Integrand size = 13, antiderivative size = 43 \[ \int \frac {1}{x^{19} \sqrt {-1+x^6}} \, dx=\frac {\sqrt {-1+x^6} \left (8+10 x^6+15 x^{12}\right )}{144 x^{18}}+\frac {5}{48} \arctan \left (\sqrt {-1+x^6}\right ) \] Output:
1/144*(x^6-1)^(1/2)*(15*x^12+10*x^6+8)/x^18+5/48*arctan((x^6-1)^(1/2))
Time = 0.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^{19} \sqrt {-1+x^6}} \, dx=\frac {\sqrt {-1+x^6} \left (8+10 x^6+15 x^{12}\right )}{144 x^{18}}+\frac {5}{48} \arctan \left (\sqrt {-1+x^6}\right ) \] Input:
Integrate[1/(x^19*Sqrt[-1 + x^6]),x]
Output:
(Sqrt[-1 + x^6]*(8 + 10*x^6 + 15*x^12))/(144*x^18) + (5*ArcTan[Sqrt[-1 + x ^6]])/48
Time = 0.17 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.63, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {798, 52, 52, 52, 73, 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.
\(\displaystyle \int \frac {1}{x^{19} \sqrt {x^6-1}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{6} \int \frac {1}{x^{24} \sqrt {x^6-1}}dx^6\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{6} \left (\frac {5}{6} \int \frac {1}{x^{18} \sqrt {x^6-1}}dx^6+\frac {\sqrt {x^6-1}}{3 x^{18}}\right )\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{6} \left (\frac {5}{6} \left (\frac {3}{4} \int \frac {1}{x^{12} \sqrt {x^6-1}}dx^6+\frac {\sqrt {x^6-1}}{2 x^{12}}\right )+\frac {\sqrt {x^6-1}}{3 x^{18}}\right )\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{6} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{x^6 \sqrt {x^6-1}}dx^6+\frac {\sqrt {x^6-1}}{x^6}\right )+\frac {\sqrt {x^6-1}}{2 x^{12}}\right )+\frac {\sqrt {x^6-1}}{3 x^{18}}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{6} \left (\frac {5}{6} \left (\frac {3}{4} \left (\int \frac {1}{x^{12}+1}d\sqrt {x^6-1}+\frac {\sqrt {x^6-1}}{x^6}\right )+\frac {\sqrt {x^6-1}}{2 x^{12}}\right )+\frac {\sqrt {x^6-1}}{3 x^{18}}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{6} \left (\frac {5}{6} \left (\frac {3}{4} \left (\arctan \left (\sqrt {x^6-1}\right )+\frac {\sqrt {x^6-1}}{x^6}\right )+\frac {\sqrt {x^6-1}}{2 x^{12}}\right )+\frac {\sqrt {x^6-1}}{3 x^{18}}\right )\) |
Input:
Int[1/(x^19*Sqrt[-1 + x^6]),x]
Output:
(Sqrt[-1 + x^6]/(3*x^18) + (5*(Sqrt[-1 + x^6]/(2*x^12) + (3*(Sqrt[-1 + x^6 ]/x^6 + ArcTan[Sqrt[-1 + x^6]]))/4))/6)/6
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.31 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.93
method | result | size |
pseudoelliptic | \(\frac {-15 \arctan \left (\frac {1}{\sqrt {x^{6}-1}}\right ) x^{18}+\left (15 x^{12}+10 x^{6}+8\right ) \sqrt {x^{6}-1}}{144 x^{18}}\) | \(40\) |
trager | \(\frac {\sqrt {x^{6}-1}\, \left (15 x^{12}+10 x^{6}+8\right )}{144 x^{18}}-\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{48}\) | \(55\) |
risch | \(\frac {15 x^{18}-5 x^{12}-2 x^{6}-8}{144 x^{18} \sqrt {x^{6}-1}}+\frac {5 \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \left (\left (-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )\right )}{96 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}\) | \(91\) |
meijerg | \(-\frac {\sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \left (\frac {\sqrt {\pi }}{3 x^{18}}+\frac {\sqrt {\pi }}{4 x^{12}}+\frac {3 \sqrt {\pi }}{8 x^{6}}-\frac {5 \left (\frac {37}{30}-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{16}-\frac {\sqrt {\pi }\, \left (-148 x^{18}+144 x^{12}+96 x^{6}+128\right )}{384 x^{18}}+\frac {\sqrt {\pi }\, \left (240 x^{12}+160 x^{6}+128\right ) \sqrt {-x^{6}+1}}{384 x^{18}}+\frac {5 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )}{8}\right )}{6 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}\) | \(141\) |
Input:
int(1/x^19/(x^6-1)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/144*(-15*arctan(1/(x^6-1)^(1/2))*x^18+(15*x^12+10*x^6+8)*(x^6-1)^(1/2))/ x^18
Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x^{19} \sqrt {-1+x^6}} \, dx=\frac {15 \, x^{18} \arctan \left (\sqrt {x^{6} - 1}\right ) + {\left (15 \, x^{12} + 10 \, x^{6} + 8\right )} \sqrt {x^{6} - 1}}{144 \, x^{18}} \] Input:
integrate(1/x^19/(x^6-1)^(1/2),x, algorithm="fricas")
Output:
1/144*(15*x^18*arctan(sqrt(x^6 - 1)) + (15*x^12 + 10*x^6 + 8)*sqrt(x^6 - 1 ))/x^18
Result contains complex when optimal does not.
