\(\int \frac {(1+x^4) \sqrt {x+\sqrt {1+x^2}}}{-1+x^4} \, dx\) [2866]

Optimal result

Integrand size = 28, antiderivative size = 304 \[ \int \frac {\left (1+x^4\right ) \sqrt {x+\sqrt {1+x^2}}}{-1+x^4} \, dx=-\frac {1}{\sqrt {x+\sqrt {1+x^2}}}+\frac {1}{3} \left (x+\sqrt {1+x^2}\right )^{3/2}-\sqrt {-1+\sqrt {2}} \arctan \left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-1+\sqrt {2}}}\right )+\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {1+\sqrt {2}}}\right )-\sqrt {2} \arctan \left (\frac {-\frac {1}{\sqrt {2}}+\frac {x}{\sqrt {2}}+\frac {\sqrt {1+x^2}}{\sqrt {2}}}{\sqrt {x+\sqrt {1+x^2}}}\right )+\sqrt {-1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-1+\sqrt {2}}}\right )-\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {1+\sqrt {2}}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\frac {1}{\sqrt {2}}+\frac {x}{\sqrt {2}}+\frac {\sqrt {1+x^2}}{\sqrt {2}}}{\sqrt {x+\sqrt {1+x^2}}}\right ) \]

[Out]

-1/(x+(x^2+1)^(1/2))^(1/2)+1/3*(x+(x^2+1)^(1/2))^(3/2)-(2^(1/2)-1)^(1/2)*arctan((x+(x^2+1)^(1/2))^(1/2)/(2^(1/
2)-1)^(1/2))+(1+2^(1/2))^(1/2)*arctan((x+(x^2+1)^(1/2))^(1/2)/(1+2^(1/2))^(1/2))-2^(1/2)*arctan((-1/2*2^(1/2)+
1/2*x*2^(1/2)+1/2*(x^2+1)^(1/2)*2^(1/2))/(x+(x^2+1)^(1/2))^(1/2))+(2^(1/2)-1)^(1/2)*arctanh((x+(x^2+1)^(1/2))^
(1/2)/(2^(1/2)-1)^(1/2))-(1+2^(1/2))^(1/2)*arctanh((x+(x^2+1)^(1/2))^(1/2)/(1+2^(1/2))^(1/2))+2^(1/2)*arctanh(
(1/2*2^(1/2)+1/2*x*2^(1/2)+1/2*(x^2+1)^(1/2)*2^(1/2))/(x+(x^2+1)^(1/2))^(1/2))

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.13, number of steps used = 34, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6857, 2142, 14, 2144, 1642, 840, 1180, 210, 212, 213, 209, 2147, 335, 303, 1176, 631, 1179, 642} \[ \int \frac {\left (1+x^4\right ) \sqrt {x+\sqrt {1+x^2}}}{-1+x^4} \, dx=\sqrt {1+\sqrt {2}} \arctan \left (\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+x}\right )-\sqrt {\sqrt {2}-1} \arctan \left (\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+x}\right )+\sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\sqrt {x^2+1}+x}\right )-\sqrt {2} \arctan \left (\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+1\right )-\sqrt {1+\sqrt {2}} \text {arctanh}\left (\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+x}\right )+\sqrt {\sqrt {2}-1} \text {arctanh}\left (\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+x}\right )+\frac {1}{3} \left (\sqrt {x^2+1}+x\right )^{3/2}-\frac {1}{\sqrt {\sqrt {x^2+1}+x}}-\frac {\log \left (\sqrt {x^2+1}-\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+x+1\right )}{\sqrt {2}}+\frac {\log \left (\sqrt {x^2+1}+\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+x+1\right )}{\sqrt {2}} \]

[In]

Int[((1 + x^4)*Sqrt[x + Sqrt[1 + x^2]])/(-1 + x^4),x]

[Out]

