Integrand size = 25, antiderivative size = 52 \[ \int \frac {\left (1+x^3\right ) \sqrt {-1+x^6}}{x^{13} \left (-1+x^3\right )} \, dx=\frac {\sqrt {-1+x^6} \left (6+16 x^3+21 x^6+32 x^9\right )}{72 x^{12}}+\frac {7}{12} \arctan \left (x^3+\sqrt {-1+x^6}\right ) \]
Time = 0.14 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.04 \[ \int \frac {\left (1+x^3\right ) \sqrt {-1+x^6}}{x^{13} \left (-1+x^3\right )} \, dx=\frac {\sqrt {-1+x^6} \left (6+16 x^3+21 x^6+32 x^9\right )}{72 x^{12}}-\frac {7}{12} \arctan \left (x^3-\sqrt {-1+x^6}\right ) \]
(Sqrt[-1 + x^6]*(6 + 16*x^3 + 21*x^6 + 32*x^9))/(72*x^12) - (7*ArcTan[x^3 - Sqrt[-1 + x^6]])/12
Leaf count is larger than twice the leaf count of optimal. \(167\) vs. \(2(52)=104\).
Time = 0.27 (sec) , antiderivative size = 167, normalized size of antiderivative = 3.21, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {1396, 948, 109, 168, 168, 168, 27, 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.
| \(\displaystyle \int \frac {\left (x^3+1\right ) \sqrt {x^6-1}}{x^{13} \left (x^3-1\right )} \, dx\) |
| \(\Big \downarrow \) 1396 |
| \(\displaystyle \frac {\sqrt {x^6-1} \int \frac {\left (x^3+1\right )^{3/2}}{x^{13} \sqrt {x^3-1}}dx}{\sqrt {x^3-1} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle \frac {\sqrt {x^6-1} \int \frac {\left (x^3+1\right )^{3/2}}{x^{15} \sqrt {x^3-1}}dx^3}{3 \sqrt {x^3-1} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {\sqrt {x^6-1} \left (\frac {1}{4} \int \frac {7 x^3+8}{x^{12} \sqrt {x^3-1} \sqrt {x^3+1}}dx^3+\frac {\sqrt {x^3-1} \sqrt {x^3+1}}{4 x^{12}}\right )}{3 \sqrt {x^3-1} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {\sqrt {x^6-1} \left (\frac {1}{4} \left (\frac {1}{3} \int \frac {16 x^3+21}{x^9 \sqrt {x^3-1} \sqrt {x^3+1}}dx^3+\frac {8 \sqrt {x^3-1} \sqrt {x^3+1}}{3 x^9}\right )+\frac {\sqrt {x^3-1} \sqrt {x^3+1}}{4 x^{12}}\right )}{3 \sqrt {x^3-1} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {\sqrt {x^6-1} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {21 x^3+32}{x^6 \sqrt {x^3-1} \sqrt {x^3+1}}dx^3+\frac {21 \sqrt {x^3-1} \sqrt {x^3+1}}{2 x^6}\right )+\frac {8 \sqrt {x^3-1} \sqrt {x^3+1}}{3 x^9}\right )+\frac {\sqrt {x^3-1} \sqrt {x^3+1}}{4 x^{12}}\right )}{3 \sqrt {x^3-1} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {\sqrt {x^6-1} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int \frac {21}{x^3 \sqrt {x^3-1} \sqrt {x^3+1}}dx^3+\frac {32 \sqrt {x^3-1} \sqrt {x^3+1}}{x^3}\right )+\frac {21 \sqrt {x^3-1} \sqrt {x^3+1}}{2 x^6}\right )+\frac {8 \sqrt {x^3-1} \sqrt {x^3+1}}{3 x^9}\right )+\frac {\sqrt {x^3-1} \sqrt {x^3+1}}{4 x^{12}}\right )}{3 \sqrt {x^3-1} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {x^6-1} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (21 \int \frac {1}{x^3 \sqrt {x^3-1} \sqrt {x^3+1}}dx^3+\frac {32 \sqrt {x^3-1} \sqrt {x^3+1}}{x^3}\right )+\frac {21 \sqrt {x^3-1} \sqrt {x^3+1}}{2 x^6}\right )+\frac {8 \sqrt {x^3-1} \sqrt {x^3+1}}{3 x^9}\right )+\frac {\sqrt {x^3-1} \sqrt {x^3+1}}{4 x^{12}}\right )}{3 \sqrt {x^3-1} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle \frac {\sqrt {x^6-1} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (21 \int \frac {1}{x^6+1}d\left (\sqrt {x^3-1} \sqrt {x^3+1}\right )+\frac {32 \sqrt {x^3-1} \sqrt {x^3+1}}{x^3}\right )+\frac {21 \sqrt {x^3-1} \sqrt {x^3+1}}{2 x^6}\right )+\frac {8 \sqrt {x^3-1} \sqrt {x^3+1}}{3 x^9}\right )+\frac {\sqrt {x^3-1} \sqrt {x^3+1}}{4 x^{12}}\right )}{3 \sqrt {x^3-1} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\sqrt {x^6-1} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (21 \arctan \left (\sqrt {x^3-1} \sqrt {x^3+1}\right )+\frac {32 \sqrt {x^3-1} \sqrt {x^3+1}}{x^3}\right )+\frac {21 \sqrt {x^3-1} \sqrt {x^3+1}}{2 x^6}\right )+\frac {8 \sqrt {x^3-1} \sqrt {x^3+1}}{3 x^9}\right )+\frac {\sqrt {x^3-1} \sqrt {x^3+1}}{4 x^{12}}\right )}{3 \sqrt {x^3-1} \sqrt {x^3+1}}\) |
