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

Optimal. Leaf size=184 \[ \sqrt {2} \text {RootSum}\left [\text {$\#$1}^4+16 \text {$\#$1}^3-80 \text {$\#$1}^2+128 \text {$\#$1}-64\& ,\frac {\text {$\#$1}^2 \log \left (-\text {$\#$1}+4 x^4+2 \sqrt {2} \sqrt {2 x^4-1} x^2\right )-2 \text {$\#$1} \log \left (-\text {$\#$1}+4 x^4+2 \sqrt {2} \sqrt {2 x^4-1} x^2\right )+2 \log \left (-\text {$\#$1}+4 x^4+2 \sqrt {2} \sqrt {2 x^4-1} x^2\right )}{\text {$\#$1}^3+12 \text {$\#$1}^2-40 \text {$\#$1}+32}\& \right ]+\frac {\sqrt {2 x^4-1} \left (-4 x^4-1\right )}{6 x^6} \]

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Rubi [A]  time = 0.91, antiderivative size = 226, normalized size of antiderivative = 1.23, number of steps used = 20, number of rules used = 11, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.324, Rules used = {6728, 264, 275, 277, 217, 206, 6715, 1692, 402, 377, 204} \begin {gather*} \frac {\left (2 x^4-1\right )^{3/2}}{6 x^6}-\frac {\sqrt {2 x^4-1}}{x^2}-\frac {1}{4} \sqrt {\frac {1}{2} \left (5 \sqrt {2}-1\right )} \tan ^{-1}\left (\frac {\sqrt {\sqrt {2}-1} x^2}{\sqrt {2 x^4-1}}\right )-\frac {1}{4} \left (1+2 \sqrt {2}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} x^2}{\sqrt {2 x^4-1}}\right )+\frac {1}{4} \left (1-2 \sqrt {2}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} x^2}{\sqrt {2 x^4-1}}\right )+\sqrt {2} \tanh ^{-1}\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 )} \tanh ^{-1}\left (\frac {\sqrt {1+\sqrt {2}} x^2}{\sqrt {2 x^4-1}}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[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 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

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 217

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 264

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 275

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 277

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 377

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 402

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

Rule 1692

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 6715

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 6728

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*} \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 \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} \operatorname {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 )+\operatorname {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} \operatorname {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 \operatorname {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 \operatorname {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 ) \operatorname {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 ) \operatorname {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} \tanh ^{-1}\left (\frac {\sqrt {2} x^2}{\sqrt {-1+2 x^4}}\right )+\frac {1}{4} \left (8-5 \sqrt {2}\right ) \operatorname {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 ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+2 x^2}} \, dx,x,x^2\right )-\frac {1}{4} \left (4+\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+2 x^2}} \, dx,x,x^2\right )+\frac {1}{4} \left (8+5 \sqrt {2}\right ) \operatorname {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} \tanh ^{-1}\left (\frac {\sqrt {2} x^2}{\sqrt {-1+2 x^4}}\right )+\frac {1}{4} \left (8-5 \sqrt {2}\right ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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}} \tan ^{-1}\left (\frac {\sqrt {-1+\sqrt {2}} x^2}{\sqrt {-1+2 x^4}}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x^2}{\sqrt {-1+2 x^4}}\right )+\frac {1}{4} \left (1-2 \sqrt {2}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} x^2}{\sqrt {-1+2 x^4}}\right )-\frac {1}{4} \left (1+2 \sqrt {2}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} x^2}{\sqrt {-1+2 x^4}}\right )+\frac {1}{8} \sqrt {2+10 \sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {1+\sqrt {2}} x^2}{\sqrt {-1+2 x^4}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.67, size = 116, normalized size = 0.63 \begin {gather*} \frac {1}{8} \left (\sqrt {2+10 \sqrt {2}} \tanh ^{-1}\left (\frac {x^2}{\sqrt {\sqrt {2}-1} \sqrt {2 x^4-1}}\right )-\sqrt {10 \sqrt {2}-2} \tan ^{-1}\left (\frac {x^2}{\sqrt {1+\sqrt {2}} \sqrt {2 x^4-1}}\right )\right )+\sqrt {2 x^4-1} \left (-\frac {1}{6 x^6}-\frac {2}{3 x^2}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-1/6*1/x^6 - 2/(3*x^2))*Sqrt[-1 + 2*x^4] + (-(Sqrt[-2 + 10*Sqrt[2]]*ArcTan[x^2/(Sqrt[1 + Sqrt[2]]*Sqrt[-1 + 2
*x^4])]) + Sqrt[2 + 10*Sqrt[2]]*ArcTanh[x^2/(Sqrt[-1 + Sqrt[2]]*Sqrt[-1 + 2*x^4])])/8

