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

Optimal result

Integrand size = 22, antiderivative size = 127 \[ \int \frac {x}{\sqrt {1-x^3} \left (4-x^3\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {1-x^3}}\right )}{3\ 2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {\sqrt {1-x^3}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}-\frac {\text {arctanh}\left (\frac {1+\sqrt [3]{2} x}{\sqrt {1-x^3}}\right )}{3\ 2^{2/3}}+\frac {\text {arctanh}\left (\sqrt {1-x^3}\right )}{9\ 2^{2/3}} \] Output:

-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)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 10.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.22 \[ \int \frac {x}{\sqrt {1-x^3} \left (4-x^3\right )} \, dx=\frac {1}{8} x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},x^3,\frac {x^3}{4}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {986}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 986

\(\displaystyle -\frac {\arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {1-x^3}}\right )}{3\ 2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {\sqrt {1-x^3}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}-\frac {\text {arctanh}\left (\frac {\sqrt [3]{2} x+1}{\sqrt {1-x^3}}\right )}{3\ 2^{2/3}}+\frac {\text {arctanh}\left (\sqrt {1-x^3}\right )}{9\ 2^{2/3}}\)

Input:

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

Output:

-1/3*ArcTan[(Sqrt[3]*(1 - 2^(1/3)*x))/Sqrt[1 - x^3]]/(2^(2/3)*Sqrt[3]) + A 
rcTan[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))
 

Defintions of rubi rules used

rule 986

Int[(x_)/(((a_) + (b_.)*(x_)^3)*Sqrt[(c_) + (d_.)*(x_)^3]), x_Symbol] :> Wi 
th[{q = Rt[d/c, 3]}, Simp[q*(ArcTanh[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]*R 
t[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]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 11.85 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.29

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

Input:

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

Output:

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=RootO 
f(_Z^3-4))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1019 vs. \(2 (92) = 184\).

Time = 0.14 (sec) , antiderivative size = 1019, normalized size of antiderivative = 8.02 \[ \int \frac {x}{\sqrt {1-x^3} \left (4-x^3\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/36*(-1/432)^(1/6)*(sqrt(-3) + 1)*log(-(x^9 + 66*x^6 - 72*x^3 - 24*(-1/2 
)^(2/3)*(x^7 + x^4 + sqrt(-3)*(x^7 + x^4 - 2*x) - 2*x) + 6*sqrt(-x^3 + 1)* 
(648*(-1/432)^(5/6)*(sqrt(-3)*x^5 - x^5) - sqrt(-1/3)*(5*x^6 + 20*x^3 - 16 
) - (-1/432)^(1/6)*(x^7 + 16*x^4 + sqrt(-3)*(x^7 + 16*x^4 - 8*x) - 8*x)) + 
 6*(-1/2)^(1/3)*(x^8 + 7*x^5 - 8*x^2 - sqrt(-3)*(x^8 + 7*x^5 - 8*x^2)) + 3 
2)/(x^9 - 12*x^6 + 48*x^3 - 64)) + 1/36*(-1/432)^(1/6)*(sqrt(-3) + 1)*log( 
-(x^9 + 66*x^6 - 72*x^3 - 24*(-1/2)^(2/3)*(x^7 + x^4 + sqrt(-3)*(x^7 + x^4 
 - 2*x) - 2*x) - 6*sqrt(-x^3 + 1)*(648*(-1/432)^(5/6)*(sqrt(-3)*x^5 - x^5) 
 - sqrt(-1/3)*(5*x^6 + 20*x^3 - 16) - (-1/432)^(1/6)*(x^7 + 16*x^4 + sqrt( 
-3)*(x^7 + 16*x^4 - 8*x) - 8*x)) + 6*(-1/2)^(1/3)*(x^8 + 7*x^5 - 8*x^2 - s 
qrt(-3)*(x^8 + 7*x^5 - 8*x^2)) + 32)/(x^9 - 12*x^6 + 48*x^3 - 64)) - 1/36* 
(-1/432)^(1/6)*(sqrt(-3) - 1)*log(-(x^9 + 66*x^6 - 72*x^3 - 24*(-1/2)^(2/3 
)*(x^7 + x^4 - sqrt(-3)*(x^7 + x^4 - 2*x) - 2*x) + 6*sqrt(-x^3 + 1)*(648*( 
-1/432)^(5/6)*(sqrt(-3)*x^5 + x^5) + sqrt(-1/3)*(5*x^6 + 20*x^3 - 16) + (- 
1/432)^(1/6)*(x^7 + 16*x^4 - sqrt(-3)*(x^7 + 16*x^4 - 8*x) - 8*x)) + 6*(-1 
/2)^(1/3)*(x^8 + 7*x^5 - 8*x^2 + sqrt(-3)*(x^8 + 7*x^5 - 8*x^2)) + 32)/(x^ 
9 - 12*x^6 + 48*x^3 - 64)) + 1/36*(-1/432)^(1/6)*(sqrt(-3) - 1)*log(-(x^9 
+ 66*x^6 - 72*x^3 - 24*(-1/2)^(2/3)*(x^7 + x^4 - sqrt(-3)*(x^7 + x^4 - 2*x 
) - 2*x) - 6*sqrt(-x^3 + 1)*(648*(-1/432)^(5/6)*(sqrt(-3)*x^5 + x^5) + sqr 
t(-1/3)*(5*x^6 + 20*x^3 - 16) + (-1/432)^(1/6)*(x^7 + 16*x^4 - sqrt(-3)...
 

Sympy [F]

\[ \int \frac {x}{\sqrt {1-x^3} \left (4-x^3\right )} \, dx=- \int \frac {x}{x^{3} \sqrt {1 - x^{3}} - 4 \sqrt {1 - x^{3}}}\, dx \] Input:

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

Output:

-Integral(x/(x**3*sqrt(1 - x**3) - 4*sqrt(1 - x**3)), x)
 

Maxima [F]

\[ \int \frac {x}{\sqrt {1-x^3} \left (4-x^3\right )} \, dx=\int { -\frac {x}{{\left (x^{3} - 4\right )} \sqrt {-x^{3} + 1}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {x}{\sqrt {1-x^3} \left (4-x^3\right )} \, dx=\int { -\frac {x}{{\left (x^{3} - 4\right )} \sqrt {-x^{3} + 1}} \,d x } \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 653, normalized size of antiderivative = 5.14 \[ \int \frac {x}{\sqrt {1-x^3} \left (4-x^3\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

- (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)*1 
i)/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)*(...
 

Reduce [F]

\[ \int \frac {x}{\sqrt {1-x^3} \left (4-x^3\right )} \, dx=\int \frac {\sqrt {-x^{3}+1}\, x}{x^{6}-5 x^{3}+4}d x \] Input:

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

Output:

int((sqrt( - x**3 + 1)*x)/(x**6 - 5*x**3 + 4),x)