\(\int \frac {1}{\sqrt [3]{c^2-d^2 x^2}} \, dx\) [259]

Optimal result

Integrand size = 16, antiderivative size = 613 \[ \int \frac {1}{\sqrt [3]{c^2-d^2 x^2}} \, dx=-\frac {3 x}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}-\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}+\left (c^2-d^2 x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right )|-7+4 \sqrt {3}\right )}{2 d^2 x \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}}+\frac {\sqrt {2} 3^{3/4} c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}+\left (c^2-d^2 x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right ),-7+4 \sqrt {3}\right )}{d^2 x \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}} \] Output:

-3*x/((1-3^(1/2))*c^(2/3)-(-d^2*x^2+c^2)^(1/3))-3/2*3^(1/4)*(1/2*6^(1/2)+1 
/2*2^(1/2))*c^(2/3)*(c^(2/3)-(-d^2*x^2+c^2)^(1/3))*((c^(4/3)+c^(2/3)*(-d^2 
*x^2+c^2)^(1/3)+(-d^2*x^2+c^2)^(2/3))/((1-3^(1/2))*c^(2/3)-(-d^2*x^2+c^2)^ 
(1/3))^2)^(1/2)*EllipticE(((1+3^(1/2))*c^(2/3)-(-d^2*x^2+c^2)^(1/3))/((1-3 
^(1/2))*c^(2/3)-(-d^2*x^2+c^2)^(1/3)),2*I-I*3^(1/2))/d^2/x/(-c^(2/3)*(c^(2 
/3)-(-d^2*x^2+c^2)^(1/3))/((1-3^(1/2))*c^(2/3)-(-d^2*x^2+c^2)^(1/3))^2)^(1 
/2)+2^(1/2)*3^(3/4)*c^(2/3)*(c^(2/3)-(-d^2*x^2+c^2)^(1/3))*((c^(4/3)+c^(2/ 
3)*(-d^2*x^2+c^2)^(1/3)+(-d^2*x^2+c^2)^(2/3))/((1-3^(1/2))*c^(2/3)-(-d^2*x 
^2+c^2)^(1/3))^2)^(1/2)*EllipticF(((1+3^(1/2))*c^(2/3)-(-d^2*x^2+c^2)^(1/3 
))/((1-3^(1/2))*c^(2/3)-(-d^2*x^2+c^2)^(1/3)),2*I-I*3^(1/2))/d^2/x/(-c^(2/ 
3)*(c^(2/3)-(-d^2*x^2+c^2)^(1/3))/((1-3^(1/2))*c^(2/3)-(-d^2*x^2+c^2)^(1/3 
))^2)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 4.80 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.09 \[ \int \frac {1}{\sqrt [3]{c^2-d^2 x^2}} \, dx=\frac {x \sqrt [3]{1-\frac {d^2 x^2}{c^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},\frac {d^2 x^2}{c^2}\right )}{\sqrt [3]{c^2-d^2 x^2}} \] Input:

Integrate[(c^2 - d^2*x^2)^(-1/3),x]
 

Output:

(x*(1 - (d^2*x^2)/c^2)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, (d^2*x^2)/c^ 
2])/(c^2 - d^2*x^2)^(1/3)
 

Rubi [A] (warning: unable to verify)

Time = 0.77 (sec) , antiderivative size = 671, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {233, 833, 760, 2418}

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.

\(\displaystyle \int \frac {1}{\sqrt [3]{c^2-d^2 x^2}} \, dx\)

\(\Big \downarrow \) 233

\(\displaystyle -\frac {3 \sqrt {-d^2 x^2} \int \frac {\sqrt [3]{c^2-d^2 x^2}}{\sqrt {-d^2 x^2}}d\sqrt [3]{c^2-d^2 x^2}}{2 d^2 x}\)

\(\Big \downarrow \) 833

\(\displaystyle -\frac {3 \sqrt {-d^2 x^2} \left (\left (1+\sqrt {3}\right ) c^{2/3} \int \frac {1}{\sqrt {-d^2 x^2}}d\sqrt [3]{c^2-d^2 x^2}-\int \frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\sqrt {-d^2 x^2}}d\sqrt [3]{c^2-d^2 x^2}\right )}{2 d^2 x}\)

\(\Big \downarrow \) 760

\(\displaystyle -\frac {3 \sqrt {-d^2 x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\sqrt {-d^2 x^2}}d\sqrt [3]{c^2-d^2 x^2}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+\left (c^2-d^2 x^2\right )^{2/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-d^2 x^2} \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}}\right )}{2 d^2 x}\)

\(\Big \downarrow \) 2418

\(\displaystyle -\frac {3 \sqrt {-d^2 x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+\left (c^2-d^2 x^2\right )^{2/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-d^2 x^2} \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+\left (c^2-d^2 x^2\right )^{2/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-d^2 x^2} \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}}-\frac {2 \sqrt {-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right )}{2 d^2 x}\)

Input:

Int[(c^2 - d^2*x^2)^(-1/3),x]
 

Output:

