3.22.93 \(\int \frac {(-1+x^4) \sqrt [4]{x^2+x^6}}{1+x^8} \, dx\) [2193]

Optimal. Leaf size=162 \[ \frac {\text {ArcTan}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{2\ 2^{3/8}}+\frac {\text {ArcTan}\left (\frac {2^{5/8} x \sqrt [4]{x^2+x^6}}{\sqrt [4]{2} x^2-\sqrt {x^2+x^6}}\right )}{2\ 2^{7/8}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{2\ 2^{3/8}}+\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{2^{3/8}}+\frac {\sqrt {x^2+x^6}}{2^{5/8}}}{x \sqrt [4]{x^2+x^6}}\right )}{2\ 2^{7/8}} \]

[Out]

1/4*arctan(2^(1/8)*x/(x^6+x^2)^(1/4))*2^(5/8)+1/4*arctan(2^(5/8)*x*(x^6+x^2)^(1/4)/(x^2*2^(1/4)-(x^6+x^2)^(1/2
)))*2^(1/8)-1/4*arctanh(2^(1/8)*x/(x^6+x^2)^(1/4))*2^(5/8)+1/4*arctanh((1/2*x^2*2^(5/8)+1/2*(x^6+x^2)^(1/2)*2^
(3/8))/x/(x^6+x^2)^(1/4))*2^(1/8)

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Rubi [C] Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
time = 0.26, antiderivative size = 105, normalized size of antiderivative = 0.65, number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2081, 6857, 477, 524} \begin {gather*} -\frac {\left (\frac {1}{3}-\frac {i}{3}\right ) x \sqrt [4]{x^6+x^2} F_1\left (\frac {3}{8};1,-\frac {1}{4};\frac {11}{8};-i x^4,-x^4\right )}{\sqrt [4]{x^4+1}}-\frac {\left (\frac {1}{3}+\frac {i}{3}\right ) x \sqrt [4]{x^6+x^2} F_1\left (\frac {3}{8};1,-\frac {1}{4};\frac {11}{8};i x^4,-x^4\right )}{\sqrt [4]{x^4+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1/3 + I/3)*x*(x^2 + x^6)^(1/4)*AppellF1[3/8, 1, -1/4, 11/8, (-I)*x^4, -x^4])/(1 + x^4)^(1/4) - ((1/3 + I/3)
*x*(x^2 + x^6)^(1/4)*AppellF1[3/8, 1, -1/4, 11/8, I*x^4, -x^4])/(1 + x^4)^(1/4)

Rule 477

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 524

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)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], 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 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 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 {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{1+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^8} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}\\ &=\frac {\sqrt [4]{x^2+x^6} \int \left (-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {x} \sqrt [4]{1+x^4}}{i-x^4}+\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {x} \sqrt [4]{1+x^4}}{i+x^4}\right ) \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}\\ &=-\frac {\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{x^2+x^6}\right ) \int \frac {\sqrt {x} \sqrt [4]{1+x^4}}{i-x^4} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}+\frac {\left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt [4]{x^2+x^6}\right ) \int \frac {\sqrt {x} \sqrt [4]{1+x^4}}{i+x^4} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}\\ &=-\frac {\left ((1+i) \sqrt [4]{x^2+x^6}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^8}}{i-x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1+x^4}}+\frac {\left ((1-i) \sqrt [4]{x^2+x^6}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^8}}{i+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1+x^4}}\\ &=-\frac {\left (\frac {1}{3}-\frac {i}{3}\right ) x \sqrt [4]{x^2+x^6} F_1\left (\frac {3}{8};1,-\frac {1}{4};\frac {11}{8};-i x^4,-x^4\right )}{\sqrt [4]{1+x^4}}-\frac {\left (\frac {1}{3}+\frac {i}{3}\right ) x \sqrt [4]{x^2+x^6} F_1\left (\frac {3}{8};1,-\frac {1}{4};\frac {11}{8};i x^4,-x^4\right )}{\sqrt [4]{1+x^4}}\\ \end {align*}

