3.11.38 \(\int \frac {-1+x^8}{\sqrt [4]{1+x^4} (1+x^8)} \, dx\)
Optimal. Leaf size=78 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4+2\& ,\frac {\log \left (\sqrt [4]{x^4+1}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]+\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right ) \]
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Rubi [C] time = 0.29, antiderivative size = 153, normalized size of antiderivative = 1.96,
number of steps used = 21, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used
= {1586, 6725, 408, 240, 212, 206, 203, 377} \begin {gather*} \frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{1-i} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{1-i}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{1+i} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{1+i}}+\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{1-i} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{1-i}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{1+i} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{1+i}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(-1 + x^8)/((1 + x^4)^(1/4)*(1 + x^8)),x]
[Out]
ArcTan[x/(1 + x^4)^(1/4)]/2 - ArcTan[((1 - I)^(1/4)*x)/(1 + x^4)^(1/4)]/(2*(1 - I)^(1/4)) - ArcTan[((1 + I)^(1
/4)*x)/(1 + x^4)^(1/4)]/(2*(1 + I)^(1/4)) + ArcTanh[x/(1 + x^4)^(1/4)]/2 - ArcTanh[((1 - I)^(1/4)*x)/(1 + x^4)
^(1/4)]/(2*(1 - I)^(1/4)) - ArcTanh[((1 + I)^(1/4)*x)/(1 + x^4)^(1/4)]/(2*(1 + I)^(1/4))
Rule 203
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 212
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]
Rule 240
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x
, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p
+ 1/n]
Rule 377
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
Rule 408
Int[((a_) + (b_.)*(x_)^4)^(p_)/((c_) + (d_.)*(x_)^4), x_Symbol] :> Dist[b/d, Int[(a + b*x^4)^(p - 1), x], x] -
Dist[(b*c - a*d)/d, Int[(a + b*x^4)^(p - 1)/(c + d*x^4), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0
] && (EqQ[p, 3/4] || EqQ[p, 5/4])
Rule 1586
Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
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 {-1+x^8}{\sqrt [4]{1+x^4} \left (1+x^8\right )} \, dx &=\int \frac {\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}}{1+x^8} \, dx\\ &=\int \left (-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+x^4\right )^{3/4}}{i-x^4}+\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (1+x^4\right )^{3/4}}{i+x^4}\right ) \, dx\\ &=\left (-\frac {1}{2}-\frac {i}{2}\right ) \int \frac {\left (1+x^4\right )^{3/4}}{i-x^4} \, dx+\left (\frac {1}{2}-\frac {i}{2}\right ) \int \frac {\left (1+x^4\right )^{3/4}}{i+x^4} \, dx\\ &=-\left (i \int \frac {1}{\left (i-x^4\right ) \sqrt [4]{1+x^4}} \, dx\right )-i \int \frac {1}{\left (i+x^4\right ) \sqrt [4]{1+x^4}} \, dx+\left (\frac {1}{2}-\frac {i}{2}\right ) \int \frac {1}{\sqrt [4]{1+x^4}} \, dx+\left (\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{\sqrt [4]{1+x^4}} \, dx\\ &=-\left (i \operatorname {Subst}\left (\int \frac {1}{i-(1+i) x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\right )-i \operatorname {Subst}\left (\int \frac {1}{i+(1-i) x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\left (\frac {1}{2}-\frac {i}{2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\left (\frac {1}{4}-\frac {i}{4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\left (\frac {1}{4}-\frac {i}{4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\left (\frac {1}{4}+\frac {i}{4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\left (\frac {1}{4}+\frac {i}{4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {1-i} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {1-i} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {1+i} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {1+i} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{1-i} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{1-i}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{1+i} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{1+i}}+\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{1-i} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{1-i}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{1+i} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{1+i}}\\ \end {align*}
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Mathematica [F] time = 0.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+x^8}{\sqrt [4]{1+x^4} \left (1+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[(-1 + x^8)/((1 + x^4)^(1/4)*(1 + x^8)),x]
[Out]
Integrate[(-1 + x^8)/((1 + x^4)^(1/4)*(1 + x^8)), x]
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IntegrateAlgebraic [A] time = 0.24, size = 78, normalized size = 1.00 \begin {gather*} \frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{4} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{1+x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(-1 + x^8)/((1 + x^4)^(1/4)*(1 + x^8)),x]
[Out]
ArcTan[x/(1 + x^4)^(1/4)]/2 + ArcTanh[x/(1 + x^4)^(1/4)]/2 + RootSum[2 - 2*#1^4 + #1^8 & , (-Log[x] + Log[(1 +
x^4)^(1/4) - x*#1])/#1 & ]/4
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fricas [B] time = 0.79, size = 2014, normalized size = 25.82
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^8-1)/(x^4+1)^(1/4)/(x^8+1),x, algorithm="fricas")
[Out]
1/16*2^(7/8)*sqrt(2*sqrt(2)*(3*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 4) + 8*sqrt(2) + 16)*sqrt(-2*sqrt(2) + 4)*arctan
(1/8*(2^(1/8)*sqrt(1/2)*sqrt(2*sqrt(2)*(3*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 4) + 8*sqrt(2) + 16)*(2^(3/4)*(sqrt(2
)*x + 2*x) - 2*2^(1/4)*(sqrt(2)*x + x)*sqrt(-2*sqrt(2) + 4))*sqrt((8*2^(1/4)*x^2 + 2^(3/8)*(2^(1/4)*(x^4 + 1)^
