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

Optimal. Leaf size=151 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^5+x^3}}\right )}{\sqrt [8]{2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^5+x^3}}\right )}{\sqrt [8]{2}}-\frac {\tan ^{-1}\left (\frac {2^{5/8} x \sqrt [4]{x^5+x^3}}{\sqrt [4]{2} x^2-\sqrt {x^5+x^3}}\right )}{2^{5/8}}+\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{2^{3/8}}+\frac {\sqrt {x^5+x^3}}{2^{5/8}}}{x \sqrt [4]{x^5+x^3}}\right )}{2^{5/8}} \]

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Rubi [C]  time = 0.66, antiderivative size = 97, normalized size of antiderivative = 0.64, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2056, 1586, 6715, 6725, 429} \begin {gather*} \frac {(2-2 i) x \sqrt [4]{x^2+1} 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}}+\frac {(2+2 i) x \sqrt [4]{x^2+1} 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, 1, -3/4, 9/8, (-I)*x^2, -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 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 {(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}}+\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 = 0.85, 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 = 0.57, size = 151, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^3+x^5}}\right )}{\sqrt [8]{2}}-\frac {\tan ^{-1}\left (\frac {2^{5/8} x \sqrt [4]{x^3+x^5}}{\sqrt [4]{2} x^2-\sqrt {x^3+x^5}}\right )}{2^{5/8}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^3+x^5}}\right )}{\sqrt [8]{2}}+\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{2^{3/8}}+\frac {\sqrt {x^3+x^5}}{2^{5/8}}}{x \sqrt [4]{x^3+x^5}}\right )}{2^{5/8}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 23.49, size = 2372, normalized size = 15.71

result too large to display

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]

