Integrand size = 23, antiderivative size = 120 \[ \int \frac {\sqrt {1-x^6}}{\sqrt {1-x^3}} \, dx=\frac {2}{5} x \sqrt {1+x^3}+\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{5 \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}} \] Output:
2/5*x*(x^3+1)^(1/2)+2/5*3^(3/4)*(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)),I*3^(1/2)+2* I)/((1+x)/(1+x+3^(1/2))^2)^(1/2)/(x^3+1)^(1/2)
\[ \int \frac {\sqrt {1-x^6}}{\sqrt {1-x^3}} \, dx=\int \frac {\sqrt {1-x^6}}{\sqrt {1-x^3}} \, dx \] Input:
Integrate[Sqrt[1 - x^6]/Sqrt[1 - x^3],x]
Output:
Integrate[Sqrt[1 - x^6]/Sqrt[1 - x^3], x]
Time = 0.22 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1386, 748, 759}
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.
\(\displaystyle \int \frac {\sqrt {1-x^6}}{\sqrt {1-x^3}} \, dx\) |
\(\Big \downarrow \) 1386 |
\(\displaystyle \int \sqrt {x^3+1}dx\) |
\(\Big \downarrow \) 748 |
\(\displaystyle \frac {3}{5} \int \frac {1}{\sqrt {x^3+1}}dx+\frac {2}{5} \sqrt {x^3+1} x\) |
\(\Big \downarrow \) 759 |
\(\displaystyle \frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{5 \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+\frac {2}{5} \sqrt {x^3+1} x\) |
Input:
Int[Sqrt[1 - x^6]/Sqrt[1 - x^3],x]
Output:
(2*x*Sqrt[1 + x^3])/5 + (2*3^(3/4)*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]])/(5*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3 ])
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Simp[a*n*(p/(n*p + 1)) Int[(a + b*x^n)^(p - 1), x], x] /; Fre eQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || LtQ[Denominat or[p + 1/n], Denominator[p]])
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] & & PosQ[a]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Simp[(-e^2/c)^q Int[u*(d - e*x^n)^p, x], x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && EqQ[p + q, 0 ] && GtQ[d, 0] && LtQ[c, 0] && GtQ[e^2, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (97 ) = 194\).
Time = 0.27 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.84
method | result | size |
risch | \(-\frac {2 x \sqrt {x^{3}+1}\, \sqrt {\frac {\left (-x^{6}+1\right ) \left (-x^{3}+1\right )}{\left (x^{3}-1\right )^{2}}}\, \left (x^{3}-1\right )}{5 \sqrt {-x^{6}+1}\, \sqrt {-x^{3}+1}}-\frac {6 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right ) \sqrt {\frac {\left (-x^{6}+1\right ) \left (-x^{3}+1\right )}{\left (x^{3}-1\right )^{2}}}\, \left (x^{3}-1\right )}{5 \sqrt {x^{3}+1}\, \sqrt {-x^{6}+1}\, \sqrt {-x^{3}+1}}\) | \(221\) |
default | \(-\frac {\sqrt {-x^{6}+1}\, \left (3 i \sqrt {3}\, \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-3}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right )-9 \sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{i \sqrt {3}-3}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {2 \left (x +1\right )}{i \sqrt {3}-3}}, \sqrt {-\frac {i \sqrt {3}-3}{i \sqrt {3}+3}}\right )-2 x^{4}-2 x \right )}{5 \sqrt {-x^{3}+1}\, \left (x^{3}+1\right )}\) | \(255\) |
Input:
int((-x^6+1)^(1/2)/(-x^3+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/5*x*(x^3+1)^(1/2)*((-x^6+1)*(-x^3+1)/(x^3-1)^2)^(1/2)/(-x^6+1)^(1/2)/(- x^3+1)^(1/2)*(x^3-1)-6/5*(3/2-1/2*I*3^(1/2))*((x+1)/(3/2-1/2*I*3^(1/2)))^( 1/2)*((x-1/2-1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2+1/2*I*3^(1 /2))/(-3/2+1/2*I*3^(1/2)))^(1/2)/(x^3+1)^(1/2)*EllipticF(((x+1)/(3/2-1/2*I *3^(1/2)))^(1/2),((-3/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2))*((-x^6 +1)*(-x^3+1)/(x^3-1)^2)^(1/2)/(-x^6+1)^(1/2)/(-x^3+1)^(1/2)*(x^3-1)
Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.34 \[ \int \frac {\sqrt {1-x^6}}{\sqrt {1-x^3}} \, dx=-\frac {2 \, {\left (\sqrt {-x^{6} + 1} \sqrt {-x^{3} + 1} x + 3 \, {\left (x^{3} - 1\right )} {\rm weierstrassPInverse}\left (0, -4, x\right )\right )}}{5 \, {\left (x^{3} - 1\right )}} \] Input:
integrate((-x^6+1)^(1/2)/(-x^3+1)^(1/2),x, algorithm="fricas")
Output:
-2/5*(sqrt(-x^6 + 1)*sqrt(-x^3 + 1)*x + 3*(x^3 - 1)*weierstrassPInverse(0, -4, x))/(x^3 - 1)
\[ \int \frac {\sqrt {1-x^6}}{\sqrt {1-x^3}} \, dx=\int \frac {\sqrt {- \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}}{\sqrt {- \left (x - 1\right ) \left (x^{2} + x + 1\right )}}\, dx \] Input:
integrate((-x**6+1)**(1/2)/(-x**3+1)**(1/2),x)
Output:
Integral(sqrt(-(x - 1)*(x + 1)*(x**2 - x + 1)*(x**2 + x + 1))/sqrt(-(x - 1 )*(x**2 + x + 1)), x)
\[ \int \frac {\sqrt {1-x^6}}{\sqrt {1-x^3}} \, dx=\int { \frac {\sqrt {-x^{6} + 1}}{\sqrt {-x^{3} + 1}} \,d x } \] Input:
integrate((-x^6+1)^(1/2)/(-x^3+1)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(-x^6 + 1)/sqrt(-x^3 + 1), x)
\[ \int \frac {\sqrt {1-x^6}}{\sqrt {1-x^3}} \, dx=\int { \frac {\sqrt {-x^{6} + 1}}{\sqrt {-x^{3} + 1}} \,d x } \] Input:
integrate((-x^6+1)^(1/2)/(-x^3+1)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(-x^6 + 1)/sqrt(-x^3 + 1), x)
Timed out. \[ \int \frac {\sqrt {1-x^6}}{\sqrt {1-x^3}} \, dx=\int \frac {\sqrt {1-x^6}}{\sqrt {1-x^3}} \,d x \] Input:
int((1 - x^6)^(1/2)/(1 - x^3)^(1/2),x)
Output:
int((1 - x^6)^(1/2)/(1 - x^3)^(1/2), x)
\[ \int \frac {\sqrt {1-x^6}}{\sqrt {1-x^3}} \, dx=-\left (\int \frac {\sqrt {-x^{3}+1}\, \sqrt {-x^{6}+1}}{x^{3}-1}d x \right ) \] Input:
int((-x^6+1)^(1/2)/(-x^3+1)^(1/2),x)
Output:
- int((sqrt( - x**3 + 1)*sqrt( - x**6 + 1))/(x**3 - 1),x)