3.24.63 \(\int \frac {(-1+x^4) \sqrt [4]{x^2+x^6}}{1-x^4+x^8} \, dx\)
Optimal. Leaf size=187 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [8]{3} x}{\sqrt [4]{x^6+x^2}}\right )}{2\ 3^{3/8}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} 3^{7/8} x \sqrt [4]{x^6+x^2}}{3^{3/4} \sqrt {x^6+x^2}-3 x^2}\right )}{2 \sqrt {2} 3^{3/8}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{3} x}{\sqrt [4]{x^6+x^2}}\right )}{2\ 3^{3/8}}+\frac {\tanh ^{-1}\left (\frac {\frac {\sqrt [8]{3} x^2}{\sqrt {2}}+\frac {\sqrt {x^6+x^2}}{\sqrt {2} \sqrt [8]{3}}}{x \sqrt [4]{x^6+x^2}}\right )}{2 \sqrt {2} 3^{3/8}} \]
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Rubi [C] time = 0.44, antiderivative size = 163, normalized size of antiderivative = 0.87,
number of steps used = 7, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used =
{2056, 6728, 466, 510} \begin {gather*} -\frac {2 \left (\sqrt {3}+3 i\right ) x \sqrt [4]{x^6+x^2} F_1\left (\frac {3}{8};-\frac {1}{4},1;\frac {11}{8};-x^4,\frac {2 x^4}{1-i \sqrt {3}}\right )}{9 \left (\sqrt {3}+i\right ) \sqrt [4]{x^4+1}}-\frac {2 \left (-\sqrt {3}+3 i\right ) x \sqrt [4]{x^6+x^2} F_1\left (\frac {3}{8};-\frac {1}{4},1;\frac {11}{8};-x^4,\frac {2 x^4}{1+i \sqrt {3}}\right )}{9 \left (-\sqrt {3}+i\right ) \sqrt [4]{x^4+1}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Int[((-1 + x^4)*(x^2 + x^6)^(1/4))/(1 - x^4 + x^8),x]
[Out]
(-2*(3*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])])/(9*(I + S
qrt[3])*(1 + x^4)^(1/4)) - (2*(3*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])])/(9*(I - Sqrt[3])*(1 + x^4)^(1/4))
Rule 466
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 510
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)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), 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 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 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 {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1-x^4+x^8} \, dx &=\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+\frac {i}{\sqrt {3}}\right ) \sqrt {x} \sqrt [4]{1+x^4}}{-1-i \sqrt {3}+2 x^4}+\frac {\left (1-\frac {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 (3-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}{3 \sqrt {x} \sqrt [4]{1+x^4}}+\frac {\left (\left (3+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}{3 \sqrt {x} \sqrt [4]{1+x^4}}\\ &=\frac {\left (2 \left (3-i \sqrt {3}\right ) \sqrt [4]{x^2+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^8}}{-1+i \sqrt {3}+2 x^8} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt [4]{1+x^4}}+\frac {\left (2 \left (3+i \sqrt {3}\right ) \sqrt [4]{x^2+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^8}}{-1-i \sqrt {3}+2 x^8} \, dx,x,\sqrt {x}\right )}{3 \sqrt {x} \sqrt [4]{1+x^4}}\\ &=-\frac {2 \left (3 i+\sqrt {3}\right ) x \sqrt [4]{x^2+x^6} F_1\left (\frac {3}{8};-\frac {1}{4},1;\frac {11}{8};-x^4,\frac {2 x^4}{1-i \sqrt {3}}\right )}{9 \left (i+\sqrt {3}\right ) \sqrt [4]{1+x^4}}-\frac {2 \left (3 i-\sqrt {3}\right ) x \sqrt [4]{x^2+x^6} F_1\left (\frac {3}{8};-\frac {1}{4},1;\frac {11}{8};-x^4,\frac {2 x^4}{1+i \sqrt {3}}\right )}{9 \left (i-\sqrt {3}\right ) \sqrt [4]{1+x^4}}\\ \end {align*}
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Mathematica [F] time = 0.07, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1-x^4+x^8} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[((-1 + x^4)*(x^2 + x^6)^(1/4))/(1 - x^4 + x^8),x]
[Out]
Integrate[((-1 + x^4)*(x^2 + x^6)^(1/4))/(1 - x^4 + x^8), x]
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IntegrateAlgebraic [A] time = 0.00, size = 187, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [8]{3} x}{\sqrt [4]{x^2+x^6}}\right )}{2\ 3^{3/8}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} 3^{7/8} x \sqrt [4]{x^2+x^6}}{-3 x^2+3^{3/4} \sqrt {x^2+x^6}}\right )}{2 \sqrt {2} 3^{3/8}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{3} x}{\sqrt [4]{x^2+x^6}}\right )}{2\ 3^{3/8}}+\frac {\tanh ^{-1}\left (\frac {\frac {\sqrt [8]{3} x^2}{\sqrt {2}}+\frac {\sqrt {x^2+x^6}}{\sqrt {2} \sqrt [8]{3}}}{x \sqrt [4]{x^2+x^6}}\right )}{2 \sqrt {2} 3^{3/8}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[((-1 + x^4)*(x^2 + x^6)^(1/4))/(1 - x^4 + x^8),x]
[Out]
ArcTan[(3^(1/8)*x)/(x^2 + x^6)^(1/4)]/(2*3^(3/8)) - ArcTan[(Sqrt[2]*3^(7/8)*x*(x^2 + x^6)^(1/4))/(-3*x^2 + 3^(
3/4)*Sqrt[x^2 + x^6])]/(2*Sqrt[2]*3^(3/8)) - ArcTanh[(3^(1/8)*x)/(x^2 + x^6)^(1/4)]/(2*3^(3/8)) + ArcTanh[((3^
(1/8)*x^2)/Sqrt[2] + Sqrt[x^2 + x^6]/(Sqrt[2]*3^(1/8)))/(x*(x^2 + x^6)^(1/4))]/(2*Sqrt[2]*3^(3/8))
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fricas [B] time = 18.27, size = 1348, normalized size = 7.21
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^4-1)*(x^6+x^2)^(1/4)/(x^8-x^4+1),x, algorithm="fricas")