-1/2*2^(7/8)*arctan(-1/2*(4*(x^5 + x^3)^(3/4)*(2^(5/8)*x + 2^(1/8)*(x^2 + 1)) - (2^(5/8)*(x^6 + 4*x^5 + 4*x^4
+ 4*x^3 + x^2) + 2*sqrt(x^5 + x^3)*(2^(7/8)*(x^3 + 2*x^2 + x) + 2*2^(3/8)*(x^3 + x^2 + x)) + 2*2^(1/8)*(x^6 +
2*x^5 + 4*x^4 + 2*x^3 + x^2))*sqrt(3*2^(3/4) - 4*2^(1/4)) + 4*(x^5 + x^3)^(1/4)*(2^(7/8)*x^3 + 2^(3/8)*(x^4 +
x^2)))/(x^6 + x^2)) + 1/8*2^(7/8)*log((2^(7/8)*(x^6 - 2*x^5 + 4*x^4 - 2*x^3 + x^2) + 2*(x^5 + x^3)^(3/4)*(2*x^
2 - sqrt(2)*(x^2 - 2*x + 1) - 2*x + 2) - 2*sqrt(x^5 + x^3)*(2^(5/8)*(x^3 - 2*x^2 + x) - 2*2^(1/8)*(x^3 - x^2 +
x)) - 2^(3/8)*(x^6 - 4*x^5 + 4*x^4 - 4*x^3 + x^2) - 2*(x^5 + x^3)^(1/4)*(2^(3/4)*(x^4 - 2*x^3 + x^2) - 2*2^(1
/4)*(x^4 - x^3 + x^2)))/(x^6 + x^2)) - 1/8*2^(7/8)*log(-(2^(7/8)*(x^6 - 2*x^5 + 4*x^4 - 2*x^3 + x^2) - 2*(x^5
+ x^3)^(3/4)*(2*x^2 - sqrt(2)*(x^2 - 2*x + 1) - 2*x + 2) - 2*sqrt(x^5 + x^3)*(2^(5/8)*(x^3 - 2*x^2 + x) - 2*2^
(1/8)*(x^3 - x^2 + x)) - 2^(3/8)*(x^6 - 4*x^5 + 4*x^4 - 4*x^3 + x^2) + 2*(x^5 + x^3)^(1/4)*(2^(3/4)*(x^4 - 2*x
^3 + x^2) - 2*2^(1/4)*(x^4 - x^3 + x^2)))/(x^6 + x^2)) - 1/2*2^(3/8)*arctan(-(x^10 + 64*x^8 + 130*x^6 + 64*x^4
+ x^2 + 2*(x^5 + x^3)^(3/4)*(2^(5/8)*(x^6 - 79*x^4 - 79*x^2 + 1) + 2*2^(1/8)*(11*x^5 + 16*x^3 + 11*x)) + 16*s
qrt(2)*(x^9 + 5*x^7 + 5*x^5 + x^3) + 4*sqrt(x^5 + x^3)*(2^(3/4)*(15*x^6 + 32*x^4 + 15*x^2) + 2^(1/4)*(x^7 + 33
*x^5 + 33*x^3 + x)) - (16*(x^5 + x^3)^(3/4)*(2^(3/4)*(x^5 - 14*x^4 + 4*x^3 - 14*x^2 + x) - 2^(1/4)*(x^5 - 28*x
^4 + 4*x^3 - 28*x^2 + x)) + 2^(5/8)*(x^10 - 6*x^9 - 220*x^8 + 26*x^7 - 446*x^6 + 26*x^5 - 220*x^4 - 6*x^3 + x^
2) + 2*sqrt(x^5 + x^3)*(2^(7/8)*(x^7 - 11*x^6 - 79*x^5 - 16*x^4 - 79*x^3 - 11*x^2 + x) - 2^(3/8)*(x^7 - 22*x^6
- 79*x^5 - 32*x^4 - 79*x^3 - 22*x^2 + x)) - 4*(x^8 - 30*x^7 + 33*x^6 - 64*x^5 + 33*x^4 - 30*x^3 + x^2 - sqrt(
2)*(x^8 - 15*x^7 + 33*x^6 - 32*x^5 + 33*x^4 - 15*x^3 + x^2))*(x^5 + x^3)^(1/4) - 2^(1/8)*(x^10 - 12*x^9 - 220*
x^8 + 52*x^7 - 446*x^6 + 52*x^5 - 220*x^4 - 12*x^3 + x^2))*sqrt((3*2^(3/4)*(x^6 + x^2) + 4*(x^5 + x^3)^(3/4)*(
2^(7/8)*(2*x^2 - 3*x + 2) + 2^(3/8)*(3*x^2 - 4*x + 3)) + 8*sqrt(x^5 + x^3)*(3*x^3 - 4*x^2 + sqrt(2)*(2*x^3 - 3
*x^2 + 2*x) + 3*x) + 4*2^(1/4)*(x^6 + x^2) + 4*(x^5 + x^3)^(1/4)*(2^(5/8)*(3*x^4 - 4*x^3 + 3*x^2) + 2*2^(1/8)*
