\(\int \frac {-1+x^4}{(1+x^4) \sqrt {x+\sqrt {1+x^2}}} \, dx\) [1948]
Optimal result
Integrand size = 28, antiderivative size = 136 \[
\int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=-\frac {1}{3 \left (x+\sqrt {1+x^2}\right )^{3/2}}+\sqrt {x+\sqrt {1+x^2}}-2 \text {RootSum}\left [1-4 \text {$\#$1}^4+22 \text {$\#$1}^8-4 \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}^5}{-1+11 \text {$\#$1}^4-3 \text {$\#$1}^8+\text {$\#$1}^{12}}\&\right ]
\]
[Out]
Unintegrable
Rubi [C] (verified)
Result contains complex when optimal does not.
Time = 1.04 (sec) , antiderivative size = 511, normalized size of antiderivative = 3.76, number of
steps used = 43, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6857, 2142, 14, 2144, 1642,
842, 840, 1180, 210, 212, 213, 209} \[
\int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=\frac {(-1)^{3/4} \arctan \left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {-(-1)^{3/4}+\sqrt {1-i}}}\right )}{\sqrt {-(-1)^{3/4}+\sqrt {1-i}}}-\frac {(-1)^{3/4} \arctan \left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {(-1)^{3/4}+\sqrt {1-i}}}\right )}{\sqrt {(-1)^{3/4}+\sqrt {1-i}}}+\frac {\sqrt [4]{-1} \arctan \left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {-\sqrt [4]{-1}+\sqrt {1+i}}}\right )}{\sqrt {-\sqrt [4]{-1}+\sqrt {1+i}}}-\frac {\sqrt [4]{-1} \arctan \left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {\sqrt [4]{-1}+\sqrt {1+i}}}\right )}{\sqrt {\sqrt [4]{-1}+\sqrt {1+i}}}+\frac {(-1)^{3/4} \text {arctanh}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {-(-1)^{3/4}+\sqrt {1-i}}}\right )}{\sqrt {-(-1)^{3/4}+\sqrt {1-i}}}-\frac {(-1)^{3/4} \text {arctanh}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {(-1)^{3/4}+\sqrt {1-i}}}\right )}{\sqrt {(-1)^{3/4}+\sqrt {1-i}}}+\frac {\sqrt [4]{-1} \text {arctanh}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {-\sqrt [4]{-1}+\sqrt {1+i}}}\right )}{\sqrt {-\sqrt [4]{-1}+\sqrt {1+i}}}-\frac {\sqrt [4]{-1} \text {arctanh}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {\sqrt [4]{-1}+\sqrt {1+i}}}\right )}{\sqrt {\sqrt [4]{-1}+\sqrt {1+i}}}+\sqrt {\sqrt {x^2+1}+x}-\frac {1}{3 \left (\sqrt {x^2+1}+x\right )^{3/2}}
\]
[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]] + ((-1)^(3/4)*ArcTan[Sqrt[x + Sqrt[1 + x^2]]/Sqrt[-
(-1)^(3/4) + Sqrt[1 - I]]])/Sqrt[-(-1)^(3/4) + Sqrt[1 - I]] - ((-1)^(3/4)*ArcTan[Sqrt[x + Sqrt[1 + x^2]]/Sqrt[
(-1)^(3/4) + Sqrt[1 - I]]])/Sqrt[(-1)^(3/4) + Sqrt[1 - I]] + ((-1)^(1/4)*ArcTan[Sqrt[x + Sqrt[1 + x^2]]/Sqrt[-
(-1)^(1/4) + Sqrt[1 + I]]])/Sqrt[-(-1)^(1/4) + Sqrt[1 + I]] - ((-1)^(1/4)*ArcTan[Sqrt[x + Sqrt[1 + x^2]]/Sqrt[
(-1)^(1/4) + Sqrt[1 + I]]])/Sqrt[(-1)^(1/4) + Sqrt[1 + I]] + ((-1)^(3/4)*ArcTanh[Sqrt[x + Sqrt[1 + x^2]]/Sqrt[
-(-1)^(3/4) + Sqrt[1 - I]]])/Sqrt[-(-1)^(3/4) + Sqrt[1 - I]] - ((-1)^(3/4)*ArcTanh[Sqrt[x + Sqrt[1 + x^2]]/Sqr
t[(-1)^(3/4) + Sqrt[1 - I]]])/Sqrt[(-1)^(3/4) + Sqrt[1 - I]] + ((-1)^(1/4)*ArcTanh[Sqrt[x + Sqrt[1 + x^2]]/Sqr
t[-(-1)^(1/4) + Sqrt[1 + I]]])/Sqrt[-(-1)^(1/4) + Sqrt[1 + I]] - ((-1)^(1/4)*ArcTanh[Sqrt[x + Sqrt[1 + x^2]]/S
qrt[(-1)^(1/4) + Sqrt[1 + I]]])/Sqrt[(-1)^(1/4) + Sqrt[1 + I]]
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 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 842
