∫ 1−x4 __________________________________(1+x2 +x4) 4√____

3.30.42 \(\int \frac {1-x^4}{(1+x^2+x^4) \sqrt [4]{-x^3+x^5}} \, dx\)

Optimal. Leaf size=351 \[ \frac {\tan ^{-1}\left (\frac {3^{7/8} \sqrt {2-\sqrt {2}} x \sqrt [4]{x^5-x^3}}{3^{3/4} \sqrt {x^5-x^3}-3 x^2}\right )}{2^{3/4} \sqrt [8]{3 \left (17+12 \sqrt {2}\right )}}+\frac {\sqrt [8]{\frac {1}{3} \left (17+12 \sqrt {2}\right )} \tan ^{-1}\left (\frac {3^{7/8} \sqrt {2+\sqrt {2}} x \sqrt [4]{x^5-x^3}}{3^{3/4} \sqrt {x^5-x^3}-3 x^2}\right )}{2^{3/4}}+\frac {\tanh ^{-1}\left (\frac {\frac {\sqrt [8]{3} x^2}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {x^5-x^3}}{\sqrt [8]{3} \sqrt {2-\sqrt {2}}}}{x \sqrt [4]{x^5-x^3}}\right )}{2^{3/4} \sqrt [8]{3 \left (17+12 \sqrt {2}\right )}}+\frac {\sqrt [8]{\frac {1}{3} \left (17+12 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\frac {\sqrt [8]{3} x^2}{\sqrt {2+\sqrt {2}}}+\frac {\sqrt {x^5-x^3}}{\sqrt [8]{3} \sqrt {2+\sqrt {2}}}}{x \sqrt [4]{x^5-x^3}}\right )}{2^{3/4}} \]

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Rubi [C] time = 0.76, antiderivative size = 167, normalized size of antiderivative = 0.48, number of steps used = 9, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2056, 1586, 6715, 6728, 430, 429} \begin {gather*} \frac {4 \left (\sqrt {3}+3 i\right ) x \sqrt [4]{1-x^2} F_1\left (\frac {1}{8};-\frac {3}{4},1;\frac {9}{8};x^2,-\frac {2 x^2}{1-i \sqrt {3}}\right )}{3 \left (\sqrt {3}+i\right ) \sqrt [4]{x^5-x^3}}+\frac {4 \left (-\sqrt {3}+3 i\right ) x \sqrt [4]{1-x^2} F_1\left (\frac {1}{8};-\frac {3}{4},1;\frac {9}{8};x^2,-\frac {2 x^2}{1+i \sqrt {3}}\right )}{3 \left (-\sqrt {3}+i\right ) \sqrt [4]{x^5-x^3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[(1 - x^4)/((1 + x^2 + x^4)*(-x^3 + x^5)^(1/4)),x]

[Out]

(4*(3*I + Sqrt[3])*x*(1 - x^2)^(1/4)*AppellF1[1/8, -3/4, 1, 9/8, x^2, (-2*x^2)/(1 - I*Sqrt[3])])/(3*(I + Sqrt[

3])*(-x^3 + x^5)^(1/4)) + (4*(3*I - Sqrt[3])*x*(1 - x^2)^(1/4)*AppellF1[1/8, -3/4, 1, 9/8, x^2, (-2*x^2)/(1 +

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

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,

-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n

, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 430

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^F

racPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,

p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] && !(IntegerQ[p] || GtQ[a, 0])

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 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart

[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] && !IntegerQ[p

] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] && !PolyQ[P, x, 2]

