )-\left (3+\sqrt {5}\right ) \operatorname {PolyLog}\left (2,-\frac {1-\sqrt {5}+2 \sqrt {1+x}}{2 \sqrt {5}}\right )-\left (3-\sqrt {5}\right ) \operatorname {PolyLog}\left (2,\frac {1+\sqrt {5}+2 \sqrt {1+x}}{2 \sqrt {5}}\right )-6 \operatorname {PolyLog}\left (2,1+\frac {2 \sqrt {1+x}}{1-\sqrt {5}}\right ) \]

[Out]

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))*ln(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)

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 555, normalized size of antiderivative = 1.00, number of steps used = 35, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, Rules used = {2605, 2608, 814, 646, 31, 2604, 2404, 2353, 2352, 2354, 2438, 2465, 2437, 2338, 2441, 2440} \[ \int \frac {\log ^2\left (x+\sqrt {1+x}\right )}{(1+x)^2} \, dx=6 \operatorname {PolyLog}\left (2,-\frac {2 \sqrt {x+1}}{1+\sqrt {5}}\right )-\left (3+\sqrt {5}\right ) \operatorname {PolyLog}\left (2,-\frac {2 \sqrt {x+1}-\sqrt {5}+1}{2 \sqrt {5}}\right )-\left (3-\sqrt {5}\right ) \operatorname {PolyLog}\left (2,\frac {2 \sqrt {x+1}+\sqrt {5}+1}{2 \sqrt {5}}\right )-6 \operatorname {PolyLog}\left (2,\frac {2 \sqrt {x+1}}{1-\sqrt {5}}+1\right )-\frac {\log ^2\left (x+\sqrt {x+1}\right )}{x+1}-\frac {1}{2} \left (3+\sqrt {5}\right ) \log ^2\left (2 \sqrt {x+1}-\sqrt {5}+1\right )-\frac {1}{2} \left (3-\sqrt {5}\right ) \log ^2\left (2 \sqrt {x+1}+\sqrt {5}+1\right )-6 \log \left (\sqrt {x+1}\right ) \log \left (x+\sqrt {x+1}\right )+\left (3+\sqrt {5}\right ) \log \left (2 \sqrt {x+1}-\sqrt {5}+1\right ) \log \left (x+\sqrt {x+1}\right )+\left (3-\sqrt {5}\right ) \log \left (2 \sqrt {x+1}+\sqrt {5}+1\right ) \log \left (x+\sqrt {x+1}\right )+\frac {2 \log \left (x+\sqrt {x+1}\right )}{\sqrt {x+1}}+\log (x+1)+6 \log \left (\frac {1}{2} \left (\sqrt {5}-1\right )\right ) \log \left (2 \sqrt {x+1}-\sqrt {5}+1\right )-\left (1+\sqrt {5}\right ) \log \left (2 \sqrt {x+1}-\sqrt {5}+1\right )-\left (3-\sqrt {5}\right ) \log \left (-\frac {2 \sqrt {x+1}-\sqrt {5}+1}{2 \sqrt {5}}\right ) \log \left (2 \sqrt {x+1}+\sqrt {5}+1\right )-\left (1-\sqrt {5}\right ) \log \left (2 \sqrt {x+1}+\sqrt {5}+1\right )-\left (3+\sqrt {5}\right ) \log \left (2 \sqrt {x+1}-\sqrt {5}+1\right ) \log \left (\frac {2 \sqrt {x+1}+\sqrt {5}+1}{2 \sqrt {5}}\right )+6 \log \left (\sqrt {x+1}\right ) \log \left (\frac {2 \sqrt {x+1}}{1+\sqrt {5}}+1\right ) \]

[In]

Int[Log[x + Sqrt[1 + x]]^2/(1 + x)^2,x]

[Out]

