\(\int \frac {\sqrt {-1+2 x^4} (-1+2 x^8)}{x^7 (-1+2 x^4+x^8)} \, dx\) [2335]

Optimal result

Integrand size = 34, antiderivative size = 184 \[ \int \frac {\sqrt {-1+2 x^4} \left (-1+2 x^8\right )}{x^7 \left (-1+2 x^4+x^8\right )} \, dx=\frac {\left (-1-4 x^4\right ) \sqrt {-1+2 x^4}}{6 x^6}+\sqrt {2} \text {RootSum}\left [-64+128 \text {$\#$1}-80 \text {$\#$1}^2+16 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {2 \log \left (4 x^4+2 \sqrt {2} x^2 \sqrt {-1+2 x^4}-\text {$\#$1}\right )-2 \log \left (4 x^4+2 \sqrt {2} x^2 \sqrt {-1+2 x^4}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (4 x^4+2 \sqrt {2} x^2 \sqrt {-1+2 x^4}-\text {$\#$1}\right ) \text {$\#$1}^2}{32-40 \text {$\#$1}+12 \text {$\#$1}^2+\text {$\#$1}^3}\&\right ] \]

[Out]

Unintegrable

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.23, number of steps used = 20, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.324, Rules used = {6860, 270, 281, 283, 223, 212, 6847, 1706, 399, 385, 210} \[ \int \frac {\sqrt {-1+2 x^4} \left (-1+2 x^8\right )}{x^7 \left (-1+2 x^4+x^8\right )} \, dx=-\frac {1}{4} \sqrt {\frac {1}{2} \left (5 \sqrt {2}-1\right )} \arctan \left (\frac {\sqrt {\sqrt {2}-1} x^2}{\sqrt {2 x^4-1}}\right )-\frac {1}{4} \left (1+2 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {2} x^2}{\sqrt {2 x^4-1}}\right )+\frac {1}{4} \left (1-2 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {2} x^2}{\sqrt {2 x^4-1}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x^2}{\sqrt {2 x^4-1}}\right )+\frac {1}{4} \sqrt {\frac {1}{2} \left (1+5 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {1+\sqrt {2}} x^2}{\sqrt {2 x^4-1}}\right )+\frac {\left (2 x^4-1\right )^{3/2}}{6 x^6}-\frac {\sqrt {2 x^4-1}}{x^2} \]

[In]

Int[(Sqrt[-1 + 2*x^4]*(-1 + 2*x^8))/(x^7*(-1 + 2*x^4 + x^8)),x]

[Out]

-(Sqrt[-1 + 2*x^4]/x^2) + (-1 + 2*x^4)^(3/2)/(6*x^6) - (Sqrt[(-1 + 5*Sqrt[2])/2]*ArcTan[(Sqrt[-1 + Sqrt[2]]*x^
2)/Sqrt[-1 + 2*x^4]])/4 + Sqrt[2]*ArcTanh[(Sqrt[2]*x^2)/Sqrt[-1 + 2*x^4]] + ((1 - 2*Sqrt[2])*ArcTanh[(Sqrt[2]*
x^2)/Sqrt[-1 + 2*x^4]])/4 - ((1 + 2*Sqrt[2])*ArcTanh[(Sqrt[2]*x^2)/Sqrt[-1 + 2*x^4]])/4 + (Sqrt[(1 + 5*Sqrt[2]
)/2]*ArcTanh[(Sqrt[1 + Sqrt[2]]*x^2)/Sqrt[-1 + 2*x^4]])/4

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 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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 270

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

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 283

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1
))), x] - Dist[b*n*(p/(c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

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 399

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

Rule 1706

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandInteg
rand[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, q}, x] && PolyQ[Px, x^2] && NeQ[b
^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]

