3.24.25 \(\int \frac {\sqrt [4]{x^2+x^6} (1+x^8)}{x^4 (-1+x^4)} \, dx\) [2325]

Optimal. Leaf size=180 \[ \frac {2 \left (1+x^4\right ) \sqrt [4]{x^2+x^6}}{5 x^3}+\frac {\text {ArcTan}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{2^{3/4}}+\frac {\text {ArcTan}\left (\frac {2^{3/4} x \sqrt [4]{x^2+x^6}}{\sqrt {2} x^2-\sqrt {x^2+x^6}}\right )}{2 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^6}}\right )}{2^{3/4}}+\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^2+x^6}}{2^{3/4}}}{x \sqrt [4]{x^2+x^6}}\right )}{2 \sqrt [4]{2}} \]

[Out]

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

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Rubi [C] Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
time = 0.29, antiderivative size = 121, normalized size of antiderivative = 0.67, number of steps used = 11, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2081, 6857, 283, 335, 371, 285, 477, 524} \begin {gather*} \frac {4 \sqrt [4]{x^6+x^2} F_1\left (-\frac {5}{8};1,-\frac {1}{4};\frac {3}{8};x^4,-x^4\right )}{5 \sqrt [4]{x^4+1} x^3}+\frac {8 \sqrt [4]{x^6+x^2} x \, _2F_1\left (\frac {3}{8},\frac {3}{4};\frac {11}{8};-x^4\right )}{15 \sqrt [4]{x^4+1}}+\frac {2}{5} \sqrt [4]{x^6+x^2} x-\frac {2 \sqrt [4]{x^6+x^2}}{5 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(x^2 + x^6)^(1/4))/(5*x^3) + (2*x*(x^2 + x^6)^(1/4))/5 + (4*(x^2 + x^6)^(1/4)*AppellF1[-5/8, 1, -1/4, 3/8,
x^4, -x^4])/(5*x^3*(1 + x^4)^(1/4)) + (8*x*(x^2 + x^6)^(1/4)*Hypergeometric2F1[3/8, 3/4, 11/8, -x^4])/(15*(1
+ x^4)^(1/4))

Rule 283

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 285

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + n
*p + 1))), x] + Dist[a*n*(p/(m + n*p + 1)), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

