\(\int \frac {-2+x^4}{\sqrt [4]{1+x^4} (-1+x^4+2 x^8)} \, dx\) [850]

Optimal result

Integrand size = 27, antiderivative size = 64 \[ \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-1+x^4+2 x^8\right )} \, dx=\frac {x}{\sqrt [4]{1+x^4}}+\frac {\arctan \left (\frac {\sqrt [4]{3} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{3}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{3}} \]

[Out]

x/(x^4+1)^(1/4)+1/6*arctan(3^(1/4)*x/(x^4+1)^(1/4))*3^(3/4)+1/6*arctanh(3^(1/4)*x/(x^4+1)^(1/4))*3^(3/4)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1468, 541, 12, 385, 218, 212, 209} \[ \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-1+x^4+2 x^8\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{3} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{3}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{3}}+\frac {x}{\sqrt [4]{x^4+1}} \]

[In]

Int[(-2 + x^4)/((1 + x^4)^(1/4)*(-1 + x^4 + 2*x^8)),x]

[Out]

x/(1 + x^4)^(1/4) + ArcTan[(3^(1/4)*x)/(1 + x^4)^(1/4)]/(2*3^(1/4)) + ArcTanh[(3^(1/4)*x)/(1 + x^4)^(1/4)]/(2*
3^(1/4))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

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 541

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 1468

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((f_) + (g_.)*(x_)^(n_))^(r_.)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^
(p_.), x_Symbol] :> Int[(d + e*x^n)^(p + q)*(f + g*x^n)^r*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, b, c, d, e, f,
g, n, q, r}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-2+x^4}{\left (1+x^4\right )^{5/4} \left (-1+2 x^4\right )} \, dx \\ & = \frac {x}{\sqrt [4]{1+x^4}}+\frac {1}{3} \int -\frac {3}{\sqrt [4]{1+x^4} \left (-1+2 x^4\right )} \, dx \\ & = \frac {x}{\sqrt [4]{1+x^4}}-\int \frac {1}{\sqrt [4]{1+x^4} \left (-1+2 x^4\right )} \, dx \\ & = \frac {x}{\sqrt [4]{1+x^4}}-\text {Subst}\left (\int \frac {1}{-1+3 x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right ) \\ & = \frac {x}{\sqrt [4]{1+x^4}}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-\sqrt {3} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+\sqrt {3} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right ) \\ & = \frac {x}{\sqrt [4]{1+x^4}}+\frac {\arctan \left (\frac {\sqrt [4]{3} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{3}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00 \[ \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-1+x^4+2 x^8\right )} \, dx=\frac {x}{\sqrt [4]{1+x^4}}+\frac {\arctan \left (\frac {\sqrt [4]{3} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{3}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{3}} \]

[In]

Integrate[(-2 + x^4)/((1 + x^4)^(1/4)*(-1 + x^4 + 2*x^8)),x]

[Out]

x/(1 + x^4)^(1/4) + ArcTan[(3^(1/4)*x)/(1 + x^4)^(1/4)]/(2*3^(1/4)) + ArcTanh[(3^(1/4)*x)/(1 + x^4)^(1/4)]/(2*
3^(1/4))

Maple [A] (verified)

Time = 5.87 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.31

method	result	size
pseudoelliptic	\(\frac {-2 \arctan \left (\frac {3^{\frac {3}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}}{3 x}\right ) 3^{\frac {3}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}+\ln \left (\frac {3^{\frac {1}{4}} x +\left (x^{4}+1\right )^{\frac {1}{4}}}{-3^{\frac {1}{4}} x +\left (x^{4}+1\right )^{\frac {1}{4}}}\right ) 3^{\frac {3}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}+12 x}{12 \left (x^{4}+1\right )^{\frac {1}{4}}}\)	\(84\)
trager	\(\frac {x}{\left (x^{4}+1\right )^{\frac {1}{4}}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right ) \ln \left (\frac {2 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{3} x^{2}-6 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2} x^{3}+12 \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right ) x^{4}-18 \left (x^{4}+1\right )^{\frac {3}{4}} x +3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )}{2 x^{4}-1}\right )}{12}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2}\right ) \ln \left (\frac {2 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2} x^{2}+6 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2} x^{3}-12 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2}\right ) x^{4}-18 \left (x^{4}+1\right )^{\frac {3}{4}} x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2}\right )}{2 x^{4}-1}\right )}{12}\)	\(223\)
risch	\(\frac {x}{\left (x^{4}+1\right )^{\frac {1}{4}}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right ) \ln \left (-\frac {2 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{3} x^{2}+6 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2} x^{3}+12 \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right ) x^{4}+18 \left (x^{4}+1\right )^{\frac {3}{4}} x +3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )}{2 x^{4}-1}\right )}{12}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2}\right ) \ln \left (\frac {2 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2} x^{2}+6 \left (x^{4}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2} x^{3}-12 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2}\right ) x^{4}-18 \left (x^{4}+1\right )^{\frac {3}{4}} x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2}\right )}{2 x^{4}-1}\right )}{12}\)	\(224\)

[In]

int((x^4-2)/(x^4+1)^(1/4)/(2*x^8+x^4-1),x,method=_RETURNVERBOSE)

[Out]

