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

Optimal. Leaf size=469 \[ \frac {1}{2} \sqrt [4]{3 \sqrt {2}-4} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x}{2^{7/8} \sqrt [4]{x^5-x^3}-\sqrt {2+\sqrt {2}} x}\right )+\frac {1}{2} \sqrt [4]{3 \sqrt {2}-4} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x}{2^{7/8} \sqrt [4]{x^5-x^3}+\sqrt {2+\sqrt {2}} x}\right )-\frac {1}{4} \sqrt [4]{4+3 \sqrt {2}} \log \left (-2 x^2+2^{7/8} \sqrt {2+\sqrt {2}} \sqrt [4]{x^5-x^3} x-2^{3/4} \sqrt {x^5-x^3}\right )+\frac {1}{4} \sqrt [4]{4+3 \sqrt {2}} \log \left (2 \sqrt {2-\sqrt {2}} x^2+2\ 2^{3/8} \sqrt [4]{x^5-x^3} x+2^{3/4} \sqrt {2-\sqrt {2}} \sqrt {x^5-x^3}\right )+\frac {1}{2} \sqrt [4]{4+3 \sqrt {2}} \tan ^{-1}\left (\frac {2^{7/8} \sqrt {2+\sqrt {2}} x \sqrt [4]{x^5-x^3}}{2^{3/4} \sqrt {x^5-x^3}-2 x^2}\right )+\frac {1}{2} \sqrt [4]{3 \sqrt {2}-4} \tanh ^{-1}\left (\frac {\frac {\sqrt [8]{2} x^2}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {x^5-x^3}}{\sqrt [8]{2} \sqrt {2-\sqrt {2}}}}{x \sqrt [4]{x^5-x^3}}\right ) \]

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Rubi [C]  time = 0.65, antiderivative size = 101, normalized size of antiderivative = 0.22, number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2056, 1586, 6715, 6725, 430, 429} \begin {gather*} \frac {(2-2 i) x \sqrt [4]{1-x^2} F_1\left (\frac {1}{8};-\frac {3}{4},1;\frac {9}{8};x^2,i x^2\right )}{\sqrt [4]{x^5-x^3}}+\frac {(2+2 i) x \sqrt [4]{1-x^2} F_1\left (\frac {1}{8};1,-\frac {3}{4};\frac {9}{8};-i x^2,x^2\right )}{\sqrt [4]{x^5-x^3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((2 - 2*I)*x*(1 - x^2)^(1/4)*AppellF1[1/8, -3/4, 1, 9/8, x^2, I*x^2])/(-x^3 + x^5)^(1/4) + ((2 + 2*I)*x*(1 - x
^2)^(1/4)*AppellF1[1/8, 1, -3/4, 9/8, (-I)*x^2, x^2])/(-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 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^4}{\left (1+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^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^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^{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 (\frac {1}{2}-\frac {i}{2}\right ) \left (-1+x^8\right )^{3/4}}{i-x^8}-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-1+x^8\right )^{3/4}}{i+x^8}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}}\\ &=-\frac {\left ((2+2 i) x^{3/4} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^8\right )^{3/4}}{i+x^8} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}}+\frac {\left ((2-2 i) x^{3/4} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^8\right )^{3/4}}{i-x^8} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}}\\ &=-\frac {\left ((2+2 i) x^{3/4} \left (-1+x^2\right )\right ) \operatorname {Subst}\left (\int \frac {\left (1-x^8\right )^{3/4}}{i+x^8} \, dx,x,\sqrt [4]{x}\right )}{\left (1-x^2\right )^{3/4} \sqrt [4]{-x^3+x^5}}+\frac {\left ((2-2 