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 {\sqrt [4]{x^2+x^6} \left (1+x^8\right )}{x^4 \left (-1+x^4\right )} \, dx &=\frac {\sqrt [4]{x^2+x^6} \int \frac {\sqrt [4]{1+x^4} \left (1+x^8\right )}{x^{7/2} \left (-1+x^4\right )} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}\\ &=\frac {\sqrt [4]{x^2+x^6} \int \left (\frac {\sqrt [4]{1+x^4}}{x^{7/2}}+\sqrt {x} \sqrt [4]{1+x^4}+\frac {2 \sqrt [4]{1+x^4}}{x^{7/2} \left (-1+x^4\right )}\right ) \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}\\ &=\frac {\sqrt [4]{x^2+x^6} \int \frac {\sqrt [4]{1+x^4}}{x^{7/2}} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}+\frac {\sqrt [4]{x^2+x^6} \int \sqrt {x} \sqrt [4]{1+x^4} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}+\frac {\left (2 \sqrt [4]{x^2+x^6}\right ) \int \frac {\sqrt [4]{1+x^4}}{x^{7/2} \left (-1+x^4\right )} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}\\ &=-\frac {2 \sqrt [4]{x^2+x^6}}{5 x^3}+\frac {2}{5} x \sqrt [4]{x^2+x^6}+2 \frac {\left (2 \sqrt [4]{x^2+x^6}\right ) \int \frac {\sqrt {x}}{\left (1+x^4\right )^{3/4}} \, dx}{5 \sqrt {x} \sqrt [4]{1+x^4}}+\frac {\left (4 \sqrt [4]{x^2+x^6}\right ) \text {Subst}\left (\int \frac {\sqrt [4]{1+x^8}}{x^6 \left (-1+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1+x^4}}\\ &=-\frac {2 \sqrt [4]{x^2+x^6}}{5 x^3}+\frac {2}{5} x \sqrt [4]{x^2+x^6}+\frac {4 \sqrt [4]{x^2+x^6} F_1\left (-\frac {5}{8};1,-\frac {1}{4};\frac {3}{8};x^4,-x^4\right )}{5 x^3 \sqrt [4]{1+x^4}}+2 \frac {\left (4 \sqrt [4]{x^2+x^6}\right ) \text {Subst}\left (\int \frac {x^2}{\left (1+x^8\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{5 \sqrt {x} \sqrt [4]{1+x^4}}\\ &=-\frac {2 \sqrt [4]{x^2+x^6}}{5 x^3}+\frac {2}{5} x \sqrt [4]{x^2+x^6}+\frac {4 \sqrt [4]{x^2+x^6} F_1\left (-\frac {5}{8};1,-\frac {1}{4};\frac {3}{8};x^4,-x^4\right )}{5 x^3 \sqrt [4]{1+x^4}}+\frac {8 x \sqrt [4]{x^2+x^6} \, _2F_1\left (\frac {3}{8},\frac {3}{4};\frac {11}{8};-x^4\right )}{15 \sqrt [4]{1+x^4}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 228, normalized size = 1.27 \begin {gather*} \frac {\sqrt [4]{x^2+x^6} \left (8 \sqrt [4]{1+x^4}+8 x^4 \sqrt [4]{1+x^4}+10 \sqrt [4]{2} x^{5/2} \text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )+5\ 2^{3/4} x^{5/2} \text {ArcTan}\left (\frac {2^{3/4} \sqrt {x} \sqrt [4]{1+x^4}}{\sqrt {2} x-\sqrt {1+x^4}}\right )-10 \sqrt [4]{2} x^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^4}}\right )+5\ 2^{3/4} x^{5/2} \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 )}{20 x^3 \sqrt [4]{1+x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((x^2 + x^6)^(1/4)*(8*(1 + x^4)^(1/4) + 8*x^4*(1 + x^4)^(1/4) + 10*2^(1/4)*x^(5/2)*ArcTan[(2^(1/4)*Sqrt[x])/(1
+ x^4)^(1/4)] + 5*2^(3/4)*x^(5/2)*ArcTan[(2^(3/4)*Sqrt[x]*(1 + x^4)^(1/4))/(Sqrt[2]*x - Sqrt[1 + x^4])] - 10*
2^(1/4)*x^(5/2)*ArcTanh[(2^(1/4)*Sqrt[x])/(1 + x^4)^(1/4)] + 5*2^(3/4)*x^(5/2)*ArcTanh[(2*2^(1/4)*Sqrt[x]*(1 +
x^4)^(1/4))/(2*x + Sqrt[2]*Sqrt[1 + x^4])]))/(20*x^3*(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 = 54.55, size = 631, normalized size = 3.51