Rule 6847

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

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}& = \int \left (\frac {\sqrt {-1+2 x^4}}{x^7}+\frac {2 \sqrt {-1+2 x^4}}{x^3}-\frac {x \sqrt {-1+2 x^4} \left (3+2 x^4\right )}{-1+2 x^4+x^8}\right ) \, dx \\ & = 2 \int \frac {\sqrt {-1+2 x^4}}{x^3} \, dx+\int \frac {\sqrt {-1+2 x^4}}{x^7} \, dx-\int \frac {x \sqrt {-1+2 x^4} \left (3+2 x^4\right )}{-1+2 x^4+x^8} \, dx \\ & = \frac {\left (-1+2 x^4\right )^{3/2}}{6 x^6}-\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {-1+2 x^2} \left (3+2 x^2\right )}{-1+2 x^2+x^4} \, dx,x,x^2\right )+\text {Subst}\left (\int \frac {\sqrt {-1+2 x^2}}{x^2} \, dx,x,x^2\right ) \\ & = -\frac {\sqrt {-1+2 x^4}}{x^2}+\frac {\left (-1+2 x^4\right )^{3/2}}{6 x^6}-\frac {1}{2} \text {Subst}\left (\int \left (\frac {\left (2+\frac {1}{\sqrt {2}}\right ) \sqrt {-1+2 x^2}}{2-2 \sqrt {2}+2 x^2}+\frac {\left (2-\frac {1}{\sqrt {2}}\right ) \sqrt {-1+2 x^2}}{2+2 \sqrt {2}+2 x^2}\right ) \, dx,x,x^2\right )+2 \text {Subst}\left (\int \frac {1}{\sqrt {-1+2 x^2}} \, dx,x,x^2\right ) \\ & = -\frac {\sqrt {-1+2 x^4}}{x^2}+\frac {\left (-1+2 x^4\right )^{3/2}}{6 x^6}+2 \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x^2}{\sqrt {-1+2 x^4}}\right )-\frac {1}{4} \left (4-\sqrt {2}\right ) \text {Subst}\left (\int \frac {\sqrt {-1+2 x^2}}{2+2 \sqrt {2}+2 x^2} \, dx,x,x^2\right )-\frac {1}{4} \left (4+\sqrt {2}\right ) \text {Subst}\left (\int \frac {\sqrt {-1+2 x^2}}{2-2 \sqrt {2}+2 x^2} \, dx,x,x^2\right ) \\ & = -\frac {\sqrt {-1+2 x^4}}{x^2}+\frac {\left (-1+2 x^4\right )^{3/2}}{6 x^6}+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x^2}{\sqrt {-1+2 x^4}}\right )+\frac {1}{4} \left (8-5 \sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+2 x^2} \left (2-2 \sqrt {2}+2 x^2\right )} \, dx,x,x^2\right )-\frac {1}{4} \left (4-\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+2 x^2}} \, dx,x,x^2\right )-\frac {1}{4} \left (4+\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+2 x^2}} \, dx,x,x^2\right )+\frac {1}{4} \left (8+5 \sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+2 x^2} \left (2+2 \sqrt {2}+2 x^2\right )} \, dx,x,x^2\right ) \\ & = -\frac {\sqrt {-1+2 x^4}}{x^2}+\frac {\left (-1+2 x^4\right )^{3/2}}{6 x^6}+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x^2}{\sqrt {-1+2 x^4}}\right )+\frac {1}{4} \left (8-5 \sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{2-2 \sqrt {2}-\left (2+2 \left (2-2 \sqrt {2}\right )\right ) x^2} \, dx,x,\frac {x^2}{\sqrt {-1+2 x^4}}\right )-\frac {1}{4} \left (4-\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x^2}{\sqrt {-1+2 x^4}}\right )-\frac {1}{4} \left (4+\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x^2}{\sqrt {-1+2 x^4}}\right )+\frac {1}{4} \left (8+5 \sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{2+2 \sqrt {2}-\left (2+2 \left (2+2 \sqrt {2}\right )\right ) x^2} \, dx,x,\frac {x^2}{\sqrt {-1+2 x^4}}\right ) \\ & = -\frac {\sqrt {-1+2 x^4}}{x^2}+\frac {\left (-1+2 x^4\right )^{3/2}}{6 x^6}-\frac {1}{8} \sqrt {-2+10 \sqrt {2}} \arctan \left (\frac {\sqrt {-1+\sqrt {2}} x^2}{\sqrt {-1+2 x^4}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x^2}{\sqrt {-1+2 x^4}}\right )+\frac {1}{4} \left (1-2 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {2} x^2}{\sqrt {-1+2 x^4}}\right )-\frac {1}{4} \left (1+2 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {2} x^2}{\sqrt {-1+2 x^4}}\right )+\frac {1}{8} \sqrt {2+10 \sqrt {2}} \text {arctanh}\left (\frac {\sqrt {1+\sqrt {2}} x^2}{\sqrt {-1+2 x^4}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {-1+2 x^4} \left (-1+2 x^8\right )}{x^7 \left (-1+2 x^4+x^8\right )} \, dx=\frac {\left (-1-4 x^4\right ) \sqrt {-1+2 x^4}}{6 x^6}+\sqrt {2} \text {RootSum}\left [1+20 \text {$\#$1}^2-26 \text {$\#$1}^4+20 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {2} x^2+\sqrt {-1+2 x^4}-\text {$\#$1}\right )+\log \left (\sqrt {2} x^2+\sqrt {-1+2 x^4}-\text {$\#$1}\right ) \text {$\#$1}^4}{5-13 \text {$\#$1}^2+15 \text {$\#$1}^4+\text {$\#$1}^6}\&\right ] \]