i) x^{3/4} \left (-1+x^2\right )\right ) \operatorname {Subst}\left (\int \frac {\left (1-x^8\right )^{3/4}}{i-x^8} \, dx,x,\sqrt [4]{x}\right )}{\left (1-x^2\right )^{3/4} \sqrt [4]{-x^3+x^5}}\\ &=\frac {(2-2 i) x \sqrt [4]{1-x^2} F_1\left (\frac {1}{8};-\frac {3}{4},1;\frac {9}{8};x^2,i x^2\right )}{\sqrt [4]{-x^3+x^5}}+\frac {(2+2 i) x \sqrt [4]{1-x^2} F_1\left (\frac {1}{8};1,-\frac {3}{4};\frac {9}{8};-i x^2,x^2\right )}{\sqrt [4]{-x^3+x^5}}\\ \end {align*}

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Mathematica [F]  time = 0.83, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-x^4}{\left (1+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^4)*(-x^3 + x^5)^(1/4)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 23.57, size = 394, normalized size = 0.84 \begin {gather*} -\frac {1}{2} \sqrt [4]{-4+3 \sqrt {2}} \tan ^{-1}\left (\frac {\sqrt [4]{-8+6 \sqrt {2}} x \sqrt [4]{-x^3+x^5}}{\sqrt [4]{2} x^2-\sqrt {-x^3+x^5}}\right )-\frac {1}{2} \sqrt [4]{4+3 \sqrt {2}} \tan ^{-1}\left (\frac {\sqrt [4]{8+6 \sqrt {2}} x \sqrt [4]{-x^3+x^5}}{\sqrt [4]{2} x^2-\sqrt {-x^3+x^5}}\right )+\frac {1}{2} \sqrt [4]{-4+3 \sqrt {2}} \tanh ^{-1}\left (\frac {2 \sqrt [4]{\frac {1}{8}+\frac {3}{16 \sqrt {2}}} x^2+2^{3/4} \sqrt [4]{\frac {1}{8}+\frac {3}{16 \sqrt {2}}} \sqrt {-x^3+x^5}}{x \sqrt [4]{-x^3+x^5}}\right )-\frac {1}{4} \sqrt [4]{4+3 \sqrt {2}} \log \left (2 x^2-2 \sqrt [4]{4+3 \sqrt {2}} x \sqrt [4]{-x^3+x^5}+2^{3/4} \sqrt {-x^3+x^5}\right )+\frac {1}{4} \sqrt [4]{4+3 \sqrt {2}} \log \left (\sqrt {2-\sqrt {2}} x^2+2^{3/8} x \sqrt [4]{-x^3+x^5}+\sqrt {-1+\sqrt {2}} \sqrt {-x^3+x^5}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/2*((-4 + 3*Sqrt[2])^(1/4)*ArcTan[((-8 + 6*Sqrt[2])^(1/4)*x*(-x^3 + x^5)^(1/4))/(2^(1/4)*x^2 - Sqrt[-x^3 + x
^5])]) - ((4 + 3*Sqrt[2])^(1/4)*ArcTan[((8 + 6*Sqrt[2])^(1/4)*x*(-x^3 + x^5)^(1/4))/(2^(1/4)*x^2 - Sqrt[-x^3 +
x^5])])/2 + ((-4 + 3*Sqrt[2])^(1/4)*ArcTanh[(2*(1/8 + 3/(16*Sqrt[2]))^(1/4)*x^2 + 2^(3/4)*(1/8 + 3/(16*Sqrt[2
]))^(1/4)*Sqrt[-x^3 + x^5])/(x*(-x^3 + x^5)^(1/4))])/2 - ((4 + 3*Sqrt[2])^(1/4)*Log[2*x^2 - 2*(4 + 3*Sqrt[2])^
(1/4)*x*(-x^3 + x^5)^(1/4) + 2^(3/4)*Sqrt[-x^3 + x^5]])/4 + ((4 + 3*Sqrt[2])^(1/4)*Log[Sqrt[2 - Sqrt[2]]*x^2 +
2^(3/8)*x*(-x^3 + x^5)^(1/4) + Sqrt[-1 + Sqrt[2]]*Sqrt[-x^3 + x^5]])/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+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} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 93.09, size = 4195, normalized size = 8.94

method	result	size
trager	\(\text {Expression too large to display}\)	\(4195\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*ln((-16*RootOf(_Z^8+128)^9*x^4-3072*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*x^2-56*Ro
otOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*RootOf(_Z^8+128)^4*x^4-800*RootOf(-_Z*RootOf(_Z^8+128
