3.20.48 \(\int \frac {-1+x^4}{(1+x^4) \sqrt {x+\sqrt {1+x^2}}} \, dx\) [1948]

Optimal. Leaf size=136 \[ -\frac {1}{3 \left (x+\sqrt {1+x^2}\right )^{3/2}}+\sqrt {x+\sqrt {1+x^2}}-2 \text {RootSum}\left [1-4 \text {$\#$1}^4+22 \text {$\#$1}^8-4 \text {$\#$1}^{12}+\text {$\#$1}^{16}\& ,\frac {\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}^5}{-1+11 \text {$\#$1}^4-3 \text {$\#$1}^8+\text {$\#$1}^{12}}\& \right ] \]

[Out]

Unintegrable

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Rubi [C] Result contains complex when optimal does not.
time = 1.01, antiderivative size = 511, normalized size of antiderivative = 3.76, number of steps used = 43, number of rules used = 12, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6857, 2142, 14, 2144, 1642, 842, 840, 1180, 210, 212, 213, 209} \begin {gather*} \frac {(-1)^{3/4} \text {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} \text {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} \text {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} \text {ArcTan}\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}}+\frac {(-1)^{3/4} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {\sqrt [4]{-1}+\sqrt {1+i}}}\right )}{\sqrt {\sqrt [4]{-1}+\sqrt {1+i}}} \end {gather*}

Antiderivative was successfully verified.

[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*} \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\\ &=-\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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*}

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Mathematica [A]
time = 0.14, size = 136, normalized size = 1.00 \begin {gather*} -\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 ] \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*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) & ]

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Maple [F]
time = 0.02, 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*} \text {Failed to integrate} \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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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 1.78, size = 3496, normalized size = 25.71 \begin {gather*} \text {Too large to display} \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)*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)*
arctan(1/16*sqrt(2)*(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)*(8*sqrt(1/16*I - 1/16) + 2*I + 1)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) + (8*sqrt(1/16*I - 1/16) + 2*I
+ 1)*(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 - 4)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) + 2)*sqrt(((16*sqrt(1/16*I - 1/16) + 4*I + 1)*(8*sqrt(-1/16*I - 1/16
) - 2*I + 1)^2 - (8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 + 2*((8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 - 32*sqrt(1/16*I
- 1/16) - 8*I - 2)*(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)*((2*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)) + 32*sqrt(1/16*I - 1/16) + 8*I + 2)*sqrt(-sqr
t(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*x + 8*sqrt(x^2 + 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)^(3/4
) - 1/4*(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*sq
rt(-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)*sq
rt(x + sqrt(x^2 + 1))*(8*sqrt(1/16*I - 1/16) + 2*I + 1)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) + sqrt(x + sqrt(x^2
+ 1))*(8*sqrt(1/16*I - 1/16) + 2*I + 1)*(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 - 4)*sqrt(x + sqrt(x^2 + 1))*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) + 2*sqrt
(x + sqrt(x^2 + 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)^(3/4)) + 4*(-1/2*sqrt(-1/16*I - 1/16) + 1/8*I
- 1/16)^(1/4)*arctan(-4*((8*sqrt(1/16*I - 1/16) + 2*I + 1)^3 + (8*sqrt(1/16*I - 1/16) + 2*I + 1)*(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*sq
rt(1/16*I - 1/16) - 8*I - 4)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) + 176*sqrt(1/16*I - 1/16) + 44*I + 21)*sqrt(-(
2*(8*sqrt(1/16*I - 1/16) + 2*I + 1)^3 + (16*sqrt(1/16*I - 1/16) + 4*I + 1)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1)^
2 - 8*(8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 + 2*((8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 - 32*sqrt(1/16*I - 1/16) -
8*I - 2)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) + 352*sqrt(1/16*I - 1/16) + 88*I + 23)*sqrt(-1/2*sqrt(-1/16*I - 1/
16) + 1/8*I - 1/16) + x + sqrt(x^2 + 1))*(-1/2*sqrt(-1/16*I - 1/16) + 1/8*I - 1/16)^(3/4) + 4*(sqrt(x + sqrt(x
^2 + 1))*(8*sqrt(1/16*I - 1/16) + 2*I + 1)*(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 - 4)*sqrt(x + sqrt(x^2 + 1))*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) + ((8*
sqrt(1/16*I - 1/16) + 2*I + 1)^3 - 4*(8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 + 176*sqrt(1/16*I - 1/16) + 44*I + 21
)*sqrt(x + sqrt(x^2 + 1)))*(-1/2*sqrt(-1/16*I - 1/16) + 1/8*I - 1/16)^(3/4)) - 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)*arctan(1/16*(sqrt(2)*(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*s
qrt(1/16*I - 1/16) + 2*I - 8)*(8*sqrt(1/16*I - 1/16) + 2*I + 1)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - (8*sqrt(1
/16*I - 1/16) + 2*I + 1)*(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 - 4)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - 2)*sqrt(((16*sqrt(1/16*I - 1/16) + 4*I + 1)*(8
*sqrt(-1/16*I - 1/16) - 2*I + 1)^2 - (8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 + 2*((8*sqrt(1/16*I - 1/16) + 2*I + 1
)^2 - 32*sqrt(1/16*I - 1/16) - 8*I - 2)*(8*sqrt...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \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 - 1)*(x + 1)*(x**2 + 1)/(sqrt(x + sqrt(x**2 + 1))*(x**4 + 1)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \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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