(2*x^4 - 3*x^3 + 2*x^2)))/(x^6 + x^2)) + 2*(x^5 + x^3)^(1/4)*(2^(7/8)*(3*x^8 - 13*x^6 - 13*x^4 + 3*x^2) + 2*2^
(3/8)*(41*x^7 + 80*x^5 + 41*x^3)))/(x^10 - 384*x^8 - 766*x^6 - 384*x^4 + x^2)) + 1/2*2^(3/8)*arctan(-(x^10 + 6
4*x^8 + 130*x^6 + 64*x^4 + x^2 - 2*(x^5 + x^3)^(3/4)*(2^(5/8)*(x^6 - 79*x^4 - 79*x^2 + 1) + 2*2^(1/8)*(11*x^5
+ 16*x^3 + 11*x)) + 16*sqrt(2)*(x^9 + 5*x^7 + 5*x^5 + x^3) + 4*sqrt(x^5 + x^3)*(2^(3/4)*(15*x^6 + 32*x^4 + 15*
x^2) + 2^(1/4)*(x^7 + 33*x^5 + 33*x^3 + x)) - (16*(x^5 + x^3)^(3/4)*(2^(3/4)*(x^5 - 14*x^4 + 4*x^3 - 14*x^2 +
x) - 2^(1/4)*(x^5 - 28*x^4 + 4*x^3 - 28*x^2 + x)) - 2^(5/8)*(x^10 - 6*x^9 - 220*x^8 + 26*x^7 - 446*x^6 + 26*x^
5 - 220*x^4 - 6*x^3 + x^2) - 2*sqrt(x^5 + x^3)*(2^(7/8)*(x^7 - 11*x^6 - 79*x^5 - 16*x^4 - 79*x^3 - 11*x^2 + x)
- 2^(3/8)*(x^7 - 22*x^6 - 79*x^5 - 32*x^4 - 79*x^3 - 22*x^2 + x)) - 4*(x^8 - 30*x^7 + 33*x^6 - 64*x^5 + 33*x^
4 - 30*x^3 + x^2 - sqrt(2)*(x^8 - 15*x^7 + 33*x^6 - 32*x^5 + 33*x^4 - 15*x^3 + x^2))*(x^5 + x^3)^(1/4) + 2^(1/
8)*(x^10 - 12*x^9 - 220*x^8 + 52*x^7 - 446*x^6 + 52*x^5 - 220*x^4 - 12*x^3 + x^2))*sqrt((3*2^(3/4)*(x^6 + x^2)
- 4*(x^5 + x^3)^(3/4)*(2^(7/8)*(2*x^2 - 3*x + 2) + 2^(3/8)*(3*x^2 - 4*x + 3)) + 8*sqrt(x^5 + x^3)*(3*x^3 - 4*
x^2 + sqrt(2)*(2*x^3 - 3*x^2 + 2*x) + 3*x) + 4*2^(1/4)*(x^6 + x^2) - 4*(x^5 + x^3)^(1/4)*(2^(5/8)*(3*x^4 - 4*x
^3 + 3*x^2) + 2*2^(1/8)*(2*x^4 - 3*x^3 + 2*x^2)))/(x^6 + x^2)) - 2*(x^5 + x^3)^(1/4)*(2^(7/8)*(3*x^8 - 13*x^6
- 13*x^4 + 3*x^2) + 2*2^(3/8)*(41*x^7 + 80*x^5 + 41*x^3)))/(x^10 - 384*x^8 - 766*x^6 - 384*x^4 + x^2)) + 1/8*2
^(3/8)*log(4*(3*2^(3/4)*(x^6 + x^2) + 4*(x^5 + x^3)^(3/4)*(2^(7/8)*(2*x^2 - 3*x + 2) + 2^(3/8)*(3*x^2 - 4*x +
3)) + 8*sqrt(x^5 + x^3)*(3*x^3 - 4*x^2 + sqrt(2)*(2*x^3 - 3*x^2 + 2*x) + 3*x) + 4*2^(1/4)*(x^6 + x^2) + 4*(x^5
+ x^3)^(1/4)*(2^(5/8)*(3*x^4 - 4*x^3 + 3*x^2) + 2*2^(1/8)*(2*x^4 - 3*x^3 + 2*x^2)))/(x^6 + x^2)) - 1/8*2^(3/8
)*log(4*(3*2^(3/4)*(x^6 + x^2) - 4*(x^5 + x^3)^(3/4)*(2^(7/8)*(2*x^2 - 3*x + 2) + 2^(3/8)*(3*x^2 - 4*x + 3)) +
8*sqrt(x^5 + x^3)*(3*x^3 - 4*x^2 + sqrt(2)*(2*x^3 - 3*x^2 + 2*x) + 3*x) + 4*2^(1/4)*(x^6 + x^2) - 4*(x^5 + x^
3)^(1/4)*(2^(5/8)*(3*x^4 - 4*x^3 + 3*x^2) + 2*2^(1/8)*(2*x^4 - 3*x^3 + 2*x^2)))/(x^6 + x^2))

