3.26 \(\int \log (x^2+\sqrt{1-x^2}) \, dx\)

Optimal. Leaf size=185 \[ x \log \left (x^2+\sqrt{1-x^2}\right )+\sqrt{\frac{1}{2} \left (1+\sqrt{5}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{1}{2} \left (1+\sqrt{5}\right )} x}{\sqrt{1-x^2}}\right )-\sqrt{\frac{1}{2} \left (\sqrt{5}-1\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{2} \left (\sqrt{5}-1\right )} x}{\sqrt{1-x^2}}\right )-2 x-\sin ^{-1}(x)+\sqrt{\frac{1}{2} \left (1+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )+\sqrt{\frac{1}{2} \left (\sqrt{5}-1\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right ) \]

[Out]

-2*x - ArcSin[x] + Sqrt[(1 + Sqrt[5])/2]*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x] + Sqrt[(1 + Sqrt[5])/2]*ArcTan[(Sqrt[
(1 + Sqrt[5])/2]*x)/Sqrt[1 - x^2]] + Sqrt[(-1 + Sqrt[5])/2]*ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x] - Sqrt[(-1 + Sqr
t[5])/2]*ArcTanh[(Sqrt[(-1 + Sqrt[5])/2]*x)/Sqrt[1 - x^2]] + x*Log[x^2 + Sqrt[1 - x^2]]

________________________________________________________________________________________

Rubi [A]  time = 0.976096, antiderivative size = 349, normalized size of antiderivative = 1.89, number of steps used = 31, number of rules used = 12, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {2548, 6742, 1293, 216, 1692, 377, 207, 203, 1166, 1130, 1174, 402} \[ x \log \left (x^2+\sqrt{1-x^2}\right )+2 \sqrt{\frac{1}{5} \left (2+\sqrt{5}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{1}{2} \left (1+\sqrt{5}\right )} x}{\sqrt{1-x^2}}\right )-\sqrt{\frac{1}{10} \left (1+\sqrt{5}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{1}{2} \left (1+\sqrt{5}\right )} x}{\sqrt{1-x^2}}\right )-\sqrt{\frac{1}{10} \left (\sqrt{5}-1\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{2} \left (\sqrt{5}-1\right )} x}{\sqrt{1-x^2}}\right )-2 \sqrt{\frac{1}{5} \left (\sqrt{5}-2\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{2} \left (\sqrt{5}-1\right )} x}{\sqrt{1-x^2}}\right )-2 x-\sin ^{-1}(x)+2 \sqrt{\frac{1}{5} \left (2+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )-\sqrt{\frac{1}{10} \left (1+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )+\sqrt{\frac{1}{10} \left (\sqrt{5}-1\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )+2 \sqrt{\frac{1}{5} \left (\sqrt{5}-2\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right ) \]

Warning: Unable to verify antiderivative.

[In]

Int[Log[x^2 + Sqrt[1 - x^2]],x]

[Out]

-2*x - ArcSin[x] - Sqrt[(1 + Sqrt[5])/10]*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x] + 2*Sqrt[(2 + Sqrt[5])/5]*ArcTan[Sqr
t[2/(1 + Sqrt[5])]*x] - Sqrt[(1 + Sqrt[5])/10]*ArcTan[(Sqrt[(1 + Sqrt[5])/2]*x)/Sqrt[1 - x^2]] + 2*Sqrt[(2 + S
qrt[5])/5]*ArcTan[(Sqrt[(1 + Sqrt[5])/2]*x)/Sqrt[1 - x^2]] + 2*Sqrt[(-2 + Sqrt[5])/5]*ArcTanh[Sqrt[2/(-1 + Sqr
t[5])]*x] + Sqrt[(-1 + Sqrt[5])/10]*ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x] - 2*Sqrt[(-2 + Sqrt[5])/5]*ArcTanh[(Sqrt
[(-1 + Sqrt[5])/2]*x)/Sqrt[1 - x^2]] - Sqrt[(-1 + Sqrt[5])/10]*ArcTanh[(Sqrt[(-1 + Sqrt[5])/2]*x)/Sqrt[1 - x^2
]] + x*Log[x^2 + Sqrt[1 - x^2]]

Rule 2548

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/u, x], x] /; InverseFunctionFr
eeQ[u, x]