[In]

Integrate[(Sqrt[-1 + 2*x^4]*(-1 + 2*x^8))/(x^7*(-1 + 2*x^4 + x^8)),x]

[Out]

((-1 - 4*x^4)*Sqrt[-1 + 2*x^4])/(6*x^6) + Sqrt[2]*RootSum[1 + 20*#1^2 - 26*#1^4 + 20*#1^6 + #1^8 & , (Log[Sqrt
[2]*x^2 + Sqrt[-1 + 2*x^4] - #1] + Log[Sqrt[2]*x^2 + Sqrt[-1 + 2*x^4] - #1]*#1^4)/(5 - 13*#1^2 + 15*#1^4 + #1^
6) & ]

Maple [N/A] (verified)

Time = 6.90 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.83

method	result	size
risch	\(-\frac {8 x^{8}-2 x^{4}-1}{6 x^{6} \sqrt {2 x^{4}-1}}-8 \sqrt {2}\, \left (-\frac {\left (\frac {1}{2}+\frac {5 \sqrt {2}}{16}\right ) \operatorname {arctanh}\left (\frac {2 \left (\sqrt {2 x^{4}-1}-\sqrt {2}\, x^{2}\right )^{2}+10+8 \sqrt {2}}{4 \sqrt {14+10 \sqrt {2}}}\right )}{4 \sqrt {14+10 \sqrt {2}}}-\frac {\left (-\frac {1}{2}+\frac {5 \sqrt {2}}{16}\right ) \arctan \left (\frac {2 \left (\sqrt {2 x^{4}-1}-\sqrt {2}\, x^{2}\right )^{2}-8 \sqrt {2}+10}{4 \sqrt {-14+10 \sqrt {2}}}\right )}{4 \sqrt {-14+10 \sqrt {2}}}\right )\)	\(152\)
elliptic	\(-\frac {\sqrt {2 x^{4}-1}}{6 x^{6}}-\frac {2 \sqrt {2 x^{4}-1}}{3 x^{2}}-8 \sqrt {2}\, \left (-\frac {\left (\frac {1}{2}+\frac {5 \sqrt {2}}{16}\right ) \operatorname {arctanh}\left (\frac {2 \left (\sqrt {2 x^{4}-1}-\sqrt {2}\, x^{2}\right )^{2}+10+8 \sqrt {2}}{4 \sqrt {14+10 \sqrt {2}}}\right )}{4 \sqrt {14+10 \sqrt {2}}}-\frac {\left (-\frac {1}{2}+\frac {5 \sqrt {2}}{16}\right ) \arctan \left (\frac {2 \left (\sqrt {2 x^{4}-1}-\sqrt {2}\, x^{2}\right )^{2}-8 \sqrt {2}+10}{4 \sqrt {-14+10 \sqrt {2}}}\right )}{4 \sqrt {-14+10 \sqrt {2}}}\right )\)	\(154\)
default	\(\frac {\left (2 x^{4}-1\right )^{\frac {3}{2}}}{6 x^{6}}+\frac {\left (2 x^{4}-1\right )^{\frac {3}{2}}}{x^{2}}-2 \sqrt {2 x^{4}-1}\, x^{2}-8 \sqrt {2}\, \left (-\frac {\left (\frac {1}{2}+\frac {5 \sqrt {2}}{16}\right ) \operatorname {arctanh}\left (\frac {2 \left (\sqrt {2 x^{4}-1}-\sqrt {2}\, x^{2}\right )^{2}+10+8 \sqrt {2}}{4 \sqrt {14+10 \sqrt {2}}}\right )}{4 \sqrt {14+10 \sqrt {2}}}-\frac {\left (-\frac {1}{2}+\frac {5 \sqrt {2}}{16}\right ) \arctan \left (\frac {2 \left (\sqrt {2 x^{4}-1}-\sqrt {2}\, x^{2}\right )^{2}-8 \sqrt {2}+10}{4 \sqrt {-14+10 \sqrt {2}}}\right )}{4 \sqrt {-14+10 \sqrt {2}}}\right )\)	\(167\)
pseudoelliptic	\(\frac {\left (16 \ln \left (\sqrt {2 x^{4}-1}-\sqrt {2}\, x^{2}\right ) x^{6}+16 \ln \left (\sqrt {2}\, x^{2}+\sqrt {2 x^{4}-1}\right ) x^{6}+\frac {4 \sqrt {2}\, \left (-4 x^{4}-1\right ) \sqrt {2 x^{4}-1}}{3}+\left (\operatorname {arctanh}\left (\frac {-\sqrt {2}\, x^{2} \sqrt {2 x^{4}-1}+2 x^{4}+2 \sqrt {2}+2}{\sqrt {14+10 \sqrt {2}}}\right ) \left (8+5 \sqrt {2}\right ) \sqrt {-14+10 \sqrt {2}}-8 \sqrt {7+5 \sqrt {2}}\, \left (\sqrt {2}-\frac {5}{4}\right ) \arctan \left (\frac {-\sqrt {2}\, x^{2} \sqrt {2 x^{4}-1}+2 x^{4}-2 \sqrt {2}+2}{\sqrt {-14+10 \sqrt {2}}}\right )\right ) x^{6}\right ) \sqrt {2}}{16 x^{6}}\)	\(192\)
trager	\(\text {Expression too large to display}\)	\(760\)