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /

; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +

b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {1-x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{-x^3+x^5}} \, dx &=\frac {\left (x^{3/4} \sqrt [4]{-1+x^2}\right ) \int \frac {1-x^4}{x^{3/4} \sqrt [4]{-1+x^2} \left (1+x^2+x^4\right )} \, dx}{\sqrt [4]{-x^3+x^5}}\\ &=\frac {\left (x^{3/4} \sqrt [4]{-1+x^2}\right ) \int \frac {\left (-1-x^2\right ) \left (-1+x^2\right )^{3/4}}{x^{3/4} \left (1+x^2+x^4\right )} \, dx}{\sqrt [4]{-x^3+x^5}}\\ &=\frac {\left (4 x^{3/4} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-1-x^8\right ) \left (-1+x^8\right )^{3/4}}{1+x^8+x^{16}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}}\\ &=\frac {\left (4 x^{3/4} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {\left (-1+\frac {i}{\sqrt {3}}\right ) \left (-1+x^8\right )^{3/4}}{1-i \sqrt {3}+2 x^8}+\frac {\left (-1-\frac {i}{\sqrt {3}}\right ) \left (-1+x^8\right )^{3/4}}{1+i \sqrt {3}+2 x^8}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}}\\ &=\frac {\left (4 \left (-3+i \sqrt {3}\right ) x^{3/4} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^8\right )^{3/4}}{1-i \sqrt {3}+2 x^8} \, dx,x,\sqrt [4]{x}\right )}{3 \sqrt [4]{-x^3+x^5}}-\frac {\left (4 \left (3+i \sqrt {3}\right ) x^{3/4} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^8\right )^{3/4}}{1+i \sqrt {3}+2 x^8} \, dx,x,\sqrt [4]{x}\right )}{3 \sqrt [4]{-x^3+x^5}}\\ &=\frac {\left (4 \left (-3+i \sqrt {3}\right ) x^{3/4} \left (-1+x^2\right )\right ) \operatorname {Subst}\left (\int \frac {\left (1-x^8\right )^{3/4}}{1-i \sqrt {3}+2 x^8} \, dx,x,\sqrt [4]{x}\right )}{3 \left (1-x^2\right )^{3/4} \sqrt [4]{-x^3+x^5}}-\frac {\left (4 \left (3+i \sqrt {3}\right ) x^{3/4} \left (-1+x^2\right )\right ) \operatorname {Subst}\left (\int \frac {\left (1-x^8\right )^{3/4}}{1+i \sqrt {3}+2 x^8} \, dx,x,\sqrt [4]{x}\right )}{3 \left (1-x^2\right )^{3/4} \sqrt [4]{-x^3+x^5}}\\ &=\frac {4 \left (3 i+\sqrt {3}\right ) x \sqrt [4]{1-x^2} F_1\left (\frac {1}{8};-\frac {3}{4},1;\frac {9}{8};x^2,-\frac {2 x^2}{1-i \sqrt {3}}\right )}{3 \left (i+\sqrt {3}\right ) \sqrt [4]{-x^3+x^5}}+\frac {4 \left (3 i-\sqrt {3}\right ) x \sqrt [4]{1-x^2} F_1\left (\frac {1}{8};-\frac {3}{4},1;\frac {9}{8};x^2,-\frac {2 x^2}{1+i \sqrt {3}}\right )}{3 \left (i-\sqrt {3}\right ) \sqrt [4]{-x^3+x^5}}\\ \end {align*}

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Mathematica [F] time = 0.25, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-x^4}{\left (1+x^2+x^4\right ) \sqrt [4]{-x^3+x^5}} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

Integrate[(1 - x^4)/((1 + x^2 + x^4)*(-x^3 + x^5)^(1/4)),x]

[Out]

Integrate[(1 - x^4)/((1 + x^2 + x^4)*(-x^3 + x^5)^(1/4)), x]