Rule 6742

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

Rule 1293

Int[(((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(q_))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Dist[(
e*f^2)/c, Int[(f*x)^(m - 2)*(d + e*x^2)^(q - 1), x], x] - Dist[f^2/c, Int[((f*x)^(m - 2)*(d + e*x^2)^(q - 1)*S
imp[a*e - (c*d - b*e)*x^2, x])/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c,
0] &&  !IntegerQ[q] && GtQ[q, 0] && GtQ[m, 1] && LeQ[m, 3]

Rule 216

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

Rule 1692

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandInteg
rand[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, q}, x] && PolyQ[Px, x^2] && NeQ[b
^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 207

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

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 1166

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

Rule 1130

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

Rule 1174

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

Rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Dist[b/d, Int[(a + b*x^2)^(p - 1), x], x]
- Dist[(b*c - a*d)/d, Int[(a + b*x^2)^(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d,
0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4])

Rubi steps

\begin{align*} \int \log \left (x^2+\sqrt{1-x^2}\right ) \, dx &=x \log \left (x^2+\sqrt{1-x^2}\right )-\int \frac{x^2 \left (2-\frac{1}{\sqrt{1-x^2}}\right )}{x^2+\sqrt{1-x^2}} \, dx\\ &=x \log \left (x^2+\sqrt{1-x^2}\right )-\int \left (\frac{2 x^2}{x^2+\sqrt{1-x^2}}-\frac{x^2}{1-x^2+x^2 \sqrt{1-x^2}}\right ) \, dx\\ &=x \log \left (x^2+\sqrt{1-x^2}\right )-2 \int \frac{x^2}{x^2+\sqrt{1-x^2}} \, dx+\int \frac{x^2}{1-x^2+x^2 \sqrt{1-x^2}} \, dx\\ &=x \log \left (x^2+\sqrt{1-x^2}\right )-2 \int \left (1-\frac{x^2 \sqrt{1-x^2}}{-1+x^2+x^4}+\frac{1-x^2}{-1+x^2+x^4}\right ) \, dx+\int \left (\frac{1}{\sqrt{1-x^2}}-\frac{x^2}{-1+x^2+x^4}+\frac{\sqrt{1-x^2}}{-1+x^2+x^4}\right ) \, dx\\ &=-2 x+x \log \left (x^2+\sqrt{1-x^2}\right )+2 \int \frac{x^2 \sqrt{1-x^2}}{-1+x^2+x^4} \, dx-2 \int \frac{1-x^2}{-1+x^2+x^4} \, dx+\int \frac{1}{\sqrt{1-x^2}} \, dx-\int \frac{x^2}{-1+x^2+x^4} \, dx+\int \frac{\sqrt{1-x^2}}{-1+x^2+x^4} \, dx\\ &=-2 x+\sin ^{-1}(x)+x \log \left (x^2+\sqrt{1-x^2}\right )-2 \int \frac{1}{\sqrt{1-x^2}} \, dx-2 \int \frac{1-2 x^2}{\sqrt{1-x^2} \left (-1+x^2+x^4\right )} \, dx+\frac{2 \int \frac{\sqrt{1-x^2}}{1-\sqrt{5}+2 x^2} \, dx}{\sqrt{5}}-\frac{2 \int \frac{\sqrt{1-x^2}}{1+\sqrt{5}+2 x^2} \, dx}{\sqrt{5}}+\frac{1}{10} \left (-5+\sqrt{5}\right ) \int \frac{1}{\frac{1}{2}-\frac{\sqrt{5}}{2}+x^2} \, dx-\frac{1}{10} \left (5+\sqrt{5}\right ) \int \frac{1}{\frac{1}{2}+\frac{\sqrt{5}}{2}+x^2} \, dx-\frac{1}{5} \left (-5+3 \sqrt{5}\right ) \int \frac{1}{\frac{1}{2}-\frac{\sqrt{5}}{2}+x^2} \, dx+\frac{1}{5} \left (5+3 \sqrt{5}\right ) \int \frac{1}{\frac{1}{2}+\frac{\sqrt{5}}{2}+x^2} \, dx\\ &=-2 x-\sin ^{-1}(x)-\sqrt{\frac{1}{10} \left (1+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )+2 \sqrt{\frac{1}{5} \left (2+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )+2 \sqrt{\frac{1}{5} \left (-2+\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{-1+\sqrt{5}}} x\right )+\sqrt{\frac{1}{10} \left (-1+\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{-1+\sqrt{5}}} x\right )+x \log \left (x^2+\sqrt{1-x^2}\right )-2 \int \left (\frac{-2+\frac{4}{\sqrt{5}}}{\sqrt{1-x^2} \left (1-\sqrt{5}+2 x^2\right )}+\frac{-2-\frac{4}{\sqrt{5}}}{\sqrt{1-x^2} \left (1+\sqrt{5}+2 x^2\right )}\right ) \, dx-\frac{1}{5} \left (5-3 \sqrt{5}\right ) \int \frac{1}{\sqrt{1-x^2} \left (1-\sqrt{5}+2 x^2\right )} \, dx-\frac{1}{5} \left (5+3 \sqrt{5}\right ) \int \frac{1}{\sqrt{1-x^2} \left (1+\sqrt{5}+2 x^2\right )} \, dx\\ &=-2 x-\sin ^{-1}(x)-\sqrt{\frac{1}{10} \left (1+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )+2 \sqrt{\frac{1}{5} \left (2+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )+2 \sqrt{\frac{1}{5} \left (-2+\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{-1+\sqrt{5}}} x\right )+\sqrt{\frac{1}{10} \left (-1+\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{-1+\sqrt{5}}} x\right )+x \log \left (x^2+\sqrt{1-x^2}\right )-\frac{1}{5} \left (5-3 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{5}-\left (-3+\sqrt{5}\right ) x^2} \, dx,x,\frac{x}{\sqrt{1-x^2}}\right )+\frac{1}{5} \left (4 \left (5-2 \sqrt{5}\right )\right ) \int \frac{1}{\sqrt{1-x^2} \left (1-\sqrt{5}+2 x^2\right )} \, dx+\frac{1}{5} \left (4 \left (5+2 \sqrt{5}\right )\right ) \int \frac{1}{\sqrt{1-x^2} \left (1+\sqrt{5}+2 x^2\right )} \, dx-\frac{1}{5} \left (5+3 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{5}-\left (-3-\sqrt{5}\right ) x^2} \, dx,x,\frac{x}{\sqrt{1-x^2}}\right )\\ &=-2 x-\sin ^{-1}(x)-\sqrt{\frac{1}{10} \left (1+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )+2 \sqrt{\frac{1}{5} \left (2+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )-\sqrt{\frac{1}{10} \left (1+\sqrt{5}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{1}{2} \left (1+\sqrt{5}\right )} x}{\sqrt{1-x^2}}\right )+2 \sqrt{\frac{1}{5} \left (-2+\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{-1+\sqrt{5}}} x\right )+\sqrt{\frac{1}{10} \left (-1+\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{-1+\sqrt{5}}} x\right )-\sqrt{\frac{1}{10} \left (-1+\sqrt{5}\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{2} \left (-1+\sqrt{5}\right )} x}{\sqrt{1-x^2}}\right )+x \log \left (x^2+\sqrt{1-x^2}\right )+\frac{1}{5} \left (4 \left (5-2 \sqrt{5}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{5}-\left (-3+\sqrt{5}\right ) x^2} \, dx,x,\frac{x}{\sqrt{1-x^2}}\right )+\frac{1}{5} \left (4 \left (5+2 \sqrt{5}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{5}-\left (-3-\sqrt{5}\right ) x^2} \, dx,x,\frac{x}{\sqrt{1-x^2}}\right )\\ &=-2 x-\sin ^{-1}(x)-\sqrt{\frac{1}{10} \left (1+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )+2 \sqrt{\frac{1}{5} \left (2+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )-\sqrt{\frac{1}{10} \left (1+\sqrt{5}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{1}{2} \left (1+\sqrt{5}\right )} x}{\sqrt{1-x^2}}\right )+2 \sqrt{\frac{1}{5} \left (2+\sqrt{5}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{1}{2} \left (1+\sqrt{5}\right )} x}{\sqrt{1-x^2}}\right )+2 \sqrt{\frac{1}{5} \left (-2+\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{-1+\sqrt{5}}} x\right )+\sqrt{\frac{1}{10} \left (-1+\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{-1+\sqrt{5}}} x\right )-2 \sqrt{\frac{1}{5} \left (-2+\sqrt{5}\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{2} \left (-1+\sqrt{5}\right )} x}{\sqrt{1-x^2}}\right )-\sqrt{\frac{1}{10} \left (-1+\sqrt{5}\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{2} \left (-1+\sqrt{5}\right )} x}{\sqrt{1-x^2}}\right )+x \log \left (x^2+\sqrt{1-x^2}\right )\\ \end{align*}

