3.1.73 \(\int \frac {x}{\sqrt {1-x^3} (4-x^3)} \, dx\) [73]

Optimal. Leaf size=127 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {1-x^3}}\right )}{3\ 2^{2/3} \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {1-x^3}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {1+\sqrt [3]{2} x}{\sqrt {1-x^3}}\right )}{3\ 2^{2/3}}+\frac {\tanh ^{-1}\left (\sqrt {1-x^3}\right )}{9\ 2^{2/3}} \]

[Out]

-1/6*arctanh((1+2^(1/3)*x)/(-x^3+1)^(1/2))*2^(1/3)+1/18*arctanh((-x^3+1)^(1/2))*2^(1/3)-1/18*arctan((1-2^(1/3)
*x)*3^(1/2)/(-x^3+1)^(1/2))*2^(1/3)*3^(1/2)+1/18*arctan(1/3*(-x^3+1)^(1/2)*3^(1/2))*2^(1/3)*3^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {497} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {1-x^3}}\right )}{3\ 2^{2/3} \sqrt {3}}+\frac {\text {ArcTan}\left (\frac {\sqrt {1-x^3}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{2} x+1}{\sqrt {1-x^3}}\right )}{3\ 2^{2/3}}+\frac {\tanh ^{-1}\left (\sqrt {1-x^3}\right )}{9\ 2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(Sqrt[1 - x^3]*(4 - x^3)),x]

[Out]

-1/3*ArcTan[(Sqrt[3]*(1 - 2^(1/3)*x))/Sqrt[1 - x^3]]/(2^(2/3)*Sqrt[3]) + ArcTan[Sqrt[1 - x^3]/Sqrt[3]]/(3*2^(2
/3)*Sqrt[3]) - ArcTanh[(1 + 2^(1/3)*x)/Sqrt[1 - x^3]]/(3*2^(2/3)) + ArcTanh[Sqrt[1 - x^3]]/(9*2^(2/3))

Rule 497

Int[(x_)/(((a_) + (b_.)*(x_)^3)*Sqrt[(c_) + (d_.)*(x_)^3]), x_Symbol] :> With[{q = Rt[d/c, 3]}, Simp[q*(ArcTan
h[Sqrt[c + d*x^3]/Rt[c, 2]]/(9*2^(2/3)*b*Rt[c, 2])), x] + (-Simp[q*(ArcTanh[Rt[c, 2]*((1 - 2^(1/3)*q*x)/Sqrt[c
+ d*x^3])]/(3*2^(2/3)*b*Rt[c, 2])), x] + Simp[q*(ArcTan[Sqrt[c + d*x^3]/(Sqrt[3]*Rt[c, 2])]/(3*2^(2/3)*Sqrt[3
]*b*Rt[c, 2])), x] - Simp[q*(ArcTan[Sqrt[3]*Rt[c, 2]*((1 + 2^(1/3)*q*x)/Sqrt[c + d*x^3])]/(3*2^(2/3)*Sqrt[3]*b
*Rt[c, 2])), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[4*b*c - a*d, 0] && PosQ[c]

Rubi steps

\begin {align*} \int \frac {x}{\sqrt {1-x^3} \left (4-x^3\right )} \, dx &=-\frac {\tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {1-x^3}}\right )}{3\ 2^{2/3} \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {1-x^3}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {1+\sqrt [3]{2} x}{\sqrt {1-x^3}}\right )}{3\ 2^{2/3}}+\frac {\tanh ^{-1}\left (\sqrt {1-x^3}\right )}{9\ 2^{2/3}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
time = 10.03, size = 28, normalized size = 0.22 \begin {gather*} \frac {1}{8} x^2 F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};x^3,\frac {x^3}{4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*AppellF1[2/3, 1/2, 1, 5/3, x^3, x^3/4])/8

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 5.86, size = 164, normalized size = 1.29