method	result	size
trager	\(\text {Expression too large to display}\)	\(631\)
risch	\(\text {Expression too large to display}\)	\(1594\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(x^4+1)*(x^6+x^2)^(1/4)/x^3-1/4*RootOf(_Z^2+RootOf(_Z^4+2)^2)*ln(-(-RootOf(_Z^4+2)^2*x^5+2*RootOf(_Z^2+Roo
tOf(_Z^4+2)^2)*RootOf(_Z^4+2)^2*(x^6+x^2)^(3/4)+2*RootOf(_Z^4+2)^2*x^3+4*RootOf(_Z^2+RootOf(_Z^4+2)^2)*(x^6+x^
2)^(1/4)*x^2-RootOf(_Z^4+2)^2*x-4*(x^6+x^2)^(1/2)*x)/x/(x^2+1)^2)+1/4*RootOf(_Z^4+2)*ln(-(RootOf(_Z^4+2)^3*x^5
-2*RootOf(_Z^4+2)^3*x^3-4*RootOf(_Z^4+2)^2*(x^6+x^2)^(1/4)*x^2+RootOf(_Z^4+2)^3*x-4*(x^6+x^2)^(1/2)*RootOf(_Z^
4+2)*x-4*(x^6+x^2)^(3/4))/x/(x^2+1)^2)+1/4*ln(-(RootOf(_Z^4+2)^2*(x^6+x^2)^(1/2)+2*(x^6+x^2)^(1/4)*RootOf(_Z^4
+2)*x+2*x^2)/(-1+x)/x/(1+x))*RootOf(_Z^4+2)^3+1/4*ln(-(RootOf(_Z^4+2)^2*(x^6+x^2)^(1/2)+2*(x^6+x^2)^(1/4)*Root
Of(_Z^4+2)*x+2*x^2)/(-1+x)/x/(1+x))*RootOf(_Z^4+2)^2*RootOf(_Z^2+RootOf(_Z^4+2)^2)-1/4*RootOf(_Z^4+2)^2*RootOf
(_Z^2+RootOf(_Z^4+2)^2)*ln(-(2*RootOf(_Z^2+RootOf(_Z^4+2)^2)*RootOf(_Z^4+2)^2*(x^6+x^2)^(1/2)*x+RootOf(_Z^2+Ro
otOf(_Z^4+2)^2)*x^5-2*RootOf(_Z^4+2)^3*(x^6+x^2)^(1/2)*x+RootOf(_Z^4+2)*x^5+4*RootOf(_Z^4+2)*(x^6+x^2)^(1/4)*R
ootOf(_Z^2+RootOf(_Z^4+2)^2)*x^2+2*RootOf(_Z^2+RootOf(_Z^4+2)^2)*x^3+2*RootOf(_Z^4+2)*x^3+4*(x^6+x^2)^(3/4)+Ro
otOf(_Z^2+RootOf(_Z^4+2)^2)*x+RootOf(_Z^4+2)*x)/(-1+x)^2/(1+x)^2/x)

