3.20.84 \(\int \frac {(-1+x^4) \sqrt {x+\sqrt {1+x^2}}}{1+x^4} \, dx\)
Optimal. Leaf size=140 \[ -2 \text {RootSum}\left [\text {$\#$1}^{16}-4 \text {$\#$1}^{12}+22 \text {$\#$1}^8-4 \text {$\#$1}^4+1\& ,\frac {\text {$\#$1}^7 \log \left (\sqrt {\sqrt {x^2+1}+x}-\text {$\#$1}\right )+\text {$\#$1}^3 \log \left (\sqrt {\sqrt {x^2+1}+x}-\text {$\#$1}\right )}{\text {$\#$1}^{12}-3 \text {$\#$1}^8+11 \text {$\#$1}^4-1}\& \right ]+\frac {1}{3} \left (\sqrt {x^2+1}+x\right )^{3/2}-\frac {1}{\sqrt {\sqrt {x^2+1}+x}} \]
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Rubi [C] time = 1.32, antiderivative size = 513, normalized size of antiderivative =
3.66, number of steps used = 39, number of rules used = 11, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\)
= 0.393, Rules used = {6725, 2117, 14, 2119, 1628, 826, 1166, 204, 206, 207, 203} \begin {gather*} \frac {1}{3} \left (\sqrt {x^2+1}+x\right )^{3/2}-\frac {1}{\sqrt {\sqrt {x^2+1}+x}}-(-1)^{3/4} \sqrt {-(-1)^{3/4}+\sqrt {1-i}} \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {-(-1)^{3/4}+\sqrt {1-i}}}\right )+(-1)^{3/4} \sqrt {(-1)^{3/4}+\sqrt {1-i}} \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {(-1)^{3/4}+\sqrt {1-i}}}\right )-\sqrt [4]{-1} \sqrt {-\sqrt [4]{-1}+\sqrt {1+i}} \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {-\sqrt [4]{-1}+\sqrt {1+i}}}\right )+\sqrt [4]{-1} \sqrt {\sqrt [4]{-1}+\sqrt {1+i}} \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {\sqrt [4]{-1}+\sqrt {1+i}}}\right )+(-1)^{3/4} \sqrt {-(-1)^{3/4}+\sqrt {1-i}} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {-(-1)^{3/4}+\sqrt {1-i}}}\right )-(-1)^{3/4} \sqrt {(-1)^{3/4}+\sqrt {1-i}} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {(-1)^{3/4}+\sqrt {1-i}}}\right )+\sqrt [4]{-1} \sqrt {-\sqrt [4]{-1}+\sqrt {1+i}} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {-\sqrt [4]{-1}+\sqrt {1+i}}}\right )-\sqrt [4]{-1} \sqrt {\sqrt [4]{-1}+\sqrt {1+i}} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {\sqrt [4]{-1}+\sqrt {1+i}}}\right ) \end {gather*}
Antiderivative was successfully verified.
[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 - (-1)^(3/4)*Sqrt[-(-1)^(3/4) + Sqrt[1 - I]]*ArcTan
[Sqrt[x + Sqrt[1 + x^2]]/Sqrt[-(-1)^(3/4) + Sqrt[1 - I]]] + (-1)^(3/4)*Sqrt[(-1)^(3/4) + Sqrt[1 - I]]*ArcTan[S
qrt[x + Sqrt[1 + x^2]]/Sqrt[(-1)^(3/4) + Sqrt[1 - I]]] - (-1)^(1/4)*Sqrt[-(-1)^(1/4) + Sqrt[1 + I]]*ArcTan[Sqr
t[x + Sqrt[1 + x^2]]/Sqrt[-(-1)^(1/4) + Sqrt[1 + I]]] + (-1)^(1/4)*Sqrt[(-1)^(1/4) + Sqrt[1 + I]]*ArcTan[Sqrt[
x + Sqrt[1 + x^2]]/Sqrt[(-1)^(1/4) + Sqrt[1 + I]]] + (-1)^(3/4)*Sqrt[-(-1)^(3/4) + Sqrt[1 - I]]*ArcTanh[Sqrt[x
+ Sqrt[1 + x^2]]/Sqrt[-(-1)^(3/4) + Sqrt[1 - I]]] - (-1)^(3/4)*Sqrt[(-1)^(3/4) + Sqrt[1 - I]]*ArcTanh[Sqrt[x
+ Sqrt[1 + x^2]]/Sqrt[(-1)^(3/4) + Sqrt[1 - I]]] + (-1)^(1/4)*Sqrt[-(-1)^(1/4) + Sqrt[1 + I]]*ArcTanh[Sqrt[x +
Sqrt[1 + x^2]]/Sqrt[-(-1)^(1/4) + Sqrt[1 + I]]] - (-1)^(1/4)*Sqrt[(-1)^(1/4) + Sqrt[1 + I]]*ArcTanh[Sqrt[x +
Sqrt[1 + x^2]]/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 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 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 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 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 {\left (-1+x^4\right ) \sqrt {x+\sqrt {1+x^2}}}{1+x^4} \, dx &=\int \left (\sqrt {x+\sqrt {1+x^2}}-\frac {2 \sqrt {x+\sqrt {1+x^2}}}{1+x^4}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {x+\sqrt {1+x^2}}}{1+x^4} \, dx\right )+\int \sqrt {x+\sqrt {1+x^2}} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1+x^2}{x^{3/2}} \, dx,x,x+\sqrt {1+x^2}\right )-2 \int \left (\frac {i \sqrt {x+\sqrt {1+x^2}}}{2 \left (i-x^2\right )}+\frac {i \sqrt {x+\sqrt {1+x^2}}}{2 \left (i+x^2\right )}\right ) \, dx\\ &=-\left (i \int \frac {\sqrt {x+\sqrt {1+x^2}}}{i-x^2} \, dx\right )-i \int \frac {\sqrt {x+\sqrt {1+x^2}}}{i+x^2} \, dx+\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{x^{3/2}}+\sqrt {x}\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}-i \int \left (-\frac {(-1)^{3/4} \sqrt {x+\sqrt {1+x^2}}}{2 \left (\sqrt [4]{-1}-x\right )}-\frac {(-1)^{3/4} \sqrt {x+\sqrt {1+x^2}}}{2 \left (\sqrt [4]{-1}+x\right )}\right ) \, dx-i \int \left (-\frac {\sqrt [4]{-1} \sqrt {x+\sqrt {1+x^2}}}{2 \left (-(-1)^{3/4}-x\right )}-\frac {\sqrt [4]{-1} \sqrt {x+\sqrt {1+x^2}}}{2 \left (-(-1)^{3/4}+x\right )}\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} \sqrt [4]{-1} \int \frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt [4]{-1}-x} \, dx-\frac {1}{2} \sqrt [4]{-1} \int \frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt [4]{-1}+x} \, dx+\frac {1}{2} (-1)^{3/4} \int \frac {\sqrt {x+\sqrt {1+x^2}}}{-(-1)^{3/4}-x} \, dx+\frac {1}{2} (-1)^{3/4} \int \frac {\sqrt {x+\sqrt {1+x^2}}}{-(-1)^{3/4}+x} \, dx\\ &=-\frac {1}{\sqrt {x+\sqrt {1+x^2}}}+\frac {1}{3} \left (x+\sqrt {1+x^2}\right )^{3/2}-\frac {1}{2} \sqrt [4]{-1} \operatorname {Subst}\left (\int \frac {1+x^2}{\sqrt {x} \left (1+2 \sqrt [4]{-1} x-x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )-\frac {1}{2} \sqrt [4]{-1} \operatorname {Subst}\left (\int \frac {1+x^2}{\sqrt {x} \left (-1+2 \sqrt [4]{-1} x+x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )+\frac {1}{2} (-1)^{3/4} \operatorname {Subst}\left (\int \frac {1+x^2}{\sqrt {x} \left (1-2 (-1)^{3/4} x-x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )+\frac {1}{2} (-1)^{3/4} \operatorname {Subst}\left (\int \frac {1+x^2}{\sqrt {x} \left (-1-2 (-1)^{3/4} 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}-\frac {1}{2} \sqrt [4]{-1} \operatorname {Subst}\left (\int \left (-\frac {1}{\sqrt {x}}+\frac {2 \left (1+\sqrt [4]{-1} x\right )}{\sqrt {x} \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} \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {x}}+\frac {2 \left (1-\sqrt [4]{-1} x\right )}{\sqrt {x} \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} \operatorname {Subst}\left (\int \left (-\frac {1}{\sqrt {x}}+\frac {2 \left (1-(-1)^{3/4} x\right )}{\sqrt {x} \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} \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {x}}+\frac {2 \left (1+(-1)^{3/4} x\right )}{\sqrt {x} \left (-1-2 (-1)^{3/4} x+x^2\right )}\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 [4]{-1} \operatorname {Subst}\left (\int \frac {1+\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} \operatorname {Subst}\left (\int \frac {1-\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} \operatorname {Subst}\left (\int \frac {1-(-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} \operatorname {Subst}\left (\int \frac {1+(-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}{\sqrt {x+\sqrt {1+x^2}}}+\frac {1}{3} \left (x+\sqrt {1+x^2}\right )^{3/2}-\left (2 \sqrt [4]{-1}\right ) \operatorname {Subst}\left (\int \frac {1+\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 ) \operatorname {Subst}\left (\int \frac {1-\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 ) \operatorname {Subst}\left (\int \frac {1-(-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 ) \operatorname {Subst}\left (\int \frac {1+(-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}{\sqrt {x+\sqrt {1+x^2}}}+\frac {1}{3} \left (x+\sqrt {1+x^2}\right )^{3/2}+\left ((-1)^{3/4} \left (-(-1)^{3/4}-\sqrt {1-i}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-(-1)^{3/4}-\sqrt {1-i}-x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )+\left ((-1)^{3/4} \left ((-1)^{3/4}+\sqrt {1-i}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-(-1)^{3/4}-\sqrt {1-i}+x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )+\left (i+(-1)^{3/4} \sqrt {1-i}\right ) \operatorname {Subst}\left (\int \frac {1}{-(-1)^{3/4}+\sqrt {1-i}-x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )-\left (\sqrt [4]{-1} \left (-\sqrt [4]{-1}-\sqrt {1+i}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1}+\sqrt {1+i}+x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )-\left (\sqrt [4]{-1} \left (\sqrt [4]{-1}-\sqrt {1+i}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1}-\sqrt {1+i}-x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )-\left (\sqrt [4]{-1} \left (-\sqrt [4]{-1}+\sqrt {1+i}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1}-\sqrt {1+i}+x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )-\left (\sqrt [4]{-1} \left (\sqrt [4]{-1}+\sqrt {1+i}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1}+\sqrt {1+i}-x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )+\left (\left (\frac {1}{2}-\frac {i}{2}\right ) \left ((1-i)+\sqrt {2-2 i}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-(-1)^{3/4}+\sqrt {1-i}+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}+(1-i) \sqrt {\frac {1}{2} \left (-(-1)^{3/4}+\sqrt {1-i}\right )} \tan ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-(-1)^{3/4}+\sqrt {1-i}}}\right )+(-1)^{3/4} \sqrt {(-1)^{3/4}+\sqrt {1-i}} \tan ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {(-1)^{3/4}+\sqrt {1-i}}}\right )-(1+i) \sqrt {\frac {1}{2} \left (-\sqrt [4]{-1}+\sqrt {1+i}\right )} \tan ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-\sqrt [4]{-1}+\sqrt {1+i}}}\right )+\sqrt [4]{-1} \sqrt {\sqrt [4]{-1}+\sqrt {1+i}} \tan ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {\sqrt [4]{-1}+\sqrt {1+i}}}\right )+(-1)^{3/4} \sqrt {-(-1)^{3/4}+\sqrt {1-i}} \tanh ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-(-1)^{3/4}+\sqrt {1-i}}}\right )-(-1)^{3/4} \sqrt {(-1)^{3/4}+\sqrt {1-i}} \tanh ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {(-1)^{3/4}+\sqrt {1-i}}}\right )+\sqrt [4]{-1} \sqrt {-\sqrt [4]{-1}+\sqrt {1+i}} \tanh ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-\sqrt [4]{-1}+\sqrt {1+i}}}\right )-\sqrt [4]{-1} \sqrt {\sqrt [4]{-1}+\sqrt {1+i}} \tanh ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {\sqrt [4]{-1}+\sqrt {1+i}}}\right )\\ \end {align*}
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Mathematica [A] time = 0.28, size = 140, normalized size = 1.00 \begin {gather*} -2 \text {RootSum}\left [\text {$\#$1}^{16}-4 \text {$\#$1}^{12}+22 \text {$\#$1}^8-4 \text {$\#$1}^4+1\&,\frac {\text {$\#$1}^7 \log \left (\sqrt {\sqrt {x^2+1}+x}-\text {$\#$1}\right )+\text {$\#$1}^3 \log \left (\sqrt {\sqrt {x^2+1}+x}-\text {$\#$1}\right )}{\text {$\#$1}^{12}-3 \text {$\#$1}^8+11 \text {$\#$1}^4-1}\&\right ]+\frac {1}{3} \left (\sqrt {x^2+1}+x\right )^{3/2}-\frac {1}{\sqrt {\sqrt {x^2+1}+x}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((-1 + x^4)*Sqrt[x + Sqrt[1 + x^2]])/(1 + x^4),x]
[Out]
-(1/Sqrt[x + Sqrt[1 + x^2]]) + (x + Sqrt[1 + x^2])^(3/2)/3 - 2*RootSum[1 - 4*#1^4 + 22*#1^8 - 4*#1^12 + #1^16
& , (Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1^3 + Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1^7)/(-1 + 11*#1^4 - 3*#1^8 +
#1^12) & ]
