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

Optimal. Leaf size=162 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^6+x^2}}\right )}{2 \sqrt [8]{2}}+\frac {\tan ^{-1}\left (\frac {2^{5/8} x \sqrt [4]{x^6+x^2}}{\sqrt [4]{2} x^2-\sqrt {x^6+x^2}}\right )}{2\ 2^{5/8}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^6+x^2}}\right )}{2 \sqrt [8]{2}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{2^{3/8}}+\frac {\sqrt {x^6+x^2}}{2^{5/8}}}{x \sqrt [4]{x^6+x^2}}\right )}{2\ 2^{5/8}} \]

________________________________________________________________________________________

Rubi [C]  time = 0.53, antiderivative size = 97, normalized size of antiderivative = 0.60, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2056, 1586, 6715, 6725, 429} \begin {gather*} -\frac {(1-i) x \sqrt [4]{x^4+1} F_1\left (\frac {1}{8};1,-\frac {3}{4};\frac {9}{8};-i x^4,-x^4\right )}{\sqrt [4]{x^6+x^2}}-\frac {(1+i) x \sqrt [4]{x^4+1} F_1\left (\frac {1}{8};1,-\frac {3}{4};\frac {9}{8};i x^4,-x^4\right )}{\sqrt [4]{x^6+x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-1 + I)*x*(1 + x^4)^(1/4)*AppellF1[1/8, 1, -3/4, 9/8, (-I)*x^4, -x^4])/(x^2 + x^6)^(1/4) - ((1 + I)*x*(1 + x
^4)^(1/4)*AppellF1[1/8, 1, -3/4, 9/8, I*x^4, -x^4])/(x^2 + x^6)^(1/4)

Rule 429

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

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 2056

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 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 6725

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

Rubi steps

\begin {align*} \int \frac {-1+x^8}{\sqrt [4]{x^2+x^6} \left (1+x^8\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{1+x^4}\right ) \int \frac {-1+x^8}{\sqrt {x} \sqrt [4]{1+x^4} \left (1+x^8\right )} \, dx}{\sqrt [4]{x^2+x^6}}\\ &=\frac {\left (\sqrt {x} \sqrt [4]{1+x^4}\right ) \int \frac {\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}}{\sqrt {x} \left (1+x^8\right )} \, dx}{\sqrt [4]{x^2+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^8\right ) \left (1+x^8\right )^{3/4}}{1+x^{16}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \left (-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+x^8\right )^{3/4}}{i-x^8}+\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (1+x^8\right )^{3/4}}{i+x^8}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=-\frac {\left ((1+i) \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^8\right )^{3/4}}{i-x^8} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}+\frac {\left ((1-i) \sqrt {x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^8\right )^{3/4}}{i+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=-\frac {(1-i) x \sqrt [4]{1+x^4} F_1\left (\frac {1}{8};1,-\frac {3}{4};\frac {9}{8};-i x^4,-x^4\right )}{\sqrt [4]{x^2+x^6}}-\frac {(1+i) x \sqrt [4]{1+x^4} F_1\left (\frac {1}{8};1,-\frac {3}{4};\frac {9}{8};i x^4,-x^4\right )}{\sqrt [4]{x^2+x^6}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 1.32, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+x^8}{\sqrt [4]{x^2+x^6} \left (1+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.69, size = 162, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{2 \sqrt [8]{2}}+\frac {\tan ^{-1}\left (\frac {2^{5/8} x \sqrt [4]{x^2+x^6}}{\sqrt [4]{2} x^2-\sqrt {x^2+x^6}}\right )}{2\ 2^{5/8}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{2 \sqrt [8]{2}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{2^{3/8}}+\frac {\sqrt {x^2+x^6}}{2^{5/8}}}{x \sqrt [4]{x^2+x^6}}\right )}{2\ 2^{5/8}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*ArcTan[(2^(1/8)*x)/(x^2 + x^6)^(1/4)]/2^(1/8) + ArcTan[(2^(5/8)*x*(x^2 + x^6)^(1/4))/(2^(1/4)*x^2 - Sqrt[
x^2 + x^6])]/(2*2^(5/8)) - ArcTanh[(2^(1/8)*x)/(x^2 + x^6)^(1/4)]/(2*2^(1/8)) - ArcTanh[(x^2/2^(3/8) + Sqrt[x^
2 + x^6]/2^(5/8))/(x*(x^2 + x^6)^(1/4))]/(2*2^(5/8))