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 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 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 (\frac {1}{\sqrt {x+\sqrt {1+x^2}}}-\frac {2}{\left (1+x^4\right ) \sqrt {x+\sqrt {1+x^2}}}\right ) \, dx \\ & = -\left (2 \int \frac {1}{\left (1+x^4\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx\right )+\int \frac {1}{\sqrt {x+\sqrt {1+x^2}}} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1+x^2}{x^{5/2}} \, dx,x,x+\sqrt {1+x^2}\right )-2 \int \left (\frac {i}{2 \left (i-x^2\right ) \sqrt {x+\sqrt {1+x^2}}}+\frac {i}{2 \left (i+x^2\right ) \sqrt {x+\sqrt {1+x^2}}}\right ) \, dx \\ & = -\left (i \int \frac {1}{\left (i-x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx\right )-i \int \frac {1}{\left (i+x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx+\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{x^{5/2}}+\frac {1}{\sqrt {x}}\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}}-i \int \left (-\frac {(-1)^{3/4}}{2 \left (\sqrt [4]{-1}-x\right ) \sqrt {x+\sqrt {1+x^2}}}-\frac {(-1)^{3/4}}{2 \left (\sqrt [4]{-1}+x\right ) \sqrt {x+\sqrt {1+x^2}}}\right ) \, dx-i \int \left (-\frac {\sqrt [4]{-1}}{2 \left (-(-1)^{3/4}-x\right ) \sqrt {x+\sqrt {1+x^2}}}-\frac {\sqrt [4]{-1}}{2 \left (-(-1)^{3/4}+x\right ) \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} \sqrt [4]{-1} \int \frac {1}{\left (\sqrt [4]{-1}-x\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx-\frac {1}{2} \sqrt [4]{-1} \int \frac {1}{\left (\sqrt [4]{-1}+x\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx+\frac {1}{2} (-1)^{3/4} \int \frac {1}{\left (-(-1)^{3/4}-x\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx+\frac {1}{2} (-1)^{3/4} \int \frac {1}{\left (-(-1)^{3/4}+x\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}}-\frac {1}{2} \sqrt [4]{-1} \text {Subst}\left (\int \frac {1+x^2}{x^{3/2} \left (1+2 \sqrt [4]{-1} x-x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )-\frac {1}{2} \sqrt [4]{-1} \text {Subst}\left (\int \frac {1+x^2}{x^{3/2} \left (-1+2 \sqrt [4]{-1} x+x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )+\frac {1}{2} (-1)^{3/4} \text {Subst}\left (\int \frac {1+x^2}{x^{3/2} \left (1-2 (-1)^{3/4} x-x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )+\frac {1}{2} (-1)^{3/4} \text {Subst}\left (\int \frac {1+x^2}{x^{3/2} \left (-1-2 (-1)^{3/4} 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}}-\frac {1}{2} \sqrt [4]{-1} \text {Subst}\left (\int \left (-\frac {1}{x^{3/2}}+\frac {2 \left (1+\sqrt [4]{-1} x\right )}{x^{3/2} \left (1+2 \sqrt [4]{-1} x-x^2\right )}\right ) \, dx,x,x+\sqrt {1+x^2}\right )-\frac {1}{2} \sqrt [4]{-1} \text {Subst}\left (\int \left (\frac {1}{x^{3/2}}+\frac {2 \left (1-\sqrt [4]{-1} x\right )}{x^{3/2} \left (-1+2 \sqrt [4]{-1} x+x^2\right )}\right ) \, dx,x,x+\sqrt {1+x^2}\right )+\frac {1}{2} (-1)^{3/4} \text {Subst}\left (\int \left (-\frac {1}{x^{3/2}}+\frac {2 \left (1-(-1)^{3/4} x\right )}{x^{3/2} \left (1-2 (-1)^{3/4} x-x^2\right )}\right ) \, dx,x,x+\sqrt {1+x^2}\right )+\frac {1}{2} (-1)^{3/4} \text {Subst}\left (\int \left (\frac {1}{x^{3/2}}+\frac {2 \left (1+(-1)^{3/4} x\right )}{x^{3/2} \left (-1-2 (-1)^{3/4} x+x^2\right )}\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 [4]{-1} \text {Subst}\left (\int \frac {1+\sqrt [4]{-1} x}{x^{3/2} \left (1+2 \sqrt [4]{-1} x-x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )-\sqrt [4]{-1} \text {Subst}\left (\int \frac {1-\sqrt [4]{-1} x}{x^{3/2} \left (-1+2 \sqrt [4]{-1} x+x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )+(-1)^{3/4} \text {Subst}\left (\int \frac {1-(-1)^{3/4} x}{x^{3/2} \left (1-2 (-1)^{3/4} x-x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )+(-1)^{3/4} \text {Subst}\left (\int \frac {1+(-1)^{3/4} x}{x^{3/2} \left (-1-2 (-1)^{3/4} 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 [4]{-1} \text {Subst}\left (\int \frac {-\sqrt [4]{-1}+x}{\sqrt {x} \left (1+2 \sqrt [4]{-1} x-x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )+\sqrt [4]{-1} \text {Subst}\left (\int \frac {-\sqrt [4]{-1}-x}{\sqrt {x} \left (-1+2 \sqrt [4]{-1} x+x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )+(-1)^{3/4} \text {Subst}\left (\int \frac {(-1)^{3/4}+x}{\sqrt {x} \left (1-2 (-1)^{3/4} x-x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )-(-1)^{3/4} \text {Subst}\left (\int \frac {(-1)^{3/4}-x}{\sqrt {x} \left (-1-2 (-1)^{3/4} 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}}-\left (2 \sqrt [4]{-1}\right ) \text {Subst}\left (\int \frac {-\sqrt [4]{-1}+x^2}{1+2 \sqrt [4]{-1} x^2-x^4} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )+\left (2 \sqrt [4]{-1}\right ) \text {Subst}\left (\int \frac {-\sqrt [4]{-1}-x^2}{-1+2 \sqrt [4]{-1} x^2+x^4} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )+\left (2 (-1)^{3/4}\right ) \text {Subst}\left (\int \frac {(-1)^{3/4}+x^2}{1-2 (-1)^{3/4} x^2-x^4} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )-\left (2 (-1)^{3/4}\right ) \text {Subst}\left (\int \frac {(-1)^{3/4}-x^2}{-1-2 (-1)^{3/4} 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 [4]{-1} \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1}-\sqrt {1+i}-x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )-\sqrt [4]{-1} \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1}+\sqrt {1+i}-x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )-\sqrt [4]{-1} \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1}-\sqrt {1+i}+x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )-\sqrt [4]{-1} \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1}+\sqrt {1+i}+x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )+(-1)^{3/4} \text {Subst}\left (\int \frac {1}{-(-1)^{3/4}-\sqrt {1-i}-x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )+(-1)^{3/4} \text {Subst}\left (\int \frac {1}{-(-1)^{3/4}+\sqrt {1-i}-x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )+(-1)^{3/4} \text {Subst}\left (\int \frac {1}{-(-1)^{3/4}-\sqrt {1-i}+x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )+(-1)^{3/4} \text {Subst}\left (\int \frac {1}{-(-1)^{3/4}+\sqrt {1-i}+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 {(-1)^{3/4} \arctan \left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-(-1)^{3/4}+\sqrt {1-i}}}\right )}{\sqrt {-(-1)^{3/4}+\sqrt {1-i}}}-\frac {(-1)^{3/4} \arctan \left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {(-1)^{3/4}+\sqrt {1-i}}}\right )}{\sqrt {(-1)^{3/4}+\sqrt {1-i}}}+\frac {\sqrt [4]{-1} \arctan \left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-\sqrt [4]{-1}+\sqrt {1+i}}}\right )}{\sqrt {-\sqrt [4]{-1}+\sqrt {1+i}}}-\frac {\sqrt [4]{-1} \arctan \left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {\sqrt [4]{-1}+\sqrt {1+i}}}\right )}{\sqrt {\sqrt [4]{-1}+\sqrt {1+i}}}+\frac {(-1)^{3/4} \text {arctanh}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-(-1)^{3/4}+\sqrt {1-i}}}\right )}{\sqrt {-(-1)^{3/4}+\sqrt {1-i}}}-\frac {(-1)^{3/4} \text {arctanh}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {(-1)^{3/4}+\sqrt {1-i}}}\right )}{\sqrt {(-1)^{3/4}+\sqrt {1-i}}}+\frac {\sqrt [4]{-1} \text {arctanh}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-\sqrt [4]{-1}+\sqrt {1+i}}}\right )}{\sqrt {-\sqrt [4]{-1}+\sqrt {1+i}}}-\frac {\sqrt [4]{-1} \text {arctanh}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {\sqrt [4]{-1}+\sqrt {1+i}}}\right )}{\sqrt {\sqrt [4]{-1}+\sqrt {1+i}}} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.16 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00
\[
\int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=-\frac {1}{3 \left (x+\sqrt {1+x^2}\right )^{3/2}}+\sqrt {x+\sqrt {1+x^2}}-2 \text {RootSum}\left [1-4 \text {$\#$1}^4+22 \text {$\#$1}^8-4 \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}^5}{-1+11 \text {$\#$1}^4-3 \text {$\#$1}^8+\text {$\#$1}^{12}}\&\right ]
\]
[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*RootSum[1 - 4*#1^4 + 22*#1^8 - 4*#1^12 + #1^16
& , (Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1 + Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1^5)/(-1 + 11*#1^4 - 3*#1^8 + #
1^12) & ]
Maple [N/A] (verified)
Not integrable
Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.18
\[\int \frac {x^{4}-1}{\left (x^{4}+1\right ) \sqrt {x +\sqrt {x^{2}+1}}}d x\]
[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)
Fricas [C] (verification not implemented)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 1.13 (sec) , antiderivative size = 3169, normalized size of antiderivative = 23.30
\[
\int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=\text {Too large to display}
\]
[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)) - 1/2*sqrt(-sqrt(sqrt(2)*sqrt(-3/8*(8*sqrt(1/16*I - 1
/16) + 2*I + 1)^2 - 1/4*(8*sqrt(1/16*I - 1/16) + 2*I - 3)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - 3/8*(8*sqrt(-1/
16*I - 1/16) - 2*I + 1)^2 + 8*sqrt(1/16*I - 1/16) + 2*I - 8) + 4*sqrt(1/16*I - 1/16) + 4*sqrt(-1/16*I - 1/16)
- 1))*log(1/2*(2*(4*sqrt(1/16*I - 1/16) + I)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1)^2 - (8*sqrt(1/16*I - 1/16) + 2
*I + 1)^2 + ((8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 - 32*sqrt(1/16*I - 1/16) - 8*I - 1)*(8*sqrt(-1/16*I - 1/16) -
2*I + 1) - 2*sqrt(-3/8*(8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 - 1/4*(8*sqrt(1/16*I - 1/16) + 2*I - 3)*(8*sqrt(-1
/16*I - 1/16) - 2*I + 1) - 3/8*(8*sqrt(-1/16*I - 1/16) - 2*I + 1)^2 + 8*sqrt(1/16*I - 1/16) + 2*I - 8)*((sqrt(
2)*(8*sqrt(1/16*I - 1/16) + 2*I + 1) - sqrt(2))*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - sqrt(2)*(8*sqrt(1/16*I -
1/16) + 2*I + 1) + sqrt(2)) + 24*sqrt(1/16*I - 1/16) + 6*I + 1)*sqrt(-sqrt(sqrt(2)*sqrt(-3/8*(8*sqrt(1/16*I -
1/16) + 2*I + 1)^2 - 1/4*(8*sqrt(1/16*I - 1/16) + 2*I - 3)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - 3/8*(8*sqrt(-1
/16*I - 1/16) - 2*I + 1)^2 + 8*sqrt(1/16*I - 1/16) + 2*I - 8) + 4*sqrt(1/16*I - 1/16) + 4*sqrt(-1/16*I - 1/16)
- 1)) + 8*sqrt(x + sqrt(x^2 + 1))) + 1/2*sqrt(-sqrt(sqrt(2)*sqrt(-3/8*(8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 - 1
/4*(8*sqrt(1/16*I - 1/16) + 2*I - 3)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - 3/8*(8*sqrt(-1/16*I - 1/16) - 2*I +
