Integrand size = 15, antiderivative size = 151 \[ \int \frac {1}{x^6 \sqrt {-1-x^3}} \, dx=\frac {\sqrt {-1-x^3}}{5 x^5}-\frac {7 \sqrt {-1-x^3}}{20 x^2}+\frac {7 \sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right ),-7+4 \sqrt {3}\right )}{20 \sqrt [4]{3} \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}} \] Output:
1/5*(-x^3-1)^(1/2)/x^5-7/20*(-x^3-1)^(1/2)/x^2+7/60*(1/2*6^(1/2)-1/2*2^(1/ 2))*(1+x)*((x^2-x+1)/(1+x-3^(1/2))^2)^(1/2)*EllipticF((1+x+3^(1/2))/(1+x-3 ^(1/2)),2*I-I*3^(1/2))*3^(3/4)/(-(1+x)/(1+x-3^(1/2))^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 = 42, normalized size of antiderivative = 0.28 \[ \int \frac {1}{x^6 \sqrt {-1-x^3}} \, dx=-\frac {\sqrt {1+x^3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{3},\frac {1}{2},-\frac {2}{3},-x^3\right )}{5 x^5 \sqrt {-1-x^3}} \] Input:
Integrate[1/(x^6*Sqrt[-1 - x^3]),x]
Output:
-1/5*(Sqrt[1 + x^3]*Hypergeometric2F1[-5/3, 1/2, -2/3, -x^3])/(x^5*Sqrt[-1 - x^3])
Time = 0.37 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {847, 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.
\(\displaystyle \int \frac {1}{x^6 \sqrt {-x^3-1}} \, dx\) |
\(\Big \downarrow \) 847 |
\(\displaystyle \frac {\sqrt {-x^3-1}}{5 x^5}-\frac {7}{10} \int \frac {1}{x^3 \sqrt {-x^3-1}}dx\) |
\(\Big \downarrow \) 847 |
\(\displaystyle \frac {\sqrt {-x^3-1}}{5 x^5}-\frac {7}{10} \left (\frac {\sqrt {-x^3-1}}{2 x^2}-\frac {1}{4} \int \frac {1}{\sqrt {-x^3-1}}dx\right )\) |
\(\Big \downarrow \) 760 |
\(\displaystyle \frac {\sqrt {-x^3-1}}{5 x^5}-\frac {7}{10} \left (\frac {\sqrt {-x^3-1}}{2 x^2}-\frac {\sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{2 \sqrt [4]{3} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}\right )\) |
Input:
Int[1/(x^6*Sqrt[-1 - x^3]),x]
Output:
Sqrt[-1 - x^3]/(5*x^5) - (7*(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])))/10
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 5 vs. order 4.
Time = 0.63 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.12
method | result | size |
meijerg | \(\frac {i \operatorname {hypergeom}\left (\left [-\frac {5}{3}, \frac {1}{2}\right ], \left [-\frac {2}{3}\right ], -x^{3}\right )}{5 x^{5}}\) | \(18\) |
risch | \(\frac {7 x^{6}+3 x^{3}-4}{20 x^{5} \sqrt {-x^{3}-1}}-\frac {7 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{60 \sqrt {-x^{3}-1}}\) | \(134\) |
default | \(\frac {\sqrt {-x^{3}-1}}{5 x^{5}}-\frac {7 \sqrt {-x^{3}-1}}{20 x^{2}}-\frac {7 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{60 \sqrt {-x^{3}-1}}\) | \(136\) |
elliptic | \(\frac {\sqrt {-x^{3}-1}}{5 x^{5}}-\frac {7 \sqrt {-x^{3}-1}}{20 x^{2}}-\frac {7 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{60 \sqrt {-x^{3}-1}}\) | \(136\) |
Input:
int(1/x^6/(-x^3-1)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/5*I/x^5*hypergeom([-5/3,1/2],[-2/3],-x^3)
Time = 0.08 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.22 \[ \int \frac {1}{x^6 \sqrt {-1-x^3}} \, dx=\frac {-7 i \, x^{5} {\rm weierstrassPInverse}\left (0, -4, x\right ) - {\left (7 \, x^{3} - 4\right )} \sqrt {-x^{3} - 1}}{20 \, x^{5}} \] Input:
integrate(1/x^6/(-x^3-1)^(1/2),x, algorithm="fricas")
Output:
1/20*(-7*I*x^5*weierstrassPInverse(0, -4, x) - (7*x^3 - 4)*sqrt(-x^3 - 1)) /x^5
Time = 0.55 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.26 \[ \int \frac {1}{x^6 \sqrt {-1-x^3}} \, dx=- \frac {i \Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, \frac {1}{2} \\ - \frac {2}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 x^{5} \Gamma \left (- \frac {2}{3}\right )} \] Input:
integrate(1/x**6/(-x**3-1)**(1/2),x)
Output:
-I*gamma(-5/3)*hyper((-5/3, 1/2), (-2/3,), x**3*exp_polar(I*pi))/(3*x**5*g amma(-2/3))
\[ \int \frac {1}{x^6 \sqrt {-1-x^3}} \, dx=\int { \frac {1}{\sqrt {-x^{3} - 1} x^{6}} \,d x } \] Input:
integrate(1/x^6/(-x^3-1)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(-x^3 - 1)*x^6), x)
\[ \int \frac {1}{x^6 \sqrt {-1-x^3}} \, dx=\int { \frac {1}{\sqrt {-x^{3} - 1} x^{6}} \,d x } \] Input:
integrate(1/x^6/(-x^3-1)^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(-x^3 - 1)*x^6), x)
Time = 0.04 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.33 \[ \int \frac {1}{x^6 \sqrt {-1-x^3}} \, dx=\frac {\sqrt {-x^3-1}}{5\,x^5}-\frac {7\,\sqrt {-x^3-1}}{20\,x^2}+\frac {7\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {x^3+1}\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{20\,\sqrt {-x^3-1}\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \] Input:
int(1/(x^6*(- x^3 - 1)^(1/2)),x)
Output:
(- x^3 - 1)^(1/2)/(5*x^5) - (7*(- x^3 - 1)^(1/2))/(20*x^2) + (7*((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 + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((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)))/(20*(- x^3 - 1)^(1/2)*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1 ) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2))
\[ \int \frac {1}{x^6 \sqrt {-1-x^3}} \, dx=-\left (\int \frac {\sqrt {x^{3}+1}}{x^{9}+x^{6}}d x \right ) i \] Input:
int(1/x^6/(-x^3-1)^(1/2),x)
Output:
- int(sqrt(x**3 + 1)/(x**9 + x**6),x)*i