∫ 1_______ 4√___ 1+x4 (−1+x4+x8) dx

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

Optimal. Leaf size=193 \[ -\frac {1}{2} \sqrt {\frac {1}{10} \left (\sqrt {5}-1\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}} x}{\sqrt [4]{x^4+1}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^4+1}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (\sqrt {5}-1\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}} x}{\sqrt [4]{x^4+1}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^4+1}}\right ) \]

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Rubi [A] time = 0.12, antiderivative size = 209, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1428, 377, 212, 206, 203} \begin {gather*} -\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt {5}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/2*(((3 - Sqrt[5])/2)^(1/4)*ArcTan[((2/(3 + Sqrt[5]))^(1/4)*x)/(1 + x^4)^(1/4)])/Sqrt[5] - (((3 + Sqrt[5])/2

)^(1/4)*ArcTan[(((3 + Sqrt[5])/2)^(1/4)*x)/(1 + x^4)^(1/4)])/(2*Sqrt[5]) - (((3 - Sqrt[5])/2)^(1/4)*ArcTanh[((

2/(3 + Sqrt[5]))^(1/4)*x)/(1 + x^4)^(1/4)])/(2*Sqrt[5]) - (((3 + Sqrt[5])/2)^(1/4)*ArcTanh[(((3 + Sqrt[5])/2)^

(1/4)*x)/(1 + x^4)^(1/4)])/(2*Sqrt[5])

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 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 1428

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{r = Rt[b^2 -

4*a*c, 2]}, Dist[(2*c)/r, Int[(d + e*x^n)^q/(b - r + 2*c*x^n), x], x] - Dist[(2*c)/r, Int[(d + e*x^n)^q/(b +

r + 2*c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -

b*d*e + a*e^2, 0] && !IntegerQ[q]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt [4]{1+x^4} \left (-1+x^4+x^8\right )} \, dx &=\frac {2 \int \frac {1}{\sqrt [4]{1+x^4} \left (1-\sqrt {5}+2 x^4\right )} \, dx}{\sqrt {5}}-\frac {2 \int \frac {1}{\sqrt [4]{1+x^4} \left (1+\sqrt {5}+2 x^4\right )} \, dx}{\sqrt {5}}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {5}-\left (-1-\sqrt {5}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {5}}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {5}-\left (-1+\sqrt {5}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {5}}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {10}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {10}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {3-\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {10}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {3+\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{\sqrt {10}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{1+x^4}}\right )}{2^{3/4} \sqrt {5} \sqrt [4]{3+\sqrt {5}}}-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt {5}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{1+x^4}}\right )}{2^{3/4} \sqrt {5} \sqrt [4]{3+\sqrt {5}}}-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt {5}}\\ \end {align*}

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Mathematica [A] time = 0.30, size = 214, normalized size = 1.11 \begin {gather*} \frac {-\frac {\sqrt [4]{3+\sqrt {5}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{x^4+1}}\right )}{1+\sqrt {5}}+\frac {\sqrt [4]{3-\sqrt {5}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{x^4+1}}\right )}{1-\sqrt {5}}-\frac {\sqrt [4]{3+\sqrt {5}} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{x^4+1}}\right )}{1+\sqrt {5}}+\frac {\sqrt [4]{3-\sqrt {5}} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{x^4+1}}\right )}{1-\sqrt {5}}}{\sqrt [4]{2} \sqrt {5}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-(((3 + Sqrt[5])^(1/4)*ArcTan[((2/(3 + Sqrt[5]))^(1/4)*x)/(1 + x^4)^(1/4)])/(1 + Sqrt[5])) + ((3 - Sqrt[5])^(

1/4)*ArcTan[(((3 + Sqrt[5])/2)^(1/4)*x)/(1 + x^4)^(1/4)])/(1 - Sqrt[5]) - ((3 + Sqrt[5])^(1/4)*ArcTanh[((2/(3

+ Sqrt[5]))^(1/4)*x)/(1 + x^4)^(1/4)])/(1 + Sqrt[5]) + ((3 - Sqrt[5])^(1/4)*ArcTanh[(((3 + Sqrt[5])/2)^(1/4)*x

)/(1 + x^4)^(1/4)])/(1 - Sqrt[5]))/(2^(1/4)*Sqrt[5])

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IntegrateAlgebraic [A] time = 0.58, size = 193, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{1+x^4}}\right ) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-1/2*(Sqrt[(-1 + Sqrt[5])/10]*ArcTan[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(1 + x^4)^(1/4)]) - (Sqrt[(1 + Sqrt[5])/10]*Ar

cTan[(Sqrt[1/2 + Sqrt[5]/2]*x)/(1 + x^4)^(1/4)])/2 - (Sqrt[(-1 + Sqrt[5])/10]*ArcTanh[(Sqrt[-1/2 + Sqrt[5]/2]*

x)/(1 + x^4)^(1/4)])/2 - (Sqrt[(1 + Sqrt[5])/10]*ArcTanh[(Sqrt[1/2 + Sqrt[5]/2]*x)/(1 + x^4)^(1/4)])/2