________________________________________________________________________________________

fricas [B]  time = 120.67, size = 2347, normalized size = 14.49

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*2^(7/8)*arctan(1/2*(4*(x^6 + x^2)^(3/4)*(2^(5/8)*x^2 + 2^(1/8)*(x^4 + 1)) + (2^(5/8)*(x^9 + 4*x^7 + 4*x^5
+ 4*x^3 + x) + 2*sqrt(x^6 + x^2)*(2^(7/8)*(x^5 + 2*x^3 + x) + 2*2^(3/8)*(x^5 + x^3 + x)) + 2*2^(1/8)*(x^9 + 2
*x^7 + 4*x^5 + 2*x^3 + x))*sqrt(3*2^(3/4) - 4*2^(1/4)) + 4*(x^6 + x^2)^(1/4)*(2^(7/8)*x^4 + 2^(3/8)*(x^6 + x^2
)))/(x^9 + x)) - 1/16*2^(7/8)*log(-(2^(7/8)*(x^9 - 2*x^7 + 4*x^5 - 2*x^3 + x) + 2*(x^6 + x^2)^(3/4)*(2*x^4 - 2
*x^2 - sqrt(2)*(x^4 - 2*x^2 + 1) + 2) - 2*sqrt(x^6 + x^2)*(2^(5/8)*(x^5 - 2*x^3 + x) - 2*2^(1/8)*(x^5 - x^3 +
x)) - 2^(3/8)*(x^9 - 4*x^7 + 4*x^5 - 4*x^3 + x) - 2*(x^6 + x^2)^(1/4)*(2^(3/4)*(x^6 - 2*x^4 + x^2) - 2*2^(1/4)
*(x^6 - x^4 + x^2)))/(x^9 + x)) + 1/16*2^(7/8)*log((2^(7/8)*(x^9 - 2*x^7 + 4*x^5 - 2*x^3 + x) - 2*(x^6 + x^2)^
(3/4)*(2*x^4 - 2*x^2 - sqrt(2)*(x^4 - 2*x^2 + 1) + 2) - 2*sqrt(x^6 + x^2)*(2^(5/8)*(x^5 - 2*x^3 + x) - 2*2^(1/
8)*(x^5 - x^3 + x)) - 2^(3/8)*(x^9 - 4*x^7 + 4*x^5 - 4*x^3 + x) + 2*(x^6 + x^2)^(1/4)*(2^(3/4)*(x^6 - 2*x^4 +
x^2) - 2*2^(1/4)*(x^6 - x^4 + x^2)))/(x^9 + x)) - 1/4*2^(3/8)*arctan((x^17 + 64*x^13 + 130*x^9 + 64*x^5 + 2*(x
^6 + x^2)^(3/4)*(2^(5/8)*(x^12 - 79*x^8 - 79*x^4 + 1) + 2*2^(1/8)*(11*x^10 + 16*x^6 + 11*x^2)) + 16*sqrt(2)*(x
^15 + 5*x^11 + 5*x^7 + x^3) + 4*sqrt(x^6 + x^2)*(2^(3/4)*(15*x^11 + 32*x^7 + 15*x^3) + 2^(1/4)*(x^13 + 33*x^9
+ 33*x^5 + x)) + (16*(x^6 + x^2)^(3/4)*(2^(3/4)*(x^10 - 14*x^8 + 4*x^6 - 14*x^4 + x^2) - 2^(1/4)*(x^10 - 28*x^