(Sqrt[-1 + x^6]*((Sqrt[-1 + x^3]*Sqrt[1 + x^3])/(4*x^12) + ((8*Sqrt[-1 + x ^3]*Sqrt[1 + x^3])/(3*x^9) + ((21*Sqrt[-1 + x^3]*Sqrt[1 + x^3])/(2*x^6) + ((32*Sqrt[-1 + x^3]*Sqrt[1 + x^3])/x^3 + 21*ArcTan[Sqrt[-1 + x^3]*Sqrt[1 + x^3]])/2)/3)/4))/(3*Sqrt[-1 + x^3]*Sqrt[1 + x^3])
3.7.57.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(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] + S imp[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*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[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 = 1.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.87
| method | result | size |
| pseudoelliptic | \(\frac {-21 \arctan \left (\frac {1}{\sqrt {x^{6}-1}}\right ) x^{12}+\left (32 x^{9}+21 x^{6}+16 x^{3}+6\right ) \sqrt {x^{6}-1}}{72 x^{12}}\) | \(45\) |
| trager | \(\frac {\sqrt {x^{6}-1}\, \left (32 x^{9}+21 x^{6}+16 x^{3}+6\right )}{72 x^{12}}+\frac {7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{24}\) | \(58\) |
| risch | \(\frac {32 x^{15}+21 x^{12}-16 x^{9}-15 x^{6}-16 x^{3}-6}{72 x^{12} \sqrt {x^{6}-1}}+\frac {7 \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )+\left (-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }\right )}{48 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}\) | \(101\) |
Time = 0.25 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.06 \[ \int \frac {\left (1+x^3\right ) \sqrt {-1+x^6}}{x^{13} \left (-1+x^3\right )} \, dx=\frac {42 \, x^{12} \arctan \left (-x^{3} + \sqrt {x^{6} - 1}\right ) + 32 \, x^{12} + {\left (32 \, x^{9} + 21 \, x^{6} + 16 \, x^{3} + 6\right )} \sqrt {x^{6} - 1}}{72 \, x^{12}} \]
1/72*(42*x^12*arctan(-x^3 + sqrt(x^6 - 1)) + 32*x^12 + (32*x^9 + 21*x^6 + 16*x^3 + 6)*sqrt(x^6 - 1))/x^12
\[ \int \frac {\left (1+x^3\right ) \sqrt {-1+x^6}}{x^{13} \left (-1+x^3\right )} \, dx=\int \frac {\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}{x^{13} \left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \]
Integral(sqrt((x - 1)*(x + 1)*(x**2 - x + 1)*(x**2 + x + 1))*(x + 1)*(x**2 - x + 1)/(x**13*(x - 1)*(x**2 + x + 1)), x)
\[ \int \frac {\left (1+x^3\right ) \sqrt {-1+x^6}}{x^{13} \left (-1+x^3\right )} \, dx=\int { \frac {\sqrt {x^{6} - 1} {\left (x^{3} + 1\right )}}{{\left (x^{3} - 1\right )} x^{13}} \,d x } \]
Exception generated. \[ \int \frac {\left (1+x^3\right ) \sqrt {-1+x^6}}{x^{13} \left (-1+x^3\right )} \, dx=\text {Exception raised: NotImplementedError} \]
Exception raised: NotImplementedError >> unable to parse Giac output: -4/9 *sign(sageVARx)+2*((1/sageVARx)^3*((1/sageVARx)^3*(1/24*(1/sageVARx)^3/sig n(sageVARx)+1/9/sign(sageVARx))+7/48/sign(sageVARx))+2/9/sign(sageVARx))*s qrt(-(1/sageVARx)^6
Timed out. \[ \int \frac {\left (1+x^3\right ) \sqrt {-1+x^6}}{x^{13} \left (-1+x^3\right )} \, dx=\int \frac {\left (x^3+1\right )\,\sqrt {x^6-1}}{x^{13}\,\left (x^3-1\right )} \,d x \]