1)^2 + 8*sqrt(1/16*I - 1/16) + 2*I - 8) + 4*sqrt(1/16*I - 1/16) + 4*sqrt(-1/16*I - 1/16) - 1))*log(-1/2*(2*(4*
sqrt(1/16*I - 1/16) + I)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1)^2 - (8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 + ((8*sqrt
(1/16*I - 1/16) + 2*I + 1)^2 - 32*sqrt(1/16*I - 1/16) - 8*I - 1)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - 2*sqrt(-
3/8*(8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 - 1/4*(8*sqrt(1/16*I - 1/16) + 2*I - 3)*(8*sqrt(-1/16*I - 1/16) - 2*I
+ 1) - 3/8*(8*sqrt(-1/16*I - 1/16) - 2*I + 1)^2 + 8*sqrt(1/16*I - 1/16) + 2*I - 8)*((sqrt(2)*(8*sqrt(1/16*I -
1/16) + 2*I + 1) - sqrt(2))*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - sqrt(2)*(8*sqrt(1/16*I - 1/16) + 2*I + 1) + s
qrt(2)) + 24*sqrt(1/16*I - 1/16) + 6*I + 1)*sqrt(-sqrt(sqrt(2)*sqrt(-3/8*(8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 -
1/4*(8*sqrt(1/16*I - 1/16) + 2*I - 3)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - 3/8*(8*sqrt(-1/16*I - 1/16) - 2*I
+ 1)^2 + 8*sqrt(1/16*I - 1/16) + 2*I - 8) + 4*sqrt(1/16*I - 1/16) + 4*sqrt(-1/16*I - 1/16) - 1)) + 8*sqrt(x +
sqrt(x^2 + 1))) - 1/2*sqrt(-sqrt(-sqrt(2)*sqrt(-3/8*(8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 - 1/4*(8*sqrt(1/16*I -
1/16) + 2*I - 3)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - 3/8*(8*sqrt(-1/16*I - 1/16) - 2*I + 1)^2 + 8*sqrt(1/16*
I - 1/16) + 2*I - 8) + 4*sqrt(1/16*I - 1/16) + 4*sqrt(-1/16*I - 1/16) - 1))*log(1/2*(2*(4*sqrt(1/16*I - 1/16)
+ I)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1)^2 - (8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 + ((8*sqrt(1/16*I - 1/16) + 2*
I + 1)^2 - 32*sqrt(1/16*I - 1/16) - 8*I - 1)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) + 2*sqrt(-3/8*(8*sqrt(1/16*I -
1/16) + 2*I + 1)^2 - 1/4*(8*sqrt(1/16*I - 1/16) + 2*I - 3)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - 3/8*(8*sqrt(-
1/16*I - 1/16) - 2*I + 1)^2 + 8*sqrt(1/16*I - 1/16) + 2*I - 8)*((sqrt(2)*(8*sqrt(1/16*I - 1/16) + 2*I + 1) - s
qrt(2))*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - sqrt(2)*(8*sqrt(1/16*I - 1/16) + 2*I + 1) + sqrt(2)) + 24*sqrt(1/
16*I - 1/16) + 6*I + 1)*sqrt(-sqrt(-sqrt(2)*sqrt(-3/8*(8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 - 1/4*(8*sqrt(1/16*I
- 1/16) + 2*I - 3)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - 3/8*(8*sqrt(-1/16*I - 1/16) - 2*I + 1)^2 + 8*sqrt(1/1
6*I - 1/16) + 2*I - 8) + 4*sqrt(1/16*I - 1/16) + 4*sqrt(-1/16*I - 1/16) - 1)) + 8*sqrt(x + sqrt(x^2 + 1))) + 1
/2*sqrt(-sqrt(-sqrt(2)*sqrt(-3/8*(8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 - 1/4*(8*sqrt(1/16*I - 1/16) + 2*I - 3)*(
8*sqrt(-1/16*I - 1/16) - 2*I + 1) - 3/8*(8*sqrt(-1/16*I - 1/16) - 2*I + 1)^2 + 8*sqrt(1/16*I - 1/16) + 2*I - 8
) + 4*sqrt(1/16*I - 1/16) + 4*sqrt(-1/16*I - 1/16) - 1))*log(-1/2*(2*(4*sqrt(1/16*I - 1/16) + I)*(8*sqrt(-1/16
*I - 1/16) - 2*I + 1)^2 - (8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 + ((8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 - 32*sqrt
(1/16*I - 1/16) - 8*I - 1)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) + 2*sqrt(-3/8*(8*sqrt(1/16*I - 1/16) + 2*I + 1)^
2 - 1/4*(8*sqrt(1/16*I - 1/16) + 2*I - 3)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - 3/8*(8*sqrt(-1/16*I - 1/16) - 2
*I + 1)^2 + 8*sqrt(1/16*I - 1/16) + 2*I - 8)*((sqrt(2)*(8*sqrt(1/16*I - 1/16) + 2*I + 1) - sqrt(2))*(8*sqrt(-1