Log[1 + x] + (2*Log[x + Sqrt[1 + x]])/Sqrt[1 + x] - 6*Log[Sqrt[1 + x]]*Log[x + Sqrt[1 + x]] - Log[x + Sqrt[1 +
x]]^2/(1 + x) - (1 + Sqrt[5])*Log[1 - Sqrt[5] + 2*Sqrt[1 + x]] + 6*Log[(-1 + Sqrt[5])/2]*Log[1 - Sqrt[5] + 2*
Sqrt[1 + x]] + (3 + Sqrt[5])*Log[x + Sqrt[1 + x]]*Log[1 - Sqrt[5] + 2*Sqrt[1 + x]] - ((3 + Sqrt[5])*Log[1 - Sq
rt[5] + 2*Sqrt[1 + x]]^2)/2 - (1 - Sqrt[5])*Log[1 + Sqrt[5] + 2*Sqrt[1 + x]] + (3 - Sqrt[5])*Log[x + Sqrt[1 +
x]]*Log[1 + Sqrt[5] + 2*Sqrt[1 + x]] - (3 - Sqrt[5])*Log[-1/2*(1 - Sqrt[5] + 2*Sqrt[1 + x])/Sqrt[5]]*Log[1 + S
qrt[5] + 2*Sqrt[1 + x]] - ((3 - Sqrt[5])*Log[1 + Sqrt[5] + 2*Sqrt[1 + x]]^2)/2 - (3 + Sqrt[5])*Log[1 - Sqrt[5]
+ 2*Sqrt[1 + x]]*Log[(1 + Sqrt[5] + 2*Sqrt[1 + x])/(2*Sqrt[5])] + 6*Log[Sqrt[1 + x]]*Log[1 + (2*Sqrt[1 + x])/
(1 + Sqrt[5])] + 6*PolyLog[2, (-2*Sqrt[1 + x])/(1 + Sqrt[5])] - (3 + Sqrt[5])*PolyLog[2, -1/2*(1 - Sqrt[5] + 2
*Sqrt[1 + x])/Sqrt[5]] - (3 - Sqrt[5])*PolyLog[2, (1 + Sqrt[5] + 2*Sqrt[1 + x])/(2*Sqrt[5])] - 6*PolyLog[2, 1
+ (2*Sqrt[1 + x])/(1 - Sqrt[5])]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 646

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rule 814

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x)^m*((f + g*x)/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2353

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(a + b*Log[(-c)*(d/e)])*(Log[d + e*
x]/e), x] + Dist[b, Int[Log[(-e)*(x/d)]/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[(-c)*(d/e), 0]

Rule 2354

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[Log[1 + e*(x/d)]*((a +
b*Log[c*x^n])^p/e), x] - Dist[b*n*(p/e), Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2404

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rule 2437

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[(f*(x/d))^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2441

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((f + g
*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])/g), x] - Dist[b*e*(n/g), Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2465

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2604

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[Log[d + e*x]*((a + b
*Log[c*RFx^p])^n/e), x] - Dist[b*n*(p/e), Int[Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*(D[RFx, x]/RFx), x], x
] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2605

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] - Dist[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] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 2608