)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*RootOf(_Z^8+128)^4*x^3+56*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+
8*_Z^2)*RootOf(_Z^8+128)^4*x^2+26*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*RootOf(_Z^8+128)^
8*x^2+400*(x^5-x^3)^(1/2)*RootOf(_Z^8+128)^6*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*x+56*R
ootOf(_Z^8+128)^5*x^3+400*RootOf(_Z^8+128)^5*x^2-400*RootOf(_Z^8+128)^5*x^4+4096*RootOf(-_Z*RootOf(_Z^8+128)^5
-8*RootOf(_Z^8+128)^2+8*_Z^2)*x^3+3072*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*x^4-2496*Roo
tOf(_Z^8+128)*x^4-3328*RootOf(_Z^8+128)*x^3+2496*RootOf(_Z^8+128)*x^2+24*RootOf(_Z^8+128)^9*x^3+16*RootOf(_Z^8
+128)^9*x^2-26*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*RootOf(_Z^8+128)^8*x^4+39*RootOf(-_Z
*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*RootOf(_Z^8+128)^8*x^3+1792*(x^5-x^3)^(3/4)-28*(x^5-x^3)^(1/2
)*RootOf(_Z^8+128)^7*x+56*(x^5-x^3)^(1/4)*RootOf(_Z^8+128)^6*x^2+3200*(x^5-x^3)^(1/2)*RootOf(_Z^8+128)^3*x-640
0*(x^5-x^3)^(1/4)*RootOf(_Z^8+128)^2*x^2+448*(x^5-x^3)^(1/2)*RootOf(_Z^8+128)^2*RootOf(-_Z*RootOf(_Z^8+128)^5-
8*RootOf(_Z^8+128)^2+8*_Z^2)*x+1600*RootOf(_Z^8+128)^4*(x^5-x^3)^(3/4))/x^2/(2*RootOf(_Z^8+128)^4*x^2-3*RootOf
(_Z^8+128)^4*x-2*RootOf(_Z^8+128)^4-24*x^2-32*x+24))*RootOf(_Z^8+128)^4*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf
(_Z^8+128)^2+8*_Z^2)-1/4*ln((-16*RootOf(_Z^8+128)^9*x^4-3072*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^
2+8*_Z^2)*x^2-56*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*RootOf(_Z^8+128)^4*x^4-800*RootOf(
-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*RootOf(_Z^8+128)^4*x^3+56*RootOf(-_Z*RootOf(_Z^8+128)^5-8*
RootOf(_Z^8+128)^2+8*_Z^2)*RootOf(_Z^8+128)^4*x^2+26*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2
)*RootOf(_Z^8+128)^8*x^2+400*(x^5-x^3)^(1/2)*RootOf(_Z^8+128)^6*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+12
8)^2+8*_Z^2)*x+56*RootOf(_Z^8+128)^5*x^3+400*RootOf(_Z^8+128)^5*x^2-400*RootOf(_Z^8+128)^5*x^4+4096*RootOf(-_Z
*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*x^3+3072*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8
*_Z^2)*x^4-2496*RootOf(_Z^8+128)*x^4-3328*RootOf(_Z^8+128)*x^3+2496*RootOf(_Z^8+128)*x^2+24*RootOf(_Z^8+128)^9
*x^3+16*RootOf(_Z^8+128)^9*x^2-26*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*RootOf(_Z^8+128)^
8*x^4+39*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*RootOf(_Z^8+128)^8*x^3+1792*(x^5-x^3)^(3/4
)-28*(x^5-x^3)^(1/2)*RootOf(_Z^8+128)^7*x+56*(x^5-x^3)^(1/4)*RootOf(_Z^8+128)^6*x^2+3200*(x^5-x^3)^(1/2)*RootO
f(_Z^8+128)^3*x-6400*(x^5-x^3)^(1/4)*RootOf(_Z^8+128)^2*x^2+448*(x^5-x^3)^(1/2)*RootOf(_Z^8+128)^2*RootOf(-_Z*
RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*x+1600*RootOf(_Z^8+128)^4*(x^5-x^3)^(3/4))/x^2/(2*RootOf(_Z^8+
128)^4*x^2-3*RootOf(_Z^8+128)^4*x-2*RootOf(_Z^8+128)^4-24*x^2-32*x+24))*RootOf(_Z^8+128)+1/4*RootOf(-_Z*RootOf