/16*I - 1/16) - 2*I + 1) - sqrt(2)*(8*sqrt(1/16*I - 1/16) + 2*I + 1) + sqrt(2)) + 24*sqrt(1/16*I - 1/16) + 6*I
+ 1)*sqrt(-sqrt(-sqrt(2)*sqrt(-3/8*(8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 - 1/4*(8*sqrt(1/16*I - 1/16) + 2*I - 3
)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - 3/8*(8*sqrt(-1/16*I - 1/16) - 2*I + 1)^2 + 8*sqrt(1/16*I - 1/16) + 2*I
- 8) + 4*sqrt(1/16*I - 1/16) + 4*sqrt(-1/16*I - 1/16) - 1)) + 8*sqrt(x + sqrt(x^2 + 1))) - sqrt(-sqrt(-1/2*sqr
t(1/16*I - 1/16) - 1/8*I - 1/16))*log(2*((8*sqrt(1/16*I - 1/16) + 2*I + 1)^3 - 3*(8*sqrt(1/16*I - 1/16) + 2*I
+ 1)^2 + 152*sqrt(1/16*I - 1/16) + 38*I + 34)*sqrt(-sqrt(-1/2*sqrt(1/16*I - 1/16) - 1/8*I - 1/16)) + 8*sqrt(x
+ sqrt(x^2 + 1))) + sqrt(-sqrt(-1/2*sqrt(1/16*I - 1/16) - 1/8*I - 1/16))*log(-2*((8*sqrt(1/16*I - 1/16) + 2*I
+ 1)^3 - 3*(8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 + 152*sqrt(1/16*I - 1/16) + 38*I + 34)*sqrt(-sqrt(-1/2*sqrt(1/1
6*I - 1/16) - 1/8*I - 1/16)) + 8*sqrt(x + sqrt(x^2 + 1))) + sqrt(-sqrt(-1/2*sqrt(-1/16*I - 1/16) + 1/8*I - 1/1
6))*log(2*((8*sqrt(1/16*I - 1/16) + 2*I + 1)^3 + 2*(4*sqrt(1/16*I - 1/16) + I)*(8*sqrt(-1/16*I - 1/16) - 2*I +
1)^2 - 4*(8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 + ((8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 - 32*sqrt(1/16*I - 1/16)
- 8*I - 1)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) + 176*sqrt(1/16*I - 1/16) + 44*I + 3)*sqrt(-sqrt(-1/2*sqrt(-1/16
*I - 1/16) + 1/8*I - 1/16)) + 8*sqrt(x + sqrt(x^2 + 1))) - sqrt(-sqrt(-1/2*sqrt(-1/16*I - 1/16) + 1/8*I - 1/16
))*log(-2*((8*sqrt(1/16*I - 1/16) + 2*I + 1)^3 + 2*(4*sqrt(1/16*I - 1/16) + I)*(8*sqrt(-1/16*I - 1/16) - 2*I +
1)^2 - 4*(8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 + ((8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 - 32*sqrt(1/16*I - 1/16)
- 8*I - 1)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) + 176*sqrt(1/16*I - 1/16) + 44*I + 3)*sqrt(-sqrt(-1/2*sqrt(-1/16
*I - 1/16) + 1/8*I - 1/16)) + 8*sqrt(x + sqrt(x^2 + 1))) - 1/2*(sqrt(2)*sqrt(-3/8*(8*sqrt(1/16*I - 1/16) + 2*I
+ 1)^2 - 1/4*(8*sqrt(1/16*I - 1/16) + 2*I - 3)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - 3/8*(8*sqrt(-1/16*I - 1/1
6) - 2*I + 1)^2 + 8*sqrt(1/16*I - 1/16) + 2*I - 8) + 4*sqrt(1/16*I - 1/16) + 4*sqrt(-1/16*I - 1/16) - 1)^(1/4)
*log(1/2*(2*(4*sqrt(1/16*I - 1/16) + I)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1)^2 - (8*sqrt(1/16*I - 1/16) + 2*I +
1)^2 + ((8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 - 32*sqrt(1/16*I - 1/16) - 8*I - 1)*(8*sqrt(-1/16*I - 1/16) - 2*I
+ 1) - 2*sqrt(-3/8*(8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 - 1/4*(8*sqrt(1/16*I - 1/16) + 2*I - 3)*(8*sqrt(-1/16*I
- 1/16) - 2*I + 1) - 3/8*(8*sqrt(-1/16*I - 1/16) - 2*I + 1)^2 + 8*sqrt(1/16*I - 1/16) + 2*I - 8)*((sqrt(2)*(8
*sqrt(1/16*I - 1/16) + 2*I + 1) - sqrt(2))*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - sqrt(2)*(8*sqrt(1/16*I - 1/16)
+ 2*I + 1) + sqrt(2)) + 24*sqrt(1/16*I - 1/16) + 6*I + 1)*(sqrt(2)*sqrt(-3/8*(8*sqrt(1/16*I - 1/16) + 2*I + 1
)^2 - 1/4*(8*sqrt(1/16*I - 1/16) + 2*I - 3)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - 3/8*(8*sqrt(-1/16*I - 1/16) -
2*I + 1)^2 + 8*sqrt(1/16*I - 1/16) + 2*I - 8) + 4*sqrt(1/16*I - 1/16) + 4*sqrt(-1/16*I - 1/16) - 1)^(1/4) + 8
*sqrt(x + sqrt(x^2 + 1))) + 1/2*(sqrt(2)*sqrt(-3/8*(8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 - 1/4*(8*sqrt(1/16*I -