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] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {\log ^2\left (-1+x+x^2\right )}{x^3} \, dx,x,\sqrt {1+x}\right ) \\ & = -\frac {\log ^2\left (x+\sqrt {1+x}\right )}{1+x}+2 \text {Subst}\left (\int \frac {(1+2 x) \log \left (-1+x+x^2\right )}{x^2 \left (-1+x+x^2\right )} \, dx,x,\sqrt {1+x}\right ) \\ & = -\frac {\log ^2\left (x+\sqrt {1+x}\right )}{1+x}+2 \text {Subst}\left (\int \left (-\frac {\log \left (-1+x+x^2\right )}{x^2}-\frac {3 \log \left (-1+x+x^2\right )}{x}+\frac {(4+3 x) \log \left (-1+x+x^2\right )}{-1+x+x^2}\right ) \, dx,x,\sqrt {1+x}\right ) \\ & = -\frac {\log ^2\left (x+\sqrt {1+x}\right )}{1+x}-2 \text {Subst}\left (\int \frac {\log \left (-1+x+x^2\right )}{x^2} \, dx,x,\sqrt {1+x}\right )+2 \text {Subst}\left (\int \frac {(4+3 x) \log \left (-1+x+x^2\right )}{-1+x+x^2} \, dx,x,\sqrt {1+x}\right )-6 \text {Subst}\left (\int \frac {\log \left (-1+x+x^2\right )}{x} \, dx,x,\sqrt {1+x}\right ) \\ & = \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}-2 \text {Subst}\left (\int \frac {1+2 x}{x \left (-1+x+x^2\right )} \, dx,x,\sqrt {1+x}\right )+2 \text {Subst}\left (\int \left (\frac {\left (3+\sqrt {5}\right ) \log \left (-1+x+x^2\right )}{1-\sqrt {5}+2 x}+\frac {\left (3-\sqrt {5}\right ) \log \left (-1+x+x^2\right )}{1+\sqrt {5}+2 x}\right ) \, dx,x,\sqrt {1+x}\right )+6 \text {Subst}\left (\int \frac {(1+2 x) \log (x)}{-1+x+x^2} \, dx,x,\sqrt {1+x}\right ) \\ & = \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}-2 \text {Subst}\left (\int \left (-\frac {1}{x}+\frac {3+x}{-1+x+x^2}\right ) \, dx,x,\sqrt {1+x}\right )+6 \text {Subst}\left (\int \left (\frac {2 \log (x)}{1-\sqrt {5}+2 x}+\frac {2 \log (x)}{1+\sqrt {5}+2 x}\right ) \, dx,x,\sqrt {1+x}\right )+\left (2 \left (3-\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {\log \left (-1+x+x^2\right )}{1+\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )+\left (2 \left (3+\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {\log \left (-1+x+x^2\right )}{1-\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right ) \\ & = \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 (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 (x+\sqrt {1+x}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+x}\right )-2 \text {Subst}\left (\int \frac {3+x}{-1+x+x^2} \, dx,x,\sqrt {1+x}\right )+12 \text {Subst}\left (\int \frac {\log (x)}{1-\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )+12 \text {Subst}\left (\int \frac {\log (x)}{1+\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )+\left (-3-\sqrt {5}\right ) \text {Subst}\left (\int \frac {(1+2 x) \log \left (1-\sqrt {5}+2 x\right )}{-1+x+x^2} \, dx,x,\sqrt {1+x}\right )+\left (-3+\sqrt {5}\right ) \text {Subst}\left (\int \frac {(1+2 x) \log \left (1+\sqrt {5}+2 x\right )}{-1+x+x^2} \, dx,x,\sqrt {1+x}\right ) \\ & = \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}+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 )+\left (3-\sqrt {5}\right ) \log \left (x+\sqrt {1+x}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+x}\right )+6 \log \left (\sqrt {1+x}\right ) \log \left (1+\frac {2 \sqrt {1+x}}{1+\sqrt {5}}\right )-6 \text {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{1+\sqrt {5}}\right )}{x} \, dx,x,\sqrt {1+x}\right )+12 \text {Subst}\left (\int \frac {\log \left (-\frac {2 x}{1-\sqrt {5}}\right )}{1-\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )+\left (-3-\sqrt {5}\right ) \text {Subst}\left (\int \left (\frac {2 \log \left (1-\sqrt {5}+2 x\right )}{1-\sqrt {5}+2 x}+\frac {2 \log \left (1-\sqrt {5}+2 x\right )}{1+\sqrt {5}+2 x}\right ) \, dx,x,\sqrt {1+x}\right )-\left (1-\sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2}+\frac {\sqrt {5}}{2}+x} \, dx,x,\sqrt {1+x}\right )+\left (-3+\sqrt {5}\right ) \text {Subst}\left (\int \left (\frac {2 \log \left (1+\sqrt {5}+2 x\right )}{1-\sqrt {5}+2 x}+\frac {2 \log \left (1+\sqrt {5}+2 x\right )}{1+\sqrt {5}+2 x}\right ) \, dx,x,\sqrt {1+x}\right )-\left (1+\sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2}-\frac {\sqrt {5}}{2}+x} \, dx,x,\sqrt {1+x}\right ) \\ & = \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 )-\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 )+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 )-6 \operatorname {PolyLog}\left (2,1+\frac {2 \sqrt {1+x}}{1-\sqrt {5}}\right )-\left (2 \left (3-\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {\log \left (1+\sqrt {5}+2 x\right )}{1-\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )-\left (2 \left (3-\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {\log \left (1+\sqrt {5}+2 x\right )}{1+\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )-\left (2 \left (3+\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {\log \left (1-\sqrt {5}+2 x\right )}{1-\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )-\left (2 \left (3+\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {\log \left (1-\sqrt {5}+2 x\right )}{1+\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right ) \\ & = \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 )-\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 )-\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 )-6 \operatorname {PolyLog}\left (2,1+\frac {2 \sqrt {1+x}}{1-\sqrt {5}}\right )-\left (3-\sqrt {5}\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1+\sqrt {5}+2 \sqrt {1+x}\right )+\left (2 \left (3-\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2 \left (1-\sqrt {5}+2 x\right )}{2 \left (1-\sqrt {5}\right )-2 \left (1+\sqrt {5}\right )}\right )}{1+\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )-\left (3+\sqrt {5}\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1-\sqrt {5}+2 \sqrt {1+x}\right )+\left (2 \left (3+\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2 \left (1+\sqrt {5}+2 x\right )}{-2 \left (1-\sqrt {5}\right )+2 \left (1+\sqrt {5}\right )}\right )}{1-\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right ) \\ & = \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 )-6 \operatorname {PolyLog}\left (2,1+\frac {2 \sqrt {1+x}}{1-\sqrt {5}}\right )+\left (3-\sqrt {5}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{2 \left (1-\sqrt {5}\right )-2 \left (1+\sqrt {5}\right )}\right )}{x} \, dx,x,1+\sqrt {5}+2 \sqrt {1+x}\right )+\left (3+\sqrt {5}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{-2 \left (1-\sqrt {5}\right )+2 \left (1+\sqrt {5}\right )}\right )}{x} \, dx,x,1-\sqrt {5}+2 \sqrt {1+x}\right ) \\ & = \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 ) \\ \end{align*}

