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

Optimal. Leaf size=224 \[ \frac {-\left (\left (x^2-1\right ) x^2\right )-\sqrt {x^4+1} x^2}{x \left (x^2+1\right ) \sqrt {\sqrt {x^4+1}+x^2}}+\sqrt {\sqrt {2}-1} \tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {2}-1\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right ) \]

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Rubi [C]  time = 2.61, antiderivative size = 426, normalized size of antiderivative = 1.90, number of steps used = 44, number of rules used = 7, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {6742, 2132, 206, 2133, 731, 725, 6725} \begin {gather*} \frac {i \sqrt {1-i x^2}}{2 (-x+i)}-\frac {i \sqrt {1-i x^2}}{2 (x+i)}-\frac {i \sqrt {1+i x^2}}{2 (-x+i)}+\frac {i \sqrt {1+i x^2}}{2 (x+i)}+\frac {1}{2} \sqrt {1+i} \tanh ^{-1}\left (\frac {1-x}{\sqrt {1+i} \sqrt {1-i x^2}}\right )+\frac {\tanh ^{-1}\left (\frac {1-x}{\sqrt {1+i} \sqrt {1-i x^2}}\right )}{(1+i)^{5/2}}-\frac {1}{2} \sqrt {1+i} \tanh ^{-1}\left (\frac {x+1}{\sqrt {1+i} \sqrt {1-i x^2}}\right )-\frac {\tanh ^{-1}\left (\frac {x+1}{\sqrt {1+i} \sqrt {1-i x^2}}\right )}{(1+i)^{5/2}}+\frac {1}{2} \sqrt {1-i} \tanh ^{-1}\left (\frac {1-x}{\sqrt {1-i} \sqrt {1+i x^2}}\right )+\frac {\tanh ^{-1}\left (\frac {1-x}{\sqrt {1-i} \sqrt {1+i x^2}}\right )}{(1-i)^{5/2}}-\frac {1}{2} \sqrt {1-i} \tanh ^{-1}\left (\frac {x+1}{\sqrt {1-i} \sqrt {1+i x^2}}\right )-\frac {\tanh ^{-1}\left (\frac {x+1}{\sqrt {1-i} \sqrt {1+i x^2}}\right )}{(1-i)^{5/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {\sqrt {x^4+1}+x^2}}\right )}{\sqrt {2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((I/2)*Sqrt[1 - I*x^2])/(I - x) - ((I/2)*Sqrt[1 - I*x^2])/(I + x) - ((I/2)*Sqrt[1 + I*x^2])/(I - x) + ((I/2)*S
qrt[1 + I*x^2])/(I + x) + ArcTanh[(1 - x)/(Sqrt[1 + I]*Sqrt[1 - I*x^2])]/(1 + I)^(5/2) + (Sqrt[1 + I]*ArcTanh[
(1 - x)/(Sqrt[1 + I]*Sqrt[1 - I*x^2])])/2 - ArcTanh[(1 + x)/(Sqrt[1 + I]*Sqrt[1 - I*x^2])]/(1 + I)^(5/2) - (Sq
rt[1 + I]*ArcTanh[(1 + x)/(Sqrt[1 + I]*Sqrt[1 - I*x^2])])/2 + ArcTanh[(1 - x)/(Sqrt[1 - I]*Sqrt[1 + I*x^2])]/(
1 - I)^(5/2) + (Sqrt[1 - I]*ArcTanh[(1 - x)/(Sqrt[1 - I]*Sqrt[1 + I*x^2])])/2 - ArcTanh[(1 + x)/(Sqrt[1 - I]*S
qrt[1 + I*x^2])]/(1 - I)^(5/2) - (Sqrt[1 - I]*ArcTanh[(1 + x)/(Sqrt[1 - I]*Sqrt[1 + I*x^2])])/2 + ArcTanh[(Sqr
t[2]*x)/Sqrt[x^2 + Sqrt[1 + x^4]]]/Sqrt[2]

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 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 731

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1)*(a + c*x^2)^(p
+ 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d)/(c*d^2 + a*e^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x]
/; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 3, 0]

Rule 2132

Int[Sqrt[(c_.)*(x_)^2 + (d_.)*Sqrt[(a_) + (b_.)*(x_)^4]]/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[d, Subst
[Int[1/(1 - 2*c*x^2), x], x, x/Sqrt[c*x^2 + d*Sqrt[a + b*x^4]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[c^2 - b*d
^2, 0]

