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

Optimal. Leaf size=97 \[ x \log \left (\sqrt{x^2+1} x+1\right )+\sqrt{2 \left (1+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\sqrt{5}-2} \left (\sqrt{x^2+1}+x\right )\right )-\sqrt{2 \left (\sqrt{5}-1\right )} \tanh ^{-1}\left (\sqrt{2+\sqrt{5}} \left (\sqrt{x^2+1}+x\right )\right )-2 x \]

[Out]

-2*x + Sqrt[2*(1 + Sqrt[5])]*ArcTan[Sqrt[-2 + Sqrt[5]]*(x + Sqrt[1 + x^2])] - Sqrt[2*(-1 + Sqrt[5])]*ArcTanh[S
qrt[2 + Sqrt[5]]*(x + Sqrt[1 + x^2])] + x*Log[1 + x*Sqrt[1 + x^2]]

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Rubi [B]  time = 0.668574, antiderivative size = 332, normalized size of antiderivative = 3.42, number of steps used = 32, number of rules used = 13, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.929, Rules used = {2548, 6742, 261, 1130, 203, 207, 1251, 824, 707, 1093, 1166, 1247, 699} \[ x \log \left (\sqrt{x^2+1} x+1\right )+\sqrt{\frac{2}{5} \left (\sqrt{5}-1\right )} \tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} \sqrt{x^2+1}\right )+\sqrt{\frac{2}{5 \left (\sqrt{5}-1\right )}} \tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} \sqrt{x^2+1}\right )-\sqrt{\frac{2}{5} \left (1+\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} \sqrt{x^2+1}\right )+\sqrt{\frac{2}{5 \left (1+\sqrt{5}\right )}} \tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} \sqrt{x^2+1}\right )-2 x+2 \sqrt{\frac{1}{5} \left (2+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )-\sqrt{\frac{1}{10} \left (1+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )+\sqrt{\frac{1}{10} \left (\sqrt{5}-1\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )+2 \sqrt{\frac{1}{5} \left (\sqrt{5}-2\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right ) \]

Warning: Unable to verify antiderivative.

[In]

Int[Log[1 + x*Sqrt[1 + x^2]],x]

[Out]

-2*x - Sqrt[(1 + Sqrt[5])/10]*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x] + 2*Sqrt[(2 + Sqrt[5])/5]*ArcTan[Sqrt[2/(1 + Sqr
t[5])]*x] + Sqrt[2/(5*(-1 + Sqrt[5]))]*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*Sqrt[1 + x^2]] + Sqrt[(2*(-1 + Sqrt[5]))/
5]*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*Sqrt[1 + x^2]] + 2*Sqrt[(-2 + Sqrt[5])/5]*ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x] +
Sqrt[(-1 + Sqrt[5])/10]*ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x] + Sqrt[2/(5*(1 + Sqrt[5]))]*ArcTanh[Sqrt[2/(1 + Sqr
t[5])]*Sqrt[1 + x^2]] - Sqrt[(2*(1 + Sqrt[5]))/5]*ArcTanh[Sqrt[2/(1 + Sqrt[5])]*Sqrt[1 + x^2]] + x*Log[1 + x*S
qrt[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 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

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 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 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 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 824

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(g
*(d + e*x)^m)/(c*m), x] + Dist[1/c, Int[((d + e*x)^(m - 1)*Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, 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] && FractionQ[m] && GtQ[m, 0]

Rule 707

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

Rule 1093

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

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 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 699

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

Rubi steps

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

Mathematica [A]  time = 0.358246, size = 194, normalized size = 2. \[ x \log \left (\sqrt{x^2+1} x+1\right )-\frac{\sqrt{2 \left (\sqrt{5}-1\right )} \tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} \sqrt{x^2+1}\right )}{1-\sqrt{5}}-\sqrt{\frac{2}{1+\sqrt{5}}} \tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} \sqrt{x^2+1}\right )-2 x+\frac{\left (5+\sqrt{5}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{10 \left (1+\sqrt{5}\right )}}-\frac{\left (\sqrt{5}-5\right ) \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{10 \left (\sqrt{5}-1\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[1 + x*Sqrt[1 + x^2]],x]