Mathematica [C]  time = 0.524187, size = 910, normalized size = 4.92 \[ \frac{4 \sqrt{5} \log \left (x^2+\sqrt{1-x^2}\right ) x-8 \sqrt{5} x-4 \sqrt{5} \sin ^{-1}(x)+\sqrt{10 \left (-1+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )+5 \sqrt{2 \left (-1+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )-\left (-5+\sqrt{5}\right ) \sqrt{2 \left (1+\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{-1+\sqrt{5}}} x\right )+3 \sqrt{5 \left (2+\sqrt{5}\right )} \log \left (x-\sqrt{\frac{1}{2} \left (-1+\sqrt{5}\right )}\right )-5 \sqrt{2+\sqrt{5}} \log \left (x-\sqrt{\frac{1}{2} \left (-1+\sqrt{5}\right )}\right )-3 \sqrt{5 \left (2+\sqrt{5}\right )} \log \left (x+\sqrt{\frac{1}{2} \left (-1+\sqrt{5}\right )}\right )+5 \sqrt{2+\sqrt{5}} \log \left (x+\sqrt{\frac{1}{2} \left (-1+\sqrt{5}\right )}\right )-3 i \sqrt{5 \left (-2+\sqrt{5}\right )} \log \left (x-i \sqrt{\frac{1}{2} \left (1+\sqrt{5}\right )}\right )-5 i \sqrt{-2+\sqrt{5}} \log \left (x-i \sqrt{\frac{1}{2} \left (1+\sqrt{5}\right )}\right )+3 i \sqrt{5 \left (-2+\sqrt{5}\right )} \log \left (x+i \sqrt{\frac{1}{2} \left (1+\sqrt{5}\right )}\right )+5 i \sqrt{-2+\sqrt{5}} \log \left (x+i \sqrt{\frac{1}{2} \left (1+\sqrt{5}\right )}\right )+3 i \sqrt{5 \left (-2+\sqrt{5}\right )} \log \left (-i \sqrt{2 \left (1+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )} \sqrt{1-x^2}+2\right )+5 i \sqrt{-2+\sqrt{5}} \log \left (-i \sqrt{2 \left (1+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )} \sqrt{1-x^2}+2\right )-3 i \sqrt{5 \left (-2+\sqrt{5}\right )} \log \left (i \sqrt{2 \left (1+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )} \sqrt{1-x^2}+2\right )-5 i \sqrt{-2+\sqrt{5}} \log \left (i \sqrt{2 \left (1+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )} \sqrt{1-x^2}+2\right )-3 \sqrt{5 \left (2+\sqrt{5}\right )} \log \left (-\sqrt{2 \left (-1+\sqrt{5}\right )} x+\sqrt{2} \sqrt{\left (-3+\sqrt{5}\right ) \left (x^2-1\right )}+2\right )+5 \sqrt{2+\sqrt{5}} \log \left (-\sqrt{2 \left (-1+\sqrt{5}\right )} x+\sqrt{2} \sqrt{\left (-3+\sqrt{5}\right ) \left (x^2-1\right )}+2\right )+3 \sqrt{5 \left (2+\sqrt{5}\right )} \log \left (\sqrt{2 \left (-1+\sqrt{5}\right )} x+\sqrt{2} \sqrt{\left (-3+\sqrt{5}\right ) \left (x^2-1\right )}+2\right )-5 \sqrt{2+\sqrt{5}} \log \left (\sqrt{2 \left (-1+\sqrt{5}\right )} x+\sqrt{2} \sqrt{\left (-3+\sqrt{5}\right ) \left (x^2-1\right )}+2\right )}{4 \sqrt{5}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Log[x^2 + Sqrt[1 - x^2]],x]