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IntegrateAlgebraic [A]  time = 0.44, size = 138, normalized size = 0.75 \begin {gather*} \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 ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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) & ]

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fricas [B]  time = 0.57, size = 292, normalized size = 1.59 \begin {gather*} -\frac {12 \, \sqrt {2} x^{6} \sqrt {5 \, \sqrt {2} - 1} \arctan \left (\frac {{\left (\sqrt {2} x^{4} + 3 \, x^{4}\right )} \sqrt {2 \, x^{4} - 1} \sqrt {5 \, \sqrt {2} - 1} \sqrt {\frac {3 \, x^{8} - 2 \, x^{4} + 2 \, \sqrt {2} {\left (x^{8} - x^{4}\right )} + 1}{x^{8}}} - {\left (7 \, x^{4} - \sqrt {2} - 3\right )} \sqrt {2 \, x^{4} - 1} \sqrt {5 \, \sqrt {2} - 1}}{14 \, {\left (2 \, x^{6} - x^{2}\right )}}\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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*(12*sqrt(2)*x^6*sqrt(5*sqrt(2) - 1)*arctan(1/14*((sqrt(2)*x^4 + 3*x^4)*sqrt(2*x^4 - 1)*sqrt(5*sqrt(2) -
1)*sqrt((3*x^8 - 2*x^4 + 2*sqrt(2)*(x^8 - x^4) + 1)/x^8) - (7*x^4 - sqrt(2) - 3)*sqrt(2*x^4 - 1)*sqrt(5*sqrt(2
) - 1))/(2*x^6 - x^2)) - 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) + 16*(4*x^4 + 1)*sqrt(2*x^4 - 1))/x
^6

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giac [B]  time = 0.47, size = 55, normalized size = 0.30 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [B]  time = 7.71, size = 241, normalized size = 1.31