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Mathematica [A]
time = 0.53, size = 175, normalized size = 1.08 \begin {gather*} \frac {\sqrt [4]{x^2+x^6} \left (\sqrt {2} \text {ArcTan}\left (\frac {\sqrt [8]{2} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )+\text {ArcTan}\left (\frac {2^{5/8} \sqrt {x} \sqrt [4]{1+x^4}}{\sqrt [4]{2} x-\sqrt {1+x^4}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt [8]{2} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )+\tanh ^{-1}\left (\frac {2\ 2^{3/8} \sqrt {x} \sqrt [4]{1+x^4}}{2 x+2^{3/4} \sqrt {1+x^4}}\right )\right )}{2\ 2^{7/8} \sqrt {x} \sqrt [4]{1+x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((x^2 + x^6)^(1/4)*(Sqrt[2]*ArcTan[(2^(1/8)*Sqrt[x])/(1 + x^4)^(1/4)] + ArcTan[(2^(5/8)*Sqrt[x]*(1 + x^4)^(1/4
))/(2^(1/4)*x - Sqrt[1 + x^4])] - Sqrt[2]*ArcTanh[(2^(1/8)*Sqrt[x])/(1 + x^4)^(1/4)] + ArcTanh[(2*2^(3/8)*Sqrt
[x]*(1 + x^4)^(1/4))/(2*x + 2^(3/4)*Sqrt[1 + x^4])]))/(2*2^(7/8)*Sqrt[x]*(1 + x^4)^(1/4))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 72.00, size = 2857, normalized size = 17.64

method	result	size
trager	\(\text {Expression too large to display}\)	\(2857\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*RootOf(_Z^8-32)*ln((RootOf(_Z^8-32)^11*x^5-2*RootOf(_Z^8-32)^11*x^3+RootOf(_Z^8-32)^11*x-8*x^3*RootOf(_Z^
8-32)^7-32*(x^6+x^2)^(1/2)*RootOf(_Z^8-32)^5*x-64*x^5*RootOf(_Z^8-32)^3+64*x^3*RootOf(_Z^8-32)^3-128*RootOf(_Z
^8-32)^2*(x^6+x^2)^(1/4)*x^2-64*x*RootOf(_Z^8-32)^3-256*(x^6+x^2)^(3/4))/x/(RootOf(_Z^8-32)^4*x^4-2*RootOf(_Z^
8-32)^4*x^2+RootOf(_Z^8-32)^4+8*x^4-8*x^2+8))-RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*ln((Ro
otOf(_Z^8-32)^11*x^5+4*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*RootOf(_Z^8-32)^10*x^5-2*Root
Of(_Z^8-32)^11*x^3-8*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*RootOf(_Z^8-32)^10*x^3+RootOf(_
Z^8-32)^11*x+4*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*RootOf(_Z^8-32)^10*x+4*x^5*RootOf(_Z^
8-32)^7+32*RootOf(_Z^8-32)^6*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*x^3-128*RootOf(_Z^8-32)
^5*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*(x^6+x^2)^(1/4)*x^2+4*x*RootOf(_Z^8-32)^7+128*Roo
tOf(_Z^8-32)^4*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*(x^6+x^2)^(1/2)*x-32*x^5*RootOf(_Z^8-
32)^3-256*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*RootOf(_Z^8-32)^2*x^5+32*x^3*RootOf(_Z^8-3
2)^3+256*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*RootOf(_Z^8-32)^2*x^3+64*RootOf(_Z^8-32)^2*
(x^6+x^2)^(1/4)*x^2-32*x*RootOf(_Z^8-32)^3-256*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*RootO
f(_Z^8-32)^2*x-128*(x^6+x^2)^(3/4))/x/(RootOf(_Z^8-32)^4*x^4-2*RootOf(_Z^8-32)^4*x^2+RootOf(_Z^8-32)^4-8*x^4+8
*x^2-8))-1/4*ln((RootOf(_Z^8-32)^11*x^5-16*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*RootOf(_Z
^8-32)^10*x^5-2*RootOf(_Z^8-32)^11*x^3+32*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*RootOf(_Z^
8-32)^10*x^3+RootOf(_Z^8-32)^11*x-16*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*RootOf(_Z^8-32)
^10*x+64*RootOf(_Z^8-32)^6*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*x^5-8*x^3*RootOf(_Z^8-32)