method	result	size
default	\(\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}-4\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {2}\, \sqrt {i \left (-i \sqrt {3}+2 x +1\right )}\, \sqrt {\frac {-1+x}{i \sqrt {3}-3}}\, \sqrt {-\frac {i \left (i \sqrt {3}+2 x +1\right )}{2}}\, \left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1+i \sqrt {3}\, \left (1-\underline {\hspace {1.25 ex}}\alpha \right )\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {\underline {\hspace {1.25 ex}}\alpha }{2}-\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{3}-\frac {1}{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{6}+\frac {i \sqrt {3}}{6}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {-x^{3}+1}}\right )}{36}\)	\(164\)
elliptic	\(\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}-4\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {2}\, \sqrt {i \left (-i \sqrt {3}+2 x +1\right )}\, \sqrt {\frac {-1+x}{i \sqrt {3}-3}}\, \sqrt {-\frac {i \left (i \sqrt {3}+2 x +1\right )}{2}}\, \left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1+i \sqrt {3}\, \left (1-\underline {\hspace {1.25 ex}}\alpha \right )\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {\underline {\hspace {1.25 ex}}\alpha }{2}-\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{3}-\frac {1}{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha \sqrt {3}}{6}+\frac {i \sqrt {3}}{6}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {-x^{3}+1}}\right )}{36}\)	\(164\)
trager	\(\text {Expression too large to display}\)	\(815\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*I*2^(1/2)*sum(_alpha^2*(1/2*I*(-I*3^(1/2)+2*x+1))^(1/2)*((-1+x)/(I*3^(1/2)-3))^(1/2)*(-1/2*I*(I*3^(1/2)+2
*x+1))^(1/2)/(-x^3+1)^(1/2)*(-2*_alpha^2+_alpha+1+I*3^(1/2)*(1-_alpha))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2-1/2*I
*3^(1/2))*3^(1/2))^(1/2),1/2*_alpha-1/3*I*_alpha^2*3^(1/2)-1/2+1/6*I*_alpha*3^(1/2)+1/6*I*3^(1/2),(I*3^(1/2)/(
-3/2+1/2*I*3^(1/2)))^(1/2)),_alpha=RootOf(_Z^3-4))