Mathematica [B] (verified)

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=\frac {2 \log (1+x)}{-1+\sqrt {5}}-\frac {2 \log (1+x)}{1+\sqrt {5}}-\frac {4 \log \left (-1+\sqrt {5}-2 \sqrt {1+x}\right )}{-1+\sqrt {5}}+\frac {\log (100) \log \left (\frac {1}{2}-\frac {\sqrt {5}}{2}+\sqrt {1+x}\right )}{\sqrt {5}}-6 \log \left (\frac {2 \sqrt {1+x}}{-1+\sqrt {5}}\right ) \log \left (\frac {1}{2}-\frac {\sqrt {5}}{2}+\sqrt {1+x}\right )+3 \log (1+x) \log \left (\frac {1}{2}-\frac {\sqrt {5}}{2}+\sqrt {1+x}\right )-3 \log \left (-1+\sqrt {5}-2 \sqrt {1+x}\right ) \log \left (\frac {1}{2}-\frac {\sqrt {5}}{2}+\sqrt {1+x}\right )-\sqrt {5} \log \left (-1+\sqrt {5}-2 \sqrt {1+x}\right ) \log \left (\frac {1}{2}-\frac {\sqrt {5}}{2}+\sqrt {1+x}\right )+\frac {3}{2} \log ^2\left (\frac {1}{2}-\frac {\sqrt {5}}{2}+\sqrt {1+x}\right )+\frac {1}{2} \sqrt {5} \log ^2\left (\frac {1}{2}-\frac {\sqrt {5}}{2}+\sqrt {1+x}\right )+\frac {\log (8) \log \left (\frac {1}{2} \left (1+\sqrt {5}\right )+\sqrt {1+x}\right )}{2 \sqrt {5}}-3 \log \left (-1+\sqrt {5}-2 \sqrt {1+x}\right ) \log \left (\frac {1}{2} \left (1+\sqrt {5}\right )+\sqrt {1+x}\right )-\sqrt {5} \log \left (-1+\sqrt {5}-2 \sqrt {1+x}\right ) \log \left (\frac {1}{2} \left (1+\sqrt {5}\right )+\sqrt {1+x}\right )+\frac {3}{2} \log ^2\left (\frac {1}{2} \left (1+\sqrt {5}\right )+\sqrt {1+x}\right )-\frac {\log ^2\left (\frac {1}{2} \left (1+\sqrt {5}\right )+\sqrt {1+x}\right )}{\sqrt {5}}+\frac {2 \log \left (x+\sqrt {1+x}\right )}{\sqrt {1+x}}-3 \log (1+x) \log \left (x+\sqrt {1+x}\right )+3 \log \left (-1+\sqrt {5}-2 \sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )+\sqrt {5} \log \left (-1+\sqrt {5}-2 \sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\frac {\log ^2\left (x+\sqrt {1+x}\right )}{1+x}+\frac {4 \log \left (1+\sqrt {5}+2 \sqrt {1+x}\right )}{1+\sqrt {5}}-3 \log \left (\frac {1}{2}-\frac {\sqrt {5}}{2}+\sqrt {1+x}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+x}\right )+\sqrt {5} \log \left (\frac {1}{2}-\frac {\sqrt {5}}{2}+\sqrt {1+x}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+x}\right )-3 \log \left (\frac {1}{2} \left (1+\sqrt {5}\right )+\sqrt {1+x}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+x}\right )+\frac {7 \log \left (\frac {1}{2} \left (1+\sqrt {5}\right )+\sqrt {1+x}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+x}\right )}{2 \sqrt {5}}+3 \log \left (x+\sqrt {1+x}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+x}\right )-\sqrt {5} \log \left (x+\sqrt {1+x}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+x}\right )+3 \log \left (\frac {1}{2}-\frac {\sqrt {5}}{2}+\sqrt {1+x}\right ) \log \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{2 \sqrt {5}}\right )-\frac {3 \log \left (\frac {1}{2}-\frac {\sqrt {5}}{2}+\sqrt {1+x}\right ) \log \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{2 \sqrt {5}}\right )}{\sqrt {5}}+3 \log \left (\frac {1}{2} \left (1+\sqrt {5}\right )+\sqrt {1+x}\right ) \log \left (\frac {1}{10} \left (5-\sqrt {5}-2 \sqrt {5} \sqrt {1+x}\right )\right )+\sqrt {5} \log \left (\frac {1}{2} \left (1+\sqrt {5}\right )+\sqrt {1+x}\right ) \log \left (\frac {1}{10} \left (5-\sqrt {5}-2 \sqrt {5} \sqrt {1+x}\right )\right )-\frac {2 \log \left (\frac {1}{2}-\frac {\sqrt {5}}{2}+\sqrt {1+x}\right ) \log \left (5+\sqrt {5}+2 \sqrt {5} \sqrt {1+x}\right )}{\sqrt {5}}+3 \log (1+x) \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 )-6 \operatorname {PolyLog}\left (2,\frac {-1+\sqrt {5}-2 \sqrt {1+x}}{-1+\sqrt {5}}\right )+3 \operatorname {PolyLog}\left (2,\frac {1+\sqrt {5}+2 \sqrt {1+x}}{2 \sqrt {5}}\right )+\sqrt {5} \operatorname {PolyLog}\left (2,\frac {1+\sqrt {5}+2 \sqrt {1+x}}{2 \sqrt {5}}\right ) \]

[In]

Integrate[Log[x + Sqrt[1 + x]]^2/(1 + x)^2,x]

[Out]