^7+64*RootOf(_Z^8-32)^6*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*x+32*(x^6+x^2)^(1/2)*RootOf(
_Z^8-32)^5*x-64*x^5*RootOf(_Z^8-32)^3+512*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*RootOf(_Z^
8-32)^2*x^5+64*x^3*RootOf(_Z^8-32)^3-512*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*RootOf(_Z^8
-32)^2*x^3+128*RootOf(_Z^8-32)^2*(x^6+x^2)^(1/4)*x^2-64*x*RootOf(_Z^8-32)^3+512*RootOf(-2*RootOf(_Z^8-32)^5*_Z
+RootOf(_Z^8-32)^2+64*_Z^2)*RootOf(_Z^8-32)^2*x-2048*(x^6+x^2)^(1/2)*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^
8-32)^2+64*_Z^2)*x-256*(x^6+x^2)^(3/4))/x/(RootOf(_Z^8-32)^4*x^4-2*RootOf(_Z^8-32)^4*x^2+RootOf(_Z^8-32)^4+8*x
^4-8*x^2+8))*RootOf(_Z^8-32)^4*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)+1/8*ln((RootOf(_Z^8-3
2)^11*x^5-16*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*RootOf(_Z^8-32)^10*x^5-2*RootOf(_Z^8-32
)^11*x^3+32*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*RootOf(_Z^8-32)^10*x^3+RootOf(_Z^8-32)^1
1*x-16*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*RootOf(_Z^8-32)^10*x+64*RootOf(_Z^8-32)^6*Roo
tOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*x^5-8*x^3*RootOf(_Z^8-32)^7+64*RootOf(_Z^8-32)^6*RootOf
(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*x+32*(x^6+x^2)^(1/2)*RootOf(_Z^8-32)^5*x-64*x^5*RootOf(_Z^
8-32)^3+512*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*RootOf(_Z^8-32)^2*x^5+64*x^3*RootOf(_Z^8
-32)^3-512*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*RootOf(_Z^8-32)^2*x^3+128*RootOf(_Z^8-32)
^2*(x^6+x^2)^(1/4)*x^2-64*x*RootOf(_Z^8-32)^3+512*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*Ro
otOf(_Z^8-32)^2*x-2048*(x^6+x^2)^(1/2)*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*x-256*(x^6+x^
2)^(3/4))/x/(RootOf(_Z^8-32)^4*x^4-2*RootOf(_Z^8-32)^4*x^2+RootOf(_Z^8-32)^4+8*x^4-8*x^2+8))*RootOf(_Z^8-32)-1
/32*ln(-(RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*RootOf(_Z^8-32)^10*x^5-2*RootOf(-2*RootOf(_
Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*RootOf(_Z^8-32)^10*x^3+RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^
2+64*_Z^2)*RootOf(_Z^8-32)^10*x+8*RootOf(_Z^8-32)^6*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*
x^3-32*RootOf(_Z^8-32)^5*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*(x^6+x^2)^(1/4)*x^2+32*Root
Of(_Z^8-32)^4*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*(x^6+x^2)^(1/2)*x-64*RootOf(-2*RootOf(
_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*RootOf(_Z^8-32)^2*x^5+64*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-3
2)^2+64*_Z^2)*RootOf(_Z^8-32)^2*x^3+16*RootOf(_Z^8-32)^2*(x^6+x^2)^(1/4)*x^2-64*RootOf(-2*RootOf(_Z^8-32)^5*_Z
+RootOf(_Z^8-32)^2+64*_Z^2)*RootOf(_Z^8-32)^2*x-32*(x^6+x^2)^(1/2)*RootOf(_Z^8-32)*x+32*(x^6+x^2)^(3/4))/x/(Ro
otOf(_Z^8-32)^4*x^4-2*RootOf(_Z^8-32)^4*x^2+RootOf(_Z^8-32)^4-8*x^4+8*x^2-8))*RootOf(_Z^8-32)^5+ln(-(RootOf(-2
*RootOf(_Z^8-32)^5*_Z+RootOf(_Z^8-32)^2+64*_Z^2)*RootOf(_Z^8-32)^10*x^5-2*RootOf(-2*RootOf(_Z^8-32)^5*_Z+RootO
f(_Z^8-32)^2+64*_Z^2)*RootOf(_Z^8-32)^10*x^3+Ro...