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IntegrateAlgebraic [A] time = 0.18, size = 140, normalized size = 1.00 \begin {gather*} -\frac {1}{\sqrt {x+\sqrt {1+x^2}}}+\frac {1}{3} \left (x+\sqrt {1+x^2}\right )^{3/2}-2 \text {RootSum}\left [1-4 \text {$\#$1}^4+22 \text {$\#$1}^8-4 \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}^3+\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}^7}{-1+11 \text {$\#$1}^4-3 \text {$\#$1}^8+\text {$\#$1}^{12}}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[((-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 - 2*RootSum[1 - 4*#1^4 + 22*#1^8 - 4*#1^12 + #1^16
& , (Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1^3 + Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1^7)/(-1 + 11*#1^4 - 3*#1^8 +
#1^12) & ]
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fricas [B] time = 2.41, size = 3304, normalized size = 23.60
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[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)) + 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(
1/32*(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*sqr
t(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(1/2*((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 - 3)*(8*sqrt(-
1/16*I - 1/16) - 2*I + 1) + 2*(sqrt(2)*(8*sqrt(1/16*I - 1/16) + 2*I + 1)*(8*sqrt(-1/16*I - 1/16) - 2*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) + 8*sqrt(1/1
6*I - 1/16) + 2*I + 1)*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) + 4*x + 4*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)^(1/4) - 1/16*(2*sqrt(x + sqrt(x^2 + 1))*(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 - 32*sqrt(1/16*I - 1/16) - 8*I - 1)*sqrt(x + sqr
t(x^2 + 1))*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) + 2*((sqrt(2)*(8*sqrt(1/16*I - 1/16) + 2*I + 1) - sqrt(2))*sqrt
(x + sqrt(x^2 + 1))*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - (sqrt(2)*(8*sqrt(1/16*I - 1/16) + 2*I + 1) - sqrt(2))
*sqrt(x + sqrt(x^2 + 1)))*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)^2 - 24*sqrt(1/16*I - 1/16) - 6*I - 1)*sqrt(x + sqrt(x^2 + 1)))*(-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)^(1/4)) + 4*(-1/2*sqrt(-1/16*I - 1/16) + 1/8*I - 1/16)^(1/4)*arctan(-1/4*
((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(-((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*sqrt(1/16*I - 1/16) - 8*I - 3)*(8*sqrt(-1/16*I - 1/16) -
2*I + 1) + 176*sqrt(1/16*I - 1/16) + 44*I + 20)*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)^(1/4) + 1/4*(2*sqrt(x + sqrt(x^2 + 1))*(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 - 32*sqrt(1/16*I - 1/16) -
8*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)^3 - 4
*(8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 + 176*sqrt(1/16*I - 1/16) + 44*I + 3)*sqrt(x + sqrt(x^2 + 1)))*(-1/2*sqrt
(-1/16*I - 1/16) + 1/8*I - 1/16)^(1/4)) - 2*(sqrt(2)*sqrt(-3/8*(8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 - 1/4*(8*sq
rt(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(1/32*(4*sqr
t(x + sqrt(x^2 + 1))*(4*sqrt(1/16*I - 1/16) + I)*(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 - 1)*sqrt(x + sqrt(x^2 + 1))*(8*sqrt(-1/16*I - 1/16) - 2*I + 1
) - 4*((sqrt(2)*(8*sqrt(1/16*I - 1/16) + 2*I + 1) - sqrt(2))*sqrt(x + sqrt(x^2 + 1))*(8*sqrt(-1/16*I - 1/16) -
2*I + 1) - (sqrt(2)*(8*sqrt(1/16*I - 1/16) + 2*I + 1) - sqrt(2))*sqrt(x + sqrt(x^2 + 1)))*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*(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 - 3
2*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 - 1/1
6) - 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*s
qrt(-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(1/2*((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 - 3)*(8*sqrt(-1/16*I - 1/16) - 2*I + 1) - 2*(sqrt(2)*(8*s
