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

Optimal. Leaf size=981 \[ \frac {1}{2} i \log \left (\sqrt {1-i}-\sqrt {x+1}\right ) \log \left (x+\sqrt {x+1}\right )-\frac {1}{2} i \log \left (\sqrt {1+i}-\sqrt {x+1}\right ) \log \left (x+\sqrt {x+1}\right )+\frac {1}{2} i \log \left (\sqrt {x+1}+\sqrt {1-i}\right ) \log \left (x+\sqrt {x+1}\right )-\frac {1}{2} i \log \left (\sqrt {x+1}+\sqrt {1+i}\right ) \log \left (x+\sqrt {x+1}\right )-\frac {1}{2} 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}{2} 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}{2} 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}{2} 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}{2} 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}{2} 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}{2} 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}{2} 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}{2} 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}{2} 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}{2} 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}{2} 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}{2} 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}{2} 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}{2} 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}{2} i \operatorname {PolyLog}\left (2,-\frac {2 \left (\sqrt {x+1}+\sqrt {1+i}\right )}{1-2 \sqrt {1+i}+\sqrt {5}}\right ) \]

[Out]

-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((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*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*ln((1+I)^(1/2)-(1+x)^(1/2))*ln(x+(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*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*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))*ln((1+I)^(1/2)+(1+x)^(1/2))

________________________________________________________________________________________

Rubi [A]  time = 1.25, antiderivative size = 981, normalized size of antiderivative = 1.00, number of steps used = 44, number of rules used = 10, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {2530, 1591, 203, 6741, 2528, 2524, 2418, 2394, 2393, 2391} \[ \frac {1}{2} i \log \left (\sqrt {1-i}-\sqrt {x+1}\right ) \log \left (x+\sqrt {x+1}\right )-\frac {1}{2} i \log \left (\sqrt {1+i}-\sqrt {x+1}\right ) \log \left (x+\sqrt {x+1}\right )+\frac {1}{2} i \log \left (\sqrt {x+1}+\sqrt {1-i}\right ) \log \left (x+\sqrt {x+1}\right )-\frac {1}{2} i \log \left (\sqrt {x+1}+\sqrt {1+i}\right ) \log \left (x+\sqrt {x+1}\right )-\frac {1}{2} 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}{2} 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}{2} 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}{2} 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}{2} 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}{2} 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}{2} 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}{2} 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}{2} i \text {PolyLog}\left (2,\frac {2 \left (\sqrt {1-i}-\sqrt {x+1}\right )}{1+2 \sqrt {1-i}-\sqrt {5}}\right )-\frac {1}{2} i \text {PolyLog}\left (2,\frac {2 \left (\sqrt {1-i}-\sqrt {x+1}\right )}{1+2 \sqrt {1-i}+\sqrt {5}}\right )+\frac {1}{2} i \text {PolyLog}\left (2,\frac {2 \left (\sqrt {1+i}-\sqrt {x+1}\right )}{1+2 \sqrt {1+i}-\sqrt {5}}\right )+\frac {1}{2} i \text {PolyLog}\left (2,\frac {2 \left (\sqrt {1+i}-\sqrt {x+1}\right )}{1+2 \sqrt {1+i}+\sqrt {5}}\right )-\frac {1}{2} i \text {PolyLog}\left (2,-\frac {2 \left (\sqrt {x+1}+\sqrt {1-i}\right )}{1-2 \sqrt {1-i}-\sqrt {5}}\right )-\frac {1}{2} i \text {PolyLog}\left (2,-\frac {2 \left (\sqrt {x+1}+\sqrt {1-i}\right )}{1-2 \sqrt {1-i}+\sqrt {5}}\right )+\frac {1}{2} i \text {PolyLog}\left (2,-\frac {2 \left (\sqrt {x+1}+\sqrt {1+i}\right )}{1-2 \sqrt {1+i}-\sqrt {5}}\right )+\frac {1}{2} i \text {PolyLog}\left (2,-\frac {2 \left (\sqrt {x+1}+\sqrt {1+i}\right )}{1-2 \sqrt {1+i}+\sqrt {5}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Log[x + Sqrt[1 + x]]/(1 + x^2),x]