method	result	size
risch	\(-\frac {8 x^{8}-2 x^{4}-1}{6 x^{6} \sqrt {2 x^{4}-1}}-\frac {\arctan \left (\frac {2 \left (\sqrt {2 x^{4}-1}-\sqrt {2}\, x^{2}\right )^{2}-8 \sqrt {2}+10}{4 \sqrt {-14+10 \sqrt {2}}}\right ) \sqrt {2}}{\sqrt {-14+10 \sqrt {2}}}+\frac {5 \arctan \left (\frac {2 \left (\sqrt {2 x^{4}-1}-\sqrt {2}\, x^{2}\right )^{2}-8 \sqrt {2}+10}{4 \sqrt {-14+10 \sqrt {2}}}\right )}{4 \sqrt {-14+10 \sqrt {2}}}+\frac {\arctanh \left (\frac {2 \left (\sqrt {2 x^{4}-1}-\sqrt {2}\, x^{2}\right )^{2}+10+8 \sqrt {2}}{4 \sqrt {14+10 \sqrt {2}}}\right ) \sqrt {2}}{\sqrt {14+10 \sqrt {2}}}+\frac {5 \arctanh \left (\frac {2 \left (\sqrt {2 x^{4}-1}-\sqrt {2}\, x^{2}\right )^{2}+10+8 \sqrt {2}}{4 \sqrt {14+10 \sqrt {2}}}\right )}{4 \sqrt {14+10 \sqrt {2}}}\)	\(241\)
elliptic	\(-\frac {\arctan \left (\frac {2 \left (\sqrt {2 x^{4}-1}-\sqrt {2}\, x^{2}\right )^{2}-8 \sqrt {2}+10}{4 \sqrt {-14+10 \sqrt {2}}}\right ) \sqrt {2}}{\sqrt {-14+10 \sqrt {2}}}+\frac {5 \arctan \left (\frac {2 \left (\sqrt {2 x^{4}-1}-\sqrt {2}\, x^{2}\right )^{2}-8 \sqrt {2}+10}{4 \sqrt {-14+10 \sqrt {2}}}\right )}{4 \sqrt {-14+10 \sqrt {2}}}+\frac {\arctanh \left (\frac {2 \left (\sqrt {2 x^{4}-1}-\sqrt {2}\, x^{2}\right )^{2}+10+8 \sqrt {2}}{4 \sqrt {14+10 \sqrt {2}}}\right ) \sqrt {2}}{\sqrt {14+10 \sqrt {2}}}+\frac {5 \arctanh \left (\frac {2 \left (\sqrt {2 x^{4}-1}-\sqrt {2}\, x^{2}\right )^{2}+10+8 \sqrt {2}}{4 \sqrt {14+10 \sqrt {2}}}\right )}{4 \sqrt {14+10 \sqrt {2}}}-\frac {\sqrt {2 x^{4}-1}}{6 x^{6}}-\frac {2 \sqrt {2 x^{4}-1}}{3 x^{2}}\)	\(243\)
default	\(-\frac {\arctan \left (\frac {2 \left (\sqrt {2 x^{4}-1}-\sqrt {2}\, x^{2}\right )^{2}-8 \sqrt {2}+10}{4 \sqrt {-14+10 \sqrt {2}}}\right ) \sqrt {2}}{\sqrt {-14+10 \sqrt {2}}}+\frac {5 \arctan \left (\frac {2 \left (\sqrt {2 x^{4}-1}-\sqrt {2}\, x^{2}\right )^{2}-8 \sqrt {2}+10}{4 \sqrt {-14+10 \sqrt {2}}}\right )}{4 \sqrt {-14+10 \sqrt {2}}}+\frac {\arctanh \left (\frac {2 \left (\sqrt {2 x^{4}-1}-\sqrt {2}\, x^{2}\right )^{2}+10+8 \sqrt {2}}{4 \sqrt {14+10 \sqrt {2}}}\right ) \sqrt {2}}{\sqrt {14+10 \sqrt {2}}}+\frac {5 \arctanh \left (\frac {2 \left (\sqrt {2 x^{4}-1}-\sqrt {2}\, x^{2}\right )^{2}+10+8 \sqrt {2}}{4 \sqrt {14+10 \sqrt {2}}}\right )}{4 \sqrt {14+10 \sqrt {2}}}+\frac {\left (2 x^{4}-1\right )^{\frac {3}{2}}}{6 x^{6}}+\frac {\left (2 x^{4}-1\right )^{\frac {3}{2}}}{x^{2}}-2 \sqrt {2 x^{4}-1}\, x^{2}\)	\(256\)
trager	\(-\frac {\left (4 x^{4}+1\right ) \sqrt {2 x^{4}-1}}{6 x^{6}}+\frac {\RootOf \left (\textit {\_Z}^{2}+64 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )^{2}-1\right ) \ln \left (-\frac {490 \left (49152 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )^{4} \RootOf \left (\textit {\_Z}^{2}+64 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )^{2}-1\right ) x^{4}+6400 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )^{2} \RootOf \left (\textit {\_Z}^{2}+64 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )^{2}-1\right ) x^{4}-49152 \RootOf \left (\textit {\_Z}^{2}+64 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )^{2}-1\right ) \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )^{4}-133 \RootOf \left (\textit {\_Z}^{2}+64 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )^{2}-1\right ) x^{4}-1792 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )^{2} \sqrt {2 x^{4}-1}\, x^{2}-2560 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )^{2} \RootOf \left (\textit {\_Z}^{2}+64 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )^{2}-1\right )-546 \sqrt {2 x^{4}-1}\, x^{2}+63 \RootOf \left (\textit {\_Z}^{2}+64 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )^{2}-1\right )\right )}{\left (4096 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )^{3} x^{2}+896 x^{2} \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )^{2}-4096 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )^{3}+48 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right ) x^{2}-896 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )^{2}+63 x^{2}-48 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )+7\right ) \left (4096 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )^{3} x^{2}-896 x^{2} \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )^{2}-4096 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )^{3}+48 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right ) x^{2}+896 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )^{2}-63 x^{2}-48 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )-7\right )}\right )}{8}-\RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right ) \ln \left (-\frac {196608 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )^{5} x^{4}-31744 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )^{3} x^{4}-196608 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )^{5}-896 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )^{2} \sqrt {2 x^{4}-1}\, x^{2}-84 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right ) x^{4}+16384 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )^{3}+287 \sqrt {2 x^{4}-1}\, x^{2}+44 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )}{128 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )^{2} x^{4}-11 x^{4}-128 \RootOf \left (16384 \textit {\_Z}^{4}-256 \textit {\_Z}^{2}-49\right )^{2}+1}\right )\)	\(761\)

Verification of antiderivative is not currently implemented for this CAS.

[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)-1/(-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))*2^(1/2)+5/4/(-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))+1/(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))*2^(1/2)+5/4/(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))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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