(1/4)*(sqrt(2)*x + 2*x)*sqrt(-2*sqrt(2) + 4) - 4*2^(1/4)*(x^4 + 1)^(1/4)*x)*sqrt(2*sqrt(2)*(3*sqrt(2) + 4)*sqr
t(-2*sqrt(2) + 4) + 8*sqrt(2) + 16) + 8*sqrt(x^4 + 1))/x^2) - 4*sqrt(2)*(sqrt(2)*x + 2*x)*sqrt(-2*sqrt(2) + 4)
- 2*2^(1/8)*(2^(3/4)*(x^4 + 1)^(1/4)*(sqrt(2) + 2) - 2*2^(1/4)*(x^4 + 1)^(1/4)*(sqrt(2) + 1)*sqrt(-2*sqrt(2)
+ 4))*sqrt(2*sqrt(2)*(3*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 4) + 8*sqrt(2) + 16) + 8*sqrt(2)*x + 8*x)/x) + 1/16*2^(
7/8)*sqrt(2*sqrt(2)*(3*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 4) + 8*sqrt(2) + 16)*sqrt(-2*sqrt(2) + 4)*arctan(1/8*(2^
(1/8)*sqrt(1/2)*sqrt(2*sqrt(2)*(3*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 4) + 8*sqrt(2) + 16)*(2^(3/4)*(sqrt(2)*x + 2*
x) - 2*2^(1/4)*(sqrt(2)*x + x)*sqrt(-2*sqrt(2) + 4))*sqrt((8*2^(1/4)*x^2 - 2^(3/8)*(2^(1/4)*(x^4 + 1)^(1/4)*(s
qrt(2)*x + 2*x)*sqrt(-2*sqrt(2) + 4) - 4*2^(1/4)*(x^4 + 1)^(1/4)*x)*sqrt(2*sqrt(2)*(3*sqrt(2) + 4)*sqrt(-2*sqr
t(2) + 4) + 8*sqrt(2) + 16) + 8*sqrt(x^4 + 1))/x^2) + 4*sqrt(2)*(sqrt(2)*x + 2*x)*sqrt(-2*sqrt(2) + 4) - 2*2^(
1/8)*(2^(3/4)*(x^4 + 1)^(1/4)*(sqrt(2) + 2) - 2*2^(1/4)*(x^4 + 1)^(1/4)*(sqrt(2) + 1)*sqrt(-2*sqrt(2) + 4))*sq
rt(2*sqrt(2)*(3*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 4) + 8*sqrt(2) + 16) - 8*sqrt(2)*x - 8*x)/x) - 1/16*2^(7/8)*sqr
t(-2*sqrt(2)*(3*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 4) + 8*sqrt(2) + 16)*sqrt(-2*sqrt(2) + 4)*arctan(1/8*(2^(1/8)*s
qrt(1/2)*sqrt(-2*sqrt(2)*(3*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 4) + 8*sqrt(2) + 16)*(2^(3/4)*(sqrt(2)*x + 2*x) + 2
*2^(1/4)*(sqrt(2)*x + x)*sqrt(-2*sqrt(2) + 4))*sqrt((8*2^(1/4)*x^2 + 2^(3/8)*(2^(1/4)*(x^4 + 1)^(1/4)*(sqrt(2)
*x + 2*x)*sqrt(-2*sqrt(2) + 4) + 4*2^(1/4)*(x^4 + 1)^(1/4)*x)*sqrt(-2*sqrt(2)*(3*sqrt(2) + 4)*sqrt(-2*sqrt(2)
+ 4) + 8*sqrt(2) + 16) + 8*sqrt(x^4 + 1))/x^2) - 4*sqrt(2)*(sqrt(2)*x + 2*x)*sqrt(-2*sqrt(2) + 4) - 2*2^(1/8)*
(2^(3/4)*(x^4 + 1)^(1/4)*(sqrt(2) + 2) + 2*2^(1/4)*(x^4 + 1)^(1/4)*(sqrt(2) + 1)*sqrt(-2*sqrt(2) + 4))*sqrt(-2
*sqrt(2)*(3*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 4) + 8*sqrt(2) + 16) - 8*sqrt(2)*x - 8*x)/x) - 1/16*2^(7/8)*sqrt(-2
*sqrt(2)*(3*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 4) + 8*sqrt(2) + 16)*sqrt(-2*sqrt(2) + 4)*arctan(1/8*(2^(1/8)*sqrt(
1/2)*sqrt(-2*sqrt(2)*(3*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 4) + 8*sqrt(2) + 16)*(2^(3/4)*(sqrt(2)*x + 2*x) + 2*2^(
1/4)*(sqrt(2)*x + x)*sqrt(-2*sqrt(2) + 4))*sqrt((8*2^(1/4)*x^2 - 2^(3/8)*(2^(1/4)*(x^4 + 1)^(1/4)*(sqrt(2)*x +
2*x)*sqrt(-2*sqrt(2) + 4) + 4*2^(1/4)*(x^4 + 1)^(1/4)*x)*sqrt(-2*sqrt(2)*(3*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 4)
+ 8*sqrt(2) + 16) + 8*sqrt(x^4 + 1))/x^2) + 4*sqrt(2)*(sqrt(2)*x + 2*x)*sqrt(-2*sqrt(2) + 4) - 2*2^(1/8)*(2^(