[Out]

(-8*Sqrt[5]*x - 4*Sqrt[5]*ArcSin[x] + 5*Sqrt[2*(-1 + Sqrt[5])]*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x] + Sqrt[10*(-1 +
Sqrt[5])]*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x] - (-5 + Sqrt[5])*Sqrt[2*(1 + Sqrt[5])]*ArcTanh[Sqrt[2/(-1 + Sqrt[5]
)]*x] - 5*Sqrt[2 + Sqrt[5]]*Log[-Sqrt[(-1 + Sqrt[5])/2] + x] + 3*Sqrt[5*(2 + Sqrt[5])]*Log[-Sqrt[(-1 + Sqrt[5]
)/2] + x] + 5*Sqrt[2 + Sqrt[5]]*Log[Sqrt[(-1 + Sqrt[5])/2] + x] - 3*Sqrt[5*(2 + Sqrt[5])]*Log[Sqrt[(-1 + Sqrt[
5])/2] + x] - (5*I)*Sqrt[-2 + Sqrt[5]]*Log[(-I)*Sqrt[(1 + Sqrt[5])/2] + x] - (3*I)*Sqrt[5*(-2 + Sqrt[5])]*Log[
(-I)*Sqrt[(1 + Sqrt[5])/2] + x] + (5*I)*Sqrt[-2 + Sqrt[5]]*Log[I*Sqrt[(1 + Sqrt[5])/2] + x] + (3*I)*Sqrt[5*(-2
+ Sqrt[5])]*Log[I*Sqrt[(1 + Sqrt[5])/2] + x] + 4*Sqrt[5]*x*Log[x^2 + Sqrt[1 - x^2]] + (5*I)*Sqrt[-2 + Sqrt[5]
]*Log[2 - I*Sqrt[2*(1 + Sqrt[5])]*x + Sqrt[2*(3 + Sqrt[5])]*Sqrt[1 - x^2]] + (3*I)*Sqrt[5*(-2 + Sqrt[5])]*Log[
2 - I*Sqrt[2*(1 + Sqrt[5])]*x + Sqrt[2*(3 + Sqrt[5])]*Sqrt[1 - x^2]] - (5*I)*Sqrt[-2 + Sqrt[5]]*Log[2 + I*Sqrt
[2*(1 + Sqrt[5])]*x + Sqrt[2*(3 + Sqrt[5])]*Sqrt[1 - x^2]] - (3*I)*Sqrt[5*(-2 + Sqrt[5])]*Log[2 + I*Sqrt[2*(1
+ Sqrt[5])]*x + Sqrt[2*(3 + Sqrt[5])]*Sqrt[1 - x^2]] + 5*Sqrt[2 + Sqrt[5]]*Log[2 - Sqrt[2*(-1 + Sqrt[5])]*x +
Sqrt[2]*Sqrt[(-3 + Sqrt[5])*(-1 + x^2)]] - 3*Sqrt[5*(2 + Sqrt[5])]*Log[2 - Sqrt[2*(-1 + Sqrt[5])]*x + Sqrt[2]*
Sqrt[(-3 + Sqrt[5])*(-1 + x^2)]] - 5*Sqrt[2 + Sqrt[5]]*Log[2 + Sqrt[2*(-1 + Sqrt[5])]*x + Sqrt[2]*Sqrt[(-3 + S
qrt[5])*(-1 + x^2)]] + 3*Sqrt[5*(2 + Sqrt[5])]*Log[2 + Sqrt[2*(-1 + Sqrt[5])]*x + Sqrt[2]*Sqrt[(-3 + Sqrt[5])*
(-1 + x^2)]])/(4*Sqrt[5])

