\(\int \frac {1}{(1+x^4) \sqrt [4]{2+x^4}} \, dx\) [315]

Optimal result

Integrand size = 17, antiderivative size = 141 \[ \int \frac {1}{\left (1+x^4\right ) \sqrt [4]{2+x^4}} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} x}{\sqrt [4]{2+x^4}}\right )}{2 \sqrt {2}}+\frac {\arctan \left (1+\frac {\sqrt {2} x}{\sqrt [4]{2+x^4}}\right )}{2 \sqrt {2}}-\frac {\log \left (1+\frac {x^2}{\sqrt {2+x^4}}-\frac {\sqrt {2} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {2}}+\frac {\log \left (1+\frac {x^2}{\sqrt {2+x^4}}+\frac {\sqrt {2} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {2}} \]

[Out]

1/4*arctan(-1+x*2^(1/2)/(x^4+2)^(1/4))*2^(1/2)+1/4*arctan(1+x*2^(1/2)/(x^4+2)^(1/4))*2^(1/2)-1/8*ln(1-x*2^(1/2
)/(x^4+2)^(1/4)+x^2/(x^4+2)^(1/2))*2^(1/2)+1/8*ln(1+x*2^(1/2)/(x^4+2)^(1/4)+x^2/(x^4+2)^(1/2))*2^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {385, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {1}{\left (1+x^4\right ) \sqrt [4]{2+x^4}} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} x}{\sqrt [4]{x^4+2}}\right )}{2 \sqrt {2}}+\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt [4]{x^4+2}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (-\frac {\sqrt {2} x}{\sqrt [4]{x^4+2}}+\frac {x^2}{\sqrt {x^4+2}}+1\right )}{4 \sqrt {2}}+\frac {\log \left (\frac {\sqrt {2} x}{\sqrt [4]{x^4+2}}+\frac {x^2}{\sqrt {x^4+2}}+1\right )}{4 \sqrt {2}} \]

[In]

Int[1/((1 + x^4)*(2 + x^4)^(1/4)),x]

[Out]

-1/2*ArcTan[1 - (Sqrt[2]*x)/(2 + x^4)^(1/4)]/Sqrt[2] + ArcTan[1 + (Sqrt[2]*x)/(2 + x^4)^(1/4)]/(2*Sqrt[2]) - L
og[1 + x^2/Sqrt[2 + x^4] - (Sqrt[2]*x)/(2 + x^4)^(1/4)]/(4*Sqrt[2]) + Log[1 + x^2/Sqrt[2 + x^4] + (Sqrt[2]*x)/
(2 + x^4)^(1/4)]/(4*Sqrt[2])

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 217

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

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right ) \\ & = \frac {1}{4} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )+\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )-\frac {\text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {2}} \\ & = -\frac {\log \left (1+\frac {x^2}{\sqrt {2+x^4}}-\frac {\sqrt {2} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {2}}+\frac {\log \left (1+\frac {x^2}{\sqrt {2+x^4}}+\frac {\sqrt {2} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} x}{\sqrt [4]{2+x^4}}\right )}{2 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} x}{\sqrt [4]{2+x^4}}\right )}{2 \sqrt {2}} \\ & = -\frac {\arctan \left (1-\frac {\sqrt {2} x}{\sqrt [4]{2+x^4}}\right )}{2 \sqrt {2}}+\frac {\arctan \left (1+\frac {\sqrt {2} x}{\sqrt [4]{2+x^4}}\right )}{2 \sqrt {2}}-\frac {\log \left (1+\frac {x^2}{\sqrt {2+x^4}}-\frac {\sqrt {2} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {2}}+\frac {\log \left (1+\frac {x^2}{\sqrt {2+x^4}}+\frac {\sqrt {2} x}{\sqrt [4]{2+x^4}}\right )}{4 \sqrt {2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.54 \[ \int \frac {1}{\left (1+x^4\right ) \sqrt [4]{2+x^4}} \, dx=\frac {\arctan \left (\frac {\sqrt {2} x \sqrt [4]{2+x^4}}{-x^2+\sqrt {2+x^4}}\right )+\text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{2+x^4}}{x^2+\sqrt {2+x^4}}\right )}{2 \sqrt {2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.81 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.71

method	result	size
pseudoelliptic	\(-\frac {\sqrt {2}\, \left (\ln \left (\frac {-\left (x^{4}+2\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{4}+2}}{\left (x^{4}+2\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{4}+2}}\right )+2 \arctan \left (\frac {\left (x^{4}+2\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right )+2 \arctan \left (\frac {\left (x^{4}+2\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right )\right )}{8}\)	\(100\)
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {\left (x^{4}+2\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-\sqrt {x^{4}+2}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}+\left (x^{4}+2\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{x^{4}+1}\right )}{4}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {-\sqrt {x^{4}+2}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-\left (x^{4}+2\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+\left (x^{4}+2\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{4}+1}\right )}{4}\)	\(149\)