8 + 4*x^6 - 28*x^4 + x^2)) + 2^(5/8)*(x^17 - 6*x^15 - 220*x^13 + 26*x^11 - 446*x^9 + 26*x^7 - 220*x^5 - 6*x^3
+ x) + 2*sqrt(x^6 + x^2)*(2^(7/8)*(x^13 - 11*x^11 - 79*x^9 - 16*x^7 - 79*x^5 - 11*x^3 + x) - 2^(3/8)*(x^13 - 2
2*x^11 - 79*x^9 - 32*x^7 - 79*x^5 - 22*x^3 + x)) - 4*(x^14 - 30*x^12 + 33*x^10 - 64*x^8 + 33*x^6 - 30*x^4 + x^
2 - sqrt(2)*(x^14 - 15*x^12 + 33*x^10 - 32*x^8 + 33*x^6 - 15*x^4 + x^2))*(x^6 + x^2)^(1/4) - 2^(1/8)*(x^17 - 1
2*x^15 - 220*x^13 + 52*x^11 - 446*x^9 + 52*x^7 - 220*x^5 - 12*x^3 + x))*sqrt((3*2^(3/4)*(x^9 + x) + 4*(x^6 + x
^2)^(3/4)*(2^(7/8)*(2*x^4 - 3*x^2 + 2) + 2^(3/8)*(3*x^4 - 4*x^2 + 3)) + 8*sqrt(x^6 + x^2)*(3*x^5 - 4*x^3 + sqr
t(2)*(2*x^5 - 3*x^3 + 2*x) + 3*x) + 4*2^(1/4)*(x^9 + x) + 4*(x^6 + x^2)^(1/4)*(2^(5/8)*(3*x^6 - 4*x^4 + 3*x^2)
+ 2*2^(1/8)*(2*x^6 - 3*x^4 + 2*x^2)))/(x^9 + x)) + 2*(x^6 + x^2)^(1/4)*(2^(7/8)*(3*x^14 - 13*x^10 - 13*x^6 +
3*x^2) + 2*2^(3/8)*(41*x^12 + 80*x^8 + 41*x^4)) + x)/(x^17 - 384*x^13 - 766*x^9 - 384*x^5 + x)) + 1/4*2^(3/8)*
arctan((x^17 + 64*x^13 + 130*x^9 + 64*x^5 - 2*(x^6 + x^2)^(3/4)*(2^(5/8)*(x^12 - 79*x^8 - 79*x^4 + 1) + 2*2^(1
/8)*(11*x^10 + 16*x^6 + 11*x^2)) + 16*sqrt(2)*(x^15 + 5*x^11 + 5*x^7 + x^3) + 4*sqrt(x^6 + x^2)*(2^(3/4)*(15*x
^11 + 32*x^7 + 15*x^3) + 2^(1/4)*(x^13 + 33*x^9 + 33*x^5 + x)) + (16*(x^6 + x^2)^(3/4)*(2^(3/4)*(x^10 - 14*x^8
+ 4*x^6 - 14*x^4 + x^2) - 2^(1/4)*(x^10 - 28*x^8 + 4*x^6 - 28*x^4 + x^2)) - 2^(5/8)*(x^17 - 6*x^15 - 220*x^13
+ 26*x^11 - 446*x^9 + 26*x^7 - 220*x^5 - 6*x^3 + x) - 2*sqrt(x^6 + x^2)*(2^(7/8)*(x^13 - 11*x^11 - 79*x^9 - 1