-(1/Sqrt[x + Sqrt[1 + x^2]]) + (x + Sqrt[1 + x^2])^(3/2)/3 + Sqrt[1 + Sqrt[2]]*ArcTan[Sqrt[-1 + Sqrt[2]]*Sqrt[
x + Sqrt[1 + x^2]]] - Sqrt[-1 + Sqrt[2]]*ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[2]*ArcTan[1
- Sqrt[2]*Sqrt[x + Sqrt[1 + x^2]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[x + Sqrt[1 + x^2]]] - Sqrt[1 + Sqrt[2]]*A
rcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[-1 + Sqrt[2]]*ArcTanh[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqr
t[1 + x^2]]] - Log[1 + x + Sqrt[1 + x^2] - Sqrt[2]*Sqrt[x + Sqrt[1 + x^2]]]/Sqrt[2] + Log[1 + x + Sqrt[1 + x^2
] + Sqrt[2]*Sqrt[x + Sqrt[1 + x^2]]]/Sqrt[2]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 209

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

Rule 210

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

Rule 212

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

Rule 213

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

Rule 303

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 631

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

Rule 642

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

Rule 840

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

Rule 1176

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

Rule 1179

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

Rule 1180

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 1642

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 2142

Int[((g_.) + (h_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_))^(p_.), x_Symbol] :> Dist[1/(2*
e), Subst[Int[(g + h*x^n)^p*((d^2 + a*f^2 - 2*d*x + x^2)/(d - x)^2), x], x, d + e*x + f*Sqrt[a + c*x^2]], x] /
; FreeQ[{a, c, d, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && IntegerQ[p]

Rule 2144

Int[((g_.) + (h_.)*(x_))^(m_.)*((e_.)*(x_) + (f_.)*Sqrt[(a_.) + (c_.)*(x_)^2])^(n_.), x_Symbol] :> Dist[1/(2^(
m + 1)*e^(m + 1)), Subst[Int[x^(n - m - 2)*(a*f^2 + x^2)*((-a)*f^2*h + 2*e*g*x + h*x^2)^m, x], x, e*x + f*Sqrt
[a + c*x^2]], x] /; FreeQ[{a, c, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && IntegerQ[m]

Rule 2147

Int[((g_) + (i_.)*(x_)^2)^(m_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_.), x_Symbol] :> Dis
t[(1/(2^(2*m + 1)*e*f^(2*m)))*(i/c)^m, Subst[Int[x^n*((d^2 + a*f^2 - 2*d*x + x^2)^(2*m + 1)/(-d + x)^(2*(m + 1
))), x], x, d + e*x + f*Sqrt[a + c*x^2]], x] /; FreeQ[{a, c, d, e, f, g, i, n}, x] && EqQ[e^2 - c*f^2, 0] && E
qQ[c*g - a*i, 0] && IntegerQ[2*m] && (IntegerQ[m] || GtQ[i/c, 0])

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

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

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.92 \[ \int \frac {\left (1+x^4\right ) \sqrt {x+\sqrt {1+x^2}}}{-1+x^4} \, dx=-\frac {1}{\sqrt {x+\sqrt {1+x^2}}}+\frac {1}{3} \left (x+\sqrt {1+x^2}\right )^{3/2}-\sqrt {2} \arctan \left (\frac {-1+x+\sqrt {1+x^2}}{\sqrt {2} \sqrt {x+\sqrt {1+x^2}}}\right )+\sqrt {1+\sqrt {2}} \arctan \left (\sqrt {-1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )-\sqrt {-1+\sqrt {2}} \arctan \left (\sqrt {1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )-\sqrt {1+\sqrt {2}} \text {arctanh}\left (\sqrt {-1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )+\sqrt {-1+\sqrt {2}} \text {arctanh}\left (\sqrt {1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )+\sqrt {2} \text {arctanh}\left (\frac {1+x+\sqrt {1+x^2}}{\sqrt {2} \sqrt {x+\sqrt {1+x^2}}}\right ) \]