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1099 vs. \(2 (142) = 284\).
time = 7.04, size = 1099, normalized size = 6.11 \begin {gather*} \frac {20 \cdot 8^{\frac {3}{4}} \sqrt {2} x^{3} \arctan \left (-\frac {8 \, x^{9} + 32 \, x^{7} + 48 \, x^{5} + 4 \cdot 8^{\frac {3}{4}} \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} {\left (x^{4} - 6 \, x^{2} + 1\right )} + 32 \, x^{3} + 16 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (3 \, x^{6} - 2 \, x^{4} + 3 \, x^{2}\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} + 32 \, \sqrt {2} \sqrt {x^{6} + x^{2}} {\left (x^{5} + 2 \, x^{3} + x\right )} - \sqrt {2} {\left (128 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} x^{2} + 8^{\frac {3}{4}} \sqrt {2} {\left (x^{9} - 16 \, x^{7} - 2 \, x^{5} - 16 \, x^{3} + x\right )} + 8 \cdot 8^{\frac {1}{4}} \sqrt {2} \sqrt {x^{6} + x^{2}} {\left (x^{5} - 6 \, x^{3} + x\right )} + 32 \, {\left (x^{6} + 2 \, x^{4} + x^{2}\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {8^{\frac {3}{4}} \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} + 8 \, \sqrt {x^{6} + x^{2}} x}{x^{5} + 2 \, x^{3} + x}} + 8 \, x}{8 \, {\left (x^{9} - 28 \, x^{7} + 6 \, x^{5} - 28 \, x^{3} + x\right )}}\right ) - 20 \cdot 8^{\frac {3}{4}} \sqrt {2} x^{3} \arctan \left (-\frac {8 \, x^{9} + 32 \, x^{7} + 48 \, x^{5} - 4 \cdot 8^{\frac {3}{4}} \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} {\left (x^{4} - 6 \, x^{2} + 1\right )} + 32 \, x^{3} - 16 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (3 \, x^{6} - 2 \, x^{4} + 3 \, x^{2}\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} + 32 \, \sqrt {2} \sqrt {x^{6} + x^{2}} {\left (x^{5} + 2 \, x^{3} + x\right )} - \sqrt {2} {\left (128 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} x^{2} - 8^{\frac {3}{4}} \sqrt {2} {\left (x^{9} - 16 \, x^{7} - 2 \, x^{5} - 16 \, x^{3} + x\right )} - 8 \cdot 8^{\frac {1}{4}} \sqrt {2} \sqrt {x^{6} + x^{2}} {\left (x^{5} - 6 \, x^{3} + x\right )} + 32 \, {\left (x^{6} + 2 \, x^{4} + x^{2}\right )} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}}\right )} \sqrt {-\frac {8^{\frac {3}{4}} \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} - \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} - 8 \, \sqrt {x^{6} + x^{2}} x}{x^{5} + 2 \, x^{3} + x}} + 8 \, x}{8 \, {\left (x^{9} - 28 \, x^{7} + 6 \, x^{5} - 28 \, x^{3} + x\right )}}\right ) + 5 \cdot 8^{\frac {3}{4}} \sqrt {2} x^{3} \log \left (\frac {8 \, {\left (8^{\frac {3}{4}} \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} + 8 \, \sqrt {x^{6} + x^{2}} x\right )}}{x^{5} + 2 \, x^{3} + x}\right ) - 5 \cdot 8^{\frac {3}{4}} \sqrt {2} x^{3} \log \left (-\frac {8 \, {\left (8^{\frac {3}{4}} \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} - \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} - 8 \, \sqrt {x^{6} + x^{2}} x\right )}}{x^{5} + 2 \, x^{3} + x}\right ) + 40 \cdot 8^{\frac {3}{4}} x^{3} \arctan \left (\frac {16 \cdot 8^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2^{\frac {3}{4}} {\left (8^{\frac {3}{4}} {\left (x^{5} + 2 \, x^{3} + x\right )} + 8 \cdot 8^{\frac {1}{4}} \sqrt {x^{6} + x^{2}} x\right )} + 4 \cdot 8^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{8 \, {\left (x^{5} - 2 \, x^{3} + x\right )}}\right ) - 10 \cdot 8^{\frac {3}{4}} x^{3} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 8^{\frac {3}{4}} \sqrt {x^{6} + x^{2}} x + 8^{\frac {1}{4}} {\left (x^{5} + 2 \, x^{3} + x\right )} + 4 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} - 2 \, x^{3} + x}\right ) + 10 \cdot 8^{\frac {3}{4}} x^{3} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} - 8^{\frac {3}{4}} \sqrt {x^{6} + x^{2}} x - 8^{\frac {1}{4}} {\left (x^{5} + 2 \, x^{3} + x\right )} + 4 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} - 2 \, x^{3} + x}\right ) + 128 \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}}{320 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/320*(20*8^(3/4)*sqrt(2)*x^3*arctan(-1/8*(8*x^9 + 32*x^7 + 48*x^5 + 4*8^(3/4)*sqrt(2)*(x^6 + x^2)^(3/4)*(x^4
- 6*x^2 + 1) + 32*x^3 + 16*8^(1/4)*sqrt(2)*(3*x^6 - 2*x^4 + 3*x^2)*(x^6 + x^2)^(1/4) + 32*sqrt(2)*sqrt(x^6 + x
^2)*(x^5 + 2*x^3 + x) - sqrt(2)*(128*sqrt(2)*(x^6 + x^2)^(3/4)*x^2 + 8^(3/4)*sqrt(2)*(x^9 - 16*x^7 - 2*x^5 - 1
6*x^3 + x) + 8*8^(1/4)*sqrt(2)*sqrt(x^6 + x^2)*(x^5 - 6*x^3 + x) + 32*(x^6 + 2*x^4 + x^2)*(x^6 + x^2)^(1/4))*s
qrt((8^(3/4)*sqrt(2)*(x^6 + x^2)^(1/4)*x^2 + 2*8^(1/4)*sqrt(2)*(x^6 + x^2)^(3/4) + sqrt(2)*(x^5 + 2*x^3 + x) +
8*sqrt(x^6 + x^2)*x)/(x^5 + 2*x^3 + x)) + 8*x)/(x^9 - 28*x^7 + 6*x^5 - 28*x^3 + x)) - 20*8^(3/4)*sqrt(2)*x^3*
arctan(-1/8*(8*x^9 + 32*x^7 + 48*x^5 - 4*8^(3/4)*sqrt(2)*(x^6 + x^2)^(3/4)*(x^4 - 6*x^2 + 1) + 32*x^3 - 16*8^(
1/4)*sqrt(2)*(3*x^6 - 2*x^4 + 3*x^2)*(x^6 + x^2)^(1/4) + 32*sqrt(2)*sqrt(x^6 + x^2)*(x^5 + 2*x^3 + x) - sqrt(2
)*(128*sqrt(2)*(x^6 + x^2)^(3/4)*x^2 - 8^(3/4)*sqrt(2)*(x^9 - 16*x^7 - 2*x^5 - 16*x^3 + x) - 8*8^(1/4)*sqrt(2)
*sqrt(x^6 + x^2)*(x^5 - 6*x^3 + x) + 32*(x^6 + 2*x^4 + x^2)*(x^6 + x^2)^(1/4))*sqrt(-(8^(3/4)*sqrt(2)*(x^6 + x
^2)^(1/4)*x^2 + 2*8^(1/4)*sqrt(2)*(x^6 + x^2)^(3/4) - sqrt(2)*(x^5 + 2*x^3 + x) - 8*sqrt(x^6 + x^2)*x)/(x^5 +
2*x^3 + x)) + 8*x)/(x^9 - 28*x^7 + 6*x^5 - 28*x^3 + x)) + 5*8^(3/4)*sqrt(2)*x^3*log(8*(8^(3/4)*sqrt(2)*(x^6 +
x^2)^(1/4)*x^2 + 2*8^(1/4)*sqrt(2)*(x^6 + x^2)^(3/4) + sqrt(2)*(x^5 + 2*x^3 + x) + 8*sqrt(x^6 + x^2)*x)/(x^5 +
2*x^3 + x)) - 5*8^(3/4)*sqrt(2)*x^3*log(-8*(8^(3/4)*sqrt(2)*(x^6 + x^2)^(1/4)*x^2 + 2*8^(1/4)*sqrt(2)*(x^6 +
x^2)^(3/4) - sqrt(2)*(x^5 + 2*x^3 + x) - 8*sqrt(x^6 + x^2)*x)/(x^5 + 2*x^3 + x)) + 40*8^(3/4)*x^3*arctan(1/8*(
16*8^(1/4)*(x^6 + x^2)^(1/4)*x^2 + 2^(3/4)*(8^(3/4)*(x^5 + 2*x^3 + x) + 8*8^(1/4)*sqrt(x^6 + x^2)*x) + 4*8^(3/
4)*(x^6 + x^2)^(3/4))/(x^5 - 2*x^3 + x)) - 10*8^(3/4)*x^3*log(-(4*sqrt(2)*(x^6 + x^2)^(1/4)*x^2 + 8^(3/4)*sqrt
(x^6 + x^2)*x + 8^(1/4)*(x^5 + 2*x^3 + x) + 4*(x^6 + x^2)^(3/4))/(x^5 - 2*x^3 + x)) + 10*8^(3/4)*x^3*log(-(4*s
qrt(2)*(x^6 + x^2)^(1/4)*x^2 - 8^(3/4)*sqrt(x^6 + x^2)*x - 8^(1/4)*(x^5 + 2*x^3 + x) + 4*(x^6 + x^2)^(3/4))/(x
^5 - 2*x^3 + x)) + 128*(x^6 + x^2)^(1/4)*(x^4 + 1))/x^3

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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^{8} + 1\right )}{x^{4} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate((x^8 + 1)*(x^6 + x^2)^(1/4)/((x^4 - 1)*x^4), 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^8+1\right )}{x^4\,\left (x^4-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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