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

Optimal. Leaf size=326 \[ \frac {4}{3} \sqrt {\sqrt {x^2+1}-x} x+\frac {2}{3} \sqrt {x^2+1} \sqrt {\sqrt {x^2+1}-x}+\sqrt {\sqrt {2}-1} \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}-x}}{\sqrt {\sqrt {2}-1}}\right )-\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}-x}}{\sqrt {1+\sqrt {2}}}\right )+\sqrt {2} \tan ^{-1}\left (\frac {\frac {\sqrt {x^2+1}}{\sqrt {2}}-\frac {x}{\sqrt {2}}-\frac {1}{\sqrt {2}}}{\sqrt {\sqrt {x^2+1}-x}}\right )-\sqrt {\sqrt {2}-1} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}-x}}{\sqrt {\sqrt {2}-1}}\right )+\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}-x}}{\sqrt {1+\sqrt {2}}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\sqrt {x^2+1}-x}}{\sqrt {x^2+1}-x+1}\right ) \]

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Rubi [A]  time = 0.82, antiderivative size = 341, normalized size of antiderivative = 1.05, number of steps used = 36, number of rules used = 19, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.679, Rules used = {6725, 2117, 14, 2119, 1628, 828, 826, 1166, 204, 206, 207, 203, 2122, 329, 211, 1165, 628, 1162, 617} \begin {gather*} \sqrt {\sqrt {x^2+1}+x}-\frac {1}{3 \left (\sqrt {x^2+1}+x\right )^{3/2}}+\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}}-\frac {\tan ^{-1}\left (\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {1+\sqrt {2}}}+\frac {\tan ^{-1}\left (\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {\sqrt {2}-1}}+\sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\sqrt {x^2+1}+x}\right )-\sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {\sqrt {x^2+1}+x}+1\right )-\frac {\tanh ^{-1}\left (\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {1+\sqrt {2}}}+\frac {\tanh ^{-1}\left (\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {\sqrt {2}-1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/3*1/(x + Sqrt[1 + x^2])^(3/2) + Sqrt[x + Sqrt[1 + x^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[-1 + Sqrt[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]]] - ArcTanh[Sqrt[-1 + S
qrt[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]] + 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 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 204

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[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 211

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

Rule 329

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 617

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 628

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 826

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 828

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[((
e*f - d*g)*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d
+ e*x)^(m + 1)*Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x])/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,
d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[m] && LtQ[m, -1]

Rule 1162

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 1165

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 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 1628

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 2117

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 2119

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 2122

Int[((g_) + (i_.)*(x_)^2)^(m_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_.), x_Symbol] :> Dis
t[(1*(i/c)^m)/(2^(2*m + 1)*e*f^(2*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 6725

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

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Mathematica [A]  time = 0.96, size = 383, normalized size = 1.17 \begin {gather*} \sqrt {\sqrt {x^2+1}+x}-\frac {1}{3 \left (\sqrt {x^2+1}+x\right )^{3/2}}+\frac {\log \left (-\frac {\sqrt {2}}{\sqrt {\sqrt {x^2+1}+x}}+\frac {1}{\sqrt {x^2+1}+x}+1\right )}{\sqrt {2}}-\frac {\log \left (\frac {\sqrt {2}}{\sqrt {\sqrt {x^2+1}+x}}+\frac {1}{\sqrt {x^2+1}+x}+1\right )}{\sqrt {2}}-\frac {\left (\sqrt {2}-2\right ) \tan ^{-1}\left (\frac {1}{\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+x}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}-\frac {\left (2+\sqrt {2}\right ) \tan ^{-1}\left (\frac {1}{\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+x}}\right )}{\sqrt {2 \left (1+\sqrt {2}\right )}}-\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2}}{\sqrt {\sqrt {x^2+1}+x}}\right )+\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2}}{\sqrt {\sqrt {x^2+1}+x}}+1\right )+\frac {\left (\sqrt {2}-2\right ) \tanh ^{-1}\left (\frac {1}{\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+x}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}+\frac {\left (2+\sqrt {2}\right ) \tanh ^{-1}\left (\frac {1}{\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+x}}\right )}{\sqrt {2 \left (1+\sqrt {2}\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.42, size = 303, normalized size = 0.93 \begin {gather*} -\frac {1}{3 \left (x+\sqrt {1+x^2}\right )^{3/2}}+\sqrt {x+\sqrt {1+x^2}}-\sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\sqrt {-1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )+\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\sqrt {1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )-\sqrt {2} \tan ^{-1}\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}} \tanh ^{-1}\left (\sqrt {-1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )+\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\sqrt {1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\frac {1}{\sqrt {2}}+\frac {x}{\sqrt {2}}+\frac {\sqrt {1+x^2}}{\sqrt {2}}}{\sqrt {x+\sqrt {1+x^2}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*1/(x + Sqrt[1 + x^2])^(3/2) + Sqrt[x + Sqrt[1 + x^2]] - 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]) + x/Sqrt[2] + Sqrt[1 + x^2]/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/Sqrt[2] + x/Sqrt[2] + Sqrt[1 + x^2]/Sqrt[2])/Sqrt[x + Sqrt[1 + x^2]]]