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Maple [B]  time = 0.129, size = 392, normalized size = 2.1 \begin{align*} x\ln \left ({x}^{2}+\sqrt{-{x}^{2}+1} \right ) +{\frac{1}{\sqrt{2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }+{\frac{\sqrt{5}}{\sqrt{2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }-{\frac{1}{\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{\sqrt{5}}{\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-2\,x-{\frac{3}{2\,\sqrt{2+\sqrt{5}}}\arctan \left ({\frac{1}{x\sqrt{2+\sqrt{5}}} \left ( \sqrt{-{x}^{2}+1}-1 \right ) } \right ) }-{\frac{\sqrt{5}}{2\,\sqrt{2+\sqrt{5}}}\arctan \left ({\frac{1}{x\sqrt{2+\sqrt{5}}} \left ( \sqrt{-{x}^{2}+1}-1 \right ) } \right ) }+{\frac{3}{2\,\sqrt{-2+\sqrt{5}}}{\it Artanh} \left ({\frac{1}{x\sqrt{-2+\sqrt{5}}} \left ( \sqrt{-{x}^{2}+1}-1 \right ) } \right ) }-{\frac{\sqrt{5}}{2\,\sqrt{-2+\sqrt{5}}}{\it Artanh} \left ({\frac{1}{x\sqrt{-2+\sqrt{5}}} \left ( \sqrt{-{x}^{2}+1}-1 \right ) } \right ) }-{\frac{\sqrt{5}}{2\,\sqrt{-2+\sqrt{5}}}\arctan \left ({\frac{1}{x\sqrt{-2+\sqrt{5}}} \left ( \sqrt{-{x}^{2}+1}-1 \right ) } \right ) }+{\frac{1}{2\,\sqrt{-2+\sqrt{5}}}\arctan \left ({\frac{1}{x\sqrt{-2+\sqrt{5}}} \left ( \sqrt{-{x}^{2}+1}-1 \right ) } \right ) }-{\frac{\sqrt{5}}{2\,\sqrt{2+\sqrt{5}}}{\it Artanh} \left ({\frac{1}{x\sqrt{2+\sqrt{5}}} \left ( \sqrt{-{x}^{2}+1}-1 \right ) } \right ) }-{\frac{1}{2\,\sqrt{2+\sqrt{5}}}{\it Artanh} \left ({\frac{1}{x\sqrt{2+\sqrt{5}}} \left ( \sqrt{-{x}^{2}+1}-1 \right ) } \right ) }-\arcsin \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*ln(x^2+(-x^2+1)^(1/2))+1/(2+2*5^(1/2))^(1/2)*arctan(2*x/(2+2*5^(1/2))^(1/2))+5^(1/2)/(2+2*5^(1/2))^(1/2)*arc
tan(2*x/(2+2*5^(1/2))^(1/2))-1/(-2+2*5^(1/2))^(1/2)*arctanh(2*x/(-2+2*5^(1/2))^(1/2))+5^(1/2)/(-2+2*5^(1/2))^(
1/2)*arctanh(2*x/(-2+2*5^(1/2))^(1/2))-2*x-3/2/(2+5^(1/2))^(1/2)*arctan(((-x^2+1)^(1/2)-1)/x/(2+5^(1/2))^(1/2)
)-1/2*5^(1/2)/(2+5^(1/2))^(1/2)*arctan(((-x^2+1)^(1/2)-1)/x/(2+5^(1/2))^(1/2))+3/2/(-2+5^(1/2))^(1/2)*arctanh(
((-x^2+1)^(1/2)-1)/x/(-2+5^(1/2))^(1/2))-1/2*5^(1/2)/(-2+5^(1/2))^(1/2)*arctanh(((-x^2+1)^(1/2)-1)/x/(-2+5^(1/
2))^(1/2))-1/2*5^(1/2)/(-2+5^(1/2))^(1/2)*arctan(((-x^2+1)^(1/2)-1)/x/(-2+5^(1/2))^(1/2))+1/2/(-2+5^(1/2))^(1/
2)*arctan(((-x^2+1)^(1/2)-1)/x/(-2+5^(1/2))^(1/2))-1/2*5^(1/2)/(2+5^(1/2))^(1/2)*arctanh(((-x^2+1)^(1/2)-1)/x/
(2+5^(1/2))^(1/2))-1/2/(2+5^(1/2))^(1/2)*arctanh(((-x^2+1)^(1/2)-1)/x/(2+5^(1/2))^(1/2))-arcsin(x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} x \log \left (x^{2} + \sqrt{x + 1} \sqrt{-x + 1}\right ) - x - \int \frac{x^{4} - 2 \, x^{2}}{x^{4} - x^{2} +{\left (x^{2} - 1\right )} e^{\left (\frac{1}{2} \, \log \left (x + 1\right ) + \frac{1}{2} \, \log \left (-x + 1\right )\right )}}\,{d x} + \frac{1}{2} \, \log \left (x + 1\right ) - \frac{1}{2} \, \log \left (-x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(x^2 + sqrt(x + 1)*sqrt(-x + 1)) - x - integrate((x^4 - 2*x^2)/(x^4 - x^2 + (x^2 - 1)*e^(1/2*log(x + 1) +
1/2*log(-x + 1))), x) + 1/2*log(x + 1) - 1/2*log(-x + 1)