3/4)*(x^4 + 1)^(1/4)*(sqrt(2) + 2) + 2*2^(1/4)*(x^4 + 1)^(1/4)*(sqrt(2) + 1)*sqrt(-2*sqrt(2) + 4))*sqrt(-2*sqr
t(2)*(3*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 4) + 8*sqrt(2) + 16) + 8*sqrt(2)*x + 8*x)/x) - 1/64*2^(3/8)*sqrt(2*sqrt
(2)*(3*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 4) + 8*sqrt(2) + 16)*(sqrt(2)*(sqrt(2) + 1)*sqrt(-2*sqrt(2) + 4) - 4)*lo
g(1/2*(8*2^(1/4)*x^2 + 2^(3/8)*(2^(1/4)*(x^4 + 1)^(1/4)*(sqrt(2)*x + 2*x)*sqrt(-2*sqrt(2) + 4) - 4*2^(1/4)*(x^
4 + 1)^(1/4)*x)*sqrt(2*sqrt(2)*(3*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 4) + 8*sqrt(2) + 16) + 8*sqrt(x^4 + 1))/x^2)
+ 1/64*2^(3/8)*sqrt(2*sqrt(2)*(3*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 4) + 8*sqrt(2) + 16)*(sqrt(2)*(sqrt(2) + 1)*sq
rt(-2*sqrt(2) + 4) - 4)*log(1/2*(8*2^(1/4)*x^2 - 2^(3/8)*(2^(1/4)*(x^4 + 1)^(1/4)*(sqrt(2)*x + 2*x)*sqrt(-2*sq
rt(2) + 4) - 4*2^(1/4)*(x^4 + 1)^(1/4)*x)*sqrt(2*sqrt(2)*(3*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 4) + 8*sqrt(2) + 16
) + 8*sqrt(x^4 + 1))/x^2) - 1/64*2^(3/8)*sqrt(-2*sqrt(2)*(3*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 4) + 8*sqrt(2) + 16
)*(sqrt(2)*(sqrt(2) + 1)*sqrt(-2*sqrt(2) + 4) + 4)*log(1/2*(8*2^(1/4)*x^2 + 2^(3/8)*(2^(1/4)*(x^4 + 1)^(1/4)*(
sqrt(2)*x + 2*x)*sqrt(-2*sqrt(2) + 4) + 4*2^(1/4)*(x^4 + 1)^(1/4)*x)*sqrt(-2*sqrt(2)*(3*sqrt(2) + 4)*sqrt(-2*s
qrt(2) + 4) + 8*sqrt(2) + 16) + 8*sqrt(x^4 + 1))/x^2) + 1/64*2^(3/8)*sqrt(-2*sqrt(2)*(3*sqrt(2) + 4)*sqrt(-2*s
qrt(2) + 4) + 8*sqrt(2) + 16)*(sqrt(2)*(sqrt(2) + 1)*sqrt(-2*sqrt(2) + 4) + 4)*log(1/2*(8*2^(1/4)*x^2 - 2^(3/8
)*(2^(1/4)*(x^4 + 1)^(1/4)*(sqrt(2)*x + 2*x)*sqrt(-2*sqrt(2) + 4) + 4*2^(1/4)*(x^4 + 1)^(1/4)*x)*sqrt(-2*sqrt(
2)*(3*sqrt(2) + 4)*sqrt(-2*sqrt(2) + 4) + 8*sqrt(2) + 16) + 8*sqrt(x^4 + 1))/x^2) - 1/2*arctan((x^4 + 1)^(1/4)
/x) + 1/4*log((x + (x^4 + 1)^(1/4))/x) - 1/4*log(-(x - (x^4 + 1)^(1/4))/x)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8} - 1}{{\left (x^{8} + 1\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^8-1)/(x^4+1)^(1/4)/(x^8+1),x, algorithm="giac")
[Out]
integrate((x^8 - 1)/((x^8 + 1)*(x^4 + 1)^(1/4)), x)
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maple [B] time = 15.28, size = 1245, normalized size = 15.96
| | |
| method |
result |
size |
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| trager |
\(\text {Expression too large to display}\) |
\(1245\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^8-1)/(x^4+1)^(1/4)/(x^8+1),x,method=_RETURNVERBOSE)
[Out]
1/4*RootOf(_Z^2+1)*ln(-2*RootOf(_Z^2+1)*(x^4+1)^(1/2)*x^2+2*RootOf(_Z^2+1)*x^4+2*(x^4+1)^(3/4)*x-2*x^3*(x^4+1)
^(1/4)+RootOf(_Z^2+1))-1/4*ln(2*(x^4+1)^(3/4)*x-2*x^2*(x^4+1)^(1/2)+2*x^3*(x^4+1)^(1/4)-2*x^4-1)-1/8*RootOf(_Z
^4-8*RootOf(_Z^2+1)-8)*ln(-((x^4+1)^(1/2)*RootOf(_Z^2+1)*RootOf(_Z^4-8*RootOf(_Z^2+1)-8)^3*x^2+3*(x^4+1)^(1/2)
*RootOf(_Z^4-8*RootOf(_Z^2+1)-8)^3*x^2+2*(x^4+1)^(1/4)*RootOf(_Z^4-8*RootOf(_Z^2+1)-8)^2*RootOf(_Z^2+1)*x^3-4*