[In]

Integrate[((1 + x^4)*Sqrt[x + Sqrt[1 + x^2]])/(-1 + x^4),x]

[Out]

-(1/Sqrt[x + Sqrt[1 + x^2]]) + (x + Sqrt[1 + x^2])^(3/2)/3 - Sqrt[2]*ArcTan[(-1 + x + Sqrt[1 + x^2])/(Sqrt[2]*
Sqrt[x + Sqrt[1 + x^2]])] + Sqrt[1 + Sqrt[2]]*ArcTan[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] - Sqrt[-1 + S
qrt[2]]*ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] - Sqrt[1 + Sqrt[2]]*ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[
x + Sqrt[1 + x^2]]] + Sqrt[-1 + Sqrt[2]]*ArcTanh[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[2]*ArcTanh[
(1 + x + Sqrt[1 + x^2])/(Sqrt[2]*Sqrt[x + Sqrt[1 + x^2]])]

Maple [F]

\[\int \frac {\left (x^{4}+1\right ) \sqrt {x +\sqrt {x^{2}+1}}}{x^{4}-1}d x\]

[In]

int((x^4+1)*(x+(x^2+1)^(1/2))^(1/2)/(x^4-1),x)

[Out]

int((x^4+1)*(x+(x^2+1)^(1/2))^(1/2)/(x^4-1),x)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.26 \[ \int \frac {\left (1+x^4\right ) \sqrt {x+\sqrt {1+x^2}}}{-1+x^4} \, dx=\frac {2}{3} \, {\left (2 \, x - \sqrt {x^{2} + 1}\right )} \sqrt {x + \sqrt {x^{2} + 1}} - \left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \log \left (\left (i + 1\right ) \, \sqrt {2} + 2 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \log \left (-\left (i - 1\right ) \, \sqrt {2} + 2 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \log \left (\left (i - 1\right ) \, \sqrt {2} + 2 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \log \left (-\left (i + 1\right ) \, \sqrt {2} + 2 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \frac {1}{2} \, \sqrt {\sqrt {2} + 1} \log \left (\sqrt {x + \sqrt {x^{2} + 1}} + \sqrt {\sqrt {2} + 1}\right ) + \frac {1}{2} \, \sqrt {\sqrt {2} + 1} \log \left (\sqrt {x + \sqrt {x^{2} + 1}} - \sqrt {\sqrt {2} + 1}\right ) + \frac {1}{2} \, \sqrt {\sqrt {2} - 1} \log \left (\sqrt {x + \sqrt {x^{2} + 1}} + \sqrt {\sqrt {2} - 1}\right ) - \frac {1}{2} \, \sqrt {\sqrt {2} - 1} \log \left (\sqrt {x + \sqrt {x^{2} + 1}} - \sqrt {\sqrt {2} - 1}\right ) - \frac {1}{2} \, \sqrt {-\sqrt {2} + 1} \log \left (\sqrt {x + \sqrt {x^{2} + 1}} + \sqrt {-\sqrt {2} + 1}\right ) + \frac {1}{2} \, \sqrt {-\sqrt {2} + 1} \log \left (\sqrt {x + \sqrt {x^{2} + 1}} - \sqrt {-\sqrt {2} + 1}\right ) + \frac {1}{2} \, \sqrt {-\sqrt {2} - 1} \log \left (\sqrt {x + \sqrt {x^{2} + 1}} + \sqrt {-\sqrt {2} - 1}\right ) - \frac {1}{2} \, \sqrt {-\sqrt {2} - 1} \log \left (\sqrt {x + \sqrt {x^{2} + 1}} - \sqrt {-\sqrt {2} - 1}\right ) \]

[In]

integrate((x^4+1)*(x+(x^2+1)^(1/2))^(1/2)/(x^4-1),x, algorithm="fricas")

[Out]

