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

Optimal. Leaf size=555 \[ 6 \text{PolyLog}\left (2,-\frac{2 \sqrt{x+1}}{1+\sqrt{5}}\right )-\left (3+\sqrt{5}\right ) \text{PolyLog}\left (2,-\frac{2 \sqrt{x+1}-\sqrt{5}+1}{2 \sqrt{5}}\right )-\left (3-\sqrt{5}\right ) \text{PolyLog}\left (2,\frac{2 \sqrt{x+1}+\sqrt{5}+1}{2 \sqrt{5}}\right )-6 \text{PolyLog}\left (2,\frac{2 \sqrt{x+1}}{1-\sqrt{5}}+1\right )-\frac{\log ^2\left (x+\sqrt{x+1}\right )}{x+1}-\frac{1}{2} \left (3+\sqrt{5}\right ) \log ^2\left (2 \sqrt{x+1}-\sqrt{5}+1\right )-\frac{1}{2} \left (3-\sqrt{5}\right ) \log ^2\left (2 \sqrt{x+1}+\sqrt{5}+1\right )-6 \log \left (\sqrt{x+1}\right ) \log \left (x+\sqrt{x+1}\right )+\left (3+\sqrt{5}\right ) \log \left (2 \sqrt{x+1}-\sqrt{5}+1\right ) \log \left (x+\sqrt{x+1}\right )+\left (3-\sqrt{5}\right ) \log \left (2 \sqrt{x+1}+\sqrt{5}+1\right ) \log \left (x+\sqrt{x+1}\right )+\frac{2 \log \left (x+\sqrt{x+1}\right )}{\sqrt{x+1}}+\log (x+1)+6 \log \left (\frac{1}{2} \left (\sqrt{5}-1\right )\right ) \log \left (2 \sqrt{x+1}-\sqrt{5}+1\right )-\left (1+\sqrt{5}\right ) \log \left (2 \sqrt{x+1}-\sqrt{5}+1\right )-\left (3-\sqrt{5}\right ) \log \left (-\frac{2 \sqrt{x+1}-\sqrt{5}+1}{2 \sqrt{5}}\right ) \log \left (2 \sqrt{x+1}+\sqrt{5}+1\right )-\left (1-\sqrt{5}\right ) \log \left (2 \sqrt{x+1}+\sqrt{5}+1\right )-\left (3+\sqrt{5}\right ) \log \left (2 \sqrt{x+1}-\sqrt{5}+1\right ) \log \left (\frac{2 \sqrt{x+1}+\sqrt{5}+1}{2 \sqrt{5}}\right )+6 \log \left (\sqrt{x+1}\right ) \log \left (\frac{2 \sqrt{x+1}}{1+\sqrt{5}}+1\right ) \]

[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 - Sqrt[5] + 2*Sqrt[1 + x])/(2*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 - Sqrt[5] + 2*Sqr
t[1 + x])/(2*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])]

________________________________________________________________________________________

Rubi [A]  time = 0.711506, antiderivative size = 555, normalized size of antiderivative = 1., number of steps used = 35, number of rules used = 16, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.889, Rules used = {2525, 2528, 800, 632, 31, 2524, 2357, 2316, 2315, 2317, 2391, 2418, 2390, 2301, 2394, 2393} \[ 6 \text{PolyLog}\left (2,-\frac{2 \sqrt{x+1}}{1+\sqrt{5}}\right )-\left (3+\sqrt{5}\right ) \text{PolyLog}\left (2,-\frac{2 \sqrt{x+1}-\sqrt{5}+1}{2 \sqrt{5}}\right )-\left (3-\sqrt{5}\right ) \text{PolyLog}\left (2,\frac{2 \sqrt{x+1}+\sqrt{5}+1}{2 \sqrt{5}}\right )-6 \text{PolyLog}\left (2,\frac{2 \sqrt{x+1}}{1-\sqrt{5}}+1\right )-\frac{\log ^2\left (x+\sqrt{x+1}\right )}{x+1}-\frac{1}{2} \left (3+\sqrt{5}\right ) \log ^2\left (2 \sqrt{x+1}-\sqrt{5}+1\right )-\frac{1}{2} \left (3-\sqrt{5}\right ) \log ^2\left (2 \sqrt{x+1}+\sqrt{5}+1\right )-6 \log \left (\sqrt{x+1}\right ) \log \left (x+\sqrt{x+1}\right )+\left (3+\sqrt{5}\right ) \log \left (2 \sqrt{x+1}-\sqrt{5}+1\right ) \log \left (x+\sqrt{x+1}\right )+\left (3-\sqrt{5}\right ) \log \left (2 \sqrt{x+1}+\sqrt{5}+1\right ) \log \left (x+\sqrt{x+1}\right )+\frac{2 \log \left (x+\sqrt{x+1}\right )}{\sqrt{x+1}}+\log (x+1)+6 \log \left (\frac{1}{2} \left (\sqrt{5}-1\right )\right ) \log \left (2 \sqrt{x+1}-\sqrt{5}+1\right )-\left (1+\sqrt{5}\right ) \log \left (2 \sqrt{x+1}-\sqrt{5}+1\right )-\left (3-\sqrt{5}\right ) \log \left (-\frac{2 \sqrt{x+1}-\sqrt{5}+1}{2 \sqrt{5}}\right ) \log \left (2 \sqrt{x+1}+\sqrt{5}+1\right )-\left (1-\sqrt{5}\right ) \log \left (2 \sqrt{x+1}+\sqrt{5}+1\right )-\left (3+\sqrt{5}\right ) \log \left (2 \sqrt{x+1}-\sqrt{5}+1\right ) \log \left (\frac{2 \sqrt{x+1}+\sqrt{5}+1}{2 \sqrt{5}}\right )+6 \log \left (\sqrt{x+1}\right ) \log \left (\frac{2 \sqrt{x+1}}{1+\sqrt{5}}+1\right ) \]

