\(\int \frac {(-1+x^4) \sqrt [4]{x^2+x^6}}{1+x^4+x^8} \, dx\) [1944]

Optimal result

Integrand size = 27, antiderivative size = 136 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^4+x^8} \, dx=\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^2+x^6}}\right )-\frac {\arctan \left (\frac {\sqrt {2} x \sqrt [4]{x^2+x^6}}{-x^2+\sqrt {x^2+x^6}}\right )}{2 \sqrt {2}}-\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^2+x^6}}\right )+\frac {\text {arctanh}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x^2+x^6}}{\sqrt {2}}}{x \sqrt [4]{x^2+x^6}}\right )}{2 \sqrt {2}} \]

[Out]

1/2*arctan(x/(x^6+x^2)^(1/4))-1/4*arctan(2^(1/2)*x*(x^6+x^2)^(1/4)/(-x^2+(x^6+x^2)^(1/2)))*2^(1/2)-1/2*arctanh
(x/(x^6+x^2)^(1/4))+1/4*arctanh((1/2*2^(1/2)*x^2+1/2*(x^6+x^2)^(1/2)*2^(1/2))/x/(x^6+x^2)^(1/4))*2^(1/2)

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 0.32 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.20, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2081, 6860, 477, 524} \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^4+x^8} \, dx=\frac {2 \left (-\sqrt {3}+i\right ) x \sqrt [4]{x^6+x^2} \operatorname {AppellF1}\left (\frac {3}{8},-\frac {1}{4},1,\frac {11}{8},-x^4,-\frac {2 x^4}{1-i \sqrt {3}}\right )}{3 \left (\sqrt {3}+i\right ) \sqrt [4]{x^4+1}}+\frac {2 \left (\sqrt {3}+i\right ) x \sqrt [4]{x^6+x^2} \operatorname {AppellF1}\left (\frac {3}{8},-\frac {1}{4},1,\frac {11}{8},-x^4,-\frac {2 x^4}{1+i \sqrt {3}}\right )}{3 \left (-\sqrt {3}+i\right ) \sqrt [4]{x^4+1}} \]

[In]

Int[((-1 + x^4)*(x^2 + x^6)^(1/4))/(1 + x^4 + x^8),x]

[Out]

(2*(I - Sqrt[3])*x*(x^2 + x^6)^(1/4)*AppellF1[3/8, -1/4, 1, 11/8, -x^4, (-2*x^4)/(1 - I*Sqrt[3])])/(3*(I + Sqr
t[3])*(1 + x^4)^(1/4)) + (2*(I + Sqrt[3])*x*(x^2 + x^6)^(1/4)*AppellF1[3/8, -1/4, 1, 11/8, -x^4, (-2*x^4)/(1 +
I*Sqrt[3])])/(3*(I - Sqrt[3])*(1 + x^4)^(1/4))

Rule 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 524

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{x^2+x^6} \int \frac {\sqrt {x} \left (-1+x^4\right ) \sqrt [4]{1+x^4}}{1+x^4+x^8} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}} \\ & = \frac {\sqrt [4]{x^2+x^6} \int \left (\frac {\left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^4}}{1-i \sqrt {3}+2 x^4}+\frac {\left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt [4]{1+x^4}}{1+i \sqrt {3}+2 x^4}\right ) \, dx}{\sqrt {x} \sqrt [4]{1+x^4}} \\ & = \frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{x^2+x^6}\right ) \int \frac {\sqrt {x} \sqrt [4]{1+x^4}}{1+i \sqrt {3}+2 x^4} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{x^2+x^6}\right ) \int \frac {\sqrt {x} \sqrt [4]{1+x^4}}{1-i \sqrt {3}+2 x^4} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}} \\ & = \frac {\left (2 \left (1-i \sqrt {3}\right ) \sqrt [4]{x^2+x^6}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^8}}{1+i \sqrt {3}+2 x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1+x^4}}+\frac {\left (2 \left (1+i \sqrt {3}\right ) \sqrt [4]{x^2+x^6}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^8}}{1-i \sqrt {3}+2 x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1+x^4}} \\ & = \frac {2 \left (i-\sqrt {3}\right ) x \sqrt [4]{x^2+x^6} \operatorname {AppellF1}\left (\frac {3}{8},-\frac {1}{4},1,\frac {11}{8},-x^4,-\frac {2 x^4}{1-i \sqrt {3}}\right )}{3 \left (i+\sqrt {3}\right ) \sqrt [4]{1+x^4}}+\frac {2 \left (i+\sqrt {3}\right ) x \sqrt [4]{x^2+x^6} \operatorname {AppellF1}\left (\frac {3}{8},-\frac {1}{4},1,\frac {11}{8},-x^4,-\frac {2 x^4}{1+i \sqrt {3}}\right )}{3 \left (i-\sqrt {3}\right ) \sqrt [4]{1+x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.09 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^4+x^8} \, dx=\frac {\sqrt [4]{x^2+x^6} \left (2 \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{1+x^4}}\right )+\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {x} \sqrt [4]{1+x^4}}{x-\sqrt {1+x^4}}\right )-2 \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{1+x^4}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {x} \sqrt [4]{1+x^4}}{x+\sqrt {1+x^4}}\right )\right )}{4 \sqrt {x} \sqrt [4]{1+x^4}} \]