[Out]

(I/2)*Log[Sqrt[1 - I] - Sqrt[1 + x]]*Log[x + Sqrt[1 + x]] - (I/2)*Log[Sqrt[1 + I] - Sqrt[1 + x]]*Log[x + Sqrt[
1 + x]] + (I/2)*Log[Sqrt[1 - I] + Sqrt[1 + x]]*Log[x + Sqrt[1 + x]] - (I/2)*Log[Sqrt[1 + I] + Sqrt[1 + x]]*Log
[x + Sqrt[1 + x]] - (I/2)*Log[Sqrt[1 - I] + Sqrt[1 + x]]*Log[(1 - Sqrt[5] + 2*Sqrt[1 + x])/(1 - 2*Sqrt[1 - I]
- Sqrt[5])] - (I/2)*Log[Sqrt[1 - I] - Sqrt[1 + x]]*Log[(1 - Sqrt[5] + 2*Sqrt[1 + x])/(1 + 2*Sqrt[1 - I] - Sqrt
[5])] + (I/2)*Log[Sqrt[1 + I] + Sqrt[1 + x]]*Log[(1 - Sqrt[5] + 2*Sqrt[1 + x])/(1 - 2*Sqrt[1 + I] - Sqrt[5])]
+ (I/2)*Log[Sqrt[1 + I] - Sqrt[1 + x]]*Log[(1 - Sqrt[5] + 2*Sqrt[1 + x])/(1 + 2*Sqrt[1 + I] - Sqrt[5])] - (I/2
)*Log[Sqrt[1 - I] + Sqrt[1 + x]]*Log[(1 + Sqrt[5] + 2*Sqrt[1 + x])/(1 - 2*Sqrt[1 - I] + Sqrt[5])] - (I/2)*Log[
Sqrt[1 - I] - Sqrt[1 + x]]*Log[(1 + Sqrt[5] + 2*Sqrt[1 + x])/(1 + 2*Sqrt[1 - I] + Sqrt[5])] + (I/2)*Log[Sqrt[1
+ I] + Sqrt[1 + x]]*Log[(1 + Sqrt[5] + 2*Sqrt[1 + x])/(1 - 2*Sqrt[1 + I] + Sqrt[5])] + (I/2)*Log[Sqrt[1 + I]
- Sqrt[1 + x]]*Log[(1 + Sqrt[5] + 2*Sqrt[1 + x])/(1 + 2*Sqrt[1 + I] + Sqrt[5])] - (I/2)*PolyLog[2, (2*(Sqrt[1
- I] - Sqrt[1 + x]))/(1 + 2*Sqrt[1 - I] - Sqrt[5])] - (I/2)*PolyLog[2, (2*(Sqrt[1 - I] - Sqrt[1 + x]))/(1 + 2*
Sqrt[1 - I] + Sqrt[5])] + (I/2)*PolyLog[2, (2*(Sqrt[1 + I] - Sqrt[1 + x]))/(1 + 2*Sqrt[1 + I] - Sqrt[5])] + (I
/2)*PolyLog[2, (2*(Sqrt[1 + I] - Sqrt[1 + x]))/(1 + 2*Sqrt[1 + I] + Sqrt[5])] - (I/2)*PolyLog[2, (-2*(Sqrt[1 -
I] + Sqrt[1 + x]))/(1 - 2*Sqrt[1 - I] - Sqrt[5])] - (I/2)*PolyLog[2, (-2*(Sqrt[1 - I] + Sqrt[1 + x]))/(1 - 2*
Sqrt[1 - I] + Sqrt[5])] + (I/2)*PolyLog[2, (-2*(Sqrt[1 + I] + Sqrt[1 + x]))/(1 - 2*Sqrt[1 + I] - Sqrt[5])] + (
I/2)*PolyLog[2, (-2*(Sqrt[1 + I] + Sqrt[1 + x]))/(1 - 2*Sqrt[1 + I] + Sqrt[5])]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 1591

Int[((a_.) + (b_.)*(Pq_)^(n_.))^(p_.)*(Qr_), x_Symbol] :> With[{q = Expon[Pq, x], r = Expon[Qr, x]}, Dist[Coef
f[Qr, x, r]/(q*Coeff[Pq, x, q]), Subst[Int[(a + b*x^n)^p, x], x, Pq], x] /; EqQ[r, q - 1] && EqQ[Coeff[Qr, x,
r]*D[Pq, x], q*Coeff[Pq, x, q]*Qr]] /; FreeQ[{a, b, n, p}, x] && PolyQ[Pq, x] && PolyQ[Qr, x]

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 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]

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 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 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 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 2530

Int[((a_.) + Log[u_]*(b_.))*(RFx_), x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[RFx*(a + b*Log[u]
), x]}, Dist[lst[[2]]*lst[[4]], Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x] /;  !FalseQ[lst]] /; Fre
eQ[{a, b}, x] && RationalFunctionQ[RFx, x]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rubi steps