1/12*(-2*arctan(1/3*3^(3/4)/x*(x^4+1)^(1/4))*3^(3/4)*(x^4+1)^(1/4)+ln((3^(1/4)*x+(x^4+1)^(1/4))/(-3^(1/4)*x+(x
^4+1)^(1/4)))*3^(3/4)*(x^4+1)^(1/4)+12*x)/(x^4+1)^(1/4)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.99 (sec) , antiderivative size = 318, normalized size of antiderivative = 4.97 \[ \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-1+x^4+2 x^8\right )} \, dx=\frac {3^{\frac {3}{4}} {\left (x^{4} + 1\right )} \log \left (\frac {6 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 6 \cdot 3^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} + 3^{\frac {3}{4}} {\left (4 \, x^{4} + 1\right )} + 6 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{2 \, x^{4} - 1}\right ) + 3^{\frac {3}{4}} {\left (i \, x^{4} + i\right )} \log \left (-\frac {6 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 6 i \cdot 3^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 3^{\frac {3}{4}} {\left (4 i \, x^{4} + i\right )} - 6 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{2 \, x^{4} - 1}\right ) + 3^{\frac {3}{4}} {\left (-i \, x^{4} - i\right )} \log \left (-\frac {6 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 6 i \cdot 3^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 3^{\frac {3}{4}} {\left (-4 i \, x^{4} - i\right )} - 6 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{2 \, x^{4} - 1}\right ) - 3^{\frac {3}{4}} {\left (x^{4} + 1\right )} \log \left (\frac {6 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 6 \cdot 3^{\frac {1}{4}} \sqrt {x^{4} + 1} x^{2} - 3^{\frac {3}{4}} {\left (4 \, x^{4} + 1\right )} + 6 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{2 \, x^{4} - 1}\right ) + 24 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{24 \, {\left (x^{4} + 1\right )}} \]

[In]

integrate((x^4-2)/(x^4+1)^(1/4)/(2*x^8+x^4-1),x, algorithm="fricas")

[Out]

1/24*(3^(3/4)*(x^4 + 1)*log((6*sqrt(3)*(x^4 + 1)^(1/4)*x^3 + 6*3^(1/4)*sqrt(x^4 + 1)*x^2 + 3^(3/4)*(4*x^4 + 1)
+ 6*(x^4 + 1)^(3/4)*x)/(2*x^4 - 1)) + 3^(3/4)*(I*x^4 + I)*log(-(6*sqrt(3)*(x^4 + 1)^(1/4)*x^3 + 6*I*3^(1/4)*s
qrt(x^4 + 1)*x^2 - 3^(3/4)*(4*I*x^4 + I) - 6*(x^4 + 1)^(3/4)*x)/(2*x^4 - 1)) + 3^(3/4)*(-I*x^4 - I)*log(-(6*sq
rt(3)*(x^4 + 1)^(1/4)*x^3 - 6*I*3^(1/4)*sqrt(x^4 + 1)*x^2 - 3^(3/4)*(-4*I*x^4 - I) - 6*(x^4 + 1)^(3/4)*x)/(2*x
^4 - 1)) - 3^(3/4)*(x^4 + 1)*log((6*sqrt(3)*(x^4 + 1)^(1/4)*x^3 - 6*3^(1/4)*sqrt(x^4 + 1)*x^2 - 3^(3/4)*(4*x^4
+ 1) + 6*(x^4 + 1)^(3/4)*x)/(2*x^4 - 1)) + 24*(x^4 + 1)^(3/4)*x)/(x^4 + 1)

Sympy [F]

\[ \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-1+x^4+2 x^8\right )} \, dx=\int \frac {x^{4} - 2}{\left (x^{4} + 1\right )^{\frac {5}{4}} \cdot \left (2 x^{4} - 1\right )}\, dx \]

[In]

integrate((x**4-2)/(x**4+1)**(1/4)/(2*x**8+x**4-1),x)

[Out]

Integral((x**4 - 2)/((x**4 + 1)**(5/4)*(2*x**4 - 1)), x)

Maxima [F]

\[ \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-1+x^4+2 x^8\right )} \, dx=\int { \frac {x^{4} - 2}{{\left (2 \, x^{8} + x^{4} - 1\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate((x^4-2)/(x^4+1)^(1/4)/(2*x^8+x^4-1),x, algorithm="maxima")

[Out]

integrate((x^4 - 2)/((2*x^8 + x^4 - 1)*(x^4 + 1)^(1/4)), x)

Giac [F]

\[ \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-1+x^4+2 x^8\right )} \, dx=\int { \frac {x^{4} - 2}{{\left (2 \, x^{8} + x^{4} - 1\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate((x^4-2)/(x^4+1)^(1/4)/(2*x^8+x^4-1),x, algorithm="giac")

[Out]

integrate((x^4 - 2)/((2*x^8 + x^4 - 1)*(x^4 + 1)^(1/4)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {-2+x^4}{\sqrt [4]{1+x^4} \left (-1+x^4+2 x^8\right )} \, dx=\int \frac {x^4-2}{{\left (x^4+1\right )}^{1/4}\,\left (2\,x^8+x^4-1\right )} \,d x \]

[In]

int((x^4 - 2)/((x^4 + 1)^(1/4)*(x^4 + 2*x^8 - 1)),x)

[Out]

int((x^4 - 2)/((x^4 + 1)^(1/4)*(x^4 + 2*x^8 - 1)), x)