Antiderivative was successfully verified.

[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 - Sqrt[5] + 2*Sqrt[1 + x])/(2*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 - Sqrt[5] + 2*Sqr
t[1 + x])/(2*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 2525

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 2528

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]

Rule 800

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 632

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 31

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

Rule 2524

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 2357

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 2316

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 2315

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

Rule 2317

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 2391

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 2418

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 2390

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 2301

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 2394

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 2393

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]

Rubi steps

\begin{align*} \int \frac{\log ^2\left (x+\sqrt{1+x}\right )}{(1+x)^2} \, dx &=2 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{Subst}\left (\int \frac{\log \left (-1+x+x^2\right )}{x^2} \, dx,x,\sqrt{1+x}\right )+2 \operatorname{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 \operatorname{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 \operatorname{Subst}\left (\int \frac{1+2 x}{x \left (-1+x+x^2\right )} \, dx,x,\sqrt{1+x}\right )+2 \operatorname{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 \operatorname{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 \operatorname{Subst}\left (\int \left (-\frac{1}{x}+\frac{3+x}{-1+x+x^2}\right ) \, dx,x,\sqrt{1+x}\right )+6 \operatorname{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 ) \operatorname{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 ) \operatorname{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 \operatorname{Subst}\left (\int \frac{3+x}{-1+x+x^2} \, dx,x,\sqrt{1+x}\right )+12 \operatorname{Subst}\left (\int \frac{\log (x)}{1-\sqrt{5}+2 x} \, dx,x,\sqrt{1+x}\right )+12 \operatorname{Subst}\left (\int \frac{\log (x)}{1+\sqrt{5}+2 x} \, dx,x,\sqrt{1+x}\right )+\left (-3-\sqrt{5}\right ) \operatorname{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 ) \operatorname{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 \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 x}{1+\sqrt{5}}\right )}{x} \, dx,x,\sqrt{1+x}\right )+12 \operatorname{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 ) \operatorname{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 ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{2}+\frac{\sqrt{5}}{2}+x} \, dx,x,\sqrt{1+x}\right )+\left (-3+\sqrt{5}\right ) \operatorname{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 ) \operatorname{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 \text{Li}_2\left (-\frac{2 \sqrt{1+x}}{1+\sqrt{5}}\right )-6 \text{Li}_2\left (1+\frac{2 \sqrt{1+x}}{1-\sqrt{5}}\right )-\left (2 \left (3-\sqrt{5}\right )\right ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 \text{Li}_2\left (-\frac{2 \sqrt{1+x}}{1+\sqrt{5}}\right )-6 \text{Li}_2\left (1+\frac{2 \sqrt{1+x}}{1-\sqrt{5}}\right )-\left (3-\sqrt{5}\right ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 \text{Li}_2\left (-\frac{2 \sqrt{1+x}}{1+\sqrt{5}}\right )-6 \text{Li}_2\left (1+\frac{2 \sqrt{1+x}}{1-\sqrt{5}}\right )+\left (3-\sqrt{5}\right ) \operatorname{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 ) \operatorname{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 \text{Li}_2\left (-\frac{2 \sqrt{1+x}}{1+\sqrt{5}}\right )-\left (3+\sqrt{5}\right ) \text{Li}_2\left (-\frac{1-\sqrt{5}+2 \sqrt{1+x}}{2 \sqrt{5}}\right )-\left (3-\sqrt{5}\right ) \text{Li}_2\left (\frac{1+\sqrt{5}+2 \sqrt{1+x}}{2 \sqrt{5}}\right )-6 \text{Li}_2\left (1+\frac{2 \sqrt{1+x}}{1-\sqrt{5}}\right )\\ \end{align*}