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1191 vs. \(2 (92) = 184\).
time = 1.60, size = 1191, normalized size = 9.38 \begin {gather*} -\frac {1}{31104} \cdot 432^{\frac {5}{6}} \sqrt {3} \log \left (\frac {144 \, {\left (36 \, x^{9} - 8208 \, x^{6} + 9504 \, x^{3} - 648 \cdot 2^{\frac {2}{3}} {\left (x^{8} - 5 \, x^{5} + 4 \, x^{2}\right )} + {\left (2592 \, x^{6} - 2592 \, x^{3} - 432^{\frac {5}{6}} \sqrt {3} {\left (x^{7} - 26 \, x^{4} + 16 \, x\right )} - 216 \cdot 432^{\frac {1}{6}} \sqrt {3} {\left (7 \, x^{5} - 4 \, x^{2}\right )}\right )} \sqrt {-x^{3} + 1} + 3888 \cdot 2^{\frac {1}{3}} {\left (x^{7} - x^{4}\right )} - 2304\right )}}{x^{9} - 12 \, x^{6} + 48 \, x^{3} - 64}\right ) - \frac {1}{31104} \cdot 432^{\frac {5}{6}} \sqrt {3} \log \left (\frac {36 \, {\left (36 \, x^{9} - 8208 \, x^{6} + 9504 \, x^{3} - 648 \cdot 2^{\frac {2}{3}} {\left (x^{8} - 5 \, x^{5} + 4 \, x^{2}\right )} + {\left (2592 \, x^{6} - 2592 \, x^{3} - 432^{\frac {5}{6}} \sqrt {3} {\left (x^{7} - 26 \, x^{4} + 16 \, x\right )} - 216 \cdot 432^{\frac {1}{6}} \sqrt {3} {\left (7 \, x^{5} - 4 \, x^{2}\right )}\right )} \sqrt {-x^{3} + 1} + 3888 \cdot 2^{\frac {1}{3}} {\left (x^{7} - x^{4}\right )} - 2304\right )}}{x^{9} - 12 \, x^{6} + 48 \, x^{3} - 64}\right ) + \frac {1}{31104} \cdot 432^{\frac {5}{6}} \sqrt {3} \log \left (\frac {144 \, {\left (36 \, x^{9} - 8208 \, x^{6} + 9504 \, x^{3} - 648 \cdot 2^{\frac {2}{3}} {\left (x^{8} - 5 \, x^{5} + 4 \, x^{2}\right )} - {\left (2592 \, x^{6} - 2592 \, x^{3} - 432^{\frac {5}{6}} \sqrt {3} {\left (x^{7} - 26 \, x^{4} + 16 \, x\right )} - 216 \cdot 432^{\frac {1}{6}} \sqrt {3} {\left (7 \, x^{5} - 4 \, x^{2}\right )}\right )} \sqrt {-x^{3} + 1} + 3888 \cdot 2^{\frac {1}{3}} {\left (x^{7} - x^{4}\right )} - 2304\right )}}{x^{9} - 12 \, x^{6} + 48 \, x^{3} - 64}\right ) + \frac {1}{31104} \cdot 432^{\frac {5}{6}} \sqrt {3} \log \left (\frac {36 \, {\left (36 \, x^{9} - 8208 \, x^{6} + 9504 \, x^{3} - 648 \cdot 2^{\frac {2}{3}} {\left (x^{8} - 5 \, x^{5} + 4 \, x^{2}\right )} - {\left (2592 \, x^{6} - 2592 \, x^{3} - 432^{\frac {5}{6}} \sqrt {3} {\left (x^{7} - 26 \, x^{4} + 16 \, x\right )} - 216 \cdot 432^{\frac {1}{6}} \sqrt {3} {\left (7 \, x^{5} - 4 \, x^{2}\right )}\right )} \sqrt {-x^{3} + 1} + 3888 \cdot 2^{\frac {1}{3}} {\left (x^{7} - x^{4}\right )} - 2304\right )}}{x^{9} - 12 \, x^{6} + 48 \, x^{3} - 64}\right ) - \frac {1}{1944} \cdot 432^{\frac {5}{6}} \arctan \left (\frac {\sqrt {-x^{3} + 1} {\left (72 \cdot 432^{\frac {1}{6}} x^{2} + 432^{\frac {5}{6}} x + 72 \, \sqrt {3}\right )}}{216 \, {\left (2 \, x^{3} - 1\right )}}\right ) + \frac {1}{3888} \cdot 432^{\frac {5}{6}} \arctan \left (-\frac {6 \, \sqrt {-x^{3} + 1} {\left (432^{\frac {5}{6}} {\left (x^{4} + 2 \, x\right )} - 36 \, \sqrt {3} {\left (x^{3} - 4\right )} + 18 \cdot 432^{\frac {1}{6}} {\left (x^{5} + 8 \, x^{2}\right )}\right )} + {\left (108 \, \sqrt {3} 2^{\frac {2}{3}} {\left (x^{5} - x^{2}\right )} - 216 \, \sqrt {3} 2^{\frac {1}{3}} {\left (x^{4} - x\right )} - 108 \, \sqrt {3} {\left (x^{6} - x^{3}\right )} - \sqrt {-x^{3} + 1} {\left (432^{\frac {5}{6}} {\left (2 \, x^{4} + x\right )} - 36 \, \sqrt {3} {\left (5 \, x^{3} - 8\right )} - 18 \cdot 432^{\frac {1}{6}} {\left (x^{5} - 10 \, x^{2}\right )}\right )}\right )} \sqrt {\frac {36 \, x^{9} - 8208 \, x^{6} + 9504 \, x^{3} - 648 \cdot 2^{\frac {2}{3}} {\left (x^{8} - 5 \, x^{5} + 4 \, x^{2}\right )} + {\left (2592 \, x^{6} - 2592 \, x^{3} - 432^{\frac {5}{6}} \sqrt {3} {\left (x^{7} - 26 \, x^{4} + 16 \, x\right )} - 216 \cdot 432^{\frac {1}{6}} \sqrt {3} {\left (7 \, x^{5} - 4 \, x^{2}\right )}\right )} \sqrt {-x^{3} + 1} + 3888 \cdot 2^{\frac {1}{3}} {\left (x^{7} - x^{4}\right )} - 2304}{x^{9} - 12 \, x^{6} + 48 \, x^{3} - 64}}}{648 \, {\left (x^{6} + 3 \, x^{3} - 4\right )}}\right ) + \frac {1}{3888} \cdot 432^{\frac {5}{6}} \arctan \left (-\frac {6 \, \sqrt {-x^{3} + 1} {\left (432^{\frac {5}{6}} {\left (x^{4} + 2 \, x\right )} - 36 \, \sqrt {3} {\left (x^{3} - 4\right )} + 18 \cdot 432^{\frac {1}{6}} {\left (x^{5} + 8 \, x^{2}\right )}\right )} - {\left (108 \, \sqrt {3} 2^{\frac {2}{3}} {\left (x^{5} - x^{2}\right )} - 216 \, \sqrt {3} 2^{\frac {1}{3}} {\left (x^{4} - x\right )} - 108 \, \sqrt {3} {\left (x^{6} - x^{3}\right )} + \sqrt {-x^{3} + 1} {\left (432^{\frac {5}{6}} {\left (2 \, x^{4} + x\right )} - 36 \, \sqrt {3} {\left (5 \, x^{3} - 8\right )} - 18 \cdot 432^{\frac {1}{6}} {\left (x^{5} - 10 \, x^{2}\right )}\right )}\right )} \sqrt {\frac {36 \, x^{9} - 8208 \, x^{6} + 9504 \, x^{3} - 648 \cdot 2^{\frac {2}{3}} {\left (x^{8} - 5 \, x^{5} + 4 \, x^{2}\right )} - {\left (2592 \, x^{6} - 2592 \, x^{3} - 432^{\frac {5}{6}} \sqrt {3} {\left (x^{7} - 26 \, x^{4} + 16 \, x\right )} - 216 \cdot 432^{\frac {1}{6}} \sqrt {3} {\left (7 \, x^{5} - 4 \, x^{2}\right )}\right )} \sqrt {-x^{3} + 1} + 3888 \cdot 2^{\frac {1}{3}} {\left (x^{7} - x^{4}\right )} - 2304}{x^{9} - 12 \, x^{6} + 48 \, x^{3} - 64}}}{648 \, {\left (x^{6} + 3 \, x^{3} - 4\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/31104*432^(5/6)*sqrt(3)*log(144*(36*x^9 - 8208*x^6 + 9504*x^3 - 648*2^(2/3)*(x^8 - 5*x^5 + 4*x^2) + (2592*x
^6 - 2592*x^3 - 432^(5/6)*sqrt(3)*(x^7 - 26*x^4 + 16*x) - 216*432^(1/6)*sqrt(3)*(7*x^5 - 4*x^2))*sqrt(-x^3 + 1
) + 3888*2^(1/3)*(x^7 - x^4) - 2304)/(x^9 - 12*x^6 + 48*x^3 - 64)) - 1/31104*432^(5/6)*sqrt(3)*log(36*(36*x^9
- 8208*x^6 + 9504*x^3 - 648*2^(2/3)*(x^8 - 5*x^5 + 4*x^2) + (2592*x^6 - 2592*x^3 - 432^(5/6)*sqrt(3)*(x^7 - 26
*x^4 + 16*x) - 216*432^(1/6)*sqrt(3)*(7*x^5 - 4*x^2))*sqrt(-x^3 + 1) + 3888*2^(1/3)*(x^7 - x^4) - 2304)/(x^9 -
12*x^6 + 48*x^3 - 64)) + 1/31104*432^(5/6)*sqrt(3)*log(144*(36*x^9 - 8208*x^6 + 9504*x^3 - 648*2^(2/3)*(x^8 -
5*x^5 + 4*x^2) - (2592*x^6 - 2592*x^3 - 432^(5/6)*sqrt(3)*(x^7 - 26*x^4 + 16*x) - 216*432^(1/6)*sqrt(3)*(7*x^
5 - 4*x^2))*sqrt(-x^3 + 1) + 3888*2^(1/3)*(x^7 - x^4) - 2304)/(x^9 - 12*x^6 + 48*x^3 - 64)) + 1/31104*432^(5/6
)*sqrt(3)*log(36*(36*x^9 - 8208*x^6 + 9504*x^3 - 648*2^(2/3)*(x^8 - 5*x^5 + 4*x^2) - (2592*x^6 - 2592*x^3 - 43
2^(5/6)*sqrt(3)*(x^7 - 26*x^4 + 16*x) - 216*432^(1/6)*sqrt(3)*(7*x^5 - 4*x^2))*sqrt(-x^3 + 1) + 3888*2^(1/3)*(
x^7 - x^4) - 2304)/(x^9 - 12*x^6 + 48*x^3 - 64)) - 1/1944*432^(5/6)*arctan(1/216*sqrt(-x^3 + 1)*(72*432^(1/6)*
x^2 + 432^(5/6)*x + 72*sqrt(3))/(2*x^3 - 1)) + 1/3888*432^(5/6)*arctan(-1/648*(6*sqrt(-x^3 + 1)*(432^(5/6)*(x^
4 + 2*x) - 36*sqrt(3)*(x^3 - 4) + 18*432^(1/6)*(x^5 + 8*x^2)) + (108*sqrt(3)*2^(2/3)*(x^5 - x^2) - 216*sqrt(3)
*2^(1/3)*(x^4 - x) - 108*sqrt(3)*(x^6 - x^3) - sqrt(-x^3 + 1)*(432^(5/6)*(2*x^4 + x) - 36*sqrt(3)*(5*x^3 - 8)
- 18*432^(1/6)*(x^5 - 10*x^2)))*sqrt((36*x^9 - 8208*x^6 + 9504*x^3 - 648*2^(2/3)*(x^8 - 5*x^5 + 4*x^2) + (2592
*x^6 - 2592*x^3 - 432^(5/6)*sqrt(3)*(x^7 - 26*x^4 + 16*x) - 216*432^(1/6)*sqrt(3)*(7*x^5 - 4*x^2))*sqrt(-x^3 +
1) + 3888*2^(1/3)*(x^7 - x^4) - 2304)/(x^9 - 12*x^6 + 48*x^3 - 64)))/(x^6 + 3*x^3 - 4)) + 1/3888*432^(5/6)*ar
ctan(-1/648*(6*sqrt(-x^3 + 1)*(432^(5/6)*(x^4 + 2*x) - 36*sqrt(3)*(x^3 - 4) + 18*432^(1/6)*(x^5 + 8*x^2)) - (1
08*sqrt(3)*2^(2/3)*(x^5 - x^2) - 216*sqrt(3)*2^(1/3)*(x^4 - x) - 108*sqrt(3)*(x^6 - x^3) + sqrt(-x^3 + 1)*(432
^(5/6)*(2*x^4 + x) - 36*sqrt(3)*(5*x^3 - 8) - 18*432^(1/6)*(x^5 - 10*x^2)))*sqrt((36*x^9 - 8208*x^6 + 9504*x^3
- 648*2^(2/3)*(x^8 - 5*x^5 + 4*x^2) - (2592*x^6 - 2592*x^3 - 432^(5/6)*sqrt(3)*(x^7 - 26*x^4 + 16*x) - 216*43
2^(1/6)*sqrt(3)*(7*x^5 - 4*x^2))*sqrt(-x^3 + 1) + 3888*2^(1/3)*(x^7 - x^4) - 2304)/(x^9 - 12*x^6 + 48*x^3 - 64
)))/(x^6 + 3*x^3 - 4))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x}{x^{3} \sqrt {1 - x^{3}} - 4 \sqrt {1 - x^{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(x/(x**3*sqrt(1 - x**3) - 4*sqrt(1 - x**3)), 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/(-x^3+4)/(-x^3+1)^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.45, size = 653, normalized size = 5.14 \begin {gather*} -\frac {2^{1/3}\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {x^3-1}\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{2^{2/3}-1};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{3\,\sqrt {1-x^3}\,\left (2^{2/3}-1\right )\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}}-\frac {2^{1/3}\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {x^3-1}\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{2^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )+1};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{3\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {1-x^3}\,\left (2^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )+1\right )\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}}-\frac {2^{1/3}\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {x^3-1}\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{2^{2/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{3\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {1-x^3}\,\left (2^{2/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x/((1 - x^3)^(1/2)*(x^3 - 4)),x)