[In]

Integrate[((-1 + x^4)*(x^2 + x^6)^(1/4))/(1 + x^4 + x^8),x]

[Out]

((x^2 + x^6)^(1/4)*(2*ArcTan[Sqrt[x]/(1 + x^4)^(1/4)] + Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[x]*(1 + x^4)^(1/4))/(x -
Sqrt[1 + x^4])] - 2*ArcTanh[Sqrt[x]/(1 + x^4)^(1/4)] + Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[x]*(1 + x^4)^(1/4))/(x +
Sqrt[1 + x^4])]))/(4*Sqrt[x]*(1 + x^4)^(1/4))

Maple [A] (verified)

Time = 5.53 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.40

method	result	size
pseudoelliptic	\(\frac {\ln \left (\frac {\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}-x}{x}\right )}{4}-\frac {\ln \left (\frac {x +\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}{x}\right )}{4}-\frac {\arctan \left (\frac {\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}{x}\right )}{2}+\frac {\ln \left (\frac {\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{2} \left (x^{4}+1\right )}}{-\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{2} \left (x^{4}+1\right )}}\right ) \sqrt {2}}{8}+\frac {\arctan \left (\frac {\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right ) \sqrt {2}}{4}+\frac {\arctan \left (\frac {\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right ) \sqrt {2}}{4}\)	\(190\)
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {2 \sqrt {x^{6}+x^{2}}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{5}-2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{3}+2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x}{\left (x^{2}+x +1\right ) \left (x^{2}-x +1\right ) x}\right )}{4}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{5}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{3}+2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+2 \sqrt {x^{6}+x^{2}}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x +2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}}{\left (x^{2}+x +1\right ) \left (x^{2}-x +1\right ) x}\right )}{4}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{5}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \sqrt {x^{6}+x^{2}}\, x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}-2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x}{\left (x^{4}-x^{2}+1\right ) x}\right )}{4}-\frac {\ln \left (-\frac {x^{5}+2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}+2 \sqrt {x^{6}+x^{2}}\, x +2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}} x^{2}+x^{3}+x}{\left (x^{4}-x^{2}+1\right ) x}\right )}{4}\)	\(413\)

[In]

int((x^4-1)*(x^6+x^2)^(1/4)/(x^8+x^4+1),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 7.73 (sec) , antiderivative size = 407, normalized size of antiderivative = 2.99 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^4+x^8} \, dx=\left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {4 i \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} - \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{6} + x^{2}} x + \sqrt {2} {\left (\left (i + 1\right ) \, x^{5} - \left (i + 1\right ) \, x^{3} + \left (i + 1\right ) \, x\right )} - 4 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + x^{3} + x}\right ) - \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {4 i \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{6} + x^{2}} x + \sqrt {2} {\left (-\left (i + 1\right ) \, x^{5} + \left (i + 1\right ) \, x^{3} - \left (i + 1\right ) \, x\right )} - 4 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + x^{3} + x}\right ) - \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {-4 i \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{6} + x^{2}} x + \sqrt {2} {\left (-\left (i - 1\right ) \, x^{5} + \left (i - 1\right ) \, x^{3} - \left (i - 1\right ) \, x\right )} - 4 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + x^{3} + x}\right ) + \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {-4 i \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} - \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{6} + x^{2}} x + \sqrt {2} {\left (\left (i - 1\right ) \, x^{5} - \left (i - 1\right ) \, x^{3} + \left (i - 1\right ) \, x\right )} - 4 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + x^{3} + x}\right ) + \frac {1}{4} \, \arctan \left (\frac {2 \, {\left ({\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}\right )}}{x^{5} - x^{3} + x}\right ) + \frac {1}{4} \, \log \left (-\frac {x^{5} + x^{3} - 2 \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{6} + x^{2}} x + x - 2 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} - x^{3} + x}\right ) \]

