\(\int \frac {\sqrt [4]{-1+x^4} (-1+x^8)}{x^6 (1+x^8)} \, dx\) [1037]

Optimal result

Integrand size = 25, antiderivative size = 78 \[ \int \frac {\sqrt [4]{-1+x^4} \left (-1+x^8\right )}{x^6 \left (1+x^8\right )} \, dx=\frac {\left (1-x^4\right ) \sqrt [4]{-1+x^4}}{5 x^5}+\frac {1}{4} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^4}\&\right ] \]

[Out]

Unintegrable

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.41 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.45, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6857, 270, 1543, 525, 524} \[ \int \frac {\sqrt [4]{-1+x^4} \left (-1+x^8\right )}{x^6 \left (1+x^8\right )} \, dx=\frac {\sqrt [4]{x^4-1} x^3 \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,-i x^4\right )}{3 \sqrt [4]{1-x^4}}+\frac {\sqrt [4]{x^4-1} x^3 \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,i x^4\right )}{3 \sqrt [4]{1-x^4}}-\frac {\left (x^4-1\right )^{5/4}}{5 x^5} \]

[In]

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

[Out]

-1/5*(-1 + x^4)^(5/4)/x^5 + (x^3*(-1 + x^4)^(1/4)*AppellF1[3/4, -1/4, 1, 7/4, x^4, (-I)*x^4])/(3*(1 - x^4)^(1/
4)) + (x^3*(-1 + x^4)^(1/4)*AppellF1[3/4, -1/4, 1, 7/4, x^4, I*x^4])/(3*(1 - x^4)^(1/4))

Rule 270

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

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 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x
] /; FreeQ[{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])

Rule 1543

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^(n2_.)), x_Symbol] :> Int[ExpandInte
grand[(d + e*x^n)^q, (f*x)^m/(a + c*x^(2*n)), x], x] /; FreeQ[{a, c, d, e, f, q, n}, x] && EqQ[n2, 2*n] && IGt
Q[n, 0] &&  !IntegerQ[q] && IntegerQ[m]

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*} \text {integral}& = \int \left (-\frac {\sqrt [4]{-1+x^4}}{x^6}+\frac {2 x^2 \sqrt [4]{-1+x^4}}{1+x^8}\right ) \, dx \\ & = 2 \int \frac {x^2 \sqrt [4]{-1+x^4}}{1+x^8} \, dx-\int \frac {\sqrt [4]{-1+x^4}}{x^6} \, dx \\ & = -\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}+2 \int \left (-\frac {i x^2 \sqrt [4]{-1+x^4}}{2 \left (-i+x^4\right )}+\frac {i x^2 \sqrt [4]{-1+x^4}}{2 \left (i+x^4\right )}\right ) \, dx \\ & = -\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}-i \int \frac {x^2 \sqrt [4]{-1+x^4}}{-i+x^4} \, dx+i \int \frac {x^2 \sqrt [4]{-1+x^4}}{i+x^4} \, dx \\ & = -\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}-\frac {\left (i \sqrt [4]{-1+x^4}\right ) \int \frac {x^2 \sqrt [4]{1-x^4}}{-i+x^4} \, dx}{\sqrt [4]{1-x^4}}+\frac {\left (i \sqrt [4]{-1+x^4}\right ) \int \frac {x^2 \sqrt [4]{1-x^4}}{i+x^4} \, dx}{\sqrt [4]{1-x^4}} \\ & = -\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}+\frac {x^3 \sqrt [4]{-1+x^4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,-i x^4\right )}{3 \sqrt [4]{1-x^4}}+\frac {x^3 \sqrt [4]{-1+x^4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,i x^4\right )}{3 \sqrt [4]{1-x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt [4]{-1+x^4} \left (-1+x^8\right )}{x^6 \left (1+x^8\right )} \, dx=\frac {-4 \left (-1+x^4\right )^{5/4}+5 x^5 \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^4}\&\right ]}{20 x^5} \]

[In]

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

[Out]

(-4*(-1 + x^4)^(5/4) + 5*x^5*RootSum[2 - 2*#1^4 + #1^8 & , (-(Log[x]*#1) + Log[(-1 + x^4)^(1/4) - x*#1]*#1)/(-
1 + #1^4) & ])/(20*x^5)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 1.