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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 = 52.17, size = 2874, normalized size = 19.03

method	result	size
trager	\(\text {Expression too large to display}\)	\(2874\)

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]

-1/4*RootOf(_Z^8-128)*ln((RootOf(_Z^8-128)^9*x^4-2*RootOf(_Z^8-128)^9*x^3+RootOf(_Z^8-128)^9*x^2-8*(x^5+x^3)^(
1/2)*RootOf(_Z^8-128)^7*x+16*(x^5+x^3)^(1/4)*RootOf(_Z^8-128)^6*x^2-16*RootOf(_Z^8-128)^5*x^3-256*RootOf(_Z^8-
128)*x^4+256*RootOf(_Z^8-128)*x^3+512*(x^5+x^3)^(3/4)-256*RootOf(_Z^8-128)*x^2)/(RootOf(_Z^8-128)^4*x^2-2*Root
Of(_Z^8-128)^4*x+RootOf(_Z^8-128)^4+16*x^2-16*x+16)/x^2)-RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8-128)^2+32*
_Z^2)*ln((RootOf(_Z^8-128)^8*x^4*RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8-128)^2+32*_Z^2)-2*RootOf(RootOf(_Z
^8-128)^5*_Z+2*RootOf(_Z^8-128)^2+32*_Z^2)*RootOf(_Z^8-128)^8*x^3+RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8-1
28)^2+32*_Z^2)*RootOf(_Z^8-128)^8*x^2+8*(x^5+x^3)^(1/2)*RootOf(_Z^8-128)^6*RootOf(RootOf(_Z^8-128)^5*_Z+2*Root
Of(_Z^8-128)^2+32*_Z^2)*x+4*(x^5+x^3)^(1/4)*RootOf(_Z^8-128)^6*x^2+16*RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z
^8-128)^2+32*_Z^2)*RootOf(_Z^8-128)^4*x^3+32*(x^5+x^3)^(1/2)*RootOf(_Z^8-128)^3*x+256*RootOf(RootOf(_Z^8-128)^
5*_Z+2*RootOf(_Z^8-128)^2+32*_Z^2)*(x^5+x^3)^(1/4)*RootOf(_Z^8-128)*x^2-256*RootOf(RootOf(_Z^8-128)^5*_Z+2*Roo
tOf(_Z^8-128)^2+32*_Z^2)*x^4+256*RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8-128)^2+32*_Z^2)*x^3-256*RootOf(Roo
tOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8-128)^2+32*_Z^2)*x^2+128*(x^5+x^3)^(3/4))/(RootOf(_Z^8-128)^4*x^2-2*RootOf(_Z^
8-128)^4*x+RootOf(_Z^8-128)^4-16*x^2+16*x-16)/x^2)-1/8*ln((8*RootOf(_Z^8-128)^8*x^4*RootOf(RootOf(_Z^8-128)^5*
_Z+2*RootOf(_Z^8-128)^2+32*_Z^2)+RootOf(_Z^8-128)^9*x^4-16*RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8-128)^2+3
2*_Z^2)*RootOf(_Z^8-128)^8*x^3-2*RootOf(_Z^8-128)^9*x^3+8*RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8-128)^2+32
*_Z^2)*RootOf(_Z^8-128)^8*x^2+RootOf(_Z^8-128)^9*x^2+8*(x^5+x^3)^(1/2)*RootOf(_Z^8-128)^7*x-64*RootOf(RootOf(_
Z^8-128)^5*_Z+2*RootOf(_Z^8-128)^2+32*_Z^2)*RootOf(_Z^8-128)^4*x^4-16*(x^5+x^3)^(1/4)*RootOf(_Z^8-128)^6*x^2-1
6*RootOf(_Z^8-128)^5*x^3-64*RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8-128)^2+32*_Z^2)*RootOf(_Z^8-128)^4*x^2+
512*(x^5+x^3)^(1/2)*RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8-128)^2+32*_Z^2)*RootOf(_Z^8-128)^2*x-1024*RootO
f(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8-128)^2+32*_Z^2)*x^4-256*RootOf(_Z^8-128)*x^4+1024*RootOf(RootOf(_Z^8-128
)^5*_Z+2*RootOf(_Z^8-128)^2+32*_Z^2)*x^3+256*RootOf(_Z^8-128)*x^3-1024*RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_
Z^8-128)^2+32*_Z^2)*x^2+512*(x^5+x^3)^(3/4)-256*RootOf(_Z^8-128)*x^2)/(RootOf(_Z^8-128)^4*x^2-2*RootOf(_Z^8-12
8)^4*x+RootOf(_Z^8-128)^4+16*x^2-16*x+16)/x^2)*RootOf(_Z^8-128)^4*RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8-1
28)^2+32*_Z^2)-1/4*ln((8*RootOf(_Z^8-128)^8*x^4*RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8-128)^2+32*_Z^2)+Roo
tOf(_Z^8-128)^9*x^4-16*RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8-128)^2+32*_Z^2)*RootOf(_Z^8-128)^8*x^3-2*Roo
tOf(_Z^8-128)^9*x^3+8*RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8-128)^2+32*_Z^2)*RootOf(_Z^8-128)^8*x^2+RootOf
(_Z^8-128)^9*x^2+8*(x^5+x^3)^(1/2)*RootOf(_Z^8-128)^7*x-64*RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8-128)^2+3
