Integrand size = 13, antiderivative size = 89 \[ \int \frac {x^{10}}{\sqrt {-1+x^4}} \, dx=\frac {7}{45} x^3 \sqrt {-1+x^4}+\frac {1}{9} x^7 \sqrt {-1+x^4}+\frac {7 \sqrt {1-x^4} E(\arcsin (x)|-1)}{15 \sqrt {-1+x^4}}-\frac {7 \sqrt {1-x^4} \operatorname {EllipticF}(\arcsin (x),-1)}{15 \sqrt {-1+x^4}} \] Output:
7/45*x^3*(x^4-1)^(1/2)+1/9*x^7*(x^4-1)^(1/2)+7/15*(-x^4+1)^(1/2)*EllipticE (x,I)/(x^4-1)^(1/2)-7/15*(-x^4+1)^(1/2)*EllipticF(x,I)/(x^4-1)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.61 \[ \int \frac {x^{10}}{\sqrt {-1+x^4}} \, dx=\frac {x^3 \left (-7+2 x^4+5 x^8+7 \sqrt {1-x^4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},x^4\right )\right )}{45 \sqrt {-1+x^4}} \] Input:
Integrate[x^10/Sqrt[-1 + x^4],x]
Output:
(x^3*(-7 + 2*x^4 + 5*x^8 + 7*Sqrt[1 - x^4]*Hypergeometric2F1[1/2, 3/4, 7/4 , x^4]))/(45*Sqrt[-1 + x^4])
Time = 0.44 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.89, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {843, 843, 835, 763, 1499}
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.
\(\displaystyle \int \frac {x^{10}}{\sqrt {x^4-1}} \, dx\) |
\(\Big \downarrow \) 843 |
\(\displaystyle \frac {7}{9} \int \frac {x^6}{\sqrt {x^4-1}}dx+\frac {1}{9} \sqrt {x^4-1} x^7\) |
\(\Big \downarrow \) 843 |
\(\displaystyle \frac {7}{9} \left (\frac {3}{5} \int \frac {x^2}{\sqrt {x^4-1}}dx+\frac {1}{5} \sqrt {x^4-1} x^3\right )+\frac {1}{9} \sqrt {x^4-1} x^7\) |
\(\Big \downarrow \) 835 |
\(\displaystyle \frac {7}{9} \left (\frac {3}{5} \left (\int \frac {1}{\sqrt {x^4-1}}dx-\int \frac {1-x^2}{\sqrt {x^4-1}}dx\right )+\frac {1}{5} \sqrt {x^4-1} x^3\right )+\frac {1}{9} \sqrt {x^4-1} x^7\) |
\(\Big \downarrow \) 763 |
\(\displaystyle \frac {7}{9} \left (\frac {3}{5} \left (\frac {\sqrt {x^2-1} \sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^4-1}}-\int \frac {1-x^2}{\sqrt {x^4-1}}dx\right )+\frac {1}{5} \sqrt {x^4-1} x^3\right )+\frac {1}{9} \sqrt {x^4-1} x^7\) |
\(\Big \downarrow \) 1499 |
\(\displaystyle \frac {7}{9} \left (\frac {3}{5} \left (\frac {\sqrt {x^2-1} \sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right )}{\sqrt {2} \sqrt {x^4-1}}-\frac {\sqrt {2} \sqrt {x^2-1} \sqrt {x^2+1} E\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{\sqrt {x^4-1}}+\frac {x \left (x^2+1\right )}{\sqrt {x^4-1}}\right )+\frac {1}{5} \sqrt {x^4-1} x^3\right )+\frac {1}{9} \sqrt {x^4-1} x^7\) |
Input:
Int[x^10/Sqrt[-1 + x^4],x]
Output:
(x^7*Sqrt[-1 + x^4])/9 + (7*((x^3*Sqrt[-1 + x^4])/5 + (3*((x*(1 + x^2))/Sq rt[-1 + x^4] - (Sqrt[2]*Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticE[ArcSin[(Sqr t[2]*x)/Sqrt[-1 + x^2]], 1/2])/Sqrt[-1 + x^4] + (Sqrt[-1 + x^2]*Sqrt[1 + x ^2]*EllipticF[ArcSin[(Sqrt[2]*x)/Sqrt[-1 + x^2]], 1/2])/(Sqrt[2]*Sqrt[-1 + x^4])))/5))/9
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[(-a)*b, 2]}, Sim p[Sqrt[-a + q*x^2]*(Sqrt[(a + q*x^2)/q]/(Sqrt[2]*Sqrt[-a]*Sqrt[a + b*x^4])) *EllipticF[ArcSin[x/Sqrt[(a + q*x^2)/(2*q)]], 1/2], x] /; IntegerQ[q]] /; F reeQ[{a, b}, x] && LtQ[a, 0] && GtQ[b, 0]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[1/q Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q Int[(1 - q*x^2)/Sqrt[ a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && LtQ[a, 0] && GtQ[b, 0]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Simp[e*x*((q + c*x^2)/(c*Sqrt[a + c*x^4])), x] - Simp[Sqrt [2]*e*q*Sqrt[-a + q*x^2]*(Sqrt[(a + q*x^2)/q]/(Sqrt[-a]*c*Sqrt[a + c*x^4])) *EllipticE[ArcSin[x/Sqrt[(a + q*x^2)/(2*q)]], 1/2], x] /; EqQ[c*d + e*q, 0] && IntegerQ[q]] /; FreeQ[{a, c, d, e}, x] && LtQ[a, 0] && GtQ[c, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.34 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.37