qrt(1/16*I - 1/16) + 2*I + 1)*(8*sqrt(-1/16*I - 1/16) - 2*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) + 8*sqrt(1/16*I - 1/16) + 2*I + 1)*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) + 4*x + 4*sqrt(x^2 + 1)) - 2*((8*sqrt(1/16*I - 1/16) + 2*I + 1)^2 - 24*sqrt(1/16*I -
1/16) - 6*I - 1)*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*sqr
t(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)) - 4*(-1/2*sqrt(1/1
6*I - 1/16) - 1/8*I - 1/16)^(1/4)*arctan(-1/4*(((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(((8*sqrt(1/16*I - 1/16) + 2*I + 1)^3 - 4*(8*sqrt(1/1
6*I - 1/16) + 2*I + 1)^2 + 168*sqrt(1/16*I - 1/16) + 42*I + 19)*sqrt(-1/2*sqrt(1/16*I - 1/16) - 1/8*I - 1/16)
+ x + sqrt(x^2 + 1)) - ((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(x + sqrt(x^2 + 1)))*(-1/2*sqrt(1/16*I - 1/16) - 1/8*I - 1/16)^(1/4)) - 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*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(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) + 2*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*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(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)^(3/4) + 2*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*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(2)*sqrt(-3/8*(8*s
qrt(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) + 2*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)*l
og(-1/2*(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*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)*(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*sq
rt(-1/16*I - 1/16) - 2*I + 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)^(3/4) + 2*sqrt(x + sqrt(x^2 + 1)
)) + (-1/2*sqrt(1/16*I - 1/16) - 1/8*I - 1/16)^(1/4)*log(4*((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 + 19)*(-1/2*sqrt(1/16*I - 1/16) - 1/8*I - 1/16)^(3/
4) + sqrt(x + sqrt(x^2 + 1))) - (-1/2*sqrt(1/16*I - 1/16) - 1/8*I - 1/16)^(1/4)*log(-4*((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 + 19)*(-1/2*sqrt(1/16*I
- 1/16) - 1/8*I - 1/16)^(3/4) + sqrt(x + sqrt(x^2 + 1))) - (-1/2*sqrt(-1/16*I - 1/16) + 1/8*I - 1/16)^(1/4)*l
og(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*sqrt(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)*(-1/2*sqrt(-1/16*I - 1/16)
+ 1/8*I - 1/16)^(3/4) + sqrt(x + sqrt(x^2 + 1))) + (-1/2*sqrt(-1/16*I - 1/16) + 1/8*I - 1/16)^(1/4)*log(-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*sqrt(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)*(-1/2*sqrt(-1/16*I - 1/16) + 1/8*
I - 1/16)^(3/4) + sqrt(x + sqrt(x^2 + 1)))
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} - 1\right )} \sqrt {x + \sqrt {x^{2} + 1}}}{x^{4} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{4}-1\right ) \sqrt {x +\sqrt {x^{2}+1}}}{x^{4}+1}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[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)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} - 1\right )} \sqrt {x + \sqrt {x^{2} + 1}}}{x^{4} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^4-1\right )\,\sqrt {x+\sqrt {x^2+1}}}{x^4+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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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 ) \sqrt {x + \sqrt {x^{2} + 1}} \left (x^{2} + 1\right )}{x^{4} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x**4-1)*(x+(x**2+1)**(1/2))**(1/2)/(x**4+1),x)
[Out]
Integral((x - 1)*(x + 1)*sqrt(x + sqrt(x**2 + 1))*(x**2 + 1)/(x**4 + 1), x)
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