2*_Z^2)*RootOf(_Z^8-128)^4*x^4-16*(x^5+x^3)^(1/4)*RootOf(_Z^8-128)^6*x^2-16*RootOf(_Z^8-128)^5*x^3-64*RootOf(R
ootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8-128)^2+32*_Z^2)*RootOf(_Z^8-128)^4*x^2+512*(x^5+x^3)^(1/2)*RootOf(RootOf(_Z
^8-128)^5*_Z+2*RootOf(_Z^8-128)^2+32*_Z^2)*RootOf(_Z^8-128)^2*x-1024*RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^
8-128)^2+32*_Z^2)*x^4-256*RootOf(_Z^8-128)*x^4+1024*RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8-128)^2+32*_Z^2)
*x^3+256*RootOf(_Z^8-128)*x^3-1024*RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8-128)^2+32*_Z^2)*x^2+512*(x^5+x^3
)^(3/4)-256*RootOf(_Z^8-128)*x^2)/(RootOf(_Z^8-128)^4*x^2-2*RootOf(_Z^8-128)^4*x+RootOf(_Z^8-128)^4+16*x^2-16*
x+16)/x^2)*RootOf(_Z^8-128)-1/32*ln((2*RootOf(_Z^8-128)^8*x^4*RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8-128)^
2+32*_Z^2)-RootOf(_Z^8-128)^9*x^4-4*RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8-128)^2+32*_Z^2)*RootOf(_Z^8-128
)^8*x^3+2*RootOf(_Z^8-128)^9*x^3+2*RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8-128)^2+32*_Z^2)*RootOf(_Z^8-128)
^8*x^2-RootOf(_Z^8-128)^9*x^2+16*(x^5+x^3)^(1/2)*RootOf(_Z^8-128)^6*RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8
-128)^2+32*_Z^2)*x-8*(x^5+x^3)^(1/4)*RootOf(_Z^8-128)^6*x^2-8*RootOf(_Z^8-128)^5*x^4+32*RootOf(RootOf(_Z^8-128
)^5*_Z+2*RootOf(_Z^8-128)^2+32*_Z^2)*RootOf(_Z^8-128)^4*x^3-8*RootOf(_Z^8-128)^5*x^2-512*RootOf(RootOf(_Z^8-12
8)^5*_Z+2*RootOf(_Z^8-128)^2+32*_Z^2)*(x^5+x^3)^(1/4)*RootOf(_Z^8-128)*x^2-512*RootOf(RootOf(_Z^8-128)^5*_Z+2*
RootOf(_Z^8-128)^2+32*_Z^2)*x^4+128*RootOf(_Z^8-128)*x^4+512*RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8-128)^2
+32*_Z^2)*x^3-128*RootOf(_Z^8-128)*x^3-512*RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8-128)^2+32*_Z^2)*x^2+256*
(x^5+x^3)^(3/4)+128*RootOf(_Z^8-128)*x^2)/(RootOf(_Z^8-128)^4*x^2-2*RootOf(_Z^8-128)^4*x+RootOf(_Z^8-128)^4-16
*x^2+16*x-16)/x^2)*RootOf(_Z^8-128)^5-ln((2*RootOf(_Z^8-128)^8*x^4*RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8-
128)^2+32*_Z^2)-RootOf(_Z^8-128)^9*x^4-4*RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8-128)^2+32*_Z^2)*RootOf(_Z^
8-128)^8*x^3+2*RootOf(_Z^8-128)^9*x^3+2*RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8-128)^2+32*_Z^2)*RootOf(_Z^8
-128)^8*x^2-RootOf(_Z^8-128)^9*x^2+16*(x^5+x^3)^(1/2)*RootOf(_Z^8-128)^6*RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf
(_Z^8-128)^2+32*_Z^2)*x-8*(x^5+x^3)^(1/4)*RootOf(_Z^8-128)^6*x^2-8*RootOf(_Z^8-128)^5*x^4+32*RootOf(RootOf(_Z^
8-128)^5*_Z+2*RootOf(_Z^8-128)^2+32*_Z^2)*RootOf(_Z^8-128)^4*x^3-8*RootOf(_Z^8-128)^5*x^2-512*RootOf(RootOf(_Z
^8-128)^5*_Z+2*RootOf(_Z^8-128)^2+32*_Z^2)*(x^5+x^3)^(1/4)*RootOf(_Z^8-128)*x^2-512*RootOf(RootOf(_Z^8-128)^5*
_Z+2*RootOf(_Z^8-128)^2+32*_Z^2)*x^4+128*RootOf(_Z^8-128)*x^4+512*RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8-1
28)^2+32*_Z^2)*x^3-128*RootOf(_Z^8-128)*x^3-512*RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8-128)^2+32*_Z^2)*x^2
+256*(x^5+x^3)^(3/4)+128*RootOf(_Z^8-128)*x^2)/(RootOf(_Z^8-128)^4*x^2-2*RootOf(_Z^8-128)^4*x+RootOf(_Z^8-128)
^4-16*x^2+16*x-16)/x^2)*RootOf(RootOf(_Z^8-128)^5*_Z+2*RootOf(_Z^8-128)^2+32*_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 = -0.01 \begin {gather*} -\int \frac {x^4-1}{{\left (x^5+x^3\right )}^{1/4}\,\left (x^4+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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