1/16) + 2*I - 3)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - 3/8*(8*sqrt(-1/16*I - 1/16) - 2*I + 1)^2 + 8*sqrt(1/16*I
- 1/16) + 2*I - 8) + 4*sqrt(1/16*I - 1/16) + 4*sqrt(-1/16*I - 1/16) - 1)^(1/4)*log(-1/2*(2*(4*sqrt(1/16*I - 1
/16) + I)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1)^2 - (8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 + ((8*sqrt(1/16*I - 1/16)
+ 2*I + 1)^2 - 32*sqrt(1/16*I - 1/16) - 8*I - 1)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - 2*sqrt(-3/8*(8*sqrt(1/1
6*I - 1/16) + 2*I + 1)^2 - 1/4*(8*sqrt(1/16*I - 1/16) + 2*I - 3)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - 3/8*(8*s
qrt(-1/16*I - 1/16) - 2*I + 1)^2 + 8*sqrt(1/16*I - 1/16) + 2*I - 8)*((sqrt(2)*(8*sqrt(1/16*I - 1/16) + 2*I + 1
) - sqrt(2))*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - sqrt(2)*(8*sqrt(1/16*I - 1/16) + 2*I + 1) + sqrt(2)) + 24*sq
rt(1/16*I - 1/16) + 6*I + 1)*(sqrt(2)*sqrt(-3/8*(8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 - 1/4*(8*sqrt(1/16*I - 1/1
6) + 2*I - 3)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - 3/8*(8*sqrt(-1/16*I - 1/16) - 2*I + 1)^2 + 8*sqrt(1/16*I -
1/16) + 2*I - 8) + 4*sqrt(1/16*I - 1/16) + 4*sqrt(-1/16*I - 1/16) - 1)^(1/4) + 8*sqrt(x + sqrt(x^2 + 1))) - 1/
2*(-sqrt(2)*sqrt(-3/8*(8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 - 1/4*(8*sqrt(1/16*I - 1/16) + 2*I - 3)*(8*sqrt(-1/1
6*I - 1/16) - 2*I + 1) - 3/8*(8*sqrt(-1/16*I - 1/16) - 2*I + 1)^2 + 8*sqrt(1/16*I - 1/16) + 2*I - 8) + 4*sqrt(
1/16*I - 1/16) + 4*sqrt(-1/16*I - 1/16) - 1)^(1/4)*log(1/2*(2*(4*sqrt(1/16*I - 1/16) + I)*(8*sqrt(-1/16*I - 1/
16) - 2*I + 1)^2 - (8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 + ((8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 - 32*sqrt(1/16*I
- 1/16) - 8*I - 1)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) + 2*sqrt(-3/8*(8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 - 1/4
*(8*sqrt(1/16*I - 1/16) + 2*I - 3)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - 3/8*(8*sqrt(-1/16*I - 1/16) - 2*I + 1)
^2 + 8*sqrt(1/16*I - 1/16) + 2*I - 8)*((sqrt(2)*(8*sqrt(1/16*I - 1/16) + 2*I + 1) - sqrt(2))*(8*sqrt(-1/16*I -
1/16) - 2*I + 1) - sqrt(2)*(8*sqrt(1/16*I - 1/16) + 2*I + 1) + sqrt(2)) + 24*sqrt(1/16*I - 1/16) + 6*I + 1)*(
-sqrt(2)*sqrt(-3/8*(8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 - 1/4*(8*sqrt(1/16*I - 1/16) + 2*I - 3)*(8*sqrt(-1/16*I
- 1/16) - 2*I + 1) - 3/8*(8*sqrt(-1/16*I - 1/16) - 2*I + 1)^2 + 8*sqrt(1/16*I - 1/16) + 2*I - 8) + 4*sqrt(1/1
6*I - 1/16) + 4*sqrt(-1/16*I - 1/16) - 1)^(1/4) + 8*sqrt(x + sqrt(x^2 + 1))) + 1/2*(-sqrt(2)*sqrt(-3/8*(8*sqrt
(1/16*I - 1/16) + 2*I + 1)^2 - 1/4*(8*sqrt(1/16*I - 1/16) + 2*I - 3)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - 3/8*
(8*sqrt(-1/16*I - 1/16) - 2*I + 1)^2 + 8*sqrt(1/16*I - 1/16) + 2*I - 8) + 4*sqrt(1/16*I - 1/16) + 4*sqrt(-1/16
*I - 1/16) - 1)^(1/4)*log(-1/2*(2*(4*sqrt(1/16*I - 1/16) + I)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1)^2 - (8*sqrt(1
/16*I - 1/16) + 2*I + 1)^2 + ((8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 - 32*sqrt(1/16*I - 1/16) - 8*I - 1)*(8*sqrt(
-1/16*I - 1/16) - 2*I + 1) + 2*sqrt(-3/8*(8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 - 1/4*(8*sqrt(1/16*I - 1/16) + 2*
I - 3)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - 3/8*(8*sqrt(-1/16*I - 1/16) - 2*I + 1)^2 + 8*sqrt(1/16*I - 1/16) +