Mathematica [A]  time = 1.42651, size = 1076, normalized size = 1.94 \[ \frac{1}{2} \left (\sqrt{5} \log ^2\left (\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right )+3 \log ^2\left (\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right )-12 \log \left (\frac{2 \sqrt{x+1}}{-1+\sqrt{5}}\right ) \log \left (\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right )+6 \log (x+1) \log \left (\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right )-2 \sqrt{5} \log \left (-2 \sqrt{x+1}+\sqrt{5}-1\right ) \log \left (\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right )-6 \log \left (-2 \sqrt{x+1}+\sqrt{5}-1\right ) \log \left (\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right )+2 \sqrt{5} \log \left (2 \sqrt{x+1}+\sqrt{5}+1\right ) \log \left (\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right )-6 \log \left (2 \sqrt{x+1}+\sqrt{5}+1\right ) \log \left (\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right )-2 \sqrt{5} \log \left (\frac{2 \sqrt{x+1}+\sqrt{5}+1}{2 \sqrt{5}}\right ) \log \left (\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right )+6 \log \left (\frac{2 \sqrt{x+1}+\sqrt{5}+1}{2 \sqrt{5}}\right ) \log \left (\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right )-\sqrt{5} \log ^2\left (\sqrt{x+1}+\frac{1}{2} \left (1+\sqrt{5}\right )\right )+3 \log ^2\left (\sqrt{x+1}+\frac{1}{2} \left (1+\sqrt{5}\right )\right )-\frac{2 \log ^2\left (x+\sqrt{x+1}\right )}{x+1}-6 \log \left (\frac{1}{2} \left (1+\sqrt{5}\right )\right ) \log (x+1)+2 \log (x+1)+6 \log (x+1) \log \left (\sqrt{x+1}+\frac{1}{2} \left (1+\sqrt{5}\right )\right )-2 \sqrt{5} \log \left (-2 \sqrt{x+1}+\sqrt{5}-1\right ) \log \left (\sqrt{x+1}+\frac{1}{2} \left (1+\sqrt{5}\right )\right )-6 \log \left (-2 \sqrt{x+1}+\sqrt{5}-1\right ) \log \left (\sqrt{x+1}+\frac{1}{2} \left (1+\sqrt{5}\right )\right )-6 \log (x+1) \log \left (x+\sqrt{x+1}\right )+2 \sqrt{5} \log \left (-2 \sqrt{x+1}+\sqrt{5}-1\right ) \log \left (x+\sqrt{x+1}\right )+6 \log \left (-2 \sqrt{x+1}+\sqrt{5}-1\right ) \log \left (x+\sqrt{x+1}\right )+\frac{4 \log \left (x+\sqrt{x+1}\right )}{\sqrt{x+1}}+\sqrt{5} \log (5) \log \left (2 \sqrt{x+1}-\sqrt{5}+1\right )+3 \log (5) \log \left (2 \sqrt{x+1}-\sqrt{5}+1\right )-2 \sqrt{5} \log \left (2 \sqrt{x+1}-\sqrt{5}+1\right )-2 \log \left (2 \sqrt{x+1}-\sqrt{5}+1\right )+2 \sqrt{5} \log \left (\sqrt{x+1}+\frac{1}{2} \left (1+\sqrt{5}\right )\right ) \log \left (2 \sqrt{x+1}+\sqrt{5}+1\right )-6 \log \left (\sqrt{x+1}+\frac{1}{2} \left (1+\sqrt{5}\right )\right ) \log \left (2 \sqrt{x+1}+\sqrt{5}+1\right )-2 \sqrt{5} \log \left (x+\sqrt{x+1}\right ) \log \left (2 \sqrt{x+1}+\sqrt{5}+1\right )+6 \log \left (x+\sqrt{x+1}\right ) \log \left (2 \sqrt{x+1}+\sqrt{5}+1\right )+2 \sqrt{5} \log \left (2 \sqrt{x+1}+\sqrt{5}+1\right )-2 \log \left (2 \sqrt{x+1}+\sqrt{5}+1\right )+12 \text{PolyLog}\left (2,-\frac{2 \sqrt{x+1}}{1+\sqrt{5}}\right )-4 \sqrt{5} \text{PolyLog}\left (2,\frac{-2 \sqrt{x+1}+\sqrt{5}-1}{2 \sqrt{5}}\right )-12 \text{PolyLog}\left (2,\frac{-2 \sqrt{x+1}+\sqrt{5}-1}{-1+\sqrt{5}}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*Log[1 + x] - 6*Log[(1 + Sqrt[5])/2]*Log[1 + x] - 12*Log[(2*Sqrt[1 + x])/(-1 + Sqrt[5])]*Log[1/2 - Sqrt[5]/2