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1273 vs. \(2 (123) = 246\).
time = 8.31, size = 1273, normalized size = 7.86 \begin {gather*} \text {Too large to display} \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+1),x, algorithm="fricas")

[Out]

-1/32*8^(7/8)*sqrt(2)*arctan(1/8*(2*(x^6 + x^2)^(3/4)*(8^(7/8)*sqrt(2)*x^2 + 2*8^(3/8)*sqrt(2)*(x^4 + 1)) + (8
^(3/4)*(x^9 + 4*x^5 + x) + 2*(x^6 + x^2)^(3/4)*(8^(5/8)*sqrt(2)*x^2 + 2*8^(1/8)*sqrt(2)*(x^4 + 1)) + 8*sqrt(x^
6 + x^2)*(2*x^3 + sqrt(2)*(x^5 + x)) + 8*8^(1/4)*(x^7 + x^3) + (x^6 + x^2)^(1/4)*(4*8^(3/8)*sqrt(2)*x^4 + 8^(7
/8)*sqrt(2)*(x^6 + x^2)))*sqrt(((x^6 + x^2)^(3/4)*(4*8^(3/8)*sqrt(2)*x^2 - 8^(7/8)*sqrt(2)*(x^4 + 1)) + 2*sqrt
(2)*(x^9 + x) - 4*sqrt(x^6 + x^2)*(8^(3/4)*x^3 - 2*8^(1/4)*(x^5 + x)) + 2*(x^6 + x^2)^(1/4)*(4*8^(1/8)*sqrt(2)
*x^4 - 8^(5/8)*sqrt(2)*(x^6 + x^2)))/(x^9 + x)) + 4*(x^6 + x^2)^(1/4)*(8^(5/8)*sqrt(2)*x^4 + 2*8^(1/8)*sqrt(2)
*(x^6 + x^2)))/(x^9 + x)) - 1/32*8^(7/8)*sqrt(2)*arctan(1/8*(2*(x^6 + x^2)^(3/4)*(8^(7/8)*sqrt(2)*x^2 + 2*8^(3
/8)*sqrt(2)*(x^4 + 1)) - (8^(3/4)*(x^9 + 4*x^5 + x) - 2*(x^6 + x^2)^(3/4)*(8^(5/8)*sqrt(2)*x^2 + 2*8^(1/8)*sqr
t(2)*(x^4 + 1)) + 8*sqrt(x^6 + x^2)*(2*x^3 + sqrt(2)*(x^5 + x)) + 8*8^(1/4)*(x^7 + x^3) - (x^6 + x^2)^(1/4)*(4
*8^(3/8)*sqrt(2)*x^4 + 8^(7/8)*sqrt(2)*(x^6 + x^2)))*sqrt(-((x^6 + x^2)^(3/4)*(4*8^(3/8)*sqrt(2)*x^2 - 8^(7/8)
*sqrt(2)*(x^4 + 1)) - 2*sqrt(2)*(x^9 + x) + 4*sqrt(x^6 + x^2)*(8^(3/4)*x^3 - 2*8^(1/4)*(x^5 + x)) + 2*(x^6 + x
^2)^(1/4)*(4*8^(1/8)*sqrt(2)*x^4 - 8^(5/8)*sqrt(2)*(x^6 + x^2)))/(x^9 + x)) + 4*(x^6 + x^2)^(1/4)*(8^(5/8)*sqr
t(2)*x^4 + 2*8^(1/8)*sqrt(2)*(x^6 + x^2)))/(x^9 + x)) - 1/128*8^(7/8)*sqrt(2)*log(4*((x^6 + x^2)^(3/4)*(4*8^(3
/8)*sqrt(2)*x^2 - 8^(7/8)*sqrt(2)*(x^4 + 1)) + 2*sqrt(2)*(x^9 + x) - 4*sqrt(x^6 + x^2)*(8^(3/4)*x^3 - 2*8^(1/4
)*(x^5 + x)) + 2*(x^6 + x^2)^(1/4)*(4*8^(1/8)*sqrt(2)*x^4 - 8^(5/8)*sqrt(2)*(x^6 + x^2)))/(x^9 + x)) + 1/128*8
^(7/8)*sqrt(2)*log(-4*((x^6 + x^2)^(3/4)*(4*8^(3/8)*sqrt(2)*x^2 - 8^(7/8)*sqrt(2)*(x^4 + 1)) - 2*sqrt(2)*(x^9
+ x) + 4*sqrt(x^6 + x^2)*(8^(3/4)*x^3 - 2*8^(1/4)*(x^5 + x)) + 2*(x^6 + x^2)^(1/4)*(4*8^(1/8)*sqrt(2)*x^4 - 8^
(5/8)*sqrt(2)*(x^6 + x^2)))/(x^9 + x)) - 1/16*8^(7/8)*arctan(1/8*(8^(5/8)*(x^6 + x^2)^(1/4)*(x^4 + 1) + 2^(3/4
)*(8^(3/8)*(x^6 + x^2)^(1/4)*(x^4 + 1) + 2*8^(1/8)*(x^6 + x^2)^(3/4)) - 2*8^(3/8)*(x^6 + x^2)^(3/4))/(x^5 + x)
) - 1/64*8^(7/8)*log((8^(3/4)*(x^7 + x^3) + (x^6 + x^2)^(3/4)*(4*8^(1/8)*x^2 + 8^(5/8)*(x^4 + 1)) + 4*sqrt(x^6
+ x^2)*(x^5 + sqrt(2)*x^3 + x) + 8^(1/4)*(x^9 + 4*x^5 + x) + (x^6 + x^2)^(1/4)*(8^(7/8)*x^4 + 2*8^(3/8)*(x^6
+ x^2)))/(x^9 + x)) + 1/64*8^(7/8)*log((8^(3/4)*(x^7 + x^3) - (x^6 + x^2)^(3/4)*(4*8^(1/8)*x^2 + 8^(5/8)*(x^4
+ 1)) + 4*sqrt(x^6 + x^2)*(x^5 + sqrt(2)*x^3 + x) + 8^(1/4)*(x^9 + 4*x^5 + x) - (x^6 + x^2)^(1/4)*(8^(7/8)*x^4
+ 2*8^(3/8)*(x^6 + x^2)))/(x^9 + 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} + 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+1),x)

[Out]

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

[Out]

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

[Out]

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

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