6*x^7 - 79*x^5 - 11*x^3 + x) - 2^(3/8)*(x^13 - 22*x^11 - 79*x^9 - 32*x^7 - 79*x^5 - 22*x^3 + x)) - 4*(x^14 - 3
0*x^12 + 33*x^10 - 64*x^8 + 33*x^6 - 30*x^4 + x^2 - sqrt(2)*(x^14 - 15*x^12 + 33*x^10 - 32*x^8 + 33*x^6 - 15*x
^4 + x^2))*(x^6 + x^2)^(1/4) + 2^(1/8)*(x^17 - 12*x^15 - 220*x^13 + 52*x^11 - 446*x^9 + 52*x^7 - 220*x^5 - 12*
x^3 + x))*sqrt((3*2^(3/4)*(x^9 + x) - 4*(x^6 + x^2)^(3/4)*(2^(7/8)*(2*x^4 - 3*x^2 + 2) + 2^(3/8)*(3*x^4 - 4*x^
2 + 3)) + 8*sqrt(x^6 + x^2)*(3*x^5 - 4*x^3 + sqrt(2)*(2*x^5 - 3*x^3 + 2*x) + 3*x) + 4*2^(1/4)*(x^9 + x) - 4*(x
^6 + x^2)^(1/4)*(2^(5/8)*(3*x^6 - 4*x^4 + 3*x^2) + 2*2^(1/8)*(2*x^6 - 3*x^4 + 2*x^2)))/(x^9 + x)) - 2*(x^6 + x
^2)^(1/4)*(2^(7/8)*(3*x^14 - 13*x^10 - 13*x^6 + 3*x^2) + 2*2^(3/8)*(41*x^12 + 80*x^8 + 41*x^4)) + x)/(x^17 - 3
84*x^13 - 766*x^9 - 384*x^5 + x)) - 1/16*2^(3/8)*log(4*(3*2^(3/4)*(x^9 + x) + 4*(x^6 + x^2)^(3/4)*(2^(7/8)*(2*
x^4 - 3*x^2 + 2) + 2^(3/8)*(3*x^4 - 4*x^2 + 3)) + 8*sqrt(x^6 + x^2)*(3*x^5 - 4*x^3 + sqrt(2)*(2*x^5 - 3*x^3 +
2*x) + 3*x) + 4*2^(1/4)*(x^9 + x) + 4*(x^6 + x^2)^(1/4)*(2^(5/8)*(3*x^6 - 4*x^4 + 3*x^2) + 2*2^(1/8)*(2*x^6 -
3*x^4 + 2*x^2)))/(x^9 + x)) + 1/16*2^(3/8)*log(4*(3*2^(3/4)*(x^9 + x) - 4*(x^6 + x^2)^(3/4)*(2^(7/8)*(2*x^4 -
3*x^2 + 2) + 2^(3/8)*(3*x^4 - 4*x^2 + 3)) + 8*sqrt(x^6 + x^2)*(3*x^5 - 4*x^3 + sqrt(2)*(2*x^5 - 3*x^3 + 2*x) +
3*x) + 4*2^(1/4)*(x^9 + x) - 4*(x^6 + x^2)^(1/4)*(2^(5/8)*(3*x^6 - 4*x^4 + 3*x^2) + 2*2^(1/8)*(2*x^6 - 3*x^4
+ 2*x^2)))/(x^9 + x))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8} - 1}{{\left (x^{8} + 1\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [C]  time = 134.56, size = 2802, normalized size = 17.30

