3.24.65 \(\int \frac {1-x^4+x^8}{\sqrt [4]{x^2+x^6} (-1+x^8)} \, dx\) [2365]

Optimal. Leaf size=187 \[ -\frac {3 \left (x^2+x^6\right )^{3/4}}{2 x \left (1+x^4\right )}-\frac {\text {ArcTan}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{8 \sqrt [4]{2}}+\frac {\text {ArcTan}\left (\frac {2^{3/4} x \sqrt [4]{x^2+x^6}}{\sqrt {2} x^2-\sqrt {x^2+x^6}}\right )}{8\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{8 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^2+x^6}}{2^{3/4}}}{x \sqrt [4]{x^2+x^6}}\right )}{8\ 2^{3/4}} \]

[Out]

-3/2*(x^6+x^2)^(3/4)/x/(x^4+1)-1/16*arctan(2^(1/4)*x/(x^6+x^2)^(1/4))*2^(3/4)+1/16*arctan(2^(3/4)*x*(x^6+x^2)^
(1/4)/(x^2*2^(1/2)-(x^6+x^2)^(1/2)))*2^(1/4)-1/16*arctanh(2^(1/4)*x/(x^6+x^2)^(1/4))*2^(3/4)-1/16*arctanh((1/2
*x^2*2^(3/4)+1/2*(x^6+x^2)^(1/2)*2^(1/4))/x/(x^6+x^2)^(1/4))*2^(1/4)

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Rubi [C] Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
time = 0.33, antiderivative size = 99, normalized size of antiderivative = 0.53, number of steps used = 10, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2081, 6847, 6857, 251, 1469, 541, 544, 440} \begin {gather*} -\frac {\sqrt [4]{x^4+1} x F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};x^4,-x^4\right )}{\sqrt [4]{x^6+x^2}}+\frac {\sqrt [4]{x^4+1} x \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^4\right )}{2 \sqrt [4]{x^6+x^2}}-\frac {3 x}{2 \sqrt [4]{x^6+x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*x)/(2*(x^2 + x^6)^(1/4)) - (x*(1 + x^4)^(1/4)*AppellF1[1/8, 1, 1/4, 9/8, x^4, -x^4])/(x^2 + x^6)^(1/4) + (
x*(1 + x^4)^(1/4)*Hypergeometric2F1[1/8, 1/4, 9/8, -x^4])/(2*(x^2 + x^6)^(1/4))

Rule 251

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rule 440

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 541

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 544

Int[(((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[f/d,
Int[(a + b*x^n)^p, x], x] + Dist[(d*e - c*f)/d, Int[(a + b*x^n)^p/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
f, p, n}, x]

Rule 1469

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((f_) + (g_.)*(x_)^(n_))^(r_.)*((a_) + (c_.)*(x_)^(n2_))^(p_.), x_Symbol] :
> Int[(d + e*x^n)^(p + q)*(f + g*x^n)^r*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, f, g, n, q, r}, x] && Eq
Q[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[p]

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 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 6857