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fricas [B] time = 25.05, size = 913, normalized size = 4.73

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*sqrt(10)*sqrt(sqrt(5) + 1)*arctan(-1/40*(sqrt(2)*(2*sqrt(10)*(5*x^6 - sqrt(5)*(x^6 - 2*x^2))*sqrt(x^4 +

1) - sqrt(10)*(5*x^8 - 5*x^4 - sqrt(5)*(5*x^8 + 3*x^4 - 1) - 5))*(sqrt(5) + 1) + 4*(sqrt(10)*(5*x^5 - sqrt(5)*

(x^5 - 2*x))*(x^4 + 1)^(3/4) + sqrt(10)*(x^4 + 1)^(1/4)*(5*x^3 + sqrt(5)*(2*x^7 + x^3)))*sqrt(sqrt(5) + 1))/(x

^8 + x^4 - 1)) + 1/20*sqrt(10)*sqrt(sqrt(5) - 1)*arctan(1/40*(sqrt(2)*(2*sqrt(10)*(5*x^6 + sqrt(5)*(x^6 - 2*x^

2))*sqrt(x^4 + 1) + sqrt(10)*(5*x^8 - 5*x^4 + sqrt(5)*(5*x^8 + 3*x^4 - 1) - 5))*(sqrt(5) - 1) - 4*(sqrt(10)*(5

*x^5 + sqrt(5)*(x^5 - 2*x))*(x^4 + 1)^(3/4) - sqrt(10)*(x^4 + 1)^(1/4)*(5*x^3 - sqrt(5)*(2*x^7 + x^3)))*sqrt(s

qrt(5) - 1))/(x^8 + x^4 - 1)) - 1/80*sqrt(10)*sqrt(sqrt(5) + 1)*log((10*(2*x^5 + sqrt(5)*x + x)*(x^4 + 1)^(3/4

) + (sqrt(10)*sqrt(x^4 + 1)*(5*x^2 + sqrt(5)*(2*x^6 + x^2)) + sqrt(10)*(5*x^8 + 5*x^4 + sqrt(5)*(2*x^4 + 1)))*

sqrt(sqrt(5) + 1) + 10*(x^7 + 3*x^3 + sqrt(5)*(x^7 + x^3))*(x^4 + 1)^(1/4))/(x^8 + x^4 - 1)) + 1/80*sqrt(10)*s

qrt(sqrt(5) + 1)*log((10*(2*x^5 + sqrt(5)*x + x)*(x^4 + 1)^(3/4) - (sqrt(10)*sqrt(x^4 + 1)*(5*x^2 + sqrt(5)*(2

*x^6 + x^2)) + sqrt(10)*(5*x^8 + 5*x^4 + sqrt(5)*(2*x^4 + 1)))*sqrt(sqrt(5) + 1) + 10*(x^7 + 3*x^3 + sqrt(5)*(

x^7 + x^3))*(x^4 + 1)^(1/4))/(x^8 + x^4 - 1)) + 1/80*sqrt(10)*sqrt(sqrt(5) - 1)*log((10*(2*x^5 - sqrt(5)*x + x

)*(x^4 + 1)^(3/4) + (sqrt(10)*sqrt(x^4 + 1)*(5*x^2 - sqrt(5)*(2*x^6 + x^2)) - sqrt(10)*(5*x^8 + 5*x^4 - sqrt(5

)*(2*x^4 + 1)))*sqrt(sqrt(5) - 1) - 10*(x^7 + 3*x^3 - sqrt(5)*(x^7 + x^3))*(x^4 + 1)^(1/4))/(x^8 + x^4 - 1)) -

1/80*sqrt(10)*sqrt(sqrt(5) - 1)*log((10*(2*x^5 - sqrt(5)*x + x)*(x^4 + 1)^(3/4) - (sqrt(10)*sqrt(x^4 + 1)*(5*

x^2 - sqrt(5)*(2*x^6 + x^2)) - sqrt(10)*(5*x^8 + 5*x^4 - sqrt(5)*(2*x^4 + 1)))*sqrt(sqrt(5) - 1) - 10*(x^7 + 3

*x^3 - sqrt(5)*(x^7 + x^3))*(x^4 + 1)^(1/4))/(x^8 + x^4 - 1))

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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab

le to convert to real 1/4 Error: Bad Argument ValueUnable to convert to real 1/4 Error: Bad Argument Value

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maple [C] time = 11.49, size = 1705, normalized size = 8.83


method	result	size

trager	\(\text {Expression too large to display}\)	\(1705\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootOf(6400*_Z^4-80*_Z^2-1)*ln(-(64000*RootOf(6400*_Z^4-80*_Z^2-1)^5*x^4+3200*RootOf(6400*_Z^4-80*_Z^2-1)^3*(x

^4+1)^(1/2)*x^2+2400*x^4*RootOf(6400*_Z^4-80*_Z^2-1)^3-320*RootOf(6400*_Z^4-80*_Z^2-1)^2*(x^4+1)^(3/4)*x-560*(

x^4+1)^(1/4)*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^3+20*RootOf(6400*_Z^4-80*_Z^2-1)*(x^4+1)^(1/2)*x^2+20*RootOf(6400

