Integrand size = 18, antiderivative size = 54 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{13}} \, dx=\frac {\sqrt {-1+x^6} \left (6-8 x^3-3 x^6+8 x^9\right )}{72 x^{12}}+\frac {1}{12} \arctan \left (\frac {1+x^3}{\sqrt {-1+x^6}}\right ) \]
Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{13}} \, dx=\frac {\sqrt {-1+x^6} \left (6-8 x^3-3 x^6+8 x^9\right )}{72 x^{12}}+\frac {1}{12} \arctan \left (\frac {\sqrt {-1+x^6}}{-1+x^3}\right ) \]
(Sqrt[-1 + x^6]*(6 - 8*x^3 - 3*x^6 + 8*x^9))/(72*x^12) + ArcTan[Sqrt[-1 + x^6]/(-1 + x^3)]/12
Leaf count is larger than twice the leaf count of optimal. \(162\) vs. \(2(54)=108\).
Time = 0.25 (sec) , antiderivative size = 162, normalized size of antiderivative = 3.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {1396, 948, 107, 105, 105, 105, 103, 216}
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.
\(\displaystyle \int \frac {\left (x^3-1\right ) \sqrt {x^6-1}}{x^{13}} \, dx\) |
\(\Big \downarrow \) 1396 |
\(\displaystyle \frac {\sqrt {x^6-1} \int \frac {\left (x^3-1\right )^{3/2} \sqrt {x^3+1}}{x^{13}}dx}{\sqrt {x^3-1} \sqrt {x^3+1}}\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {\sqrt {x^6-1} \int \frac {\left (x^3-1\right )^{3/2} \sqrt {x^3+1}}{x^{15}}dx^3}{3 \sqrt {x^3-1} \sqrt {x^3+1}}\) |
\(\Big \downarrow \) 107 |
\(\displaystyle \frac {\sqrt {x^6-1} \left (\frac {\left (x^3-1\right )^{5/2} \left (x^3+1\right )^{3/2}}{4 x^{12}}-\frac {1}{4} \int \frac {\left (x^3-1\right )^{3/2} \sqrt {x^3+1}}{x^{12}}dx^3\right )}{3 \sqrt {x^3-1} \sqrt {x^3+1}}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {\sqrt {x^6-1} \left (\frac {1}{4} \left (\frac {\left (x^3-1\right )^{3/2} \left (x^3+1\right )^{3/2}}{3 x^9}-\int \frac {\sqrt {x^3-1} \sqrt {x^3+1}}{x^9}dx^3\right )+\frac {\left (x^3+1\right )^{3/2} \left (x^3-1\right )^{5/2}}{4 x^{12}}\right )}{3 \sqrt {x^3-1} \sqrt {x^3+1}}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {\sqrt {x^6-1} \left (\frac {1}{4} \left (-\frac {1}{2} \int \frac {\sqrt {x^3+1}}{x^6 \sqrt {x^3-1}}dx^3+\frac {\left (x^3-1\right )^{3/2} \left (x^3+1\right )^{3/2}}{3 x^9}+\frac {\sqrt {x^3-1} \left (x^3+1\right )^{3/2}}{2 x^6}\right )+\frac {\left (x^3+1\right )^{3/2} \left (x^3-1\right )^{5/2}}{4 x^{12}}\right )}{3 \sqrt {x^3-1} \sqrt {x^3+1}}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {\sqrt {x^6-1} \left (\frac {1}{4} \left (\frac {1}{2} \left (-\int \frac {1}{x^3 \sqrt {x^3-1} \sqrt {x^3+1}}dx^3-\frac {\sqrt {x^3-1} \sqrt {x^3+1}}{x^3}\right )+\frac {\left (x^3-1\right )^{3/2} \left (x^3+1\right )^{3/2}}{3 x^9}+\frac {\sqrt {x^3-1} \left (x^3+1\right )^{3/2}}{2 x^6}\right )+\frac {\left (x^3+1\right )^{3/2} \left (x^3-1\right )^{5/2}}{4 x^{12}}\right )}{3 \sqrt {x^3-1} \sqrt {x^3+1}}\) |
\(\Big \downarrow \) 103 |
\(\displaystyle \frac {\sqrt {x^6-1} \left (\frac {1}{4} \left (\frac {1}{2} \left (-\int \frac {1}{x^6+1}d\left (\sqrt {x^3-1} \sqrt {x^3+1}\right )-\frac {\sqrt {x^3-1} \sqrt {x^3+1}}{x^3}\right )+\frac {\left (x^3-1\right )^{3/2} \left (x^3+1\right )^{3/2}}{3 x^9}+\frac {\sqrt {x^3-1} \left (x^3+1\right )^{3/2}}{2 x^6}\right )+\frac {\left (x^3+1\right )^{3/2} \left (x^3-1\right )^{5/2}}{4 x^{12}}\right )}{3 \sqrt {x^3-1} \sqrt {x^3+1}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\sqrt {x^6-1} \left (\frac {1}{4} \left (\frac {1}{2} \left (-\arctan \left (\sqrt {x^3-1} \sqrt {x^3+1}\right )-\frac {\sqrt {x^3-1} \sqrt {x^3+1}}{x^3}\right )+\frac {\left (x^3-1\right )^{3/2} \left (x^3+1\right )^{3/2}}{3 x^9}+\frac {\sqrt {x^3-1} \left (x^3+1\right )^{3/2}}{2 x^6}\right )+\frac {\left (x^3+1\right )^{3/2} \left (x^3-1\right )^{5/2}}{4 x^{12}}\right )}{3 \sqrt {x^3-1} \sqrt {x^3+1}}\) |
(Sqrt[-1 + x^6]*(((-1 + x^3)^(5/2)*(1 + x^3)^(3/2))/(4*x^12) + ((Sqrt[-1 + x^3]*(1 + x^3)^(3/2))/(2*x^6) + ((-1 + x^3)^(3/2)*(1 + x^3)^(3/2))/(3*x^9 ) + (-((Sqrt[-1 + x^3]*Sqrt[1 + x^3])/x^3) - ArcTan[Sqrt[-1 + x^3]*Sqrt[1 + x^3]])/2)/4))/(3*Sqrt[-1 + x^3]*Sqrt[1 + x^3])
3.7.94.3.1 Defintions of rubi rules used
Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ ))), x_] :> Simp[b*f Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq rt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d *e - f*(b*c + a*d), 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x ] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_)*((d_) + (e_.)*(x_)^(n_))^(q_.), x _Symbol] :> Simp[(a + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + c*(x^n/e))^FracPart[p]) Int[u*(d + e*x^n)^(p + q)*(a/d + (c/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] && !IntegerQ[p] && !(EqQ[q, 1] && EqQ[n, 2])
Time = 1.99 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81
method | result | size |
pseudoelliptic | \(\frac {3 \arctan \left (\frac {1}{\sqrt {x^{6}-1}}\right ) x^{12}+8 \left (x^{9}-\frac {3}{8} x^{6}-x^{3}+\frac {3}{4}\right ) \sqrt {x^{6}-1}}{72 x^{12}}\) | \(44\) |
trager | \(\frac {\sqrt {x^{6}-1}\, \left (8 x^{9}-3 x^{6}-8 x^{3}+6\right )}{72 x^{12}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{24}\) | \(60\) |
risch | \(\frac {8 x^{15}-3 x^{12}-16 x^{9}+9 x^{6}+8 x^{3}-6}{72 x^{12} \sqrt {x^{6}-1}}-\frac {\sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )+\left (-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }\right )}{48 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}\) | \(101\) |
meijerg | \(\frac {\sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, \left (-\frac {\sqrt {\pi }\, \left (x^{12}-8 x^{6}+8\right )}{8 x^{12}}+\frac {\sqrt {\pi }\, \left (-4 x^{6}+8\right ) \sqrt {-x^{6}+1}}{8 x^{12}}-\frac {\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )}{2}+\frac {\left (\frac {1}{2}-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{4}+\frac {\sqrt {\pi }}{x^{12}}-\frac {\sqrt {\pi }}{x^{6}}\right )}{12 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}}-\frac {\sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, \left (-x^{6}+1\right )^{\frac {3}{2}}}{9 \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, x^{9}}\) | \(153\) |
Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{13}} \, dx=-\frac {6 \, x^{12} \arctan \left (-x^{3} + \sqrt {x^{6} - 1}\right ) - 8 \, x^{12} - {\left (8 \, x^{9} - 3 \, x^{6} - 8 \, x^{3} + 6\right )} \sqrt {x^{6} - 1}}{72 \, x^{12}} \]
-1/72*(6*x^12*arctan(-x^3 + sqrt(x^6 - 1)) - 8*x^12 - (8*x^9 - 3*x^6 - 8*x ^3 + 6)*sqrt(x^6 - 1))/x^12
Time = 2.01 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.17 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{13}} \, dx=\frac {\begin {cases} \frac {\left (x^{6} - 1\right )^{\frac {3}{2}}}{3 x^{9}} & \text {for}\: x^{3} > -1 \wedge x^{3} < 1 \end {cases}}{3} - \frac {\begin {cases} \frac {\operatorname {acos}{\left (\frac {1}{x^{3}} \right )}}{8} - \frac {\left (-1 + \frac {2}{x^{6}}\right ) \sqrt {1 - \frac {1}{x^{6}}}}{8 x^{3}} & \text {for}\: x^{3} > -1 \wedge x^{3} < 1 \end {cases}}{3} \]
Piecewise(((x**6 - 1)**(3/2)/(3*x**9), (x**3 > -1) & (x**3 < 1)))/3 - Piec ewise((acos(x**(-3))/8 - (-1 + 2/x**6)*sqrt(1 - 1/x**6)/(8*x**3), (x**3 > -1) & (x**3 < 1)))/3
Time = 0.30 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.07 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{13}} \, dx=-\frac {{\left (x^{6} - 1\right )}^{\frac {3}{2}} - \sqrt {x^{6} - 1}}{24 \, {\left (2 \, x^{6} + {\left (x^{6} - 1\right )}^{2} - 1\right )}} + \frac {{\left (x^{6} - 1\right )}^{\frac {3}{2}}}{9 \, x^{9}} - \frac {1}{24} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \]
-1/24*((x^6 - 1)^(3/2) - sqrt(x^6 - 1))/(2*x^6 + (x^6 - 1)^2 - 1) + 1/9*(x ^6 - 1)^(3/2)/x^9 - 1/24*arctan(sqrt(x^6 - 1))
Exception generated. \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{13}} \, dx=\text {Exception raised: NotImplementedError} \]
Exception raised: NotImplementedError >> unable to parse Giac output: -1/9 *sign(sageVARx)+2*((1/sageVARx)^3*((1/sageVARx)^3*(1/24*(1/sageVARx)^3/sig n(sageVARx)-1/18/sign(sageVARx))-1/48/sign(sageVARx))+1/18/sign(sageVARx)) *sqrt(-(1/sageVARx)
Time = 6.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{13}} \, dx=\frac {\frac {\sqrt {x^6-1}}{24}-\frac {{\left (x^6-1\right )}^{3/2}}{24}}{x^{12}}-\frac {\mathrm {atan}\left (\sqrt {x^6-1}\right )}{24}+\frac {{\left (x^6-1\right )}^{3/2}}{9\,x^9} \]