\begin {align*} \int \frac {\log \left (x+\sqrt {1+x}\right )}{1+x^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {x \log \left (-1+x+x^2\right )}{1+\left (-1+x^2\right )^2} \, dx,x,\sqrt {1+x}\right )\\ &=2 \operatorname {Subst}\left (\int \frac {x \log \left (-1+x+x^2\right )}{2-2 x^2+x^4} \, dx,x,\sqrt {1+x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {i x \log \left (-1+x+x^2\right )}{(2+2 i)-2 x^2}+\frac {i x \log \left (-1+x+x^2\right )}{(-2+2 i)+2 x^2}\right ) \, dx,x,\sqrt {1+x}\right )\\ &=2 i \operatorname {Subst}\left (\int \frac {x \log \left (-1+x+x^2\right )}{(2+2 i)-2 x^2} \, dx,x,\sqrt {1+x}\right )+2 i \operatorname {Subst}\left (\int \frac {x \log \left (-1+x+x^2\right )}{(-2+2 i)+2 x^2} \, dx,x,\sqrt {1+x}\right )\\ &=2 i \operatorname {Subst}\left (\int \left (-\frac {\log \left (-1+x+x^2\right )}{4 \left (\sqrt {1-i}-x\right )}+\frac {\log \left (-1+x+x^2\right )}{4 \left (\sqrt {1-i}+x\right )}\right ) \, dx,x,\sqrt {1+x}\right )+2 i \operatorname {Subst}\left (\int \left (\frac {\log \left (-1+x+x^2\right )}{4 \left (\sqrt {1+i}-x\right )}-\frac {\log \left (-1+x+x^2\right )}{4 \left (\sqrt {1+i}+x\right )}\right ) \, dx,x,\sqrt {1+x}\right )\\ &=-\left (\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log \left (-1+x+x^2\right )}{\sqrt {1-i}-x} \, dx,x,\sqrt {1+x}\right )\right )+\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log \left (-1+x+x^2\right )}{\sqrt {1+i}-x} \, dx,x,\sqrt {1+x}\right )+\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log \left (-1+x+x^2\right )}{\sqrt {1-i}+x} \, dx,x,\sqrt {1+x}\right )-\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log \left (-1+x+x^2\right )}{\sqrt {1+i}+x} \, dx,x,\sqrt {1+x}\right )\\ &=\frac {1}{2} i \log \left (\sqrt {1-i}-\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\frac {1}{2} i \log \left (\sqrt {1+i}-\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )+\frac {1}{2} i \log \left (\sqrt {1-i}+\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\frac {1}{2} i \log \left (\sqrt {1+i}+\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\frac {1}{2} i \operatorname {Subst}\left (\int \frac {(1+2 x) \log \left (\sqrt {1-i}-x\right )}{-1+x+x^2} \, dx,x,\sqrt {1+x}\right )+\frac {1}{2} i \operatorname {Subst}\left (\int \frac {(1+2 x) \log \left (\sqrt {1+i}-x\right )}{-1+x+x^2} \, dx,x,\sqrt {1+x}\right )-\frac {1}{2} i \operatorname {Subst}\left (\int \frac {(1+2 x) \log \left (\sqrt {1-i}+x\right )}{-1+x+x^2} \, dx,x,\sqrt {1+x}\right )+\frac {1}{2} i \operatorname {Subst}\left (\int \frac {(1+2 x) \log \left (\sqrt {1+i}+x\right )}{-1+x+x^2} \, dx,x,\sqrt {1+x}\right )\\ &=\frac {1}{2} i \log \left (\sqrt {1-i}-\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\frac {1}{2} i \log \left (\sqrt {1+i}-\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )+\frac {1}{2} i \log \left (\sqrt {1-i}+\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\frac {1}{2} i \log \left (\sqrt {1+i}+\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\frac {1}{2} i \operatorname {Subst}\left (\int \left (\frac {2 \log \left (\sqrt {1-i}-x\right )}{1-\sqrt {5}+2 x}+\frac {2 \log \left (\sqrt {1-i}-x\right )}{1+\sqrt {5}+2 x}\right ) \, dx,x,\sqrt {1+x}\right )+\frac {1}{2} i \operatorname {Subst}\left (\int \left (\frac {2 \log \left (\sqrt {1+i}-x\right )}{1-\sqrt {5}+2 x}+\frac {2 \log \left (\sqrt {1+i}-x\right )}{1+\sqrt {5}+2 x}\right ) \, dx,x,\sqrt {1+x}\right )-\frac {1}{2} i \operatorname {Subst}\left (\int \left (\frac {2 \log \left (\sqrt {1-i}+x\right )}{1-\sqrt {5}+2 x}+\frac {2 \log \left (\sqrt {1-i}+x\right )}{1+\sqrt {5}+2 x}\right ) \, dx,x,\sqrt {1+x}\right )+\frac {1}{2} i \operatorname {Subst}\left (\int \left (\frac {2 \log \left (\sqrt {1+i}+x\right )}{1-\sqrt {5}+2 x}+\frac {2 \log \left (\sqrt {1+i}+x\right )}{1+\sqrt {5}+2 x}\right ) \, dx,x,\sqrt {1+x}\right )\\ &=\frac {1}{2} i \log \left (\sqrt {1-i}-\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\frac {1}{2} i \log \left (\sqrt {1+i}-\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )+\frac {1}{2} i \log \left (\sqrt {1-i}+\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\frac {1}{2} i \log \left (\sqrt {1+i}+\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-i \operatorname {Subst}\left (\int \frac {\log \left (\sqrt {1-i}-x\right )}{1-\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )-i \operatorname {Subst}\left (\int \frac {\log \left (\sqrt {1-i}-x\right )}{1+\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )+i \operatorname {Subst}\left (\int \frac {\log \left (\sqrt {1+i}-x\right )}{1-\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )+i \operatorname {Subst}\left (\int \frac {\log \left (\sqrt {1+i}-x\right )}{1+\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )-i \operatorname {Subst}\left (\int \frac {\log \left (\sqrt {1-i}+x\right )}{1-\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )-i \operatorname {Subst}\left (\int \frac {\log \left (\sqrt {1-i}+x\right )}{1+\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )+i \operatorname {Subst}\left (\int \frac {\log \left (\sqrt {1+i}+x\right )}{1-\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )+i \operatorname {Subst}\left (\int \frac {\log \left (\sqrt {1+i}+x\right )}{1+\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )\\ &=\frac {1}{2} i \log \left (\sqrt {1-i}-\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\frac {1}{2} i \log \left (\sqrt {1+i}-\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )+\frac {1}{2} i \log \left (\sqrt {1-i}+\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\frac {1}{2} i \log \left (\sqrt {1+i}+\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\frac {1}{2} i \log \left (\sqrt {1-i}+\sqrt {1+x}\right ) \log \left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1-i}-\sqrt {5}}\right )-\frac {1}{2} i \log \left (\sqrt {1-i}-\sqrt {1+x}\right ) \log \left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1-i}-\sqrt {5}}\right )+\frac {1}{2} i \log \left (\sqrt {1+i}+\sqrt {1+x}\right ) \log \left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1+i}-\sqrt {5}}\right )+\frac {1}{2} i \log \left (\sqrt {1+i}-\sqrt {1+x}\right ) \log \left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1+i}-\sqrt {5}}\right )-\frac {1}{2} i \log \left (\sqrt {1-i}+\sqrt {1+x}\right ) \log \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1-i}+\sqrt {5}}\right )-\frac {1}{2} i \log \left (\sqrt {1-i}-\sqrt {1+x}\right ) \log \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1-i}+\sqrt {5}}\right )+\frac {1}{2} i \log \left (\sqrt {1+i}+\sqrt {1+x}\right ) \log \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1+i}+\sqrt {5}}\right )+\frac {1}{2} i \log \left (\sqrt {1+i}-\sqrt {1+x}\right ) \log \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1+i}+\sqrt {5}}\right )-\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log \left (\frac {-1-\sqrt {5}-2 x}{-1-2 \sqrt {1-i}-\sqrt {5}}\right )}{\sqrt {1-i}-x} \, dx,x,\sqrt {1+x}\right )+\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log \left (\frac {-1-\sqrt {5}-2 x}{-1-2 \sqrt {1+i}-\sqrt {5}}\right )}{\sqrt {1+i}-x} \, dx,x,\sqrt {1+x}\right )-\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log \left (\frac {-1+\sqrt {5}-2 x}{-1-2 \sqrt {1-i}+\sqrt {5}}\right )}{\sqrt {1-i}-x} \, dx,x,\sqrt {1+x}\right )+\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log \left (\frac {-1+\sqrt {5}-2 x}{-1-2 \sqrt {1+i}+\sqrt {5}}\right )}{\sqrt {1+i}-x} \, dx,x,\sqrt {1+x}\right )+\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log \left (\frac {1-\sqrt {5}+2 x}{1-2 \sqrt {1-i}-\sqrt {5}}\right )}{\sqrt {1-i}+x} \, dx,x,\sqrt {1+x}\right )-\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log \left (\frac {1-\sqrt {5}+2 x}{1-2 \sqrt {1+i}-\sqrt {5}}\right )}{\sqrt {1+i}+x} \, dx,x,\sqrt {1+x}\right )+\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log \left (\frac {1+\sqrt {5}+2 x}{1-2 \sqrt {1-i}+\sqrt {5}}\right )}{\sqrt {1-i}+x} \, dx,x,\sqrt {1+x}\right )-\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log \left (\frac {1+\sqrt {5}+2 x}{1-2 \sqrt {1+i}+\sqrt {5}}\right )}{\sqrt {1+i}+x} \, dx,x,\sqrt {1+x}\right )\\ &=\frac {1}{2} i \log \left (\sqrt {1-i}-\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\frac {1}{2} i \log \left (\sqrt {1+i}-\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )+\frac {1}{2} i \log \left (\sqrt {1-i}+\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\frac {1}{2} i \log \left (\sqrt {1+i}+\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\frac {1}{2} i \log \left (\sqrt {1-i}+\sqrt {1+x}\right ) \log \left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1-i}-\sqrt {5}}\right )-\frac {1}{2} i \log \left (\sqrt {1-i}-\sqrt {1+x}\right ) \log \left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1-i}-\sqrt {5}}\right )+\frac {1}{2} i \log \left (\sqrt {1+i}+\sqrt {1+x}\right ) \log \left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1+i}-\sqrt {5}}\right )+\frac {1}{2} i \log \left (\sqrt {1+i}-\sqrt {1+x}\right ) \log \left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1+i}-\sqrt {5}}\right )-\frac {1}{2} i \log \left (\sqrt {1-i}+\sqrt {1+x}\right ) \log \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1-i}+\sqrt {5}}\right )-\frac {1}{2} i \log \left (\sqrt {1-i}-\sqrt {1+x}\right ) \log \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1-i}+\sqrt {5}}\right )+\frac {1}{2} i \log \left (\sqrt {1+i}+\sqrt {1+x}\right ) \log \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1+i}+\sqrt {5}}\right )+\frac {1}{2} i \log \left (\sqrt {1+i}-\sqrt {1+x}\right ) \log \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1+i}+\sqrt {5}}\right )+\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{-1-2 \sqrt {1-i}-\sqrt {5}}\right )}{x} \, dx,x,\sqrt {1-i}-\sqrt {1+x}\right )+\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{1-2 \sqrt {1-i}-\sqrt {5}}\right )}{x} \, dx,x,\sqrt {1-i}+\sqrt {1+x}\right )-\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{-1-2 \sqrt {1+i}-\sqrt {5}}\right )}{x} \, dx,x,\sqrt {1+i}-\sqrt {1+x}\right )-\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{1-2 \sqrt {1+i}-\sqrt {5}}\right )}{x} \, dx,x,\sqrt {1+i}+\sqrt {1+x}\right )+\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{-1-2 \sqrt {1-i}+\sqrt {5}}\right )}{x} \, dx,x,\sqrt {1-i}-\sqrt {1+x}\right )+\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{1-2 \sqrt {1-i}+\sqrt {5}}\right )}{x} \, dx,x,\sqrt {1-i}+\sqrt {1+x}\right )-\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{-1-2 \sqrt {1+i}+\sqrt {5}}\right )}{x} \, dx,x,\sqrt {1+i}-\sqrt {1+x}\right )-\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{1-2 \sqrt {1+i}+\sqrt {5}}\right )}{x} \, dx,x,\sqrt {1+i}+\sqrt {1+x}\right )\\ &=\frac {1}{2} i \log \left (\sqrt {1-i}-\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\frac {1}{2} i \log \left (\sqrt {1+i}-\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )+\frac {1}{2} i \log \left (\sqrt {1-i}+\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\frac {1}{2} i \log \left (\sqrt {1+i}+\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\frac {1}{2} i \log \left (\sqrt {1-i}+\sqrt {1+x}\right ) \log \left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1-i}-\sqrt {5}}\right )-\frac {1}{2} i \log \left (\sqrt {1-i}-\sqrt {1+x}\right ) \log \left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1-i}-\sqrt {5}}\right )+\frac {1}{2} i \log \left (\sqrt {1+i}+\sqrt {1+x}\right ) \log \left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1+i}-\sqrt {5}}\right )+\frac {1}{2} i \log \left (\sqrt {1+i}-\sqrt {1+x}\right ) \log \left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1+i}-\sqrt {5}}\right )-\frac {1}{2} i \log \left (\sqrt {1-i}+\sqrt {1+x}\right ) \log \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1-i}+\sqrt {5}}\right )-\frac {1}{2} i \log \left (\sqrt {1-i}-\sqrt {1+x}\right ) \log \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1-i}+\sqrt {5}}\right )+\frac {1}{2} i \log \left (\sqrt {1+i}+\sqrt {1+x}\right ) \log \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1-2 \sqrt {1+i}+\sqrt {5}}\right )+\frac {1}{2} i \log \left (\sqrt {1+i}-\sqrt {1+x}\right ) \log \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1+2 \sqrt {1+i}+\sqrt {5}}\right )-\frac {1}{2} i \text {Li}_2\left (\frac {2 \left (\sqrt {1-i}-\sqrt {1+x}\right )}{1+2 \sqrt {1-i}-\sqrt {5}}\right )-\frac {1}{2} i \text {Li}_2\left (\frac {2 \left (\sqrt {1-i}-\sqrt {1+x}\right )}{1+2 \sqrt {1-i}+\sqrt {5}}\right )+\frac {1}{2} i \text {Li}_2\left (\frac {2 \left (\sqrt {1+i}-\sqrt {1+x}\right )}{1+2 \sqrt {1+i}-\sqrt {5}}\right )+\frac {1}{2} i \text {Li}_2\left (\frac {2 \left (\sqrt {1+i}-\sqrt {1+x}\right )}{1+2 \sqrt {1+i}+\sqrt {5}}\right )-\frac {1}{2} i \text {Li}_2\left (-\frac {2 \left (\sqrt {1-i}+\sqrt {1+x}\right )}{1-2 \sqrt {1-i}-\sqrt {5}}\right )-\frac {1}{2} i \text {Li}_2\left (-\frac {2 \left (\sqrt {1-i}+\sqrt {1+x}\right )}{1-2 \sqrt {1-i}+\sqrt {5}}\right )+\frac {1}{2} i \text {Li}_2\left (-\frac {2 \left (\sqrt {1+i}+\sqrt {1+x}\right )}{1-2 \sqrt {1+i}-\sqrt {5}}\right )+\frac {1}{2} i \text {Li}_2\left (-\frac {2 \left (\sqrt {1+i}+\sqrt {1+x}\right )}{1-2 \sqrt {1+i}+\sqrt {5}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.56, size = 868, normalized size = 0.88 \[ \frac {1}{2} i \left (2 i \tan ^{-1}(x) \log \left (\sqrt {x+1}-\frac {\sqrt {5}}{2}+\frac {1}{2}\right )+\log \left (\frac {2 \left (\sqrt {1-i}-\sqrt {x+1}\right )}{1+2 \sqrt {1-i}-\sqrt {5}}\right ) \log \left (\sqrt {x+1}-\frac {\sqrt {5}}{2}+\frac {1}{2}\right )-\log \left (\frac {2 \left (\sqrt {1+i}-\sqrt {x+1}\right )}{1+2 \sqrt {1+i}-\sqrt {5}}\right ) \log \left (\sqrt {x+1}-\frac {\sqrt {5}}{2}+\frac {1}{2}\right )+\log \left (\frac {2 \left (\sqrt {x+1}+\sqrt {1-i}\right )}{-1+2 \sqrt {1-i}+\sqrt {5}}\right ) \log \left (\sqrt {x+1}-\frac {\sqrt {5}}{2}+\frac {1}{2}\right )-\log \left (\frac {2 \left (\sqrt {x+1}+\sqrt {1+i}\right )}{-1+2 \sqrt {1+i}+\sqrt {5}}\right ) \log \left (\sqrt {x+1}-\frac {\sqrt {5}}{2}+\frac {1}{2}\right )+2 i \tan ^{-1}(x) \log \left (\sqrt {x+1}+\frac {1}{2} \left (1+\sqrt {5}\right )\right )+\log \left (\frac {2 \left (\sqrt {1-i}-\sqrt {x+1}\right )}{1+2 \sqrt {1-i}+\sqrt {5}}\right ) \log \left (\sqrt {x+1}+\frac {1}{2} \left (1+\sqrt {5}\right )\right )-\log \left (\frac {2 \left (\sqrt {1+i}-\sqrt {x+1}\right )}{1+2 \sqrt {1+i}+\sqrt {5}}\right ) \log \left (\sqrt {x+1}+\frac {1}{2} \left (1+\sqrt {5}\right )\right )+\log \left (\frac {2 \left (\sqrt {x+1}+\sqrt {1-i}\right )}{-1+2 \sqrt {1-i}-\sqrt {5}}\right ) \log \left (\sqrt {x+1}+\frac {1}{2} \left (1+\sqrt {5}\right )\right )-\log \left (\frac {2 \left (\sqrt {x+1}+\sqrt {1+i}\right )}{-1+2 \sqrt {1+i}-\sqrt {5}}\right ) \log \left (\sqrt {x+1}+\frac {1}{2} \left (1+\sqrt {5}\right )\right )-2 i \tan ^{-1}(x) \log \left (x+\sqrt {x+1}\right )+\operatorname {PolyLog}\left (2,\frac {-2 \sqrt {x+1}+\sqrt {5}-1}{-1+2 \sqrt {1-i}+\sqrt {5}}\right )-\operatorname {PolyLog}\left (2,\frac {-2 \sqrt {x+1}+\sqrt {5}-1}{-1+2 \sqrt {1+i}+\sqrt {5}}\right )+\operatorname {PolyLog}\left (2,\frac {2 \sqrt {x+1}-\sqrt {5}+1}{1+2 \sqrt {1-i}-\sqrt {5}}\right )-\operatorname {PolyLog}\left (2,\frac {2 \sqrt {x+1}-\sqrt {5}+1}{1+2 \sqrt {1+i}-\sqrt {5}}\right )+\operatorname {PolyLog}\left (2,\frac {2 \sqrt {x+1}+\sqrt {5}+1}{1-2 \sqrt {1-i}+\sqrt {5}}\right )+\operatorname {PolyLog}\left (2,\frac {2 \sqrt {x+1}+\sqrt {5}+1}{1+2 \sqrt {1-i}+\sqrt {5}}\right )-\operatorname {PolyLog}\left (2,\frac {2 \sqrt {x+1}+\sqrt {5}+1}{1-2 \sqrt {1+i}+\sqrt {5}}\right )-\operatorname {PolyLog}\left (2,\frac {2 \sqrt {x+1}+\sqrt {5}+1}{1+2 \sqrt {1+i}+\sqrt {5}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(I/2)*((2*I)*ArcTan[x]*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]] + Log[(2*(Sqrt[1 - I] - Sqrt[1 + x]))/(1 + 2*Sqrt[1
- I] - Sqrt[5])]*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]] - Log[(2*(Sqrt[1 + I] - Sqrt[1 + x]))/(1 + 2*Sqrt[1 + I] -
Sqrt[5])]*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]] + Log[(2*(Sqrt[1 - I] + Sqrt[1 + x]))/(-1 + 2*Sqrt[1 - I] + Sqrt
[5])]*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]] - Log[(2*(Sqrt[1 + I] + Sqrt[1 + x]))/(-1 + 2*Sqrt[1 + I] + Sqrt[5])]
*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]] + (2*I)*ArcTan[x]*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]] + Log[(2*(Sqrt[1 - I]
- Sqrt[1 + x]))/(1 + 2*Sqrt[1 - I] + Sqrt[5])]*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]] - Log[(2*(Sqrt[1 + I] - Sqr
t[1 + x]))/(1 + 2*Sqrt[1 + I] + Sqrt[5])]*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]] + Log[(2*(Sqrt[1 - I] + Sqrt[1 +
x]))/(-1 + 2*Sqrt[1 - I] - Sqrt[5])]*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]] - Log[(2*(Sqrt[1 + I] + Sqrt[1 + x]))/
(-1 + 2*Sqrt[1 + I] - Sqrt[5])]*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]] - (2*I)*ArcTan[x]*Log[x + Sqrt[1 + x]] + Po
lyLog[2, (-1 + Sqrt[5] - 2*Sqrt[1 + x])/(-1 + 2*Sqrt[1 - I] + Sqrt[5])] - PolyLog[2, (-1 + Sqrt[5] - 2*Sqrt[1
+ x])/(-1 + 2*Sqrt[1 + I] + Sqrt[5])] + PolyLog[2, (1 - Sqrt[5] + 2*Sqrt[1 + x])/(1 + 2*Sqrt[1 - I] - Sqrt[5])
] - PolyLog[2, (1 - Sqrt[5] + 2*Sqrt[1 + x])/(1 + 2*Sqrt[1 + I] - Sqrt[5])] + PolyLog[2, (1 + Sqrt[5] + 2*Sqrt
[1 + x])/(1 - 2*Sqrt[1 - I] + Sqrt[5])] + PolyLog[2, (1 + Sqrt[5] + 2*Sqrt[1 + x])/(1 + 2*Sqrt[1 - I] + Sqrt[5
])] - PolyLog[2, (1 + Sqrt[5] + 2*Sqrt[1 + x])/(1 - 2*Sqrt[1 + I] + Sqrt[5])] - PolyLog[2, (1 + Sqrt[5] + 2*Sq
rt[1 + x])/(1 + 2*Sqrt[1 + I] + Sqrt[5])])

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fricas [F]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (x + \sqrt {x + 1}\right )}{x^{2} + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (x + \sqrt {x + 1}\right )}{x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.09, size = 730, normalized size = 0.74 \[ -\frac {i \ln \left (\frac {1-\sqrt {5}+2 \sqrt {x +1}}{1-2 \sqrt {1-i}-\sqrt {5}}\right ) \ln \left (\sqrt {1-i}+\sqrt {x +1}\right )}{2}-\frac {i \ln \left (\frac {1-\sqrt {5}+2 \sqrt {x +1}}{1+2 \sqrt {1-i}-\sqrt {5}}\right ) \ln \left (\sqrt {x +1}-\sqrt {1-i}\right )}{2}+\frac {i \ln \left (\frac {1-\sqrt {5}+2 \sqrt {x +1}}{1-2 \sqrt {1+i}-\sqrt {5}}\right ) \ln \left (\sqrt {1+i}+\sqrt {x +1}\right )}{2}+\frac {i \ln \left (\frac {1-\sqrt {5}+2 \sqrt {x +1}}{1+2 \sqrt {1+i}-\sqrt {5}}\right ) \ln \left (\sqrt {x +1}-\sqrt {1+i}\right )}{2}-\frac {i \ln \left (\frac {1+\sqrt {5}+2 \sqrt {x +1}}{1-2 \sqrt {1-i}+\sqrt {5}}\right ) \ln \left (\sqrt {1-i}+\sqrt {x +1}\right )}{2}-\frac {i \ln \left (\frac {1+\sqrt {5}+2 \sqrt {x +1}}{1+2 \sqrt {1-i}+\sqrt {5}}\right ) \ln \left (\sqrt {x +1}-\sqrt {1-i}\right )}{2}+\frac {i \ln \left (\frac {1+\sqrt {5}+2 \sqrt {x +1}}{1-2 \sqrt {1+i}+\sqrt {5}}\right ) \ln \left (\sqrt {1+i}+\sqrt {x +1}\right )}{2}+\frac {i \ln \left (\frac {1+\sqrt {5}+2 \sqrt {x +1}}{1+2 \sqrt {1+i}+\sqrt {5}}\right ) \ln \left (\sqrt {x +1}-\sqrt {1+i}\right )}{2}+\frac {i \ln \left (x +\sqrt {x +1}\right ) \ln \left (\sqrt {1-i}+\sqrt {x +1}\right )}{2}-\frac {i \ln \left (x +\sqrt {x +1}\right ) \ln \left (\sqrt {1+i}+\sqrt {x +1}\right )}{2}+\frac {i \ln \left (x +\sqrt {x +1}\right ) \ln \left (\sqrt {x +1}-\sqrt {1-i}\right )}{2}-\frac {i \ln \left (x +\sqrt {x +1}\right ) \ln \left (\sqrt {x +1}-\sqrt {1+i}\right )}{2}-\frac {i \dilog \left (\frac {1-\sqrt {5}+2 \sqrt {x +1}}{1-2 \sqrt {1-i}-\sqrt {5}}\right )}{2}-\frac {i \dilog \left (\frac {1-\sqrt {5}+2 \sqrt {x +1}}{1+2 \sqrt {1-i}-\sqrt {5}}\right )}{2}+\frac {i \dilog \left (\frac {1-\sqrt {5}+2 \sqrt {x +1}}{1-2 \sqrt {1+i}-\sqrt {5}}\right )}{2}+\frac {i \dilog \left (\frac {1-\sqrt {5}+2 \sqrt {x +1}}{1+2 \sqrt {1+i}-\sqrt {5}}\right )}{2}-\frac {i \dilog \left (\frac {1+\sqrt {5}+2 \sqrt {x +1}}{1-2 \sqrt {1-i}+\sqrt {5}}\right )}{2}-\frac {i \dilog \left (\frac {1+\sqrt {5}+2 \sqrt {x +1}}{1+2 \sqrt {1-i}+\sqrt {5}}\right )}{2}+\frac {i \dilog \left (\frac {1+\sqrt {5}+2 \sqrt {x +1}}{1-2 \sqrt {1+i}+\sqrt {5}}\right )}{2}+\frac {i \dilog \left (\frac {1+\sqrt {5}+2 \sqrt {x +1}}{1+2 \sqrt {1+i}+\sqrt {5}}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*I*dilog((1+5^(1/2)+2*(x+1)^(1/2))/(1+2*(1+I)^(1/2)+5^(1/2)))+1/2*I*ln((x+1)^(1/2)-(1+I)^(1/2))*ln((1-5^(1/
2)+2*(x+1)^(1/2))/(1+2*(1+I)^(1/2)-5^(1/2)))-1/2*I*ln((x+1)^(1/2)-(1-I)^(1/2))*ln((1+5^(1/2)+2*(x+1)^(1/2))/(1
+2*(1-I)^(1/2)+5^(1/2)))-1/2*I*dilog((1+5^(1/2)+2*(x+1)^(1/2))/(1-2*(1-I)^(1/2)+5^(1/2)))+1/2*I*ln((1+I)^(1/2)
+(x+1)^(1/2))*ln((1-5^(1/2)+2*(x+1)^(1/2))/(1-2*(1+I)^(1/2)-5^(1/2)))+1/2*I*ln((x+1)^(1/2)-(1+I)^(1/2))*ln((1+
5^(1/2)+2*(x+1)^(1/2))/(1+2*(1+I)^(1/2)+5^(1/2)))+1/2*I*dilog((1+5^(1/2)+2*(x+1)^(1/2))/(1-2*(1+I)^(1/2)+5^(1/
2)))-1/2*I*dilog((1+5^(1/2)+2*(x+1)^(1/2))/(1+2*(1-I)^(1/2)+5^(1/2)))-1/2*I*ln(x+(x+1)^(1/2))*ln((x+1)^(1/2)-(
1+I)^(1/2))-1/2*I*ln(x+(x+1)^(1/2))*ln((1+I)^(1/2)+(x+1)^(1/2))+1/2*I*ln(x+(x+1)^(1/2))*ln((1-I)^(1/2)+(x+1)^(
1/2))-1/2*I*dilog((1-5^(1/2)+2*(x+1)^(1/2))/(1-2*(1-I)^(1/2)-5^(1/2)))-1/2*I*dilog((1-5^(1/2)+2*(x+1)^(1/2))/(
1+2*(1-I)^(1/2)-5^(1/2)))-1/2*I*ln((1-I)^(1/2)+(x+1)^(1/2))*ln((1+5^(1/2)+2*(x+1)^(1/2))/(1-2*(1-I)^(1/2)+5^(1
/2)))+1/2*I*ln(x+(x+1)^(1/2))*ln((x+1)^(1/2)-(1-I)^(1/2))-1/2*I*ln((x+1)^(1/2)-(1-I)^(1/2))*ln((1-5^(1/2)+2*(x
+1)^(1/2))/(1+2*(1-I)^(1/2)-5^(1/2)))-1/2*I*ln((1-I)^(1/2)+(x+1)^(1/2))*ln((1-5^(1/2)+2*(x+1)^(1/2))/(1-2*(1-I
)^(1/2)-5^(1/2)))+1/2*I*ln((1+I)^(1/2)+(x+1)^(1/2))*ln((1+5^(1/2)+2*(x+1)^(1/2))/(1-2*(1+I)^(1/2)+5^(1/2)))+1/
2*I*dilog((1-5^(1/2)+2*(x+1)^(1/2))/(1-2*(1+I)^(1/2)-5^(1/2)))+1/2*I*dilog((1-5^(1/2)+2*(x+1)^(1/2))/(1+2*(1+I
)^(1/2)-5^(1/2)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (x + \sqrt {x + 1}\right )}{x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (x+\sqrt {x+1}\right )}{x^2+1} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (x + \sqrt {x + 1} \right )}}{x^{2} + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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