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IntegrateAlgebraic [A] time = 3.56, size = 377, normalized size = 1.07 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} 3^{7/8} x \sqrt [4]{-x^3+x^5}}{\sqrt [8]{17+12 \sqrt {2}} \left (-3 x^2+3^{3/4} \sqrt {-x^3+x^5}\right )}\right )}{2^{3/4} \sqrt [8]{3 \left (17+12 \sqrt {2}\right )}}+\frac {\sqrt [8]{\frac {1}{3} \left (17+12 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt [4]{2} 3^{7/8} \sqrt [8]{17+12 \sqrt {2}} x \sqrt [4]{-x^3+x^5}}{-3 x^2+3^{3/4} \sqrt {-x^3+x^5}}\right )}{2^{3/4}}+\frac {\tanh ^{-1}\left (\frac {3 \sqrt [8]{\frac {17}{8748}+\frac {\sqrt {2}}{729}} x^2+3^{3/4} \sqrt [8]{\frac {17}{8748}+\frac {\sqrt {2}}{729}} \sqrt {-x^3+x^5}}{x \sqrt [4]{-x^3+x^5}}\right )}{2^{3/4} \sqrt [8]{3 \left (17+12 \sqrt {2}\right )}}+\frac {\sqrt [8]{\frac {1}{3} \left (17+12 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\frac {\sqrt [8]{\frac {3}{17+12 \sqrt {2}}} x^2}{\sqrt [4]{2}}+\frac {\sqrt {-x^3+x^5}}{\sqrt [4]{2} \sqrt [8]{3 \left (17+12 \sqrt {2}\right )}}}{x \sqrt [4]{-x^3+x^5}}\right )}{2^{3/4}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

IntegrateAlgebraic[(1 - x^4)/((1 + x^2 + x^4)*(-x^3 + x^5)^(1/4)),x]

[Out]

ArcTan[(2^(1/4)*3^(7/8)*x*(-x^3 + x^5)^(1/4))/((17 + 12*Sqrt[2])^(1/8)*(-3*x^2 + 3^(3/4)*Sqrt[-x^3 + x^5]))]/(

2^(3/4)*(3*(17 + 12*Sqrt[2]))^(1/8)) + (((17 + 12*Sqrt[2])/3)^(1/8)*ArcTan[(2^(1/4)*3^(7/8)*(17 + 12*Sqrt[2])^

(1/8)*x*(-x^3 + x^5)^(1/4))/(-3*x^2 + 3^(3/4)*Sqrt[-x^3 + x^5])])/2^(3/4) + ArcTanh[(3*(17/8748 + Sqrt[2]/729)

^(1/8)*x^2 + 3^(3/4)*(17/8748 + Sqrt[2]/729)^(1/8)*Sqrt[-x^3 + x^5])/(x*(-x^3 + x^5)^(1/4))]/(2^(3/4)*(3*(17 +

12*Sqrt[2]))^(1/8)) + (((17 + 12*Sqrt[2])/3)^(1/8)*ArcTanh[(((3/(17 + 12*Sqrt[2]))^(1/8)*x^2)/2^(1/4) + Sqrt[

-x^3 + x^5]/(2^(1/4)*(3*(17 + 12*Sqrt[2]))^(1/8)))/(x*(-x^3 + x^5)^(1/4))])/2^(3/4)

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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)/(x^4+x^2+1)/(x^5-x^3)^(1/4),x, algorithm="fricas")

[Out]

Timed out

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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {x^{4} - 1}{{\left (x^{5} - x^{3}\right )}^{\frac {1}{4}} {\left (x^{4} + x^{2} + 1\right )}}\,{d x} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4+1)/(x^4+x^2+1)/(x^5-x^3)^(1/4),x, algorithm="giac")

[Out]

integrate(-(x^4 - 1)/((x^5 - x^3)^(1/4)*(x^4 + x^2 + 1)), x)

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maple [C] time = 73.11, size = 2140, normalized size = 6.10


method	result	size

trager	\(\text {Expression too large to display}\)	\(2140\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((-x^4+1)/(x^4+x^2+1)/(x^5-x^3)^(1/4),x,method=_RETURNVERBOSE)

[Out]

1/6*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*ln(-108*(16*x^4*RootOf(_Z^8+2187)^8*RootOf(_Z^2+RootOf(_Z^8+2187)^2)-24*x