Time = 16.58 (sec) , antiderivative size = 3768, normalized size of antiderivative = 48.31

\[\text {output too large to display}\]

[In]

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

[Out]

-1/5*(x^8-2*x^4+1)/x^5/(x^4-1)^(3/4)+(1/16*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)*ln(-(81
92*x^12*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^2*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-4096*
(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^
4+1)^4-512)^3*x^9-16384*x^8*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z
^4+1)^4-512)^2-6*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^2*x^12+8192*(x^12-3*x^8+3*x^4-1)^
(1/4)*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*x^5+(x^
12-3*x^8+3*x^4-1)^(1/4)*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*x^9-262144*RootOf(838860
8*_Z^8-4096*_Z^4+1)^4*(x^12-3*x^8+3*x^4-1)^(1/2)*x^6+65536*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*RootOf(_Z^4+1048
576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)*(x^12-3*x^8+3*x^4-1)^(3/4)*x^3+8192*x^4*RootOf(8388608*_Z^8-4096*_
Z^4+1)^4*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^2+14*x^8*RootOf(_Z^4+1048576*RootOf(83886
08*_Z^8-4096*_Z^4+1)^4-512)^2-4096*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*RootOf(_Z^4+1
048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*x-2*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(_Z^4+1048576*RootOf(838
8608*_Z^8-4096*_Z^4+1)^4-512)^3*x^5+262144*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*(x^12-3*x^8+3*x^4-1)^(1/2)*x^2+1
28*(x^12-3*x^8+3*x^4-1)^(1/2)*x^6-10*x^4*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^2+(x^12-3
*x^8+3*x^4-1)^(1/4)*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*x-128*(x^12-3*x^8+3*x^4-1)^(
1/2)*x^2+2*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^2)/(4096*RootOf(8388608*_Z^8-4096*_Z^4+
1)^4*x^4-x^4-1)/(4096*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x-x+1)^2/(4096*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x-x
-1)^2/(x-1)^2/(1+x)^2)-8192*ln(-(4096*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x^12-8192*RootOf(8388608*_Z^8-4096*_Z
^4+1)^4*x^8+x^12-16*RootOf(8388608*_Z^8-4096*_Z^4+1)*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9+128*RootOf(8388608*_Z^8-40
96*_Z^4+1)^2*(x^12-3*x^8+3*x^4-1)^(1/2)*x^6-1024*RootOf(8388608*_Z^8-4096*_Z^4+1)^3*(x^12-3*x^8+3*x^4-1)^(3/4)
*x^3+4096*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x^4-3*x^8+32*RootOf(8388608*_Z^8-4096*_Z^4+1)*(x^12-3*x^8+3*x^4-1
)^(1/4)*x^5-128*RootOf(8388608*_Z^8-4096*_Z^4+1)^2*(x^12-3*x^8+3*x^4-1)^(1/2)*x^2+3*x^4-16*RootOf(8388608*_Z^8
-4096*_Z^4+1)*(x^12-3*x^8+3*x^4-1)^(1/4)*x-1)/(4096*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x^4-x^4+1)/(4096*RootOf
