∫ −1+x8 _____________________________4√1 + x4 (1+x8) dx [1039]

3.11.39 $\int \frac {-1+x^8}{\sqrt [4]{1+x^4} (1+x^8)} \, dx$ [1039]

Optimal. Leaf size=78 \[ \frac {1}{2} \text {ArcTan}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{4} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\& ,\frac {-\log (x)+\log \left (\sqrt [4]{1+x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\& \right ] \]

[Out]

Unintegrable

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Rubi [C] Result contains complex when optimal does not.
time = 0.13, 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 = {1600, 6857, 399, 246, 218, 212, 209, 385} \begin {gather*} \frac {1}{2} \text {ArcTan}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )-\frac {\text {ArcTan}\left (\frac {\sqrt [4]{1-i} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{1-i}}-\frac {\text {ArcTan}\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 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;

FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]

/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 218

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] && !Gt

Q[a/b, 0]

Rule 246

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 385

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 399

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[b/d, Int[(a + b*x^n)^(p - 1), x]

, x] - Dist[(b*c - a*d)/d, Int[(a + b*x^n)^(p - 1)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c

- a*d, 0] && EqQ[n*(p - 1) + 1, 0] && IntegerQ[n]

Rule 1600

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 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 {-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 \text {Subst}\left (\int \frac {1}{i-(1+i) x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\right )-i \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-\sqrt {1-i} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+\sqrt {1-i} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-\sqrt {1+i} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \text {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 [A]
time = 0.00, size = 73, normalized size = 0.94 \begin {gather*} \frac {1}{4} \left (2 \left (\text {ArcTan}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )\right )+\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 ]\right ) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

+ x^4)^(1/4) - x*#1])/#1 & ])/4

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 1.
time = 12.51, size = 1291, normalized size = 16.55


method	result	size

trager	$\text {Expression too large to display}$	$1291$

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*ln(2*(x^4+1)^(3/4)*x+2*(x^4+1)^(1/2)*x^2+2*x^3*(x^4+1)^(1/4)+2*x^4+1)+1/4*RootOf(_Z^2+1)*ln(-2*(x^4+1)^(1/

2)*RootOf(_Z^2+1)*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))+1/8*RootOf(_Z

^4-8*RootOf(_Z^2+1)-8)*ln((13*(x^4+1)^(1/2)*RootOf(_Z^2+1)*RootOf(_Z^4-8*RootOf(_Z^2+1)-8)^3*x^2+9*(x^4+1)^(1/

2)*RootOf(_Z^4-8*RootOf(_Z^2+1)-8)^3*x^2-26*(x^4+1)^(1/4)*RootOf(_Z^4-8*RootOf(_Z^2+1)-8)^2*RootOf(_Z^2+1)*x^3

-22*RootOf(_Z^4-8*RootOf(_Z^2+1)-8)*RootOf(_Z^2+1)^2*x^4-18*(x^4+1)^(1/4)*RootOf(_Z^4-8*RootOf(_Z^2+1)-8)^2*x^

3+48*RootOf(_Z^4-8*RootOf(_Z^2+1)-8)*RootOf(_Z^2+1)*x^4-8*RootOf(_Z^4-8*RootOf(_Z^2+1)-8)*x^4-88*(x^4+1)^(3/4)

*RootOf(_Z^2+1)*x+16*(x^4+1)^(3/4)*x+22*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^2+1)*RootOf(_Z^4-8*RootOf(_Z^2+1)-8)*ln((3*(x^4+1)^(1/2)*Roo

tOf(_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+8*RootOf(_Z^4-8*RootOf(_Z^2+1)-8)*x^4-16*(x^4+1)^(3/4)*RootOf(_Z^2+1)*x-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^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*R

ootOf(_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((2*(x^4+1)^(1/2)*RootOf(_Z^2+1)*Ro

otOf(_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+4*(x^4+1)^(1/4)*RootOf

(_Z^4+8*RootOf(_Z^2+1)-8)^2*RootOf(_Z^2+1)*x^3+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+7*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+2*RootOf(_Z^4+8*RootOf(_Z^2+1)-8)*x^4+4*(x^4+1)^(3/4)*x+3*RootOf(_Z^2+1)*RootOf(_Z^4+8*

RootOf(_Z^2+1)-8)+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*} \text {Failed to integrate} \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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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.47, size = 2014, normalized size = 25.82 \begin {gather*} \text {Too large to display} \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="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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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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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^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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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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3.11.39 \(\int \frac {-1+x^8}{\sqrt [4]{1+x^4} (1+x^8)} \, dx\) [1039]