^3*RootOf(_Z^8+2187)^8*RootOf(_Z^2+RootOf(_Z^8+2187)^2)-16*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^8+2187)^

8*x^2-54*(x^5-x^3)^(1/2)*RootOf(_Z^8+2187)^6*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*x-162*(x^5-x^3)^(1/4)*RootOf(_Z^

8+2187)^6*x^2+1350*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^8+2187)^4*x^4+243*x^3*RootOf(_Z^8+2187)^4*RootOf

(_Z^2+RootOf(_Z^8+2187)^2)+8100*(x^5-x^3)^(3/4)*RootOf(_Z^8+2187)^4-1350*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*Root

Of(_Z^8+2187)^4*x^2-24300*(x^5-x^3)^(1/2)*RootOf(_Z^8+2187)^2*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*x-72900*(x^5-x^

3)^(1/4)*RootOf(_Z^8+2187)^2*x^2+28431*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*x^4+56862*RootOf(_Z^2+RootOf(_Z^8+2187

)^2)*x^3-39366*(x^5-x^3)^(3/4)-28431*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*x^2)/x^2/(x*RootOf(_Z^8+2187)^4-2*RootOf

(_Z^8+2187)^4-135*x-108)/(x*RootOf(_Z^8+2187)^4-2*RootOf(_Z^8+2187)^4-81*x))+1/6*RootOf(_Z^8+2187)*ln(-108*(16

*RootOf(_Z^8+2187)^9*x^4-24*RootOf(_Z^8+2187)^9*x^3-16*RootOf(_Z^8+2187)^9*x^2+54*(x^5-x^3)^(1/2)*RootOf(_Z^8+

2187)^7*x+162*(x^5-x^3)^(1/4)*RootOf(_Z^8+2187)^6*x^2+1350*RootOf(_Z^8+2187)^5*x^4+243*RootOf(_Z^8+2187)^5*x^3

+8100*(x^5-x^3)^(3/4)*RootOf(_Z^8+2187)^4-1350*RootOf(_Z^8+2187)^5*x^2+24300*(x^5-x^3)^(1/2)*RootOf(_Z^8+2187)

^3*x+72900*(x^5-x^3)^(1/4)*RootOf(_Z^8+2187)^2*x^2+28431*RootOf(_Z^8+2187)*x^4+56862*RootOf(_Z^8+2187)*x^3-393

66*(x^5-x^3)^(3/4)-28431*RootOf(_Z^8+2187)*x^2)/x^2/(x*RootOf(_Z^8+2187)^4-2*RootOf(_Z^8+2187)^4-135*x-108)/(x

*RootOf(_Z^8+2187)^4-2*RootOf(_Z^8+2187)^4-81*x))+1/13122*RootOf(_Z^8+2187)^7*RootOf(_Z^2+RootOf(_Z^8+2187)^2)

*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*ln(-108*(26*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*

RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)^11*x^4-39*RootOf(_Z^2+RootOf

(_Z^8+2187)^2)*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)^11*x^3-26*Roo

tOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^8+2187

)^11*x^2+243*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*

RootOf(_Z^8+2187)^7*x^4-4050*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf

(_Z^8+2187)^2))*RootOf(_Z^8+2187)^7*x^3-243*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^2+RootOf(_Z^8+2187)*Roo

tOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)^7*x^2+4374*(x^5-x^3)^(1/2)*RootOf(_Z^2+RootOf(_Z^8+2187)*Root

Of(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)^6*x+13122*(x^5-x^3)^(1/4)*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*Roo

tOf(_Z^8+2187)^5*x^2-52488*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_

Z^8+2187)^2))*RootOf(_Z^8+2187)^3*x^4-104976*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*R

ootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^8+2187)^3*x^3+656100*(x^5-x^3)^(3/4)*RootOf(_Z^8+2187)^4+52488*RootO

f(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)^

3*x^2-1968300*(x^5-x^3)^(1/2)*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187

