3.11.37 \(\int \frac {\sqrt [4]{-1+x^4} (-1+x^8)}{x^6 (1+x^8)} \, dx\)
Optimal. Leaf size=78 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4+2\& ,\frac {\text {$\#$1} \log \left (\sqrt [4]{x^4-1}-\text {$\#$1} x\right )-\text {$\#$1} \log (x)}{\text {$\#$1}^4-1}\& \right ]+\frac {\sqrt [4]{x^4-1} \left (1-x^4\right )}{5 x^5} \]
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Rubi [C] time = 0.44, antiderivative size = 113, normalized size of antiderivative = 1.45,
number of steps used = 9, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used =
{6725, 264, 1529, 511, 510} \begin {gather*} \frac {\sqrt [4]{x^4-1} x^3 F_1\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 F_1\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} \end {gather*}
Warning: Unable to verify antiderivative.
[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 264
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 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 511
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[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 1529
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 6725
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]{-1+x^4} \left (-1+x^8\right )}{x^6 \left (1+x^8\right )} \, dx &=\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} F_1\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} F_1\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*}
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Mathematica [F] time = 0.13, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{-1+x^4} \left (-1+x^8\right )}{x^6 \left (1+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[((-1 + x^4)^(1/4)*(-1 + x^8))/(x^6*(1 + x^8)),x]
[Out]
Integrate[((-1 + x^4)^(1/4)*(-1 + x^8))/(x^6*(1 + x^8)), x]
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IntegrateAlgebraic [A] time = 0.24, size = 78, normalized size = 1.00 \begin {gather*} \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 ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[((-1 + x^4)^(1/4)*(-1 + x^8))/(x^6*(1 + x^8)),x]
[Out]
((1 - x^4)*(-1 + x^4)^(1/4))/(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) & ]/4
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^4-1)^(1/4)*(x^8-1)/x^6/(x^8+1),x, algorithm="fricas")
[Out]
Timed out
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giac [B] time = 0.57, size = 290, normalized size = 3.72 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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/
536870912*(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(-(1
6777216*I + 16777216)^(1/4) - 64*(x^4 - 1)^(1/4)/x)/(sqrt(sqrt(2) + 2) + I*sqrt(-sqrt(2) + 2))^7
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maple [B] time = 17.09, size = 3771, normalized size = 48.35 \[\text {output too large to display}\]
Verification of antiderivative is not currently implemented for this CAS.
[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)+(-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*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*RootOf(_Z^4+1048576*R
ootOf(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-
(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*(x^12-3*x^
8+3*x^4-1)^(1/2)*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x^6+65536*(x^12-3*x^8+3*x^4-1)^(3/4)*RootOf(8388608*_Z^8-4
096*_Z^4+1)^4*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)*x^3+8192*x^4*RootOf(8388608*_Z^8-409
6*_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(83
88608*_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*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*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+10
48576*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*Roo
tOf(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)/(-1+x)^2/(4096*Roo
tOf(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/(1+x)^2)*RootOf(838
8608*_Z^8-4096*_Z^4+1)^4*RootOf(_Z^4+1048576*RootOf(8388608*_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*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-16384*x^8*RootOf(83
88608*_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-(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*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x^6+65536*(
x^12-3*x^8+3*x^4-1)^(3/4)*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4
+1)^4-512)*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)*R
ootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*x^5-262144*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(8388
608*_Z^8-4096*_Z^4+1)^4*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+10485
76*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)^4-5
12)^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)/(-1+x)^2/(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/(1+x)^2)*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)-8192*ln((-409
6*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^8-4096*_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)/(-1+x)^2/(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/(1+x)^2)*RootOf(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^8-4096*_Z^4+1)^4*x^8-x^12+16*RootOf(8388
608*_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^8-4
096*_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)/(-1+x)^2/(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/(1+x)^2)*RootOf(8388608*_Z^8-4096*_Z^4+1)-1/
16*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)*ln((-8192*x^12*RootOf(_Z^4+1048576*RootOf(83886
08*_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(_Z^4+1
048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x^9+16384*x^8*RootOf(83886
08*_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*Roo
tOf(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(_Z^4+1048576*RootOf(8388608
*_Z^8-4096*_Z^4+1)^4-512)^3*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x^5+(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(_Z^4+1048
576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*x^9+262144*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(8388608*_Z^8-4096*_
Z^4+1)^4*x^6+65536*(x^12-3*x^8+3*x^4-1)^(3/4)*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*RootOf(_Z^4+1048576*RootOf(83
88608*_Z^8-4096*_Z^4+1)^4-512)*x^3-8192*x^4*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*RootOf(_Z^4+1048576*RootOf(8388
608*_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)^2-4096*(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*RootOf(8388608*_Z^8-4096*_
Z^4+1)^4*x-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^5-2621
44*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x^2-128*(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(83
88608*_Z^8-4096*_Z^4+1)^4-512)^2)/(4096*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x^4-x^4-1)/(-1+x)^2/(4096*RootOf(83
88608*_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/(1+x)^2)-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^8-4096*_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)*RootO
f(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*RootOf(8388608*_Z^8-4096*_Z^4+1))/(4096*RootOf(
8388608*_Z^8-4096*_Z^4+1)^4*x^4-x^4+1)/(-1+x)^2/(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/(1+x)^2))/(x^4-1)^(3/4)*((x^4-1)^3)^(1/4)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{8} - 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{{\left (x^{8} + 1\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^4-1\right )}^{1/4}\,\left (x^8-1\right )}{x^6\,\left (x^8+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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