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^4+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^4+x^8}{\sqrt {x} \sqrt [4]{1+x^4} \left (-1+x^8\right )} \, dx}{\sqrt [4]{x^2+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \frac {1-x^8+x^{16}}{\sqrt [4]{1+x^8} \left (-1+x^{16}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \left (\frac {1}{\sqrt [4]{1+x^8}}+\frac {2-x^8}{\sqrt [4]{1+x^8} \left (-1+x^{16}\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}+\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \frac {2-x^8}{\sqrt [4]{1+x^8} \left (-1+x^{16}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=\frac {2 x \sqrt [4]{1+x^4} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^4\right )}{\sqrt [4]{x^2+x^6}}+\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \frac {2-x^8}{\left (-1+x^8\right ) \left (1+x^8\right )^{5/4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}\\ &=-\frac {3 x}{2 \sqrt [4]{x^2+x^6}}+\frac {2 x \sqrt [4]{1+x^4} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^4\right )}{\sqrt [4]{x^2+x^6}}+\frac {\left (\sqrt {x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \frac {5-3 x^8}{\left (-1+x^8\right ) \sqrt [4]{1+x^8}} \, dx,x,\sqrt {x}\right )}{2 \sqrt [4]{x^2+x^6}}\\ &=-\frac {3 x}{2 \sqrt [4]{x^2+x^6}}+\frac {2 x \sqrt [4]{1+x^4} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^4\right )}{\sqrt [4]{x^2+x^6}}+\frac {\left (\sqrt {x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+x^8\right ) \sqrt [4]{1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^6}}-\frac {\left (3 \sqrt {x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^8}} \, dx,x,\sqrt {x}\right )}{2 \sqrt [4]{x^2+x^6}}\\ &=-\frac {3 x}{2 \sqrt [4]{x^2+x^6}}-\frac {x \sqrt [4]{1+x^4} F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};x^4,-x^4\right )}{\sqrt [4]{x^2+x^6}}+\frac {x \sqrt [4]{1+x^4} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^4\right )}{2 \sqrt [4]{x^2+x^6}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 216, normalized size = 1.16 \begin {gather*} -\frac {\sqrt {x} \left (24 \sqrt {x}+2^{3/4} \sqrt [4]{1+x^4} \text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )-\sqrt [4]{2} \sqrt [4]{1+x^4} \text {ArcTan}\left (\frac {2^{3/4} \sqrt {x} \sqrt [4]{1+x^4}}{\sqrt {2} x-\sqrt {1+x^4}}\right )+2^{3/4} \sqrt [4]{1+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )+\sqrt [4]{2} \sqrt [4]{1+x^4} \tanh ^{-1}\left (\frac {2 \sqrt [4]{2} \sqrt {x} \sqrt [4]{1+x^4}}{2 x+\sqrt {2} \sqrt {1+x^4}}\right )\right )}{16 \sqrt [4]{x^2+x^6}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/16*(Sqrt[x]*(24*Sqrt[x] + 2^(3/4)*(1 + x^4)^(1/4)*ArcTan[(2^(1/4)*Sqrt[x])/(1 + x^4)^(1/4)] - 2^(1/4)*(1 +
x^4)^(1/4)*ArcTan[(2^(3/4)*Sqrt[x]*(1 + x^4)^(1/4))/(Sqrt[2]*x - Sqrt[1 + x^4])] + 2^(3/4)*(1 + x^4)^(1/4)*Arc
Tanh[(2^(1/4)*Sqrt[x])/(1 + x^4)^(1/4)] + 2^(1/4)*(1 + x^4)^(1/4)*ArcTanh[(2*2^(1/4)*Sqrt[x]*(1 + x^4)^(1/4))/
(2*x + Sqrt[2]*Sqrt[1 + x^4])]))/(x^2 + x^6)^(1/4)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 42.63, size = 648, normalized size = 3.47