Rule 2133

Int[(((c_.) + (d_.)*(x_))^(m_.)*Sqrt[(b_.)*(x_)^2 + Sqrt[(a_) + (e_.)*(x_)^4]])/Sqrt[(a_) + (e_.)*(x_)^4], x_S
ymbol] :> Dist[(1 - I)/2, Int[(c + d*x)^m/Sqrt[Sqrt[a] - I*b*x^2], x], x] + Dist[(1 + I)/2, Int[(c + d*x)^m/Sq
rt[Sqrt[a] + I*b*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[e, b^2] && GtQ[a, 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]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

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

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Mathematica [F]  time = 0.47, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^2\right )^2 \sqrt {1+x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.08, size = 295, normalized size = 1.32 \begin {gather*} \frac {-x^2 \left (-1+x^2\right )-x^2 \sqrt {1+x^4}}{x \left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}+\sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\frac {-\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}}+\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} x^2+\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \sqrt {1+x^4}}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {-\frac {1}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}+\frac {\sqrt {1+x^4}}{\sqrt {2}}}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )-\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {-\sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}}+\sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}} x^2+\sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}} \sqrt {1+x^4}}{x \sqrt {x^2+\sqrt {1+x^4}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(x^2*(-1 + x^2)) - x^2*Sqrt[1 + x^4])/(x*(1 + x^2)*Sqrt[x^2 + Sqrt[1 + x^4]]) + Sqrt[-1 + Sqrt[2]]*ArcTan[(-
Sqrt[1/2 + 1/Sqrt[2]] + Sqrt[1/2 + 1/Sqrt[2]]*x^2 + Sqrt[1/2 + 1/Sqrt[2]]*Sqrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1
+ x^4]])] + Sqrt[2]*ArcTanh[(-(1/Sqrt[2]) + x^2/Sqrt[2] + Sqrt[1 + x^4]/Sqrt[2])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])
] - Sqrt[1 + Sqrt[2]]*ArcTanh[(-Sqrt[-1/2 + 1/Sqrt[2]] + Sqrt[-1/2 + 1/Sqrt[2]]*x^2 + Sqrt[-1/2 + 1/Sqrt[2]]*S
qrt[1 + x^4])/(x*Sqrt[x^2 + Sqrt[1 + x^4]])]

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fricas [B]  time = 1.71, size = 391, normalized size = 1.75 \begin {gather*} \frac {4 \, {\left (x^{2} + 1\right )} \sqrt {\sqrt {2} - 1} \arctan \left (\frac {{\left (\sqrt {2} x^{2} + x^{2} + \sqrt {x^{4} + 1} {\left ({\left (\sqrt {2} + 1\right )} \sqrt {-2 \, \sqrt {2} + 3} - \sqrt {2} - 1\right )} - {\left (x^{2} + \sqrt {2} {\left (x^{2} + 2\right )} + 3\right )} \sqrt {-2 \, \sqrt {2} + 3} + 1\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} - 1}}{2 \, x}\right ) + \sqrt {2} {\left (x^{2} + 1\right )} \log \left (4 \, x^{4} + 4 \, \sqrt {x^{4} + 1} x^{2} + 2 \, {\left (\sqrt {2} x^{3} + \sqrt {2} \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + 1\right ) - {\left (x^{2} + 1\right )} \sqrt {\sqrt {2} + 1} \log \left (\frac {\sqrt {2} x^{2} + 2 \, x^{2} + {\left (x^{3} + \sqrt {2} x - \sqrt {x^{4} + 1} x + x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} + 1} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} + 1}\right ) + {\left (x^{2} + 1\right )} \sqrt {\sqrt {2} + 1} \log \left (\frac {\sqrt {2} x^{2} + 2 \, x^{2} - {\left (x^{3} + \sqrt {2} x - \sqrt {x^{4} + 1} x + x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} + 1} + \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} + 1}\right ) - 4 \, {\left (x^{3} - \sqrt {x^{4} + 1} x + x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{4 \, {\left (x^{2} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{2}-1\right )^{2} \sqrt {x^{2}+\sqrt {x^{4}+1}}}{\left (x^{2}+1\right )^{2} \sqrt {x^{4}+1}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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