(8388608*_Z^8-4096*_Z^4+1)^4*x-x+1)^2/(4096*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x-x-1)^2/(x-1)^2/(1+x)^2)*RootO
f(8388608*_Z^8-4096*_Z^4+1)^5+2*ln(-(4096*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x^12-8192*RootOf(8388608*_Z^8-409
6*_Z^4+1)^4*x^8+x^12-16*RootOf(8388608*_Z^8-4096*_Z^4+1)*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9+128*RootOf(8388608*_Z^
8-4096*_Z^4+1)^2*(x^12-3*x^8+3*x^4-1)^(1/2)*x^6-1024*RootOf(8388608*_Z^8-4096*_Z^4+1)^3*(x^12-3*x^8+3*x^4-1)^(
3/4)*x^3+4096*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x^4-3*x^8+32*RootOf(8388608*_Z^8-4096*_Z^4+1)*(x^12-3*x^8+3*x
^4-1)^(1/4)*x^5-128*RootOf(8388608*_Z^8-4096*_Z^4+1)^2*(x^12-3*x^8+3*x^4-1)^(1/2)*x^2+3*x^4-16*RootOf(8388608*
_Z^8-4096*_Z^4+1)*(x^12-3*x^8+3*x^4-1)^(1/4)*x-1)/(4096*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x^4-x^4+1)/(4096*Ro
otOf(8388608*_Z^8-4096*_Z^4+1)^4*x-x+1)^2/(4096*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x-x-1)^2/(x-1)^2/(1+x)^2)*R
ootOf(8388608*_Z^8-4096*_Z^4+1)-2*RootOf(8388608*_Z^8-4096*_Z^4+1)*ln((8192*x^12*RootOf(8388608*_Z^8-4096*_Z^4
+1)^5+131072*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(8388608*_Z^8-4096*_Z^4+1)^6*x^9-16384*RootOf(8388608*_Z^8-4096*
_Z^4+1)^5*x^8+2*x^12*RootOf(8388608*_Z^8-4096*_Z^4+1)-262144*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(8388608*_Z^8-40
96*_Z^4+1)^6*x^5-32*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(8388608*_Z^8-4096*_Z^4+1)^2*x^9-256*(x^12-3*x^8+3*x^4-1)
^(1/2)*RootOf(8388608*_Z^8-4096*_Z^4+1)^3*x^6-2048*(x^12-3*x^8+3*x^4-1)^(3/4)*RootOf(8388608*_Z^8-4096*_Z^4+1)
^4*x^3+8192*RootOf(8388608*_Z^8-4096*_Z^4+1)^5*x^4-6*RootOf(8388608*_Z^8-4096*_Z^4+1)*x^8+131072*(x^12-3*x^8+3
*x^4-1)^(1/4)*RootOf(8388608*_Z^8-4096*_Z^4+1)^6*x+64*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(8388608*_Z^8-4096*_Z^4
+1)^2*x^5+256*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(8388608*_Z^8-4096*_Z^4+1)^3*x^2+(x^12-3*x^8+3*x^4-1)^(3/4)*x^3
+6*RootOf(8388608*_Z^8-4096*_Z^4+1)*x^4-32*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(8388608*_Z^8-4096*_Z^4+1)^2*x-2*R
ootOf(8388608*_Z^8-4096*_Z^4+1))/(4096*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x^4-x^4+1)/(4096*RootOf(8388608*_Z^8
-4096*_Z^4+1)^4*x-x+1)^2/(4096*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x-x-1)^2/(x-1)^2/(1+x)^2)-256*ln((8192*x^12*
RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^2*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-16384*x^8*Roo
tOf(8388608*_Z^8-4096*_Z^4+1)^4*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^2-6*RootOf(_Z^4+10
48576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^2*x^12-(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(_Z^4+1048576*RootOf(838