[Out]
-1/108*27^(7/8)*sqrt(2)*arctan(1/81*(sqrt(3)*(27^(3/4)*(x^9 + 5*x^5 + x) + 3*(x^6 + x^2)^(3/4)*(27^(5/8)*sqrt(
2)*x^2 + 3*27^(1/8)*sqrt(2)*(x^4 + 1)) + 18*sqrt(x^6 + x^2)*(3*x^3 + sqrt(3)*(x^5 + x)) + 18*27^(1/4)*(x^7 + x
^3) + (x^6 + x^2)^(1/4)*(9*27^(3/8)*sqrt(2)*x^4 + 27^(7/8)*sqrt(2)*(x^6 + x^2)))*sqrt((2*(x^6 + x^2)^(3/4)*(9*
27^(3/8)*sqrt(2)*x^2 - 27^(7/8)*sqrt(2)*(x^4 + 1)) + 9*sqrt(3)*(x^9 - x^5 + x) - 12*sqrt(x^6 + x^2)*(27^(3/4)*
x^3 - 3*27^(1/4)*(x^5 + x)) + 6*(x^6 + x^2)^(1/4)*(9*27^(1/8)*sqrt(2)*x^4 - 27^(5/8)*sqrt(2)*(x^6 + x^2)))/(x^
9 - x^5 + x)) + 9*(x^6 + x^2)^(3/4)*(27^(7/8)*sqrt(2)*x^2 + 3*27^(3/8)*sqrt(2)*(x^4 + 1)) + 27*(x^6 + x^2)^(1/
4)*(27^(5/8)*sqrt(2)*x^4 + 3*27^(1/8)*sqrt(2)*(x^6 + x^2)))/(x^9 - x^5 + x)) - 1/108*27^(7/8)*sqrt(2)*arctan(-
1/81*(sqrt(3)*(27^(3/4)*(x^9 + 5*x^5 + x) - 3*(x^6 + x^2)^(3/4)*(27^(5/8)*sqrt(2)*x^2 + 3*27^(1/8)*sqrt(2)*(x^
4 + 1)) + 18*sqrt(x^6 + x^2)*(3*x^3 + sqrt(3)*(x^5 + x)) + 18*27^(1/4)*(x^7 + x^3) - (x^6 + x^2)^(1/4)*(9*27^(
3/8)*sqrt(2)*x^4 + 27^(7/8)*sqrt(2)*(x^6 + x^2)))*sqrt(-(2*(x^6 + x^2)^(3/4)*(9*27^(3/8)*sqrt(2)*x^2 - 27^(7/8
)*sqrt(2)*(x^4 + 1)) - 9*sqrt(3)*(x^9 - x^5 + x) + 12*sqrt(x^6 + x^2)*(27^(3/4)*x^3 - 3*27^(1/4)*(x^5 + x)) +
6*(x^6 + x^2)^(1/4)*(9*27^(1/8)*sqrt(2)*x^4 - 27^(5/8)*sqrt(2)*(x^6 + x^2)))/(x^9 - x^5 + x)) - 9*(x^6 + x^2)^
(3/4)*(27^(7/8)*sqrt(2)*x^2 + 3*27^(3/8)*sqrt(2)*(x^4 + 1)) - 27*(x^6 + x^2)^(1/4)*(27^(5/8)*sqrt(2)*x^4 + 3*2
7^(1/8)*sqrt(2)*(x^6 + x^2)))/(x^9 - x^5 + x)) - 1/432*27^(7/8)*sqrt(2)*log(3*(2*(x^6 + x^2)^(3/4)*(9*27^(3/8)
*sqrt(2)*x^2 - 27^(7/8)*sqrt(2)*(x^4 + 1)) + 9*sqrt(3)*(x^9 - x^5 + x) - 12*sqrt(x^6 + x^2)*(27^(3/4)*x^3 - 3*
27^(1/4)*(x^5 + x)) + 6*(x^6 + x^2)^(1/4)*(9*27^(1/8)*sqrt(2)*x^4 - 27^(5/8)*sqrt(2)*(x^6 + x^2)))/(x^9 - x^5
+ x)) + 1/432*27^(7/8)*sqrt(2)*log(-3*(2*(x^6 + x^2)^(3/4)*(9*27^(3/8)*sqrt(2)*x^2 - 27^(7/8)*sqrt(2)*(x^4 + 1
)) - 9*sqrt(3)*(x^9 - x^5 + x) + 12*sqrt(x^6 + x^2)*(27^(3/4)*x^3 - 3*27^(1/4)*(x^5 + x)) + 6*(x^6 + x^2)^(1/4
)*(9*27^(1/8)*sqrt(2)*x^4 - 27^(5/8)*sqrt(2)*(x^6 + x^2)))/(x^9 - x^5 + x)) - 1/54*27^(7/8)*arctan(1/18*(27^(5
/8)*(x^6 + x^2)^(1/4)*(x^4 + 1) + 3^(3/4)*(27^(3/8)*(x^6 + x^2)^(1/4)*(x^4 + 1) + 3*27^(1/8)*(x^6 + x^2)^(3/4)
) - 3*27^(3/8)*(x^6 + x^2)^(3/4))/(x^5 + x)) - 1/216*27^(7/8)*log((2*27^(3/4)*(x^7 + x^3) + 2*(x^6 + x^2)^(3/4
)*(9*27^(1/8)*x^2 + 27^(5/8)*(x^4 + 1)) + 18*sqrt(x^6 + x^2)*(x^5 + sqrt(3)*x^3 + x) + 3*27^(1/4)*(x^9 + 5*x^5