method	result	size
trager	\(\text {Expression too large to display}\)	\(2802\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*RootOf(_Z^8-128)*ln((RootOf(_Z^8-128)^9*x^5-2*RootOf(_Z^8-128)^9*x^3+RootOf(_Z^8-128)^9*x-8*(x^6+x^2)^(1/
2)*RootOf(_Z^8-128)^7*x-16*RootOf(_Z^8-128)^6*(x^6+x^2)^(1/4)*x^2-16*RootOf(_Z^8-128)^5*x^3-256*RootOf(_Z^8-12
8)*x^5+256*RootOf(_Z^8-128)*x^3-512*(x^6+x^2)^(3/4)-256*RootOf(_Z^8-128)*x)/x/(RootOf(_Z^8-128)^4*x^4-2*RootOf
(_Z^8-128)^4*x^2+RootOf(_Z^8-128)^4+16*x^4-16*x^2+16))+1/64*ln((4*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-12
8)^2+64*_Z^2)*RootOf(_Z^8-128)^8*x^5+RootOf(_Z^8-128)^9*x^5-8*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2
+64*_Z^2)*RootOf(_Z^8-128)^8*x^3-2*RootOf(_Z^8-128)^9*x^3+4*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+6
4*_Z^2)*RootOf(_Z^8-128)^8*x+32*(x^6+x^2)^(1/2)*RootOf(_Z^8-128)^6*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-1
28)^2+64*_Z^2)*x+RootOf(_Z^8-128)^9*x+8*RootOf(_Z^8-128)^5*x^5+8*RootOf(_Z^8-128)^6*(x^6+x^2)^(1/4)*x^2+64*Roo
tOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+64*_Z^2)*RootOf(_Z^8-128)^4*x^3-1024*RootOf(-RootOf(_Z^8-128)^5*
_Z+RootOf(_Z^8-128)^2+64*_Z^2)*x^5+8*RootOf(_Z^8-128)^5*x-128*RootOf(_Z^8-128)*x^5-1024*RootOf(-RootOf(_Z^8-12
8)^5*_Z+RootOf(_Z^8-128)^2+64*_Z^2)*RootOf(_Z^8-128)*(x^6+x^2)^(1/4)*x^2+1024*RootOf(-RootOf(_Z^8-128)^5*_Z+Ro
otOf(_Z^8-128)^2+64*_Z^2)*x^3+128*RootOf(_Z^8-128)*x^3-256*(x^6+x^2)^(3/4)-1024*RootOf(-RootOf(_Z^8-128)^5*_Z+
RootOf(_Z^8-128)^2+64*_Z^2)*x-128*RootOf(_Z^8-128)*x)/x/(RootOf(_Z^8-128)^4*x^4-2*RootOf(_Z^8-128)^4*x^2+RootO
f(_Z^8-128)^4-16*x^4+16*x^2-16))*RootOf(_Z^8-128)^5-ln((4*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+64*
_Z^2)*RootOf(_Z^8-128)^8*x^5+RootOf(_Z^8-128)^9*x^5-8*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+64*_Z^2
)*RootOf(_Z^8-128)^8*x^3-2*RootOf(_Z^8-128)^9*x^3+4*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+64*_Z^2)*
RootOf(_Z^8-128)^8*x+32*(x^6+x^2)^(1/2)*RootOf(_Z^8-128)^6*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+64
*_Z^2)*x+RootOf(_Z^8-128)^9*x+8*RootOf(_Z^8-128)^5*x^5+8*RootOf(_Z^8-128)^6*(x^6+x^2)^(1/4)*x^2+64*RootOf(-Roo
tOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+64*_Z^2)*RootOf(_Z^8-128)^4*x^3-1024*RootOf(-RootOf(_Z^8-128)^5*_Z+RootO
f(_Z^8-128)^2+64*_Z^2)*x^5+8*RootOf(_Z^8-128)^5*x-128*RootOf(_Z^8-128)*x^5-1024*RootOf(-RootOf(_Z^8-128)^5*_Z+
RootOf(_Z^8-128)^2+64*_Z^2)*RootOf(_Z^8-128)*(x^6+x^2)^(1/4)*x^2+1024*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^
8-128)^2+64*_Z^2)*x^3+128*RootOf(_Z^8-128)*x^3-256*(x^6+x^2)^(3/4)-1024*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_
Z^8-128)^2+64*_Z^2)*x-128*RootOf(_Z^8-128)*x)/x/(RootOf(_Z^8-128)^4*x^4-2*RootOf(_Z^8-128)^4*x^2+RootOf(_Z^8-1
28)^4-16*x^4+16*x^2-16))*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+64*_Z^2)-1/8*ln((16*RootOf(-RootOf(_
Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+64*_Z^2)*RootOf(_Z^8-128)^8*x^5-RootOf(_Z^8-128)^9*x^5-32*RootOf(-RootOf(_Z^8
-128)^5*_Z+RootOf(_Z^8-128)^2+64*_Z^2)*RootOf(_Z^8-128)^8*x^3+2*RootOf(_Z^8-128)^9*x^3+16*RootOf(-RootOf(_Z^8-
128)^5*_Z+RootOf(_Z^8-128)^2+64*_Z^2)*RootOf(_Z^8-128)^8*x-128*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^