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fricas [A]  time = 0.60, size = 467, normalized size = 1.43 \begin {gather*} -\frac {2}{3} \, {\left (x^{2} - \sqrt {x^{2} + 1} x - 1\right )} \sqrt {x + \sqrt {x^{2} + 1}} + 2 \, \sqrt {2} \arctan \left (\sqrt {2} \sqrt {\sqrt {2} \sqrt {x + \sqrt {x^{2} + 1}} + x + \sqrt {x^{2} + 1} + 1} - \sqrt {2} \sqrt {x + \sqrt {x^{2} + 1}} - 1\right ) + 2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-4 \, \sqrt {2} \sqrt {x + \sqrt {x^{2} + 1}} + 4 \, x + 4 \, \sqrt {x^{2} + 1} + 4} - \sqrt {2} \sqrt {x + \sqrt {x^{2} + 1}} + 1\right ) - 2 \, \sqrt {\sqrt {2} + 1} \arctan \left (\sqrt {x + \sqrt {2} + \sqrt {x^{2} + 1} - 1} \sqrt {\sqrt {2} + 1} - \sqrt {x + \sqrt {x^{2} + 1}} \sqrt {\sqrt {2} + 1}\right ) + 2 \, \sqrt {\sqrt {2} - 1} \arctan \left (\sqrt {x + \sqrt {2} + \sqrt {x^{2} + 1} + 1} \sqrt {\sqrt {2} - 1} - \sqrt {x + \sqrt {x^{2} + 1}} \sqrt {\sqrt {2} - 1}\right ) - \frac {1}{2} \, \sqrt {\sqrt {2} - 1} \log \left ({\left (\sqrt {2} + 1\right )} \sqrt {\sqrt {2} - 1} + \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \frac {1}{2} \, \sqrt {\sqrt {2} - 1} \log \left (-{\left (\sqrt {2} + 1\right )} \sqrt {\sqrt {2} - 1} + \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \frac {1}{2} \, \sqrt {\sqrt {2} + 1} \log \left (\sqrt {\sqrt {2} + 1} {\left (\sqrt {2} - 1\right )} + \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \frac {1}{2} \, \sqrt {\sqrt {2} + 1} \log \left (-\sqrt {\sqrt {2} + 1} {\left (\sqrt {2} - 1\right )} + \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \frac {1}{2} \, \sqrt {2} \log \left (4 \, \sqrt {2} \sqrt {x + \sqrt {x^{2} + 1}} + 4 \, x + 4 \, \sqrt {x^{2} + 1} + 4\right ) + \frac {1}{2} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {x + \sqrt {x^{2} + 1}} + 4 \, x + 4 \, \sqrt {x^{2} + 1} + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(x^2 - sqrt(x^2 + 1)*x - 1)*sqrt(x + sqrt(x^2 + 1)) + 2*sqrt(2)*arctan(sqrt(2)*sqrt(sqrt(2)*sqrt(x + sqrt
(x^2 + 1)) + x + sqrt(x^2 + 1) + 1) - sqrt(2)*sqrt(x + sqrt(x^2 + 1)) - 1) + 2*sqrt(2)*arctan(1/2*sqrt(2)*sqrt
(-4*sqrt(2)*sqrt(x + sqrt(x^2 + 1)) + 4*x + 4*sqrt(x^2 + 1) + 4) - sqrt(2)*sqrt(x + sqrt(x^2 + 1)) + 1) - 2*sq
rt(sqrt(2) + 1)*arctan(sqrt(x + sqrt(2) + sqrt(x^2 + 1) - 1)*sqrt(sqrt(2) + 1) - sqrt(x + sqrt(x^2 + 1))*sqrt(
sqrt(2) + 1)) + 2*sqrt(sqrt(2) - 1)*arctan(sqrt(x + sqrt(2) + sqrt(x^2 + 1) + 1)*sqrt(sqrt(2) - 1) - sqrt(x +
sqrt(x^2 + 1))*sqrt(sqrt(2) - 1)) - 1/2*sqrt(sqrt(2) - 1)*log((sqrt(2) + 1)*sqrt(sqrt(2) - 1) + sqrt(x + sqrt(
x^2 + 1))) + 1/2*sqrt(sqrt(2) - 1)*log(-(sqrt(2) + 1)*sqrt(sqrt(2) - 1) + sqrt(x + sqrt(x^2 + 1))) + 1/2*sqrt(
sqrt(2) + 1)*log(sqrt(sqrt(2) + 1)*(sqrt(2) - 1) + sqrt(x + sqrt(x^2 + 1))) - 1/2*sqrt(sqrt(2) + 1)*log(-sqrt(
sqrt(2) + 1)*(sqrt(2) - 1) + sqrt(x + sqrt(x^2 + 1))) - 1/2*sqrt(2)*log(4*sqrt(2)*sqrt(x + sqrt(x^2 + 1)) + 4*
x + 4*sqrt(x^2 + 1) + 4) + 1/2*sqrt(2)*log(-4*sqrt(2)*sqrt(x + sqrt(x^2 + 1)) + 4*x + 4*sqrt(x^2 + 1) + 4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} + 1}{{\left (x^{4} - 1\right )} \sqrt {x + \sqrt {x^{2} + 1}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {x^{4}+1}{\left (x^{4}-1\right ) \sqrt {x +\sqrt {x^{2}+1}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} + 1}{{\left (x^{4} - 1\right )} \sqrt {x + \sqrt {x^{2} + 1}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^4+1}{\left (x^4-1\right )\,\sqrt {x+\sqrt {x^2+1}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} + 1}{\left (x - 1\right ) \left (x + 1\right ) \sqrt {x + \sqrt {x^{2} + 1}} \left (x^{2} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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