[In]

int((2*x^4-1)^(1/2)*(2*x^8-1)/x^7/(x^8+2*x^4-1),x,method=_RETURNVERBOSE)

[Out]

-1/6*(8*x^8-2*x^4-1)/x^6/(2*x^4-1)^(1/2)-8*2^(1/2)*(-1/4*(1/2+5/16*2^(1/2))/(14+10*2^(1/2))^(1/2)*arctanh(1/4*
(2*((2*x^4-1)^(1/2)-2^(1/2)*x^2)^2+10+8*2^(1/2))/(14+10*2^(1/2))^(1/2))-1/4*(-1/2+5/16*2^(1/2))/(-14+10*2^(1/2
))^(1/2)*arctan(1/4*(2*((2*x^4-1)^(1/2)-2^(1/2)*x^2)^2-8*2^(1/2)+10)/(-14+10*2^(1/2))^(1/2)))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.27 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.66 \[ \int \frac {\sqrt {-1+2 x^4} \left (-1+2 x^8\right )}{x^7 \left (-1+2 x^4+x^8\right )} \, dx=\frac {3 \, \sqrt {2} x^{6} \sqrt {5 \, \sqrt {2} + 1} \log \left (\frac {7 \, \sqrt {2} x^{4} + 21 \, x^{4} + 2 \, \sqrt {2 \, x^{4} - 1} {\left (2 \, \sqrt {2} x^{2} + x^{2}\right )} \sqrt {5 \, \sqrt {2} + 1} - 7}{x^{4}}\right ) - 3 \, \sqrt {2} x^{6} \sqrt {5 \, \sqrt {2} + 1} \log \left (\frac {7 \, \sqrt {2} x^{4} + 21 \, x^{4} - 2 \, \sqrt {2 \, x^{4} - 1} {\left (2 \, \sqrt {2} x^{2} + x^{2}\right )} \sqrt {5 \, \sqrt {2} + 1} - 7}{x^{4}}\right ) - 3 \, \sqrt {2} x^{6} \sqrt {-5 \, \sqrt {2} + 1} \log \left (-\frac {7 \, \sqrt {2} x^{4} - 21 \, x^{4} + 2 \, \sqrt {2 \, x^{4} - 1} {\left (2 \, \sqrt {2} x^{2} - x^{2}\right )} \sqrt {-5 \, \sqrt {2} + 1} + 7}{x^{4}}\right ) + 3 \, \sqrt {2} x^{6} \sqrt {-5 \, \sqrt {2} + 1} \log \left (-\frac {7 \, \sqrt {2} x^{4} - 21 \, x^{4} - 2 \, \sqrt {2 \, x^{4} - 1} {\left (2 \, \sqrt {2} x^{2} - x^{2}\right )} \sqrt {-5 \, \sqrt {2} + 1} + 7}{x^{4}}\right ) - 16 \, {\left (4 \, x^{4} + 1\right )} \sqrt {2 \, x^{4} - 1}}{96 \, x^{6}} \]