8608*_Z^8-4096*_Z^4+1)^4-512)^3*x^9+262144*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*(x^12-3*x^8+3*x^4-1)^(1/2)*x^6+6
5536*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)*(x^12-3*x^
8+3*x^4-1)^(3/4)*x^3+8192*x^4*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*
_Z^4+1)^4-512)^2+14*x^8*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^2+2*(x^12-3*x^8+3*x^4-1)^(
1/4)*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*x^5-262144*RootOf(8388608*_Z^8-4096*_Z^4+1)
^4*(x^12-3*x^8+3*x^4-1)^(1/2)*x^2-128*(x^12-3*x^8+3*x^4-1)^(1/2)*x^6-32*(x^12-3*x^8+3*x^4-1)^(3/4)*RootOf(_Z^4
+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)*x^3-10*x^4*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+
1)^4-512)^2-(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*x+128*(x^
12-3*x^8+3*x^4-1)^(1/2)*x^2+2*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^2)/(4096*RootOf(8388
608*_Z^8-4096*_Z^4+1)^4*x^4-x^4-1)/(4096*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x-x+1)^2/(4096*RootOf(8388608*_Z^8
-4096*_Z^4+1)^4*x-x-1)^2/(x-1)^2/(1+x)^2)*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*RootOf(_Z^4+1048576*RootOf(838860
8*_Z^8-4096*_Z^4+1)^4-512)+1/16*ln((8192*x^12*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^2*Ro
otOf(8388608*_Z^8-4096*_Z^4+1)^4-16384*x^8*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*RootOf(_Z^4+1048576*RootOf(83886
08*_Z^8-4096*_Z^4+1)^4-512)^2-6*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^2*x^12-(x^12-3*x^8
+3*x^4-1)^(1/4)*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*x^9+262144*RootOf(8388608*_Z^8-4
096*_Z^4+1)^4*(x^12-3*x^8+3*x^4-1)^(1/2)*x^6+65536*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*RootOf(_Z^4+1048576*Root
Of(8388608*_Z^8-4096*_Z^4+1)^4-512)*(x^12-3*x^8+3*x^4-1)^(3/4)*x^3+8192*x^4*RootOf(8388608*_Z^8-4096*_Z^4+1)^4
*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^2+14*x^8*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-
4096*_Z^4+1)^4-512)^2+2*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)
^3*x^5-262144*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*(x^12-3*x^8+3*x^4-1)^(1/2)*x^2-128*(x^12-3*x^8+3*x^4-1)^(1/2)
*x^6-32*(x^12-3*x^8+3*x^4-1)^(3/4)*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)*x^3-10*x^4*Root
Of(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^2-(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(_Z^4+1048576*RootO
f(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*x+128*(x^12-3*x^8+3*x^4-1)^(1/2)*x^2+2*RootOf(_Z^4+1048576*RootOf(8388608
*_Z^8-4096*_Z^4+1)^4-512)^2)/(4096*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x^4-x^4-1)/(4096*RootOf(8388608*_Z^8-409