(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*ln(-(-4*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*R
ootOf(_Z^8+128)^8*x^4+6*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*RootOf(_Z^8+128)^8*x^3+4*Ro
otOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*RootOf(_Z^8+128)^8*x^2-100*(x^5-x^3)^(1/2)*RootOf(_Z^
8+128)^7*x+7*(x^5-x^3)^(1/2)*RootOf(_Z^8+128)^6*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*x-1
4*(x^5-x^3)^(1/4)*RootOf(_Z^8+128)^6*x^2-200*(x^5-x^3)^(1/4)*RootOf(_Z^8+128)^5*RootOf(-_Z*RootOf(_Z^8+128)^5-
8*RootOf(_Z^8+128)^2+8*_Z^2)*x^2+100*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*RootOf(_Z^8+12
8)^4*x^4-14*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*RootOf(_Z^8+128)^4*x^3+400*RootOf(_Z^8+
128)^4*(x^5-x^3)^(3/4)-100*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*RootOf(_Z^8+128)^4*x^2+1
12*(x^5-x^3)^(1/2)*RootOf(_Z^8+128)^3*x+800*(x^5-x^3)^(1/2)*RootOf(_Z^8+128)^2*RootOf(-_Z*RootOf(_Z^8+128)^5-8
*RootOf(_Z^8+128)^2+8*_Z^2)*x-1600*(x^5-x^3)^(1/4)*RootOf(_Z^8+128)^2*x^2+224*(x^5-x^3)^(1/4)*RootOf(_Z^8+128)
*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*x^2-624*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^
8+128)^2+8*_Z^2)*x^4-832*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*x^3-448*(x^5-x^3)^(3/4)+62
4*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*x^2)/x^2/(2*RootOf(_Z^8+128)^4*x^2-3*RootOf(_Z^8+
128)^4*x-2*RootOf(_Z^8+128)^4+24*x^2+32*x-24))-1/4*RootOf(_Z^8+128)*ln((-4*RootOf(_Z^8+128)^9*x^4+6*RootOf(_Z^
8+128)^9*x^3+4*RootOf(_Z^8+128)^9*x^2+7*(x^5-x^3)^(1/2)*RootOf(_Z^8+128)^7*x-14*(x^5-x^3)^(1/4)*RootOf(_Z^8+12
8)^6*x^2-100*RootOf(_Z^8+128)^5*x^4+14*RootOf(_Z^8+128)^5*x^3+400*RootOf(_Z^8+128)^4*(x^5-x^3)^(3/4)+100*RootO
f(_Z^8+128)^5*x^2-800*(x^5-x^3)^(1/2)*RootOf(_Z^8+128)^3*x+1600*(x^5-x^3)^(1/4)*RootOf(_Z^8+128)^2*x^2-624*Roo
tOf(_Z^8+128)*x^4-832*RootOf(_Z^8+128)*x^3+448*(x^5-x^3)^(3/4)+624*RootOf(_Z^8+128)*x^2)/x^2/(2*RootOf(_Z^8+12
8)^4*x^2-3*RootOf(_Z^8+128)^4*x-2*RootOf(_Z^8+128)^4-24*x^2-32*x+24))+1/32*ln(-(26*RootOf(_Z^8+128)^9*x^4-2496
*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*x^2-400*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^
8+128)^2+8*_Z^2)*RootOf(_Z^8+128)^4*x^4+56*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*RootOf(_
Z^8+128)^4*x^3+400*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*RootOf(_Z^8+128)^4*x^2-16*RootOf
(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*RootOf(_Z^8+128)^8*x^2-28*(x^5-x^3)^(1/2)*RootOf(_Z^8+128
)^6*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*x-800*RootOf(_Z^8+128)^5*x^3+56*RootOf(_Z^8+128
)^5*x^2-56*RootOf(_Z^8+128)^5*x^4+3328*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*x^3+2496*Roo
tOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*x^4-3072*RootOf(_Z^8+128)*x^4-4096*RootOf(_Z^8+128)*x^