method	result	size
risch	\(\text {Expression too large to display}\)	\(648\)
trager	\(\text {Expression too large to display}\)	\(654\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*x/(x^2*(x^4+1))^(1/4)+1/32*RootOf(_Z^4-8)*ln((-RootOf(_Z^4-8)^3*(x^6+x^2)^(1/2)*x-RootOf(_Z^4-8)*x^5+2*Ro
otOf(_Z^4-8)^2*(x^6+x^2)^(1/4)*x^2-2*RootOf(_Z^4-8)*x^3+4*(x^6+x^2)^(3/4)-RootOf(_Z^4-8)*x)/(-1+x)^2/(1+x)^2/x
)-1/32*RootOf(_Z^2+RootOf(_Z^4-8)^2)*ln(-(RootOf(_Z^2+RootOf(_Z^4-8)^2)*RootOf(_Z^4-8)^2*(x^6+x^2)^(1/2)*x-Roo
tOf(_Z^2+RootOf(_Z^4-8)^2)*x^5+2*RootOf(_Z^4-8)^2*(x^6+x^2)^(1/4)*x^2-2*RootOf(_Z^2+RootOf(_Z^4-8)^2)*x^3-4*(x
^6+x^2)^(3/4)-RootOf(_Z^2+RootOf(_Z^4-8)^2)*x)/(-1+x)^2/(1+x)^2/x)-1/64*ln((RootOf(_Z^4-8)^2*x^2+2*RootOf(_Z^4
-8)*(x^6+x^2)^(1/4)*x+2*(x^6+x^2)^(1/2))/x/(x^2+1))*RootOf(_Z^4-8)^3-1/64*ln((RootOf(_Z^4-8)^2*x^2+2*RootOf(_Z
^4-8)*(x^6+x^2)^(1/4)*x+2*(x^6+x^2)^(1/2))/x/(x^2+1))*RootOf(_Z^4-8)^2*RootOf(_Z^2+RootOf(_Z^4-8)^2)+1/64*Root
Of(_Z^4-8)^2*RootOf(_Z^2+RootOf(_Z^4-8)^2)*ln((-RootOf(_Z^2+RootOf(_Z^4-8)^2)*RootOf(_Z^4-8)^2*x^5+RootOf(_Z^4
-8)^3*x^5+2*RootOf(_Z^2+RootOf(_Z^4-8)^2)*RootOf(_Z^4-8)^2*x^3-2*RootOf(_Z^4-8)^3*x^3+8*RootOf(_Z^4-8)*(x^6+x^
2)^(1/4)*RootOf(_Z^2+RootOf(_Z^4-8)^2)*x^2+8*RootOf(_Z^2+RootOf(_Z^4-8)^2)*(x^6+x^2)^(1/2)*x+8*(x^6+x^2)^(1/2)
*RootOf(_Z^4-8)*x-RootOf(_Z^2+RootOf(_Z^4-8)^2)*RootOf(_Z^4-8)^2*x+RootOf(_Z^4-8)^3*x+16*(x^6+x^2)^(3/4))/x/(x
^2+1)^2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1055 vs. \(2 (144) = 288\).
time = 22.55, size = 1055, normalized size = 5.64 \begin {gather*} -\frac {4 \cdot 2^{\frac {3}{4}} {\left (x^{5} + x\right )} \arctan \left (\frac {4 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2^{\frac {3}{4}} {\left (2 \cdot 2^{\frac {3}{4}} \sqrt {x^{6} + x^{2}} x + 2^{\frac {1}{4}} {\left (x^{5} + 2 \, x^{3} + x\right )}\right )} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{2 \, {\left (x^{5} - 2 \, x^{3} + x\right )}}\right ) + 2^{\frac {3}{4}} {\left (x^{5} + x\right )} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2^{\frac {3}{4}} {\left (x^{5} + 2 \, x^{3} + x\right )} + 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{6} + x^{2}} x + 4 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} - 2 \, x^{3} + x}\right ) - 2^{\frac {3}{4}} {\left (x^{5} + x\right )} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} - 2^{\frac {3}{4}} {\left (x^{5} + 2 \, x^{3} + x\right )} - 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{6} + x^{2}} x + 4 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} - 2 \, x^{3} + x}\right ) - 4 \cdot 2^{\frac {1}{4}} {\left (x^{5} + x\right )} \arctan \left (-\frac {2 \, x^{9} + 8 \, x^{7} + 12 \, x^{5} + 8 \, x^{3} + 4 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} {\left (x^{4} - 6 \, x^{2} + 1\right )} + 8 \, \sqrt {2} \sqrt {x^{6} + x^{2}} {\left (x^{5} + 2 \, x^{3} + x\right )} - \sqrt {2} {\left (32 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} x^{2} + 2^{\frac {3}{4}} {\left (x^{9} - 16 \, x^{7} - 2 \, x^{5} - 16 \, x^{3} + x\right )} + 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{6} + x^{2}} {\left (x^{5} - 6 \, x^{3} + x\right )} + 8 \, {\left (x^{6} + 2 \, x^{4} + x^{2}\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {4 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} + 8 \, \sqrt {x^{6} + x^{2}} x + 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + 2 \, x^{3} + x}} + 8 \cdot 2^{\frac {1}{4}} {\left (3 \, x^{6} - 2 \, x^{4} + 3 \, x^{2}\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} + 2 \, x}{2 \, {\left (x^{9} - 28 \, x^{7} + 6 \, x^{5} - 28 \, x^{3} + x\right )}}\right ) + 4 \cdot 2^{\frac {1}{4}} {\left (x^{5} + x\right )} \arctan \left (-\frac {2 \, x^{9} + 8 \, x^{7} + 12 \, x^{5} + 8 \, x^{3} - 4 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} {\left (x^{4} - 6 \, x^{2} + 1\right )} + 8 \, \sqrt {2} \sqrt {x^{6} + x^{2}} {\left (x^{5} + 2 \, x^{3} + x\right )} - \sqrt {2} {\left (32 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} x^{2} - 2^{\frac {3}{4}} {\left (x^{9} - 16 \, x^{7} - 2 \, x^{5} - 16 \, x^{3} + x\right )} - 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{6} + x^{2}} {\left (x^{5} - 6 \, x^{3} + x\right )} + 8 \, {\left (x^{6} + 2 \, x^{4} + x^{2}\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}}\right )} \sqrt {-\frac {4 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} - \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} - 8 \, \sqrt {x^{6} + x^{2}} x + 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + 2 \, x^{3} + x}} - 8 \cdot 2^{\frac {1}{4}} {\left (3 \, x^{6} - 2 \, x^{4} + 3 \, x^{2}\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} + 2 \, x}{2 \, {\left (x^{9} - 28 \, x^{7} + 6 \, x^{5} - 28 \, x^{3} + x\right )}}\right ) + 2^{\frac {1}{4}} {\left (x^{5} + x\right )} \log \left (\frac {8 \, {\left (4 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} + 8 \, \sqrt {x^{6} + x^{2}} x + 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}\right )}}{x^{5} + 2 \, x^{3} + x}\right ) - 2^{\frac {1}{4}} {\left (x^{5} + x\right )} \log \left (-\frac {8 \, {\left (4 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} - \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} - 8 \, \sqrt {x^{6} + x^{2}} x + 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}\right )}}{x^{5} + 2 \, x^{3} + x}\right ) + 96 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{64 \, {\left (x^{5} + x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64*(4*2^(3/4)*(x^5 + x)*arctan(1/2*(4*2^(3/4)*(x^6 + x^2)^(1/4)*x^2 + 2^(3/4)*(2*2^(3/4)*sqrt(x^6 + x^2)*x
+ 2^(1/4)*(x^5 + 2*x^3 + x)) + 4*2^(1/4)*(x^6 + x^2)^(3/4))/(x^5 - 2*x^3 + x)) + 2^(3/4)*(x^5 + x)*log(-(4*sqr
t(2)*(x^6 + x^2)^(1/4)*x^2 + 2^(3/4)*(x^5 + 2*x^3 + x) + 4*2^(1/4)*sqrt(x^6 + x^2)*x + 4*(x^6 + x^2)^(3/4))/(x
^5 - 2*x^3 + x)) - 2^(3/4)*(x^5 + x)*log(-(4*sqrt(2)*(x^6 + x^2)^(1/4)*x^2 - 2^(3/4)*(x^5 + 2*x^3 + x) - 4*2^(
1/4)*sqrt(x^6 + x^2)*x + 4*(x^6 + x^2)^(3/4))/(x^5 - 2*x^3 + x)) - 4*2^(1/4)*(x^5 + x)*arctan(-1/2*(2*x^9 + 8*
x^7 + 12*x^5 + 8*x^3 + 4*2^(3/4)*(x^6 + x^2)^(3/4)*(x^4 - 6*x^2 + 1) + 8*sqrt(2)*sqrt(x^6 + x^2)*(x^5 + 2*x^3
+ x) - sqrt(2)*(32*sqrt(2)*(x^6 + x^2)^(3/4)*x^2 + 2^(3/4)*(x^9 - 16*x^7 - 2*x^5 - 16*x^3 + x) + 4*2^(1/4)*sqr
t(x^6 + x^2)*(x^5 - 6*x^3 + x) + 8*(x^6 + 2*x^4 + x^2)*(x^6 + x^2)^(1/4))*sqrt((4*2^(3/4)*(x^6 + x^2)^(1/4)*x^
2 + sqrt(2)*(x^5 + 2*x^3 + x) + 8*sqrt(x^6 + x^2)*x + 4*2^(1/4)*(x^6 + x^2)^(3/4))/(x^5 + 2*x^3 + x)) + 8*2^(1
/4)*(3*x^6 - 2*x^4 + 3*x^2)*(x^6 + x^2)^(1/4) + 2*x)/(x^9 - 28*x^7 + 6*x^5 - 28*x^3 + x)) + 4*2^(1/4)*(x^5 + x
)*arctan(-1/2*(2*x^9 + 8*x^7 + 12*x^5 + 8*x^3 - 4*2^(3/4)*(x^6 + x^2)^(3/4)*(x^4 - 6*x^2 + 1) + 8*sqrt(2)*sqrt
(x^6 + x^2)*(x^5 + 2*x^3 + x) - sqrt(2)*(32*sqrt(2)*(x^6 + x^2)^(3/4)*x^2 - 2^(3/4)*(x^9 - 16*x^7 - 2*x^5 - 16
*x^3 + x) - 4*2^(1/4)*sqrt(x^6 + x^2)*(x^5 - 6*x^3 + x) + 8*(x^6 + 2*x^4 + x^2)*(x^6 + x^2)^(1/4))*sqrt(-(4*2^
(3/4)*(x^6 + x^2)^(1/4)*x^2 - sqrt(2)*(x^5 + 2*x^3 + x) - 8*sqrt(x^6 + x^2)*x + 4*2^(1/4)*(x^6 + x^2)^(3/4))/(
x^5 + 2*x^3 + x)) - 8*2^(1/4)*(3*x^6 - 2*x^4 + 3*x^2)*(x^6 + x^2)^(1/4) + 2*x)/(x^9 - 28*x^7 + 6*x^5 - 28*x^3
+ x)) + 2^(1/4)*(x^5 + x)*log(8*(4*2^(3/4)*(x^6 + x^2)^(1/4)*x^2 + sqrt(2)*(x^5 + 2*x^3 + x) + 8*sqrt(x^6 + x^
2)*x + 4*2^(1/4)*(x^6 + x^2)^(3/4))/(x^5 + 2*x^3 + x)) - 2^(1/4)*(x^5 + x)*log(-8*(4*2^(3/4)*(x^6 + x^2)^(1/4)
*x^2 - sqrt(2)*(x^5 + 2*x^3 + x) - 8*sqrt(x^6 + x^2)*x + 4*2^(1/4)*(x^6 + x^2)^(3/4))/(x^5 + 2*x^3 + x)) + 96*
(x^6 + x^2)^(3/4))/(x^5 + x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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