6*_Z^4+1)^4*x-x+1)^2/(4096*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x-x-1)^2/(x-1)^2/(1+x)^2)*RootOf(_Z^4+1048576*Ro
otOf(8388608*_Z^8-4096*_Z^4+1)^4-512))/(x^4-1)^(3/4)*((x^4-1)^3)^(1/4)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 3.71 (sec) , antiderivative size = 863, normalized size of antiderivative = 11.06 \[ \int \frac {\sqrt [4]{-1+x^4} \left (-1+x^8\right )}{x^6 \left (1+x^8\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/40*(5*x^5*sqrt(-sqrt(I + 1))*log(-(4*sqrt(I + 1)*((3*I + 4)*x^7 + (4*I - 3)*x^3)*(x^4 - 1)^(1/4) + 4*((3*I
+ 4)*x^5 + (4*I - 3)*x)*(x^4 - 1)^(3/4) - (sqrt(I + 1)*(-(15*I - 5)*x^8 + (12*I + 16)*x^4 + I - 7) - 4*((4*I -
3)*x^6 - (3*I + 4)*x^2)*sqrt(x^4 - 1))*sqrt(-sqrt(I + 1)))/(x^8 + 1)) - 5*x^5*sqrt(-sqrt(I + 1))*log(-(4*sqrt
(I + 1)*((3*I + 4)*x^7 + (4*I - 3)*x^3)*(x^4 - 1)^(1/4) + 4*((3*I + 4)*x^5 + (4*I - 3)*x)*(x^4 - 1)^(3/4) - (s
qrt(I + 1)*((15*I - 5)*x^8 - (12*I + 16)*x^4 - I + 7) - 4*(-(4*I - 3)*x^6 + (3*I + 4)*x^2)*sqrt(x^4 - 1))*sqrt
(-sqrt(I + 1)))/(x^8 + 1)) + 5*x^5*sqrt(-sqrt(-I + 1))*log(-(4*sqrt(-I + 1)*(-(3*I - 4)*x^7 - (4*I + 3)*x^3)*(
x^4 - 1)^(1/4) + 4*(-(3*I - 4)*x^5 - (4*I + 3)*x)*(x^4 - 1)^(3/4) - (sqrt(-I + 1)*((15*I + 5)*x^8 - (12*I - 16
)*x^4 - I - 7) - 4*(-(4*I + 3)*x^6 + (3*I - 4)*x^2)*sqrt(x^4 - 1))*sqrt(-sqrt(-I + 1)))/(x^8 + 1)) - 5*x^5*sqr
t(-sqrt(-I + 1))*log(-(4*sqrt(-I + 1)*(-(3*I - 4)*x^7 - (4*I + 3)*x^3)*(x^4 - 1)^(1/4) + 4*(-(3*I - 4)*x^5 - (
4*I + 3)*x)*(x^4 - 1)^(3/4) - (sqrt(-I + 1)*(-(15*I + 5)*x^8 + (12*I - 16)*x^4 + I + 7) - 4*((4*I + 3)*x^6 - (
3*I - 4)*x^2)*sqrt(x^4 - 1))*sqrt(-sqrt(-I + 1)))/(x^8 + 1)) - 5*(-I + 1)^(1/4)*x^5*log(-(4*sqrt(-I + 1)*((3*I
- 4)*x^7 + (4*I + 3)*x^3)*(x^4 - 1)^(1/4) + 4*(-(3*I - 4)*x^5 - (4*I + 3)*x)*(x^4 - 1)^(3/4) - (-I + 1)^(1/4)
*(sqrt(-I + 1)*((15*I + 5)*x^8 - (12*I - 16)*x^4 - I - 7) - 4*((4*I + 3)*x^6 - (3*I - 4)*x^2)*sqrt(x^4 - 1)))/
(x^8 + 1)) + 5*(-I + 1)^(1/4)*x^5*log(-(4*sqrt(-I + 1)*((3*I - 4)*x^7 + (4*I + 3)*x^3)*(x^4 - 1)^(1/4) + 4*(-(
3*I - 4)*x^5 - (4*I + 3)*x)*(x^4 - 1)^(3/4) - (-I + 1)^(1/4)*(sqrt(-I + 1)*(-(15*I + 5)*x^8 + (12*I - 16)*x^4
+ I + 7) - 4*(-(4*I + 3)*x^6 + (3*I - 4)*x^2)*sqrt(x^4 - 1)))/(x^8 + 1)) - 5*(I + 1)^(1/4)*x^5*log(-(4*sqrt(I
+ 1)*(-(3*I + 4)*x^7 - (4*I - 3)*x^3)*(x^4 - 1)^(1/4) + 4*((3*I + 4)*x^5 + (4*I - 3)*x)*(x^4 - 1)^(3/4) - (I +
1)^(1/4)*(sqrt(I + 1)*(-(15*I - 5)*x^8 + (12*I + 16)*x^4 + I - 7) - 4*(-(4*I - 3)*x^6 + (3*I + 4)*x^2)*sqrt(x
^4 - 1)))/(x^8 + 1)) + 5*(I + 1)^(1/4)*x^5*log(-(4*sqrt(I + 1)*(-(3*I + 4)*x^7 - (4*I - 3)*x^3)*(x^4 - 1)^(1/4
) + 4*((3*I + 4)*x^5 + (4*I - 3)*x)*(x^4 - 1)^(3/4) - (I + 1)^(1/4)*(sqrt(I + 1)*((15*I - 5)*x^8 - (12*I + 16)
*x^4 - I + 7) - 4*((4*I - 3)*x^6 - (3*I + 4)*x^2)*sqrt(x^4 - 1)))/(x^8 + 1)) + 8*(x^4 - 1)^(5/4))/x^5