2+64*_Z^2)*RootOf(_Z^8-128)^4*x^5-RootOf(_Z^8-128)^9*x-8*(x^6+x^2)^(1/2)*RootOf(_Z^8-128)^7*x+16*RootOf(_Z^8-1
28)^6*(x^6+x^2)^(1/4)*x^2+16*RootOf(_Z^8-128)^5*x^3-128*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+64*_Z
^2)*RootOf(_Z^8-128)^4*x+1024*(x^6+x^2)^(1/2)*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+64*_Z^2)*RootOf
(_Z^8-128)^2*x-2048*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+64*_Z^2)*x^5+256*RootOf(_Z^8-128)*x^5+204
8*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+64*_Z^2)*x^3-256*RootOf(_Z^8-128)*x^3-512*(x^6+x^2)^(3/4)-2
048*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+64*_Z^2)*x+256*RootOf(_Z^8-128)*x)/x/(RootOf(_Z^8-128)^4*
x^4-2*RootOf(_Z^8-128)^4*x^2+RootOf(_Z^8-128)^4+16*x^4-16*x^2+16))*RootOf(_Z^8-128)^4*RootOf(-RootOf(_Z^8-128)
^5*_Z+RootOf(_Z^8-128)^2+64*_Z^2)+1/8*ln((16*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+64*_Z^2)*RootOf(
_Z^8-128)^8*x^5-RootOf(_Z^8-128)^9*x^5-32*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+64*_Z^2)*RootOf(_Z^
8-128)^8*x^3+2*RootOf(_Z^8-128)^9*x^3+16*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+64*_Z^2)*RootOf(_Z^8
-128)^8*x-128*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+64*_Z^2)*RootOf(_Z^8-128)^4*x^5-RootOf(_Z^8-128
)^9*x-8*(x^6+x^2)^(1/2)*RootOf(_Z^8-128)^7*x+16*RootOf(_Z^8-128)^6*(x^6+x^2)^(1/4)*x^2+16*RootOf(_Z^8-128)^5*x
^3-128*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+64*_Z^2)*RootOf(_Z^8-128)^4*x+1024*(x^6+x^2)^(1/2)*Roo
tOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+64*_Z^2)*RootOf(_Z^8-128)^2*x-2048*RootOf(-RootOf(_Z^8-128)^5*_Z
+RootOf(_Z^8-128)^2+64*_Z^2)*x^5+256*RootOf(_Z^8-128)*x^5+2048*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^
2+64*_Z^2)*x^3-256*RootOf(_Z^8-128)*x^3-512*(x^6+x^2)^(3/4)-2048*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128
)^2+64*_Z^2)*x+256*RootOf(_Z^8-128)*x)/x/(RootOf(_Z^8-128)^4*x^4-2*RootOf(_Z^8-128)^4*x^2+RootOf(_Z^8-128)^4+1
6*x^4-16*x^2+16))*RootOf(_Z^8-128)+RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+64*_Z^2)*ln(-(RootOf(-Root
Of(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+64*_Z^2)*RootOf(_Z^8-128)^8*x^5-2*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z
^8-128)^2+64*_Z^2)*RootOf(_Z^8-128)^8*x^3+RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+64*_Z^2)*RootOf(_Z^
8-128)^8*x+8*(x^6+x^2)^(1/2)*RootOf(_Z^8-128)^6*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+64*_Z^2)*x+2*
RootOf(_Z^8-128)^6*(x^6+x^2)^(1/4)*x^2+16*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+64*_Z^2)*RootOf(_Z^
8-128)^4*x^3-256*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+64*_Z^2)*x^5-16*(x^6+x^2)^(1/2)*RootOf(_Z^8-
128)^3*x-256*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+64*_Z^2)*RootOf(_Z^8-128)*(x^6+x^2)^(1/4)*x^2+25
6*RootOf(-RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+64*_Z^2)*x^3+64*(x^6+x^2)^(3/4)-256*RootOf(-RootOf(_Z^8-128
)^5*_Z+RootOf(_Z^8-128)^2+64*_Z^2)*x)/x/(RootOf(_Z^8-128)^4*x^4-2*RootOf(_Z^8-128)^4*x^2+RootOf(_Z^8-128)^4-16
*x^4+16*x^2-16))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8} - 1}{{\left (x^{8} + 1\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^8-1}{{\left (x^6+x^2\right )}^{1/4}\,\left (x^8+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________