(2*Log[1 + x])/(-1 + Sqrt[5]) - (2*Log[1 + x])/(1 + Sqrt[5]) - (4*Log[-1 + Sqrt[5] - 2*Sqrt[1 + x]])/(-1 + Sqr
t[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 - Sqrt[5]/2 + Sqrt[1 + x]]^2)/2 + (Lo
g[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
+ Sqrt[1 + x]] - Sqrt[5]*Log[-1 + Sqrt[5] - 2*Sqrt[1 + x]]*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]] + (3*Log[(1 + S
qrt[5])/2 + Sqrt[1 + x]]^2)/2 - Log[(1 + Sqrt[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]]*Log[x + Sqrt[1 + x]] + Sqrt[5]
*Log[-1 + Sqrt[5] - 2*Sqrt[1 + x]]*Log[x + Sqrt[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 + Sqrt[5] + 2*Sqrt[1 + x]] - 3*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]]
*Log[1 + Sqrt[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[5] + 2*Sqrt[1 + x]] - Sqrt[5]*Log[x + Sqrt[1 + x]]*Log[1 + Sqr
t[5] + 2*Sqrt[1 + x]] + 3*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]]*Log[(1 + Sqrt[5] + 2*Sqrt[1 + x])/(2*Sqrt[5])] -
(3*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]]*Log[(1 + Sqrt[5] + 2*Sqrt[1 + x])/(2*Sqrt[5])])/Sqrt[5] + 3*Log[(1 + Sqr
t[5])/2 + Sqrt[1 + x]]*Log[(5 - Sqrt[5] - 2*Sqrt[5]*Sqrt[1 + x])/10] + Sqrt[5]*Log[(1 + Sqrt[5])/2 + Sqrt[1 +
x]]*Log[(5 - Sqrt[5] - 2*Sqrt[5]*Sqrt[1 + x])/10] - (2*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]]*Log[5 + Sqrt[5] + 2*
Sqrt[5]*Sqrt[1 + x]])/Sqrt[5] + 3*Log[1 + x]*Log[1 + (2*Sqrt[1 + x])/(1 + Sqrt[5])] + 6*PolyLog[2, (-2*Sqrt[1
+ x])/(1 + Sqrt[5])] - (-3 + Sqrt[5])*PolyLog[2, (-1 + Sqrt[5] - 2*Sqrt[1 + x])/(2*Sqrt[5])] - 6*PolyLog[2, (-
1 + Sqrt[5] - 2*Sqrt[1 + x])/(-1 + Sqrt[5])] + 3*PolyLog[2, (1 + Sqrt[5] + 2*Sqrt[1 + x])/(2*Sqrt[5])] + Sqrt[
5]*PolyLog[2, (1 + Sqrt[5] + 2*Sqrt[1 + x])/(2*Sqrt[5])]

Maple [F]

\[\int \frac {\ln \left (x +\sqrt {1+x}\right )^{2}}{\left (1+x \right )^{2}}d x\]

[In]

int(ln(x+(1+x)^(1/2))^2/(1+x)^2,x)

[Out]

int(ln(x+(1+x)^(1/2))^2/(1+x)^2,x)

Fricas [F]

\[ \int \frac {\log ^2\left (x+\sqrt {1+x}\right )}{(1+x)^2} \, dx=\int { \frac {\log \left (x + \sqrt {x + 1}\right )^{2}}{{\left (x + 1\right )}^{2}} \,d x } \]

[In]

integrate(log(x+(1+x)^(1/2))^2/(1+x)^2,x, algorithm="fricas")

[Out]

integral(log(x + sqrt(x + 1))^2/(x^2 + 2*x + 1), x)

Sympy [F]

\[ \int \frac {\log ^2\left (x+\sqrt {1+x}\right )}{(1+x)^2} \, dx=\int \frac {\log {\left (x + \sqrt {x + 1} \right )}^{2}}{\left (x + 1\right )^{2}}\, dx \]

[In]

integrate(ln(x+(1+x)**(1/2))**2/(1+x)**2,x)

[Out]

Integral(log(x + sqrt(x + 1))**2/(x + 1)**2, x)

Maxima [F]

\[ \int \frac {\log ^2\left (x+\sqrt {1+x}\right )}{(1+x)^2} \, dx=\int { \frac {\log \left (x + \sqrt {x + 1}\right )^{2}}{{\left (x + 1\right )}^{2}} \,d x } \]

[In]

integrate(log(x+(1+x)^(1/2))^2/(1+x)^2,x, algorithm="maxima")

[Out]

-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)

Giac [F]

\[ \int \frac {\log ^2\left (x+\sqrt {1+x}\right )}{(1+x)^2} \, dx=\int { \frac {\log \left (x + \sqrt {x + 1}\right )^{2}}{{\left (x + 1\right )}^{2}} \,d x } \]

[In]

integrate(log(x+(1+x)^(1/2))^2/(1+x)^2,x, algorithm="giac")

[Out]

integrate(log(x + sqrt(x + 1))^2/(x + 1)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\log ^2\left (x+\sqrt {1+x}\right )}{(1+x)^2} \, dx=\int \frac {{\ln \left (x+\sqrt {x+1}\right )}^2}{{\left (x+1\right )}^2} \,d x \]

[In]

int(log(x + (x + 1)^(1/2))^2/(x + 1)^2,x)

[Out]

int(log(x + (x + 1)^(1/2))^2/(x + 1)^2, x)