+ x) + 2*(x^6 + x^2)^(1/4)*(27^(7/8)*x^4 + 3*27^(3/8)*(x^6 + x^2)))/(x^9 - x^5 + x)) + 1/216*27^(7/8)*log((2*
27^(3/4)*(x^7 + x^3) - 2*(x^6 + x^2)^(3/4)*(9*27^(1/8)*x^2 + 27^(5/8)*(x^4 + 1)) + 18*sqrt(x^6 + x^2)*(x^5 + s
qrt(3)*x^3 + x) + 3*27^(1/4)*(x^9 + 5*x^5 + x) - 2*(x^6 + x^2)^(1/4)*(27^(7/8)*x^4 + 3*27^(3/8)*(x^6 + x^2)))/
(x^9 - x^5 + x))
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}{x^{8} - x^{4} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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maple [C] time = 76.99, size = 1496, normalized size = 8.00
| | |
| method |
result |
size |
| | |
| trager |
\(\text {Expression too large to display}\) |
\(1496\) |
| | |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^4-1)*(x^6+x^2)^(1/4)/(x^8-x^4+1),x,method=_RETURNVERBOSE)
[Out]
-1/12*RootOf(_Z^2+RootOf(_Z^8-243)^2)*ln(-(RootOf(_Z^8-243)^10*RootOf(_Z^2+RootOf(_Z^8-243)^2)*x^5-2*RootOf(_Z
^8-243)^10*RootOf(_Z^2+RootOf(_Z^8-243)^2)*x^3+RootOf(_Z^8-243)^10*RootOf(_Z^2+RootOf(_Z^8-243)^2)*x-9*RootOf(
_Z^8-243)^6*RootOf(_Z^2+RootOf(_Z^8-243)^2)*x^3+54*RootOf(_Z^2+RootOf(_Z^8-243)^2)*(x^6+x^2)^(1/2)*RootOf(_Z^8
-243)^4*x-324*RootOf(_Z^2+RootOf(_Z^8-243)^2)*RootOf(_Z^8-243)^2*x^5+486*RootOf(_Z^2+RootOf(_Z^8-243)^2)*RootO
f(_Z^8-243)^2*x^3-486*(x^6+x^2)^(1/4)*RootOf(_Z^8-243)^2*x^2-324*RootOf(_Z^2+RootOf(_Z^8-243)^2)*RootOf(_Z^8-2
43)^2*x+1458*(x^6+x^2)^(3/4))/(RootOf(_Z^8-243)^4*x^4-2*RootOf(_Z^8-243)^4*x^2+RootOf(_Z^8-243)^4+18*x^4-27*x^
2+18)/x)-1/12*RootOf(_Z^2+RootOf(_Z^8-243)*RootOf(_Z^2+RootOf(_Z^8-243)^2))*ln((-RootOf(_Z^2+RootOf(_Z^8-243)^
2)*RootOf(_Z^2+RootOf(_Z^8-243)*RootOf(_Z^2+RootOf(_Z^8-243)^2))*RootOf(_Z^8-243)^9*x^5+2*RootOf(_Z^2+RootOf(_
Z^8-243)^2)*RootOf(_Z^2+RootOf(_Z^8-243)*RootOf(_Z^2+RootOf(_Z^8-243)^2))*RootOf(_Z^8-243)^9*x^3-RootOf(_Z^2+R
ootOf(_Z^8-243)^2)*RootOf(_Z^2+RootOf(_Z^8-243)*RootOf(_Z^2+RootOf(_Z^8-243)^2))*RootOf(_Z^8-243)^9*x-9*RootOf
(_Z^2+RootOf(_Z^8-243)^2)*RootOf(_Z^8-243)^5*RootOf(_Z^2+RootOf(_Z^8-243)*RootOf(_Z^2+RootOf(_Z^8-243)^2))*x^3
+324*RootOf(_Z^2+RootOf(_Z^8-243)^2)*RootOf(_Z^2+RootOf(_Z^8-243)*RootOf(_Z^2+RootOf(_Z^8-243)^2))*RootOf(_Z^8
-243)*x^5+54*(x^6+x^2)^(1/2)*RootOf(_Z^8-243)^4*RootOf(_Z^2+RootOf(_Z^8-243)*RootOf(_Z^2+RootOf(_Z^8-243)^2))*
x-486*RootOf(_Z^2+RootOf(_Z^8-243)^2)*RootOf(_Z^2+RootOf(_Z^8-243)*RootOf(_Z^2+RootOf(_Z^8-243)^2))*RootOf(_Z^