Sympy [N/A]

Not integrable

Time = 42.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt [4]{-1+x^4} \left (-1+x^8\right )}{x^6 \left (1+x^8\right )} \, dx=\int \frac {\sqrt [4]{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}{x^{6} \left (x^{8} + 1\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.32 \[ \int \frac {\sqrt [4]{-1+x^4} \left (-1+x^8\right )}{x^6 \left (1+x^8\right )} \, dx=\int { \frac {{\left (x^{8} - 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{{\left (x^{8} + 1\right )} x^{6}} \,d x } \]

[In]

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

[Out]

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

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.31 (sec) , antiderivative size = 290, normalized size of antiderivative = 3.72 \[ \int \frac {\sqrt [4]{-1+x^4} \left (-1+x^8\right )}{x^6 \left (1+x^8\right )} \, dx=\frac {1}{144115188075855872} i \, \left (8 i + 8\right )^{\frac {63}{4}} \log \left (\left (-281474976710656 i + 281474976710656\right )^{\frac {1}{4}} - \frac {4096 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{144115188075855872} i \, \left (8 i + 8\right )^{\frac {63}{4}} \log \left (-\left (-281474976710656 i + 281474976710656\right )^{\frac {1}{4}} - \frac {4096 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{536870912} \, \left (8 i + 8\right )^{\frac {31}{4}} \log \left (i \, \left (-16777216 i + 16777216\right )^{\frac {1}{4}} - \frac {64 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{536870912} \, \left (8 i + 8\right )^{\frac {31}{4}} \log \left (-i \, \left (-16777216 i + 16777216\right )^{\frac {1}{4}} - \frac {64 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}} {\left (\frac {1}{x^{4}} - 1\right )}}{5 \, x} - \frac {i \, \left (8 i + 8\right )^{\frac {15}{4}} \log \left (\left (16777216 i + 16777216\right )^{\frac {1}{4}} - \frac {64 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right )}{256 \, {\left (\sqrt {\sqrt {2} + 2} + i \, \sqrt {-\sqrt {2} + 2}\right )}^{7}} + \frac {\left (8 i + 8\right )^{\frac {15}{4}} \log \left (i \, \left (16777216 i + 16777216\right )^{\frac {1}{4}} - \frac {64 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right )}{256 \, {\left (\sqrt {\sqrt {2} + 2} + i \, \sqrt {-\sqrt {2} + 2}\right )}^{7}} - \frac {\left (8 i + 8\right )^{\frac {15}{4}} \log \left (-i \, \left (16777216 i + 16777216\right )^{\frac {1}{4}} - \frac {64 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right )}{256 \, {\left (\sqrt {\sqrt {2} + 2} + i \, \sqrt {-\sqrt {2} + 2}\right )}^{7}} + \frac {i \, \left (8 i + 8\right )^{\frac {15}{4}} \log \left (-\left (16777216 i + 16777216\right )^{\frac {1}{4}} - \frac {64 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right )}{256 \, {\left (\sqrt {\sqrt {2} + 2} + i \, \sqrt {-\sqrt {2} + 2}\right )}^{7}} \]

[In]

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

[Out]

1/144115188075855872*I*(8*I + 8)^(63/4)*log((-281474976710656*I + 281474976710656)^(1/4) - 4096*(x^4 - 1)^(1/4
)/x) - 1/144115188075855872*I*(8*I + 8)^(63/4)*log(-(-281474976710656*I + 281474976710656)^(1/4) - 4096*(x^4 -
1)^(1/4)/x) - 1/536870912*(8*I + 8)^(31/4)*log(I*(-16777216*I + 16777216)^(1/4) - 64*(x^4 - 1)^(1/4)/x) + 1/5
36870912*(8*I + 8)^(31/4)*log(-I*(-16777216*I + 16777216)^(1/4) - 64*(x^4 - 1)^(1/4)/x) + 1/5*(x^4 - 1)^(1/4)*
(1/x^4 - 1)/x - 1/256*I*(8*I + 8)^(15/4)*log((16777216*I + 16777216)^(1/4) - 64*(x^4 - 1)^(1/4)/x)/(sqrt(sqrt(
2) + 2) + I*sqrt(-sqrt(2) + 2))^7 + 1/256*(8*I + 8)^(15/4)*log(I*(16777216*I + 16777216)^(1/4) - 64*(x^4 - 1)^
(1/4)/x)/(sqrt(sqrt(2) + 2) + I*sqrt(-sqrt(2) + 2))^7 - 1/256*(8*I + 8)^(15/4)*log(-I*(16777216*I + 16777216)^
(1/4) - 64*(x^4 - 1)^(1/4)/x)/(sqrt(sqrt(2) + 2) + I*sqrt(-sqrt(2) + 2))^7 + 1/256*I*(8*I + 8)^(15/4)*log(-(16
777216*I + 16777216)^(1/4) - 64*(x^4 - 1)^(1/4)/x)/(sqrt(sqrt(2) + 2) + I*sqrt(-sqrt(2) + 2))^7

Mupad [N/A]

Not integrable

Time = 6.17 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.32 \[ \int \frac {\sqrt [4]{-1+x^4} \left (-1+x^8\right )}{x^6 \left (1+x^8\right )} \, dx=\int \frac {{\left (x^4-1\right )}^{1/4}\,\left (x^8-1\right )}{x^6\,\left (x^8+1\right )} \,d x \]

[In]

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

[Out]

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