*_Z^4-80*_Z^2-1)*x^4-3*(x^4+1)^(3/4)*x-4*x^3*(x^4+1)^(1/4)+800*RootOf(6400*_Z^4-80*_Z^2-1)^3+10*RootOf(6400*_Z

^4-80*_Z^2-1))/(640*RootOf(6400*_Z^4-80*_Z^2-1)^3*x^2-4*RootOf(6400*_Z^4-80*_Z^2-1)*x^2+1)/(640*RootOf(6400*_Z

^4-80*_Z^2-1)^3*x^2-4*RootOf(6400*_Z^4-80*_Z^2-1)*x^2-1))+1/20*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5

)*ln(-(6400*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*RootOf(6400*_Z^4-80*_Z^2-1)^4*x^4-160*(x^4+1)^(1/

2)*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^2-320*RootOf(_Z^2+400*Root

Of(6400*_Z^4-80*_Z^2-1)^2-5)*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^4+480*RootOf(6400*_Z^4-80*_Z^2-1)^2*(x^4+1)^(3/4)

*x+640*(x^4+1)^(1/4)*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^3+4*(x^4+1)^(1/2)*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^

2-1)^2-5)*x^2+3*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*x^4-8*(x^4+1)^(3/4)*x-14*x^3*(x^4+1)^(1/4)-80

*RootOf(6400*_Z^4-80*_Z^2-1)^2*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)+RootOf(_Z^2+400*RootOf(6400*_Z

^4-80*_Z^2-1)^2-5))/(80*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^4-x^4+1))-4*RootOf(6400*_Z^4-80*_Z^2-1)^2*RootOf(_Z^2+

400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*ln((6400*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*RootOf(6400*_Z^

4-80*_Z^2-1)^4*x^4-320*(x^4+1)^(1/2)*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*RootOf(6400*_Z^4-80*_Z^2

-1)^2*x^2+240*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^4+480*RootOf(64

00*_Z^4-80*_Z^2-1)^2*(x^4+1)^(3/4)*x-640*(x^4+1)^(1/4)*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^3-2*(x^4+1)^(1/2)*RootO

f(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*x^2+2*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*x^4+2*(x^4+

1)^(3/4)*x-6*x^3*(x^4+1)^(1/4)+80*RootOf(6400*_Z^4-80*_Z^2-1)^2*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-

5)+RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5))/(640*RootOf(6400*_Z^4-80*_Z^2-1)^3*x^2-4*RootOf(6400*_Z^4

-80*_Z^2-1)*x^2+1)/(640*RootOf(6400*_Z^4-80*_Z^2-1)^3*x^2-4*RootOf(6400*_Z^4-80*_Z^2-1)*x^2-1))+80*RootOf(6400

*_Z^4-80*_Z^2-1)^3*ln((64000*RootOf(6400*_Z^4-80*_Z^2-1)^5*x^4+3200*RootOf(6400*_Z^4-80*_Z^2-1)^3*(x^4+1)^(1/2

)*x^2-4000*x^4*RootOf(6400*_Z^4-80*_Z^2-1)^3-240*RootOf(6400*_Z^4-80*_Z^2-1)^2*(x^4+1)^(3/4)*x+320*(x^4+1)^(1/

4)*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^3-60*RootOf(6400*_Z^4-80*_Z^2-1)*(x^4+1)^(1/2)*x^2+60*RootOf(6400*_Z^4-80*_

Z^2-1)*x^4+4*(x^4+1)^(3/4)*x-7*x^3*(x^4+1)^(1/4)-800*RootOf(6400*_Z^4-80*_Z^2-1)^3+20*RootOf(6400*_Z^4-80*_Z^2

-1))/(80*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^4-x^4+1))-RootOf(6400*_Z^4-80*_Z^2-1)*ln((64000*RootOf(6400*_Z^4-80*_

Z^2-1)^5*x^4+3200*RootOf(6400*_Z^4-80*_Z^2-1)^3*(x^4+1)^(1/2)*x^2-4000*x^4*RootOf(6400*_Z^4-80*_Z^2-1)^3-240*R

ootOf(6400*_Z^4-80*_Z^2-1)^2*(x^4+1)^(3/4)*x+320*(x^4+1)^(1/4)*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^3-60*RootOf(640

0*_Z^4-80*_Z^2-1)*(x^4+1)^(1/2)*x^2+60*RootOf(6400*_Z^4-80*_Z^2-1)*x^4+4*(x^4+1)^(3/4)*x-7*x^3*(x^4+1)^(1/4)-8

00*RootOf(6400*_Z^4-80*_Z^2-1)^3+20*RootOf(6400*_Z^4-80*_Z^2-1))/(80*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^4-x^4+1))

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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{8} + x^{4} - 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(1/(x^4+1)^(1/4)/(x^8+x^4-1),x, algorithm="maxima")

[Out]

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [4]{x^{4} + 1} \left (x^{8} + x^{4} - 1\right )}\, dx \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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