Integrand size = 13, antiderivative size = 139 \[ \int \frac {1}{x^3 \sqrt {-1+x^3}} \, dx=\frac {\sqrt {-1+x^3}}{2 x^2}-\frac {\sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{2 \sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}} \] Output:
1/2*(x^3-1)^(1/2)/x^2-1/6*(1/2*6^(1/2)-1/2*2^(1/2))*(1-x)*((x^2+x+1)/(1-3^ (1/2)-x)^2)^(1/2)*EllipticF((1+3^(1/2)-x)/(1-3^(1/2)-x),2*I-I*3^(1/2))*3^( 3/4)/(-(1-x)/(1-3^(1/2)-x)^2)^(1/2)/(x^3-1)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.29 \[ \int \frac {1}{x^3 \sqrt {-1+x^3}} \, dx=-\frac {\sqrt {1-x^3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{2},\frac {1}{3},x^3\right )}{2 x^2 \sqrt {-1+x^3}} \] Input:
Integrate[1/(x^3*Sqrt[-1 + x^3]),x]
Output:
-1/2*(Sqrt[1 - x^3]*Hypergeometric2F1[-2/3, 1/2, 1/3, x^3])/(x^2*Sqrt[-1 + x^3])
Time = 0.35 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {847, 760}
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.
\(\displaystyle \int \frac {1}{x^3 \sqrt {x^3-1}} \, dx\) |
\(\Big \downarrow \) 847 |
\(\displaystyle \frac {1}{4} \int \frac {1}{\sqrt {x^3-1}}dx+\frac {\sqrt {x^3-1}}{2 x^2}\) |
\(\Big \downarrow \) 760 |
\(\displaystyle \frac {\sqrt {x^3-1}}{2 x^2}-\frac {\sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{2 \sqrt [4]{3} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}\) |
Input:
Int[1/(x^3*Sqrt[-1 + x^3]),x]
Output:
Sqrt[-1 + x^3]/(2*x^2) - (Sqrt[2 - Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/(1 - Sqrt[3] - x)^2]*EllipticF[ArcSin[(1 + Sqrt[3] - x)/(1 - Sqrt[3] - x)], - 7 + 4*Sqrt[3]])/(2*3^(1/4)*Sqrt[-((1 - x)/(1 - Sqrt[3] - x)^2)]*Sqrt[-1 + x^3])
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]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.92 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.24
method | result | size |
meijerg | \(-\frac {\sqrt {-\operatorname {signum}\left (x^{3}-1\right )}\, \operatorname {hypergeom}\left (\left [-\frac {2}{3}, \frac {1}{2}\right ], \left [\frac {1}{3}\right ], x^{3}\right )}{2 \sqrt {\operatorname {signum}\left (x^{3}-1\right )}\, x^{2}}\) | \(33\) |
default | \(\frac {\sqrt {x^{3}-1}}{2 x^{2}}+\frac {\left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {x^{3}-1}}\) | \(129\) |
risch | \(\frac {\sqrt {x^{3}-1}}{2 x^{2}}+\frac {\left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {x^{3}-1}}\) | \(129\) |
elliptic | \(\frac {\sqrt {x^{3}-1}}{2 x^{2}}+\frac {\left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {x^{3}-1}}\) | \(129\) |
Input:
int(1/x^3/(x^3-1)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/2/signum(x^3-1)^(1/2)*(-signum(x^3-1))^(1/2)/x^2*hypergeom([-2/3,1/2],[ 1/3],x^3)
Time = 0.07 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.15 \[ \int \frac {1}{x^3 \sqrt {-1+x^3}} \, dx=\frac {x^{2} {\rm weierstrassPInverse}\left (0, 4, x\right ) + \sqrt {x^{3} - 1}}{2 \, x^{2}} \] Input:
integrate(1/x^3/(x^3-1)^(1/2),x, algorithm="fricas")
Output:
1/2*(x^2*weierstrassPInverse(0, 4, x) + sqrt(x^3 - 1))/x^2
Time = 0.52 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.22 \[ \int \frac {1}{x^3 \sqrt {-1+x^3}} \, dx=- \frac {i \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{2} \\ \frac {1}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 x^{2} \Gamma \left (\frac {1}{3}\right )} \] Input:
integrate(1/x**3/(x**3-1)**(1/2),x)
Output:
-I*gamma(-2/3)*hyper((-2/3, 1/2), (1/3,), x**3)/(3*x**2*gamma(1/3))
\[ \int \frac {1}{x^3 \sqrt {-1+x^3}} \, dx=\int { \frac {1}{\sqrt {x^{3} - 1} x^{3}} \,d x } \] Input:
integrate(1/x^3/(x^3-1)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(x^3 - 1)*x^3), x)
\[ \int \frac {1}{x^3 \sqrt {-1+x^3}} \, dx=\int { \frac {1}{\sqrt {x^{3} - 1} x^{3}} \,d x } \] Input:
integrate(1/x^3/(x^3-1)^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(x^3 - 1)*x^3), x)
Time = 0.06 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x^3 \sqrt {-1+x^3}} \, dx=\frac {\sqrt {x^3-1}}{2\,x^2}-\frac {\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{2\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \] Input:
int(1/(x^3*(x^3 - 1)^(1/2)),x)
Output:
(x^3 - 1)^(1/2)/(2*x^2) - (((3^(1/2)*1i)/2 + 3/2)*(-(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)*ellipticF(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)))/(2*(((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))
\[ \int \frac {1}{x^3 \sqrt {-1+x^3}} \, dx=\int \frac {\sqrt {x^{3}-1}}{x^{6}-x^{3}}d x \] Input:
int(1/x^3/(x^3-1)^(1/2),x)
Output:
int(sqrt(x**3 - 1)/(x**6 - x**3),x)