8-243)*x^3+486*RootOf(_Z^2+RootOf(_Z^8-243)^2)*(x^6+x^2)^(1/4)*RootOf(_Z^8-243)*x^2+324*RootOf(_Z^2+RootOf(_Z^
8-243)^2)*RootOf(_Z^2+RootOf(_Z^8-243)*RootOf(_Z^2+RootOf(_Z^8-243)^2))*RootOf(_Z^8-243)*x-1458*(x^6+x^2)^(3/4
))/(RootOf(_Z^8-243)^4*x^4-2*RootOf(_Z^8-243)^4*x^2+RootOf(_Z^8-243)^4-18*x^4+27*x^2-18)/x)-1/2916*RootOf(_Z^8
-243)^7*RootOf(_Z^2+RootOf(_Z^8-243)^2)*RootOf(_Z^2+RootOf(_Z^8-243)*RootOf(_Z^2+RootOf(_Z^8-243)^2))*ln((-Roo
tOf(_Z^2+RootOf(_Z^8-243)*RootOf(_Z^2+RootOf(_Z^8-243)^2))*RootOf(_Z^8-243)^10*x^5+2*RootOf(_Z^2+RootOf(_Z^8-2
43)*RootOf(_Z^2+RootOf(_Z^8-243)^2))*RootOf(_Z^8-243)^10*x^3-RootOf(_Z^2+RootOf(_Z^8-243)*RootOf(_Z^2+RootOf(_
Z^8-243)^2))*RootOf(_Z^8-243)^10*x-9*RootOf(_Z^2+RootOf(_Z^8-243)*RootOf(_Z^2+RootOf(_Z^8-243)^2))*RootOf(_Z^8
-243)^6*x^3+54*(x^6+x^2)^(1/2)*RootOf(_Z^2+RootOf(_Z^8-243)^2)*RootOf(_Z^2+RootOf(_Z^8-243)*RootOf(_Z^2+RootOf
(_Z^8-243)^2))*RootOf(_Z^8-243)^3*x+324*RootOf(_Z^2+RootOf(_Z^8-243)*RootOf(_Z^2+RootOf(_Z^8-243)^2))*RootOf(_
Z^8-243)^2*x^5-486*RootOf(_Z^2+RootOf(_Z^8-243)*RootOf(_Z^2+RootOf(_Z^8-243)^2))*RootOf(_Z^8-243)^2*x^3-486*Ro
otOf(_Z^2+RootOf(_Z^8-243)^2)*(x^6+x^2)^(1/4)*RootOf(_Z^8-243)*x^2+324*RootOf(_Z^2+RootOf(_Z^8-243)*RootOf(_Z^
2+RootOf(_Z^8-243)^2))*RootOf(_Z^8-243)^2*x-1458*(x^6+x^2)^(3/4))/(RootOf(_Z^8-243)^4*x^4-2*RootOf(_Z^8-243)^4
*x^2+RootOf(_Z^8-243)^4-18*x^4+27*x^2-18)/x)-1/12*RootOf(_Z^8-243)*ln(-(-RootOf(_Z^8-243)^11*x^5+2*RootOf(_Z^8
-243)^11*x^3-RootOf(_Z^8-243)^11*x+9*RootOf(_Z^8-243)^7*x^3+54*RootOf(_Z^8-243)^5*(x^6+x^2)^(1/2)*x+324*RootOf
(_Z^8-243)^3*x^5-486*RootOf(_Z^8-243)^3*x^3+486*(x^6+x^2)^(1/4)*RootOf(_Z^8-243)^2*x^2+324*RootOf(_Z^8-243)^3*
x+1458*(x^6+x^2)^(3/4))/(RootOf(_Z^8-243)^4*x^4-2*RootOf(_Z^8-243)^4*x^2+RootOf(_Z^8-243)^4+18*x^4-27*x^2+18)/
x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}{x^{8} - x^{4} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^6+x^2\right )}^{1/4}\,\left (x^4-1\right )}{x^8-x^4+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((x^2 + x^6)^(1/4)*(x^4 - 1))/(x^8 - x^4 + 1),x)
[Out]
int(((x^2 + x^6)^(1/4)*(x^4 - 1))/(x^8 - x^4 + 1), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{x^{2} \left (x^{4} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{x^{8} - x^{4} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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**8 - x**4 + 1), x)
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