[In]

int(1/(x^4+1)/(x^4+2)^(1/4),x,method=_RETURNVERBOSE)

[Out]

-1/8*2^(1/2)*(ln((-(x^4+2)^(1/4)*2^(1/2)*x+x^2+(x^4+2)^(1/2))/((x^4+2)^(1/4)*2^(1/2)*x+x^2+(x^4+2)^(1/2)))+2*a
rctan(((x^4+2)^(1/4)*2^(1/2)+x)/x)+2*arctan(((x^4+2)^(1/4)*2^(1/2)-x)/x))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.39 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.60 \[ \int \frac {1}{\left (1+x^4\right ) \sqrt [4]{2+x^4}} \, dx=-\left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {\left (i + 1\right ) \, \sqrt {2} {\left (x^{4} + 2\right )}^{\frac {1}{4}} x^{3} - 2 i \, \sqrt {x^{4} + 2} x^{2} + \left (i - 1\right ) \, \sqrt {2} {\left (x^{4} + 2\right )}^{\frac {3}{4}} x + 2}{x^{4} + 1}\right ) + \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {-\left (i - 1\right ) \, \sqrt {2} {\left (x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + 2 i \, \sqrt {x^{4} + 2} x^{2} - \left (i + 1\right ) \, \sqrt {2} {\left (x^{4} + 2\right )}^{\frac {3}{4}} x + 2}{x^{4} + 1}\right ) - \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {\left (i - 1\right ) \, \sqrt {2} {\left (x^{4} + 2\right )}^{\frac {1}{4}} x^{3} + 2 i \, \sqrt {x^{4} + 2} x^{2} + \left (i + 1\right ) \, \sqrt {2} {\left (x^{4} + 2\right )}^{\frac {3}{4}} x + 2}{x^{4} + 1}\right ) + \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {-\left (i + 1\right ) \, \sqrt {2} {\left (x^{4} + 2\right )}^{\frac {1}{4}} x^{3} - 2 i \, \sqrt {x^{4} + 2} x^{2} - \left (i - 1\right ) \, \sqrt {2} {\left (x^{4} + 2\right )}^{\frac {3}{4}} x + 2}{x^{4} + 1}\right ) \]

[In]

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

[Out]

-(1/16*I + 1/16)*sqrt(2)*log(((I + 1)*sqrt(2)*(x^4 + 2)^(1/4)*x^3 - 2*I*sqrt(x^4 + 2)*x^2 + (I - 1)*sqrt(2)*(x
^4 + 2)^(3/4)*x + 2)/(x^4 + 1)) + (1/16*I - 1/16)*sqrt(2)*log((-(I - 1)*sqrt(2)*(x^4 + 2)^(1/4)*x^3 + 2*I*sqrt
(x^4 + 2)*x^2 - (I + 1)*sqrt(2)*(x^4 + 2)^(3/4)*x + 2)/(x^4 + 1)) - (1/16*I - 1/16)*sqrt(2)*log(((I - 1)*sqrt(
2)*(x^4 + 2)^(1/4)*x^3 + 2*I*sqrt(x^4 + 2)*x^2 + (I + 1)*sqrt(2)*(x^4 + 2)^(3/4)*x + 2)/(x^4 + 1)) + (1/16*I +
1/16)*sqrt(2)*log((-(I + 1)*sqrt(2)*(x^4 + 2)^(1/4)*x^3 - 2*I*sqrt(x^4 + 2)*x^2 - (I - 1)*sqrt(2)*(x^4 + 2)^(
3/4)*x + 2)/(x^4 + 1))

Sympy [F]

\[ \int \frac {1}{\left (1+x^4\right ) \sqrt [4]{2+x^4}} \, dx=\int \frac {1}{\left (x^{4} + 1\right ) \sqrt [4]{x^{4} + 2}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{\left (1+x^4\right ) \sqrt [4]{2+x^4}} \, dx=\int { \frac {1}{{\left (x^{4} + 2\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1}{\left (1+x^4\right ) \sqrt [4]{2+x^4}} \, dx=\int { \frac {1}{{\left (x^{4} + 2\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (1+x^4\right ) \sqrt [4]{2+x^4}} \, dx=\int \frac {1}{\left (x^4+1\right )\,{\left (x^4+2\right )}^{1/4}} \,d x \]

[In]

int(1/((x^4 + 1)*(x^4 + 2)^(1/4)),x)

[Out]

int(1/((x^4 + 1)*(x^4 + 2)^(1/4)), x)