3+3072*RootOf(_Z^8+128)*x^2-39*RootOf(_Z^8+128)^9*x^3-26*RootOf(_Z^8+128)^9*x^2+16*RootOf(-_Z*RootOf(_Z^8+128)
^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*RootOf(_Z^8+128)^8*x^4-24*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8
*_Z^2)*RootOf(_Z^8+128)^8*x^3-1792*(x^5-x^3)^(3/4)+56*(x^5-x^3)^(1/4)*RootOf(_Z^8+128)^6*x^2+6400*(x^5-x^3)^(1
/4)*RootOf(_Z^8+128)^2*x^2-3200*(x^5-x^3)^(1/2)*RootOf(_Z^8+128)^2*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8
+128)^2+8*_Z^2)*x+800*(x^5-x^3)^(1/4)*RootOf(_Z^8+128)^5*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*
_Z^2)*x^2-896*(x^5-x^3)^(1/4)*RootOf(_Z^8+128)*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*x^2+
1600*RootOf(_Z^8+128)^4*(x^5-x^3)^(3/4))/x^2/(2*RootOf(_Z^8+128)^4*x^2-3*RootOf(_Z^8+128)^4*x-2*RootOf(_Z^8+12
8)^4+24*x^2+32*x-24))*RootOf(_Z^8+128)^5-1/4*ln(-(26*RootOf(_Z^8+128)^9*x^4-2496*RootOf(-_Z*RootOf(_Z^8+128)^5
-8*RootOf(_Z^8+128)^2+8*_Z^2)*x^2-400*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*RootOf(_Z^8+1
28)^4*x^4+56*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*RootOf(_Z^8+128)^4*x^3+400*RootOf(-_Z*
RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*RootOf(_Z^8+128)^4*x^2-16*RootOf(-_Z*RootOf(_Z^8+128)^5-8*Root
Of(_Z^8+128)^2+8*_Z^2)*RootOf(_Z^8+128)^8*x^2-28*(x^5-x^3)^(1/2)*RootOf(_Z^8+128)^6*RootOf(-_Z*RootOf(_Z^8+128
)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*x-800*RootOf(_Z^8+128)^5*x^3+56*RootOf(_Z^8+128)^5*x^2-56*RootOf(_Z^8+128)^5*
x^4+3328*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*x^3+2496*RootOf(-_Z*RootOf(_Z^8+128)^5-8*R
ootOf(_Z^8+128)^2+8*_Z^2)*x^4-3072*RootOf(_Z^8+128)*x^4-4096*RootOf(_Z^8+128)*x^3+3072*RootOf(_Z^8+128)*x^2-39
*RootOf(_Z^8+128)^9*x^3-26*RootOf(_Z^8+128)^9*x^2+16*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2
)*RootOf(_Z^8+128)^8*x^4-24*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*RootOf(_Z^8+128)^8*x^3-
1792*(x^5-x^3)^(3/4)+56*(x^5-x^3)^(1/4)*RootOf(_Z^8+128)^6*x^2+6400*(x^5-x^3)^(1/4)*RootOf(_Z^8+128)^2*x^2-320
0*(x^5-x^3)^(1/2)*RootOf(_Z^8+128)^2*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*x+800*(x^5-x^3
)^(1/4)*RootOf(_Z^8+128)^5*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*x^2-896*(x^5-x^3)^(1/4)*
RootOf(_Z^8+128)*RootOf(-_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)*x^2+1600*RootOf(_Z^8+128)^4*(x^5-x
^3)^(3/4))/x^2/(2*RootOf(_Z^8+128)^4*x^2-3*RootOf(_Z^8+128)^4*x-2*RootOf(_Z^8+128)^4+24*x^2+32*x-24))*RootOf(-
_Z*RootOf(_Z^8+128)^5-8*RootOf(_Z^8+128)^2+8*_Z^2)

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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} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate((x^4 - 1)/((x^5 - x^3)^(1/4)*(x^4 + 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^4+1\right )\,{\left (x^5-x^3\right )}^{1/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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