[In]

integrate((x^4-1)*(x^6+x^2)^(1/4)/(x^8+x^4+1),x, algorithm="fricas")

[Out]

(1/16*I - 1/16)*sqrt(2)*log((4*I*(x^6 + x^2)^(1/4)*x^2 - (2*I - 2)*sqrt(2)*sqrt(x^6 + x^2)*x + sqrt(2)*((I + 1
)*x^5 - (I + 1)*x^3 + (I + 1)*x) - 4*(x^6 + x^2)^(3/4))/(x^5 + x^3 + x)) - (1/16*I - 1/16)*sqrt(2)*log((4*I*(x
^6 + x^2)^(1/4)*x^2 + (2*I - 2)*sqrt(2)*sqrt(x^6 + x^2)*x + sqrt(2)*(-(I + 1)*x^5 + (I + 1)*x^3 - (I + 1)*x) -
4*(x^6 + x^2)^(3/4))/(x^5 + x^3 + x)) - (1/16*I + 1/16)*sqrt(2)*log((-4*I*(x^6 + x^2)^(1/4)*x^2 + (2*I + 2)*s
qrt(2)*sqrt(x^6 + x^2)*x + sqrt(2)*(-(I - 1)*x^5 + (I - 1)*x^3 - (I - 1)*x) - 4*(x^6 + x^2)^(3/4))/(x^5 + x^3
+ x)) + (1/16*I + 1/16)*sqrt(2)*log((-4*I*(x^6 + x^2)^(1/4)*x^2 - (2*I + 2)*sqrt(2)*sqrt(x^6 + x^2)*x + sqrt(2
)*((I - 1)*x^5 - (I - 1)*x^3 + (I - 1)*x) - 4*(x^6 + x^2)^(3/4))/(x^5 + x^3 + x)) + 1/4*arctan(2*((x^6 + x^2)^
(1/4)*x^2 + (x^6 + x^2)^(3/4))/(x^5 - x^3 + x)) + 1/4*log(-(x^5 + x^3 - 2*(x^6 + x^2)^(1/4)*x^2 + 2*sqrt(x^6 +
x^2)*x + x - 2*(x^6 + x^2)^(3/4))/(x^5 - x^3 + x))

Sympy [F]

\[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^4+x^8} \, dx=\int \frac {\sqrt [4]{x^{2} \left (x^{4} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \]

[In]

integrate((x**4-1)*(x**6+x**2)**(1/4)/(x**8+x**4+1),x)

[Out]

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

Maxima [F]

\[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^4+x^8} \, dx=\int { \frac {{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}{x^{8} + x^{4} + 1} \,d x } \]

[In]

integrate((x^4-1)*(x^6+x^2)^(1/4)/(x^8+x^4+1),x, algorithm="maxima")

[Out]

integrate((x^6 + x^2)^(1/4)*(x^4 - 1)/(x^8 + x^4 + 1), x)

Giac [F]

\[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^4+x^8} \, dx=\int { \frac {{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}{x^{8} + x^{4} + 1} \,d x } \]

[In]

integrate((x^4-1)*(x^6+x^2)^(1/4)/(x^8+x^4+1),x, algorithm="giac")

[Out]

integrate((x^6 + x^2)^(1/4)*(x^4 - 1)/(x^8 + x^4 + 1), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+x^4+x^8} \, dx=\int \frac {{\left (x^6+x^2\right )}^{1/4}\,\left (x^4-1\right )}{x^8+x^4+1} \,d x \]

[In]

int(((x^2 + x^6)^(1/4)*(x^4 - 1))/(x^4 + x^8 + 1),x)

[Out]

int(((x^2 + x^6)^(1/4)*(x^4 - 1))/(x^4 + x^8 + 1), x)