[In]

integrate((2*x^4-1)^(1/2)*(2*x^8-1)/x^7/(x^8+2*x^4-1),x, algorithm="fricas")

[Out]

1/96*(3*sqrt(2)*x^6*sqrt(5*sqrt(2) + 1)*log((7*sqrt(2)*x^4 + 21*x^4 + 2*sqrt(2*x^4 - 1)*(2*sqrt(2)*x^2 + x^2)*
sqrt(5*sqrt(2) + 1) - 7)/x^4) - 3*sqrt(2)*x^6*sqrt(5*sqrt(2) + 1)*log((7*sqrt(2)*x^4 + 21*x^4 - 2*sqrt(2*x^4 -
1)*(2*sqrt(2)*x^2 + x^2)*sqrt(5*sqrt(2) + 1) - 7)/x^4) - 3*sqrt(2)*x^6*sqrt(-5*sqrt(2) + 1)*log(-(7*sqrt(2)*x
^4 - 21*x^4 + 2*sqrt(2*x^4 - 1)*(2*sqrt(2)*x^2 - x^2)*sqrt(-5*sqrt(2) + 1) + 7)/x^4) + 3*sqrt(2)*x^6*sqrt(-5*s
qrt(2) + 1)*log(-(7*sqrt(2)*x^4 - 21*x^4 - 2*sqrt(2*x^4 - 1)*(2*sqrt(2)*x^2 - x^2)*sqrt(-5*sqrt(2) + 1) + 7)/x
^4) - 16*(4*x^4 + 1)*sqrt(2*x^4 - 1))/x^6

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {-1+2 x^4} \left (-1+2 x^8\right )}{x^7 \left (-1+2 x^4+x^8\right )} \, dx=\text {Timed out} \]

[In]

integrate((2*x**4-1)**(1/2)*(2*x**8-1)/x**7/(x**8+2*x**4-1),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt {-1+2 x^4} \left (-1+2 x^8\right )}{x^7 \left (-1+2 x^4+x^8\right )} \, dx=\int { \frac {{\left (2 \, x^{8} - 1\right )} \sqrt {2 \, x^{4} - 1}}{{\left (x^{8} + 2 \, x^{4} - 1\right )} x^{7}} \,d x } \]

[In]

integrate((2*x^4-1)^(1/2)*(2*x^8-1)/x^7/(x^8+2*x^4-1),x, algorithm="maxima")

[Out]

integrate((2*x^8 - 1)*sqrt(2*x^4 - 1)/((x^8 + 2*x^4 - 1)*x^7), x)

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.31 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.30 \[ \int \frac {\sqrt {-1+2 x^4} \left (-1+2 x^8\right )}{x^7 \left (-1+2 x^4+x^8\right )} \, dx=-\frac {4 \, \sqrt {2} {\left (3 \, {\left (\sqrt {2} x^{2} - \sqrt {2 \, x^{4} - 1}\right )}^{2} + 1\right )}}{3 \, {\left ({\left (\sqrt {2} x^{2} - \sqrt {2 \, x^{4} - 1}\right )}^{2} + 1\right )}^{3}} \]

[In]

integrate((2*x^4-1)^(1/2)*(2*x^8-1)/x^7/(x^8+2*x^4-1),x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 6.62 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt {-1+2 x^4} \left (-1+2 x^8\right )}{x^7 \left (-1+2 x^4+x^8\right )} \, dx=\int \frac {\sqrt {2\,x^4-1}\,\left (2\,x^8-1\right )}{x^7\,\left (x^8+2\,x^4-1\right )} \,d x \]

[In]

int(((2*x^4 - 1)^(1/2)*(2*x^8 - 1))/(x^7*(2*x^4 + x^8 - 1)),x)

[Out]

int(((2*x^4 - 1)^(1/2)*(2*x^8 - 1))/(x^7*(2*x^4 + x^8 - 1)), x)