[Out]

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

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Maple [B]  time = 0.143, size = 426, normalized size = 4.4 \begin{align*} x\ln \left ( 1+x\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\,\sqrt{5}}{10\,\sqrt{2+\sqrt{5}}}\arctan \left ({\frac{1}{\sqrt{2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) }-{\frac{1}{2\,\sqrt{2+\sqrt{5}}}\arctan \left ({\frac{1}{\sqrt{2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) }-{\frac{3\,\sqrt{5}}{10\,\sqrt{-2+\sqrt{5}}}{\it Artanh} \left ({\frac{1}{\sqrt{-2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) }+{\frac{1}{2\,\sqrt{-2+\sqrt{5}}}{\it Artanh} \left ({\frac{1}{\sqrt{-2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) }+{\frac{1}{2\,\sqrt{-2+\sqrt{5}}}\arctan \left ({\frac{1}{\sqrt{-2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) }-{\frac{\sqrt{5}}{2\,\sqrt{-2+\sqrt{5}}}\arctan \left ({\frac{1}{\sqrt{-2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) }-{\frac{1}{2\,\sqrt{2+\sqrt{5}}}{\it Artanh} \left ({\frac{1}{\sqrt{2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) }-{\frac{\sqrt{5}}{2\,\sqrt{2+\sqrt{5}}}{\it Artanh} \left ({\frac{1}{\sqrt{2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) }+{\frac{2\,\sqrt{2+\sqrt{5}}\sqrt{5}}{5}\arctan \left ({\frac{1}{\sqrt{2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) }-{\frac{2\,\sqrt{-2+\sqrt{5}}\sqrt{5}}{5}{\it Artanh} \left ({\frac{1}{\sqrt{-2+\sqrt{5}}} \left ( \sqrt{{x}^{2}+1}-x \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*ln(1+x*(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)*arct
an(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-3/10*5^(1/2)/(2+5^(1/2))^(1/2)*arctan(((x^2+1)^(1/2)-x)/(2+5^(1/2))^
(1/2))-1/2/(2+5^(1/2))^(1/2)*arctan(((x^2+1)^(1/2)-x)/(2+5^(1/2))^(1/2))-3/10*5^(1/2)/(-2+5^(1/2))^(1/2)*arcta
nh(((x^2+1)^(1/2)-x)/(-2+5^(1/2))^(1/2))+1/2/(-2+5^(1/2))^(1/2)*arctanh(((x^2+1)^(1/2)-x)/(-2+5^(1/2))^(1/2))+
1/2/(-2+5^(1/2))^(1/2)*arctan(((x^2+1)^(1/2)-x)/(-2+5^(1/2))^(1/2))-1/2*5^(1/2)/(-2+5^(1/2))^(1/2)*arctan(((x^
2+1)^(1/2)-x)/(-2+5^(1/2))^(1/2))-1/2/(2+5^(1/2))^(1/2)*arctanh(((x^2+1)^(1/2)-x)/(2+5^(1/2))^(1/2))-1/2*5^(1/
2)/(2+5^(1/2))^(1/2)*arctanh(((x^2+1)^(1/2)-x)/(2+5^(1/2))^(1/2))+2/5*(2+5^(1/2))^(1/2)*5^(1/2)*arctan(((x^2+1
)^(1/2)-x)/(2+5^(1/2))^(1/2))-2/5*(-2+5^(1/2))^(1/2)*5^(1/2)*arctanh(((x^2+1)^(1/2)-x)/(-2+5^(1/2))^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(sqrt(x^2 + 1)*x + 1) - 2*x + arctan(x) + integrate((2*x^2 + 1)/(x^2 + (x^3 + x)*sqrt(x^2 + 1) + 1), x)