2/3*(2*x - sqrt(x^2 + 1))*sqrt(x + sqrt(x^2 + 1)) - (1/2*I - 1/2)*sqrt(2)*log((I + 1)*sqrt(2) + 2*sqrt(x + sqr
t(x^2 + 1))) + (1/2*I + 1/2)*sqrt(2)*log(-(I - 1)*sqrt(2) + 2*sqrt(x + sqrt(x^2 + 1))) - (1/2*I + 1/2)*sqrt(2)
*log((I - 1)*sqrt(2) + 2*sqrt(x + sqrt(x^2 + 1))) + (1/2*I - 1/2)*sqrt(2)*log(-(I + 1)*sqrt(2) + 2*sqrt(x + sq
rt(x^2 + 1))) - 1/2*sqrt(sqrt(2) + 1)*log(sqrt(x + sqrt(x^2 + 1)) + sqrt(sqrt(2) + 1)) + 1/2*sqrt(sqrt(2) + 1)
*log(sqrt(x + sqrt(x^2 + 1)) - sqrt(sqrt(2) + 1)) + 1/2*sqrt(sqrt(2) - 1)*log(sqrt(x + sqrt(x^2 + 1)) + sqrt(s
qrt(2) - 1)) - 1/2*sqrt(sqrt(2) - 1)*log(sqrt(x + sqrt(x^2 + 1)) - sqrt(sqrt(2) - 1)) - 1/2*sqrt(-sqrt(2) + 1)
*log(sqrt(x + sqrt(x^2 + 1)) + sqrt(-sqrt(2) + 1)) + 1/2*sqrt(-sqrt(2) + 1)*log(sqrt(x + sqrt(x^2 + 1)) - sqrt
(-sqrt(2) + 1)) + 1/2*sqrt(-sqrt(2) - 1)*log(sqrt(x + sqrt(x^2 + 1)) + sqrt(-sqrt(2) - 1)) - 1/2*sqrt(-sqrt(2)
- 1)*log(sqrt(x + sqrt(x^2 + 1)) - sqrt(-sqrt(2) - 1))

Sympy [F]

\[ \int \frac {\left (1+x^4\right ) \sqrt {x+\sqrt {1+x^2}}}{-1+x^4} \, dx=\int \frac {\sqrt {x + \sqrt {x^{2} + 1}} \left (x^{4} + 1\right )}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \]

[In]

integrate((x**4+1)*(x+(x**2+1)**(1/2))**(1/2)/(x**4-1),x)

[Out]

Integral(sqrt(x + sqrt(x**2 + 1))*(x**4 + 1)/((x - 1)*(x + 1)*(x**2 + 1)), x)

Maxima [F]

\[ \int \frac {\left (1+x^4\right ) \sqrt {x+\sqrt {1+x^2}}}{-1+x^4} \, dx=\int { \frac {{\left (x^{4} + 1\right )} \sqrt {x + \sqrt {x^{2} + 1}}}{x^{4} - 1} \,d x } \]

[In]

integrate((x^4+1)*(x+(x^2+1)^(1/2))^(1/2)/(x^4-1),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {\left (1+x^4\right ) \sqrt {x+\sqrt {1+x^2}}}{-1+x^4} \, dx=\int { \frac {{\left (x^{4} + 1\right )} \sqrt {x + \sqrt {x^{2} + 1}}}{x^{4} - 1} \,d x } \]

[In]

integrate((x^4+1)*(x+(x^2+1)^(1/2))^(1/2)/(x^4-1),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (1+x^4\right ) \sqrt {x+\sqrt {1+x^2}}}{-1+x^4} \, dx=\int \frac {\left (x^4+1\right )\,\sqrt {x+\sqrt {x^2+1}}}{x^4-1} \,d x \]

[In]

int(((x^4 + 1)*(x + (x^2 + 1)^(1/2))^(1/2))/(x^4 - 1),x)

[Out]

int(((x^4 + 1)*(x + (x^2 + 1)^(1/2))^(1/2))/(x^4 - 1), x)