+ Sqrt[1 + x]] + 6*Log[1 + x]*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]] - 6*Log[-1 + Sqrt[5] - 2*Sqrt[1 + x]]*Log[1/
2 - Sqrt[5]/2 + Sqrt[1 + x]] - 2*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 + Sqrt[5]*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]]^2 + 6*Log[1 + x]*Log[(1
+ Sqrt[5])/2 + Sqrt[1 + x]] - 6*Log[-1 + Sqrt[5] - 2*Sqrt[1 + x]]*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]] - 2*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 - Sqrt[5]*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]]^2 + (4*Log[x + Sqrt[1 + x]])/Sqrt[1 + x] - 6*Log[1 + x]*Log[x
+ Sqrt[1 + x]] + 6*Log[-1 + Sqrt[5] - 2*Sqrt[1 + x]]*Log[x + Sqrt[1 + x]] + 2*Sqrt[5]*Log[-1 + Sqrt[5] - 2*Sqr
t[1 + x]]*Log[x + Sqrt[1 + x]] - (2*Log[x + Sqrt[1 + x]]^2)/(1 + x) - 2*Log[1 - Sqrt[5] + 2*Sqrt[1 + x]] - 2*S
qrt[5]*Log[1 - Sqrt[5] + 2*Sqrt[1 + x]] + 3*Log[5]*Log[1 - Sqrt[5] + 2*Sqrt[1 + x]] + Sqrt[5]*Log[5]*Log[1 - S
qrt[5] + 2*Sqrt[1 + x]] - 2*Log[1 + Sqrt[5] + 2*Sqrt[1 + x]] + 2*Sqrt[5]*Log[1 + Sqrt[5] + 2*Sqrt[1 + x]] - 6*
Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]]*Log[1 + Sqrt[5] + 2*Sqrt[1 + x]] + 2*Sqrt[5]*Log[1/2 - Sqrt[5]/2 + Sqrt[1 +
x]]*Log[1 + Sqrt[5] + 2*Sqrt[1 + x]] - 6*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]]*Log[1 + Sqrt[5] + 2*Sqrt[1 + x]]
+ 2*Sqrt[5]*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]]*Log[1 + Sqrt[5] + 2*Sqrt[1 + x]] + 6*Log[x + Sqrt[1 + x]]*Log[1
+ Sqrt[5] + 2*Sqrt[1 + x]] - 2*Sqrt[5]*Log[x + Sqrt[1 + x]]*Log[1 + Sqrt[5] + 2*Sqrt[1 + x]] + 6*Log[1/2 - Sq
rt[5]/2 + Sqrt[1 + x]]*Log[(1 + Sqrt[5] + 2*Sqrt[1 + x])/(2*Sqrt[5])] - 2*Sqrt[5]*Log[1/2 - Sqrt[5]/2 + Sqrt[1
+ x]]*Log[(1 + Sqrt[5] + 2*Sqrt[1 + x])/(2*Sqrt[5])] + 12*PolyLog[2, (-2*Sqrt[1 + x])/(1 + Sqrt[5])] - 4*Sqrt
[5]*PolyLog[2, (-1 + Sqrt[5] - 2*Sqrt[1 + x])/(2*Sqrt[5])] - 12*PolyLog[2, (-1 + Sqrt[5] - 2*Sqrt[1 + x])/(-1
+ Sqrt[5])])/2

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Maple [F]  time = 0.01, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( 1+x \right ) ^{2}} \left ( \ln \left ( x+\sqrt{1+x} \right ) \right ) ^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{\log \left (x + \sqrt{x + 1}\right )^{2}}{x + 1} + \int \frac{{\left (2 \, x + \sqrt{x + 1} + 2\right )} \log \left (x + \sqrt{x + 1}\right )}{x^{3} + 2 \, x^{2} +{\left (x^{2} + 2 \, x + 1\right )} \sqrt{x + 1} + x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (x + \sqrt{x + 1}\right )^{2}}{x^{2} + 2 \, x + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (x + \sqrt{x + 1} \right )}^{2}}{\left (x + 1\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (x + \sqrt{x + 1}\right )^{2}}{{\left (x + 1\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)