3.73 \(\int \frac {x}{\sqrt {1-x^3} (4-x^3)} \, dx\)

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

[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 = 0.02, 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 = {484} \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {1-x^3}}\right )}{3\ 2^{2/3} \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {1-x^3}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{2} x+1}{\sqrt {1-x^3}}\right )}{3\ 2^{2/3}}+\frac {\tanh ^{-1}\left (\sqrt {1-x^3}\right )}{9\ 2^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[x/(Sqrt[1 - x^3]*(4 - x^3)),x]

[Out]

-ArcTan[(Sqrt[3]*(1 - 2^(1/3)*x))/Sqrt[1 - x^3]]/(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 484

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]  time = 0.02, size = 28, normalized size = 0.22 \[ \frac {1}{8} x^2 F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};x^3,\frac {x^3}{4}\right ) \]

Warning: Unable to verify antiderivative.

[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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fricas [B]  time = 1.26, size = 1191, normalized size = 9.38 \[ \text {result too large to display} \]

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

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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maple [C]  time = 0.19, size = 164, normalized size = 1.29 \[ \frac {i \sqrt {i \left (2 x +1-i \sqrt {3}\right )}\, \sqrt {\frac {x -1}{i \sqrt {3}-3}}\, \sqrt {-\frac {i \left (2 x +1+i \sqrt {3}\right )}{2}}\, \left (-2 \RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+\RootOf \left (\textit {\_Z}^{3}-4\right )+1+i \sqrt {3}\, \left (-\RootOf \left (\textit {\_Z}^{3}-4\right )+1\right )\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, -\frac {i \sqrt {3}\, \RootOf \left (\textit {\_Z}^{3}-4\right )^{2}}{3}+\frac {\RootOf \left (\textit {\_Z}^{3}-4\right )}{2}+\frac {i \sqrt {3}\, \RootOf \left (\textit {\_Z}^{3}-4\right )}{6}-\frac {1}{2}+\frac {i \sqrt {3}}{6}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{36 \sqrt {-x^{3}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*I*2^(1/2)*sum(_alpha^2*(1/2*I*(2*x+1-I*3^(1/2)))^(1/2)*((x-1)/(I*3^(1/2)-3))^(1/2)*(-1/2*I*(2*x+1+I*3^(1/
2)))^(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 \[ -\int \frac {x}{{\left (x^{3} - 4\right )} \sqrt {-x^{3} + 1}}\,{d x} \]

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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mupad [B]  time = 0.45, size = 653, normalized size = 5.14 \[ \text {result too large to display} \]

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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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {x}{x^{3} \sqrt {1 - x^{3}} - 4 \sqrt {1 - x^{3}}}\, dx \]

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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