RootOf(_Z^4-8*RootOf(_Z^2+1)-8)*RootOf(_Z^2+1)^2*x^4+6*(x^4+1)^(1/4)*RootOf(_Z^4-8*RootOf(_Z^2+1)-8)^2*x^3+6*R
ootOf(_Z^4-8*RootOf(_Z^2+1)-8)*RootOf(_Z^2+1)*x^4+16*(x^4+1)^(3/4)*RootOf(_Z^2+1)*x+4*RootOf(_Z^4-8*RootOf(_Z^
2+1)-8)*x^4+8*(x^4+1)^(3/4)*x+4*RootOf(_Z^2+1)*RootOf(_Z^4-8*RootOf(_Z^2+1)-8)+2*RootOf(_Z^4-8*RootOf(_Z^2+1)-
8))/(RootOf(_Z^2+1)*x^4+1))+1/8*RootOf(_Z^2+1)*RootOf(_Z^4-8*RootOf(_Z^2+1)-8)*ln(-(3*(x^4+1)^(1/2)*RootOf(_Z^
2+1)*RootOf(_Z^4-8*RootOf(_Z^2+1)-8)^3*x^2-(x^4+1)^(1/2)*RootOf(_Z^4-8*RootOf(_Z^2+1)-8)^3*x^2-2*(x^4+1)^(1/4)
*RootOf(_Z^4-8*RootOf(_Z^2+1)-8)^2*RootOf(_Z^2+1)*x^3+2*RootOf(_Z^4-8*RootOf(_Z^2+1)-8)*RootOf(_Z^2+1)^2*x^4-6
*(x^4+1)^(1/4)*RootOf(_Z^4-8*RootOf(_Z^2+1)-8)^2*x^3-8*RootOf(_Z^4-8*RootOf(_Z^2+1)-8)*RootOf(_Z^2+1)*x^4+16*(
x^4+1)^(3/4)*RootOf(_Z^2+1)*x+8*RootOf(_Z^4-8*RootOf(_Z^2+1)-8)*x^4+8*(x^4+1)^(3/4)*x-2*RootOf(_Z^2+1)*RootOf(
_Z^4-8*RootOf(_Z^2+1)-8)+4*RootOf(_Z^4-8*RootOf(_Z^2+1)-8))/(RootOf(_Z^2+1)*x^4+1))+1/8*RootOf(_Z^4+8*RootOf(_
Z^2+1)-8)*ln((-(x^4+1)^(1/2)*RootOf(_Z^2+1)*RootOf(_Z^4+8*RootOf(_Z^2+1)-8)^3*x^2+(x^4+1)^(1/2)*RootOf(_Z^4+8*
RootOf(_Z^2+1)-8)^3*x^2+2*(x^4+1)^(1/4)*RootOf(_Z^4+8*RootOf(_Z^2+1)-8)^2*RootOf(_Z^2+1)*x^3-2*RootOf(_Z^4+8*R
ootOf(_Z^2+1)-8)*RootOf(_Z^2+1)^2*x^4-2*(x^4+1)^(1/4)*RootOf(_Z^4+8*RootOf(_Z^2+1)-8)^2*x^3-4*RootOf(_Z^4+8*Ro
otOf(_Z^2+1)-8)*RootOf(_Z^2+1)*x^4+8*(x^4+1)^(3/4)*RootOf(_Z^2+1)*x-2*RootOf(_Z^2+1)*RootOf(_Z^4+8*RootOf(_Z^2
+1)-8))/(RootOf(_Z^2+1)*x^4-1))-1/8*RootOf(_Z^2+1)*RootOf(_Z^4+8*RootOf(_Z^2+1)-8)*ln(((x^4+1)^(1/2)*RootOf(_Z
^2+1)*RootOf(_Z^4+8*RootOf(_Z^2+1)-8)^3*x^2+2*(x^4+1)^(1/2)*RootOf(_Z^4+8*RootOf(_Z^2+1)-8)^3*x^2-4*(x^4+1)^(1
/4)*RootOf(_Z^4+8*RootOf(_Z^2+1)-8)^2*RootOf(_Z^2+1)*x^3+RootOf(_Z^4+8*RootOf(_Z^2+1)-8)*RootOf(_Z^2+1)^2*x^4+
2*(x^4+1)^(1/4)*RootOf(_Z^4+8*RootOf(_Z^2+1)-8)^2*x^3-RootOf(_Z^4+8*RootOf(_Z^2+1)-8)*RootOf(_Z^2+1)*x^4+12*(x
^4+1)^(3/4)*RootOf(_Z^2+1)*x-6*RootOf(_Z^4+8*RootOf(_Z^2+1)-8)*x^4+4*(x^4+1)^(3/4)*x+RootOf(_Z^2+1)*RootOf(_Z^
4+8*RootOf(_Z^2+1)-8)-3*RootOf(_Z^4+8*RootOf(_Z^2+1)-8))/(RootOf(_Z^2+1)*x^4-1))
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8} - 1}{{\left (x^{8} + 1\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^8-1)/(x^4+1)^(1/4)/(x^8+1),x, algorithm="maxima")
[Out]
integrate((x^8 - 1)/((x^8 + 1)*(x^4 + 1)^(1/4)), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^8-1}{{\left (x^4+1\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 - 1)/((x^4 + 1)^(1/4)*(x^8 + 1)),x)
[Out]
int((x^8 - 1)/((x^4 + 1)^(1/4)*(x^8 + 1)), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )^{\frac {3}{4}}}{x^{8} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x**8-1)/(x**4+1)**(1/4)/(x**8+1),x)
[Out]
Integral((x - 1)*(x + 1)*(x**2 + 1)*(x**4 + 1)**(3/4)/(x**8 + 1), x)
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