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Fricas [B]  time = 2.73356, size = 1277, normalized size = 6.9 \begin{align*} -\sqrt{2} \sqrt{\sqrt{5} + 1} \arctan \left (\frac{1}{8} \, \sqrt{4 \, x^{2} + 2 \, \sqrt{5} + 2}{\left (\sqrt{5} \sqrt{2} - \sqrt{2}\right )} \sqrt{\sqrt{5} + 1} - \frac{1}{4} \,{\left (\sqrt{5} \sqrt{2} x - \sqrt{2} x\right )} \sqrt{\sqrt{5} + 1}\right ) - \sqrt{2} \sqrt{\sqrt{5} + 1} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{-x^{2} + 1}{\left (\sqrt{5} \sqrt{2} - \sqrt{2}\right )} + \sqrt{5} \sqrt{2} - \sqrt{2}\right )} \sqrt{\sqrt{5} + 1} \sqrt{\frac{x^{4} - 4 \, x^{2} - \sqrt{5}{\left (x^{4} - 2 \, x^{2}\right )} - 2 \,{\left (\sqrt{5} x^{2} - x^{2} + 2\right )} \sqrt{-x^{2} + 1} + 4}{x^{4}}} + 2 \, \sqrt{-x^{2} + 1}{\left (\sqrt{5} \sqrt{2} - \sqrt{2}\right )} \sqrt{\sqrt{5} + 1}}{8 \, x}\right ) + x \log \left (x^{2} + \sqrt{-x^{2} + 1}\right ) + \frac{1}{4} \, \sqrt{2} \sqrt{\sqrt{5} - 1} \log \left (2 \, x + \sqrt{2} \sqrt{\sqrt{5} - 1}\right ) - \frac{1}{4} \, \sqrt{2} \sqrt{\sqrt{5} - 1} \log \left (2 \, x - \sqrt{2} \sqrt{\sqrt{5} - 1}\right ) + \frac{1}{4} \, \sqrt{2} \sqrt{\sqrt{5} - 1} \log \left (-\frac{2 \, x^{2} +{\left (\sqrt{2} \sqrt{-x^{2} + 1} x - \sqrt{2} x\right )} \sqrt{\sqrt{5} - 1} + 2 \, \sqrt{-x^{2} + 1} - 2}{x^{2}}\right ) - \frac{1}{4} \, \sqrt{2} \sqrt{\sqrt{5} - 1} \log \left (-\frac{2 \, x^{2} -{\left (\sqrt{2} \sqrt{-x^{2} + 1} x - \sqrt{2} x\right )} \sqrt{\sqrt{5} - 1} + 2 \, \sqrt{-x^{2} + 1} - 2}{x^{2}}\right ) - 2 \, x + 2 \, \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(2)*sqrt(sqrt(5) + 1)*arctan(1/8*sqrt(4*x^2 + 2*sqrt(5) + 2)*(sqrt(5)*sqrt(2) - sqrt(2))*sqrt(sqrt(5) + 1
) - 1/4*(sqrt(5)*sqrt(2)*x - sqrt(2)*x)*sqrt(sqrt(5) + 1)) - sqrt(2)*sqrt(sqrt(5) + 1)*arctan(1/8*(sqrt(2)*(sq
rt(-x^2 + 1)*(sqrt(5)*sqrt(2) - sqrt(2)) + sqrt(5)*sqrt(2) - sqrt(2))*sqrt(sqrt(5) + 1)*sqrt((x^4 - 4*x^2 - sq
rt(5)*(x^4 - 2*x^2) - 2*(sqrt(5)*x^2 - x^2 + 2)*sqrt(-x^2 + 1) + 4)/x^4) + 2*sqrt(-x^2 + 1)*(sqrt(5)*sqrt(2) -
sqrt(2))*sqrt(sqrt(5) + 1))/x) + x*log(x^2 + sqrt(-x^2 + 1)) + 1/4*sqrt(2)*sqrt(sqrt(5) - 1)*log(2*x + sqrt(2