(-3*Sqrt[-(d^2*x^2)]*((-2*Sqrt[-(d^2*x^2)])/((1 - Sqrt[3])*c^(2/3) - (c^2 
- d^2*x^2)^(1/3)) + (3^(1/4)*Sqrt[2 + Sqrt[3]]*c^(2/3)*(c^(2/3) - (c^2 - d 
^2*x^2)^(1/3))*Sqrt[(c^(4/3) + c^(2/3)*(c^2 - d^2*x^2)^(1/3) + (c^2 - d^2* 
x^2)^(2/3))/((1 - Sqrt[3])*c^(2/3) - (c^2 - d^2*x^2)^(1/3))^2]*EllipticE[A 
rcSin[((1 + Sqrt[3])*c^(2/3) - (c^2 - d^2*x^2)^(1/3))/((1 - Sqrt[3])*c^(2/ 
3) - (c^2 - d^2*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(Sqrt[-(d^2*x^2)]*Sqrt[-((c 
^(2/3)*(c^(2/3) - (c^2 - d^2*x^2)^(1/3)))/((1 - Sqrt[3])*c^(2/3) - (c^2 - 
d^2*x^2)^(1/3))^2)]) - (2*Sqrt[2 - Sqrt[3]]*(1 + Sqrt[3])*c^(2/3)*(c^(2/3) 
 - (c^2 - d^2*x^2)^(1/3))*Sqrt[(c^(4/3) + c^(2/3)*(c^2 - d^2*x^2)^(1/3) + 
(c^2 - d^2*x^2)^(2/3))/((1 - Sqrt[3])*c^(2/3) - (c^2 - d^2*x^2)^(1/3))^2]* 
EllipticF[ArcSin[((1 + Sqrt[3])*c^(2/3) - (c^2 - d^2*x^2)^(1/3))/((1 - Sqr 
t[3])*c^(2/3) - (c^2 - d^2*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(3^(1/4)*Sqrt[-( 
d^2*x^2)]*Sqrt[-((c^(2/3)*(c^(2/3) - (c^2 - d^2*x^2)^(1/3)))/((1 - Sqrt[3] 
)*c^(2/3) - (c^2 - d^2*x^2)^(1/3))^2)])))/(2*d^2*x)
 

Defintions of rubi rules used

rule 233

Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Simp[3*(Sqrt[b*x^2]/(2*b*x)) 
   Subst[Int[x/Sqrt[-a + x^3], x], x, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b 
}, x]
 

rule 760

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(- 
s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3]) 
*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x 
] && NegQ[a]
 

rule 833

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] 
], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 + Sqrt[3]))*(s/r)   Int[1/Sqrt[a + b*x 
^3], x], x] + Simp[1/r   Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x 
]] /; FreeQ[{a, b}, x] && NegQ[a]
 

rule 2418

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Simplify[(1 + Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 + Sqrt[3])*(d/c) 
]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 - Sqrt[3])*s + r*x))), x] + S 
imp[3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( 
(1 - Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - S 
qrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[ 
3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && 
 EqQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
 

Maple [F]

\[\int \frac {1}{\left (-d^{2} x^{2}+c^{2}\right )^{\frac {1}{3}}}d x\]

Input:

int(1/(-d^2*x^2+c^2)^(1/3),x)
 

Output:

int(1/(-d^2*x^2+c^2)^(1/3),x)
 

Fricas [F]

\[ \int \frac {1}{\sqrt [3]{c^2-d^2 x^2}} \, dx=\int { \frac {1}{{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {1}{3}}} \,d x } \] Input:

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

Output:

integral(-(-d^2*x^2 + c^2)^(2/3)/(d^2*x^2 - c^2), x)
 

Sympy [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.05 \[ \int \frac {1}{\sqrt [3]{c^2-d^2 x^2}} \, dx=\frac {x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {d^{2} x^{2} e^{2 i \pi }}{c^{2}}} \right )}}{c^{\frac {2}{3}}} \] Input:

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

Output:

x*hyper((1/3, 1/2), (3/2,), d**2*x**2*exp_polar(2*I*pi)/c**2)/c**(2/3)
 

Maxima [F]

\[ \int \frac {1}{\sqrt [3]{c^2-d^2 x^2}} \, dx=\int { \frac {1}{{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {1}{3}}} \,d x } \] Input:

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

Output:

integrate((-d^2*x^2 + c^2)^(-1/3), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt [3]{c^2-d^2 x^2}} \, dx=\int { \frac {1}{{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {1}{3}}} \,d x } \] Input:

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

Output:

integrate((-d^2*x^2 + c^2)^(-1/3), x)
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 6.74 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.08 \[ \int \frac {1}{\sqrt [3]{c^2-d^2 x^2}} \, dx=\frac {x\,{\left (1-\frac {d^2\,x^2}{c^2}\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {1}{2};\ \frac {3}{2};\ \frac {d^2\,x^2}{c^2}\right )}{{\left (c^2-d^2\,x^2\right )}^{1/3}} \] Input:

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

Output:

(x*(1 - (d^2*x^2)/c^2)^(1/3)*hypergeom([1/3, 1/2], 3/2, (d^2*x^2)/c^2))/(c 
^2 - d^2*x^2)^(1/3)
 

Reduce [F]

\[ \int \frac {1}{\sqrt [3]{c^2-d^2 x^2}} \, dx=\int \frac {1}{\left (-d^{2} x^{2}+c^{2}\right )^{\frac {1}{3}}}d x \] Input:

int(1/(-d^2*x^2+c^2)^(1/3),x)
 

Output:

int(1/(c**2 - d**2*x**2)**(1/3),x)