[Out]

- (2^(1/3)*((3^(1/2)*1i)/2 + 3/2)*(x^3 - 1)^(1/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*(
(x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticPi(-(
(3^(1/2)*1i)/2 + 3/2)/(2^(2/3) - 1), asin((-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((
3^(1/2)*1i)/2 - 3/2)))/(3*(1 - x^3)^(1/2)*(2^(2/3) - 1)*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) - x*(((
3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) + x^3)^(1/2)) - (2^(1/3)*((3^(1/2)*1i)/2 + 3/2)*(x^3 - 1)^(1/
2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 3/
2))^(1/2)*(-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticPi(((3^(1/2)*1i)/2 + 3/2)/(2^(2/3)*((3^(1/2)*1i)/2 +
1/2) + 1), asin((-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)))/(3
*((3^(1/2)*1i)/2 + 1/2)*(1 - x^3)^(1/2)*(2^(2/3)*((3^(1/2)*1i)/2 + 1/2) + 1)*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)
*1i)/2 + 1/2) - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) + x^3)^(1/2)) - (2^(1/3)*((3^(1/2)*1i)/2
+ 3/2)*(x^3 - 1)^(1/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)
/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticPi(-((3^(1/2)*1i)/2 + 3/2)/(2^(
2/3)*((3^(1/2)*1i)/2 - 1/2) - 1), asin((-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(
1/2)*1i)/2 - 3/2)))/(3*((3^(1/2)*1i)/2 - 1/2)*(1 - x^3)^(1/2)*(2^(2/3)*((3^(1/2)*1i)/2 - 1/2) - 1)*(((3^(1/2)*
1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) + x^3)^(1/2))

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