)*sqrt(sqrt(5) - 1)) - 1/4*sqrt(2)*sqrt(sqrt(5) - 1)*log(2*x - sqrt(2)*sqrt(sqrt(5) - 1)) + 1/4*sqrt(2)*sqrt(s
qrt(5) - 1)*log(-(2*x^2 + (sqrt(2)*sqrt(-x^2 + 1)*x - sqrt(2)*x)*sqrt(sqrt(5) - 1) + 2*sqrt(-x^2 + 1) - 2)/x^2
) - 1/4*sqrt(2)*sqrt(sqrt(5) - 1)*log(-(2*x^2 - (sqrt(2)*sqrt(-x^2 + 1)*x - sqrt(2)*x)*sqrt(sqrt(5) - 1) + 2*s
qrt(-x^2 + 1) - 2)/x^2) - 2*x + 2*arctan((sqrt(-x^2 + 1) - 1)/x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.2163, size = 406, normalized size = 2.19 \begin{align*} x \log \left (x^{2} + \sqrt{-x^{2} + 1}\right ) - \frac{1}{2} \, \pi \mathrm{sgn}\left (x\right ) + \frac{1}{2} \, \sqrt{2 \, \sqrt{5} + 2} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}}}\right ) - \frac{1}{2} \, \sqrt{2 \, \sqrt{5} + 2} \arctan \left (-\frac{\frac{x}{\sqrt{-x^{2} + 1} - 1} - \frac{\sqrt{-x^{2} + 1} - 1}{x}}{\sqrt{2 \, \sqrt{5} + 2}}\right ) + \frac{1}{4} \, \sqrt{2 \, \sqrt{5} - 2} \log \left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) - \frac{1}{4} \, \sqrt{2 \, \sqrt{5} - 2} \log \left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) - \frac{1}{4} \, \sqrt{2 \, \sqrt{5} - 2} \log \left ({\left | \sqrt{2 \, \sqrt{5} - 2} - \frac{x}{\sqrt{-x^{2} + 1} - 1} + \frac{\sqrt{-x^{2} + 1} - 1}{x} \right |}\right ) + \frac{1}{4} \, \sqrt{2 \, \sqrt{5} - 2} \log \left ({\left | -\sqrt{2 \, \sqrt{5} - 2} - \frac{x}{\sqrt{-x^{2} + 1} - 1} + \frac{\sqrt{-x^{2} + 1} - 1}{x} \right |}\right ) - 2 \, x - \arctan \left (-\frac{x{\left (\frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(x^2 + sqrt(-x^2 + 1)) - 1/2*pi*sgn(x) + 1/2*sqrt(2*sqrt(5) + 2)*arctan(x/sqrt(1/2*sqrt(5) + 1/2)) - 1/2*
sqrt(2*sqrt(5) + 2)*arctan(-(x/(sqrt(-x^2 + 1) - 1) - (sqrt(-x^2 + 1) - 1)/x)/sqrt(2*sqrt(5) + 2)) + 1/4*sqrt(
2*sqrt(5) - 2)*log(abs(x + sqrt(1/2*sqrt(5) - 1/2))) - 1/4*sqrt(2*sqrt(5) - 2)*log(abs(x - sqrt(1/2*sqrt(5) -
1/2))) - 1/4*sqrt(2*sqrt(5) - 2)*log(abs(sqrt(2*sqrt(5) - 2) - x/(sqrt(-x^2 + 1) - 1) + (sqrt(-x^2 + 1) - 1)/x
)) + 1/4*sqrt(2*sqrt(5) - 2)*log(abs(-sqrt(2*sqrt(5) - 2) - x/(sqrt(-x^2 + 1) - 1) + (sqrt(-x^2 + 1) - 1)/x))
- 2*x - arctan(-1/2*x*((sqrt(-x^2 + 1) - 1)^2/x^2 - 1)/(sqrt(-x^2 + 1) - 1))