2*I - 8)*((sqrt(2)*(8*sqrt(1/16*I - 1/16) + 2*I + 1) - sqrt(2))*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - sqrt(2)*
(8*sqrt(1/16*I - 1/16) + 2*I + 1) + sqrt(2)) + 24*sqrt(1/16*I - 1/16) + 6*I + 1)*(-sqrt(2)*sqrt(-3/8*(8*sqrt(1
/16*I - 1/16) + 2*I + 1)^2 - 1/4*(8*sqrt(1/16*I - 1/16) + 2*I - 3)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - 3/8*(8
*sqrt(-1/16*I - 1/16) - 2*I + 1)^2 + 8*sqrt(1/16*I - 1/16) + 2*I - 8) + 4*sqrt(1/16*I - 1/16) + 4*sqrt(-1/16*I
- 1/16) - 1)^(1/4) + 8*sqrt(x + sqrt(x^2 + 1))) - (-1/2*sqrt(1/16*I - 1/16) - 1/8*I - 1/16)^(1/4)*log(2*((8*s
qrt(1/16*I - 1/16) + 2*I + 1)^3 - 3*(8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 + 152*sqrt(1/16*I - 1/16) + 38*I + 34)
*(-1/2*sqrt(1/16*I - 1/16) - 1/8*I - 1/16)^(1/4) + 8*sqrt(x + sqrt(x^2 + 1))) + (-1/2*sqrt(1/16*I - 1/16) - 1/
8*I - 1/16)^(1/4)*log(-2*((8*sqrt(1/16*I - 1/16) + 2*I + 1)^3 - 3*(8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 + 152*sq
rt(1/16*I - 1/16) + 38*I + 34)*(-1/2*sqrt(1/16*I - 1/16) - 1/8*I - 1/16)^(1/4) + 8*sqrt(x + sqrt(x^2 + 1))) +
(-1/2*sqrt(-1/16*I - 1/16) + 1/8*I - 1/16)^(1/4)*log(2*((8*sqrt(1/16*I - 1/16) + 2*I + 1)^3 + 2*(4*sqrt(1/16*I
- 1/16) + I)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1)^2 - 4*(8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 + ((8*sqrt(1/16*I -
1/16) + 2*I + 1)^2 - 32*sqrt(1/16*I - 1/16) - 8*I - 1)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) + 176*sqrt(1/16*I -
1/16) + 44*I + 3)*(-1/2*sqrt(-1/16*I - 1/16) + 1/8*I - 1/16)^(1/4) + 8*sqrt(x + sqrt(x^2 + 1))) - (-1/2*sqrt(
-1/16*I - 1/16) + 1/8*I - 1/16)^(1/4)*log(-2*((8*sqrt(1/16*I - 1/16) + 2*I + 1)^3 + 2*(4*sqrt(1/16*I - 1/16) +
I)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1)^2 - 4*(8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 + ((8*sqrt(1/16*I - 1/16) + 2
*I + 1)^2 - 32*sqrt(1/16*I - 1/16) - 8*I - 1)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) + 176*sqrt(1/16*I - 1/16) + 4
4*I + 3)*(-1/2*sqrt(-1/16*I - 1/16) + 1/8*I - 1/16)^(1/4) + 8*sqrt(x + sqrt(x^2 + 1)))
Sympy [N/A]
Not integrable
Time = 11.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.23
\[
\int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{\sqrt {x + \sqrt {x^{2} + 1}} \left (x^{4} + 1\right )}\, dx
\]
[In]
integrate((x**4-1)/(x**4+1)/(x+(x**2+1)**(1/2))**(1/2),x)
[Out]
Integral((x - 1)*(x + 1)*(x**2 + 1)/(sqrt(x + sqrt(x**2 + 1))*(x**4 + 1)), x)
Maxima [N/A]
Not integrable
Time = 0.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.19
\[
\int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=\int { \frac {x^{4} - 1}{{\left (x^{4} + 1\right )} \sqrt {x + \sqrt {x^{2} + 1}}} \,d x }
\]
[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)
Giac [N/A]
Not integrable
Time = 0.41 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.19
\[
\int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=\int { \frac {x^{4} - 1}{{\left (x^{4} + 1\right )} \sqrt {x + \sqrt {x^{2} + 1}}} \,d x }
\]
[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)
Mupad [N/A]
Not integrable
Time = 6.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.19
\[
\int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=\int \frac {x^4-1}{\left (x^4+1\right )\,\sqrt {x+\sqrt {x^2+1}}} \,d x
\]
[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)