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Fricas [B]  time = 2.90299, size = 1222, normalized size = 12.6 \begin{align*} -\sqrt{2} \sqrt{\sqrt{5} + 1} \arctan \left (\frac{1}{4} \, \sqrt{2} \sqrt{4 \, x^{4} + 4 \, x^{2} + \sqrt{5}{\left (2 \, x^{2} + 1\right )} - 2 \,{\left (2 \, x^{3} + \sqrt{5} x + x\right )} \sqrt{x^{2} + 1} + 1}{\left (\sqrt{2} x + \sqrt{2} \sqrt{x^{2} + 1}\right )} \sqrt{\sqrt{5} + 1} - \frac{1}{2} \, \sqrt{2} \sqrt{x^{2} + 1} \sqrt{\sqrt{5} + 1}\right ) - \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 ) + \frac{1}{4} \, \sqrt{2} \sqrt{\sqrt{5} - 1} \log \left (4 \, x^{2} - 4 \, \sqrt{x^{2} + 1} x +{\left (\sqrt{5} \sqrt{2} x - \sqrt{x^{2} + 1}{\left (\sqrt{5} \sqrt{2} + \sqrt{2}\right )} + \sqrt{2} x\right )} \sqrt{\sqrt{5} - 1} + 4\right ) - \frac{1}{4} \, \sqrt{2} \sqrt{\sqrt{5} - 1} \log \left (4 \, x^{2} - 4 \, \sqrt{x^{2} + 1} x -{\left (\sqrt{5} \sqrt{2} x - \sqrt{x^{2} + 1}{\left (\sqrt{5} \sqrt{2} + \sqrt{2}\right )} + \sqrt{2} x\right )} \sqrt{\sqrt{5} - 1} + 4\right ) + x \log \left (\sqrt{x^{2} + 1} x + 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 ) - 2 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(2)*sqrt(sqrt(5) + 1)*arctan(1/4*sqrt(2)*sqrt(4*x^4 + 4*x^2 + sqrt(5)*(2*x^2 + 1) - 2*(2*x^3 + sqrt(5)*x
+ x)*sqrt(x^2 + 1) + 1)*(sqrt(2)*x + sqrt(2)*sqrt(x^2 + 1))*sqrt(sqrt(5) + 1) - 1/2*sqrt(2)*sqrt(x^2 + 1)*sqrt
(sqrt(5) + 1)) - 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)) + 1/4*sqrt(2)*sqrt(sqrt(5) - 1)*log
(4*x^2 - 4*sqrt(x^2 + 1)*x + (sqrt(5)*sqrt(2)*x - sqrt(x^2 + 1)*(sqrt(5)*sqrt(2) + sqrt(2)) + sqrt(2)*x)*sqrt(
sqrt(5) - 1) + 4) - 1/4*sqrt(2)*sqrt(sqrt(5) - 1)*log(4*x^2 - 4*sqrt(x^2 + 1)*x - (sqrt(5)*sqrt(2)*x - sqrt(x^
2 + 1)*(sqrt(5)*sqrt(2) + sqrt(2)) + sqrt(2)*x)*sqrt(sqrt(5) - 1) + 4) + x*log(sqrt(x^2 + 1)*x + 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)*s
qrt(sqrt(5) - 1)) - 2*x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.27549, size = 317, normalized size = 3.27 \begin{align*} x \log \left (\sqrt{x^{2} + 1} x + 1\right ) + \frac{1}{2} \, \sqrt{2 \, \sqrt{5} + 2} \arctan \left (-\frac{x - \sqrt{x^{2} + 1} + \frac{1}{x - \sqrt{x^{2} + 1}}}{\sqrt{2 \, \sqrt{5} - 2}}\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}{4} \, \sqrt{2 \, \sqrt{5} - 2} \log \left (-x + \sqrt{x^{2} + 1} + \sqrt{2 \, \sqrt{5} + 2} - \frac{1}{x - \sqrt{x^{2} + 1}}\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 | -x + \sqrt{x^{2} + 1} - \sqrt{2 \, \sqrt{5} + 2} - \frac{1}{x - \sqrt{x^{2} + 1}} \right |}\right ) - 2 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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