method | result | size |
meijerg | \(\frac {\sqrt {-\operatorname {signum}\left (x^{4}-1\right )}\, x^{11} \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {11}{4}\right ], \left [\frac {15}{4}\right ], x^{4}\right )}{11 \sqrt {\operatorname {signum}\left (x^{4}-1\right )}}\) | \(33\) |
risch | \(\frac {x^{3} \left (5 x^{4}+7\right ) \sqrt {x^{4}-1}}{45}-\frac {7 i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \left (\operatorname {EllipticF}\left (i x , i\right )-\operatorname {EllipticE}\left (i x , i\right )\right )}{15 \sqrt {x^{4}-1}}\) | \(64\) |
default | \(\frac {x^{7} \sqrt {x^{4}-1}}{9}+\frac {7 x^{3} \sqrt {x^{4}-1}}{45}-\frac {7 i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \left (\operatorname {EllipticF}\left (i x , i\right )-\operatorname {EllipticE}\left (i x , i\right )\right )}{15 \sqrt {x^{4}-1}}\) | \(69\) |
elliptic | \(\frac {x^{7} \sqrt {x^{4}-1}}{9}+\frac {7 x^{3} \sqrt {x^{4}-1}}{45}-\frac {7 i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \left (\operatorname {EllipticF}\left (i x , i\right )-\operatorname {EllipticE}\left (i x , i\right )\right )}{15 \sqrt {x^{4}-1}}\) | \(69\) |
Input:
int(x^10/(x^4-1)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/11/signum(x^4-1)^(1/2)*(-signum(x^4-1))^(1/2)*x^11*hypergeom([1/2,11/4], [15/4],x^4)
Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.49 \[ \int \frac {x^{10}}{\sqrt {-1+x^4}} \, dx=\frac {21 \, x E(\arcsin \left (\frac {1}{x}\right )\,|\,-1) - 21 \, x F(\arcsin \left (\frac {1}{x}\right )\,|\,-1) + {\left (5 \, x^{8} + 7 \, x^{4} + 21\right )} \sqrt {x^{4} - 1}}{45 \, x} \] Input:
integrate(x^10/(x^4-1)^(1/2),x, algorithm="fricas")
Output:
1/45*(21*x*elliptic_e(arcsin(1/x), -1) - 21*x*elliptic_f(arcsin(1/x), -1) + (5*x^8 + 7*x^4 + 21)*sqrt(x^4 - 1))/x
Time = 0.49 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.30 \[ \int \frac {x^{10}}{\sqrt {-1+x^4}} \, dx=- \frac {i x^{11} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {x^{4}} \right )}}{4 \Gamma \left (\frac {15}{4}\right )} \] Input:
integrate(x**10/(x**4-1)**(1/2),x)
Output:
-I*x**11*gamma(11/4)*hyper((1/2, 11/4), (15/4,), x**4)/(4*gamma(15/4))
\[ \int \frac {x^{10}}{\sqrt {-1+x^4}} \, dx=\int { \frac {x^{10}}{\sqrt {x^{4} - 1}} \,d x } \] Input:
integrate(x^10/(x^4-1)^(1/2),x, algorithm="maxima")
Output:
integrate(x^10/sqrt(x^4 - 1), x)
\[ \int \frac {x^{10}}{\sqrt {-1+x^4}} \, dx=\int { \frac {x^{10}}{\sqrt {x^{4} - 1}} \,d x } \] Input:
integrate(x^10/(x^4-1)^(1/2),x, algorithm="giac")
Output:
integrate(x^10/sqrt(x^4 - 1), x)
Timed out. \[ \int \frac {x^{10}}{\sqrt {-1+x^4}} \, dx=\int \frac {x^{10}}{\sqrt {x^4-1}} \,d x \] Input:
int(x^10/(x^4 - 1)^(1/2),x)
Output:
int(x^10/(x^4 - 1)^(1/2), x)
\[ \int \frac {x^{10}}{\sqrt {-1+x^4}} \, dx=\frac {\sqrt {x^{4}-1}\, x^{7}}{9}+\frac {7 \sqrt {x^{4}-1}\, x^{3}}{45}+\frac {7 \left (\int \frac {\sqrt {x^{4}-1}\, x^{2}}{x^{4}-1}d x \right )}{15} \] Input:
int(x^10/(x^4-1)^(1/2),x)
Output:
(5*sqrt(x**4 - 1)*x**7 + 7*sqrt(x**4 - 1)*x**3 + 21*int((sqrt(x**4 - 1)*x* *2)/(x**4 - 1),x))/45