)^2*x-5904900*(x^5-x^3)^(1/4)*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^8+2187)*x^2+3188646*(x^5-x^3)^(3/4))/

x^2/(x*RootOf(_Z^8+2187)^4-2*RootOf(_Z^8+2187)^4+81*x)/(x*RootOf(_Z^8+2187)^4-2*RootOf(_Z^8+2187)^4+135*x+108)

)+1/6*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*ln(-108*(-16*RootOf(_Z^2+RootOf(_Z^8+218

7)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)^8*x^4+24*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+Root

Of(_Z^8+2187)^2))*RootOf(_Z^8+2187)^8*x^3+16*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*R

ootOf(_Z^8+2187)^8*x^2-54*(x^5-x^3)^(1/2)*RootOf(_Z^8+2187)^5*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^2+Roo

tOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*x-162*(x^5-x^3)^(1/4)*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf

(_Z^8+2187)^5*x^2+1350*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)^4*x^4

+243*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)^4*x^3+8100*(x^5-x^3)^(3

/4)*RootOf(_Z^8+2187)^4-1350*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^8+2187)

^4*x^2+24300*(x^5-x^3)^(1/2)*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*RootOf(_Z^2+RootO

f(_Z^8+2187)^2)*RootOf(_Z^8+2187)*x+72900*(x^5-x^3)^(1/4)*RootOf(_Z^2+RootOf(_Z^8+2187)^2)*RootOf(_Z^8+2187)*x

^2-28431*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*x^4-56862*RootOf(_Z^2+RootOf(_Z^8+218

7)*RootOf(_Z^2+RootOf(_Z^8+2187)^2))*x^3+39366*(x^5-x^3)^(3/4)+28431*RootOf(_Z^2+RootOf(_Z^8+2187)*RootOf(_Z^2

+RootOf(_Z^8+2187)^2))*x^2)/x^2/(x*RootOf(_Z^8+2187)^4-2*RootOf(_Z^8+2187)^4+81*x)/(x*RootOf(_Z^8+2187)^4-2*Ro

otOf(_Z^8+2187)^4+135*x+108))

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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {x^{4} - 1}{{\left (x^{5} - x^{3}\right )}^{\frac {1}{4}} {\left (x^{4} + x^{2} + 1\right )}}\,{d x} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4+1)/(x^4+x^2+1)/(x^5-x^3)^(1/4),x, algorithm="maxima")

[Out]

-integrate((x^4 - 1)/((x^5 - x^3)^(1/4)*(x^4 + x^2 + 1)), x)

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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {x^4-1}{{\left (x^5-x^3\right )}^{1/4}\,\left (x^4+x^2+1\right )} \,d x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^4 - 1)/((x^5 - x^3)^(1/4)*(x^2 + x^4 + 1)),x)

[Out]

-int((x^4 - 1)/((x^5 - x^3)^(1/4)*(x^2 + x^4 + 1)), x)

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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x^{4}}{x^{4} \sqrt [4]{x^{5} - x^{3}} + x^{2} \sqrt [4]{x^{5} - x^{3}} + \sqrt [4]{x^{5} - x^{3}}}\, dx - \int \left (- \frac {1}{x^{4} \sqrt [4]{x^{5} - x^{3}} + x^{2} \sqrt [4]{x^{5} - x^{3}} + \sqrt [4]{x^{5} - x^{3}}}\right )\, dx \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**4+1)/(x**4+x**2+1)/(x**5-x**3)**(1/4),x)

[Out]

-Integral(x**4/(x**4*(x**5 - x**3)**(1/4) + x**2*(x**5 - x**3)**(1/4) + (x**5 - x**3)**(1/4)), x) - Integral(-

1/(x**4*(x**5 - x**3)**(1/4) + x**2*(x**5 - x**3)**(1/4) + (x**5 - x**3)**(1/4)), x)

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