Time = 8.15 (sec) , antiderivative size = 165, normalized size of antiderivative = 3.84 \[ \int \frac {1}{x^{19} \sqrt {-1+x^6}} \, dx=\begin {cases} \frac {5 i \operatorname {acosh}{\left (\frac {1}{x^{3}} \right )}}{48} - \frac {5 i}{48 x^{3} \sqrt {-1 + \frac {1}{x^{6}}}} + \frac {5 i}{144 x^{9} \sqrt {-1 + \frac {1}{x^{6}}}} + \frac {i}{72 x^{15} \sqrt {-1 + \frac {1}{x^{6}}}} + \frac {i}{18 x^{21} \sqrt {-1 + \frac {1}{x^{6}}}} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\- \frac {5 \operatorname {asin}{\left (\frac {1}{x^{3}} \right )}}{48} + \frac {5}{48 x^{3} \sqrt {1 - \frac {1}{x^{6}}}} - \frac {5}{144 x^{9} \sqrt {1 - \frac {1}{x^{6}}}} - \frac {1}{72 x^{15} \sqrt {1 - \frac {1}{x^{6}}}} - \frac {1}{18 x^{21} \sqrt {1 - \frac {1}{x^{6}}}} & \text {otherwise} \end {cases} \] Input:
integrate(1/x**19/(x**6-1)**(1/2),x)
Output:
Piecewise((5*I*acosh(x**(-3))/48 - 5*I/(48*x**3*sqrt(-1 + x**(-6))) + 5*I/ (144*x**9*sqrt(-1 + x**(-6))) + I/(72*x**15*sqrt(-1 + x**(-6))) + I/(18*x* *21*sqrt(-1 + x**(-6))), 1/Abs(x**6) > 1), (-5*asin(x**(-3))/48 + 5/(48*x* *3*sqrt(1 - 1/x**6)) - 5/(144*x**9*sqrt(1 - 1/x**6)) - 1/(72*x**15*sqrt(1 - 1/x**6)) - 1/(18*x**21*sqrt(1 - 1/x**6)), True))
Time = 0.12 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.53 \[ \int \frac {1}{x^{19} \sqrt {-1+x^6}} \, dx=\frac {15 \, {\left (x^{6} - 1\right )}^{\frac {5}{2}} + 40 \, {\left (x^{6} - 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {x^{6} - 1}}{144 \, {\left (3 \, x^{6} + {\left (x^{6} - 1\right )}^{3} + 3 \, {\left (x^{6} - 1\right )}^{2} - 2\right )}} + \frac {5}{48} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \] Input:
integrate(1/x^19/(x^6-1)^(1/2),x, algorithm="maxima")
Output:
1/144*(15*(x^6 - 1)^(5/2) + 40*(x^6 - 1)^(3/2) + 33*sqrt(x^6 - 1))/(3*x^6 + (x^6 - 1)^3 + 3*(x^6 - 1)^2 - 2) + 5/48*arctan(sqrt(x^6 - 1))
Time = 0.29 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x^{19} \sqrt {-1+x^6}} \, dx=\frac {15 \, {\left (x^{6} - 1\right )}^{\frac {5}{2}} + 40 \, {\left (x^{6} - 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {x^{6} - 1}}{144 \, x^{18}} + \frac {5}{48} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \] Input:
integrate(1/x^19/(x^6-1)^(1/2),x, algorithm="giac")
Output:
1/144*(15*(x^6 - 1)^(5/2) + 40*(x^6 - 1)^(3/2) + 33*sqrt(x^6 - 1))/x^18 + 5/48*arctan(sqrt(x^6 - 1))
Time = 7.97 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^{19} \sqrt {-1+x^6}} \, dx=\frac {5\,\mathrm {atan}\left (\sqrt {x^6-1}\right )}{48}+\frac {11\,\sqrt {x^6-1}}{48\,x^{18}}+\frac {5\,{\left (x^6-1\right )}^{3/2}}{18\,x^{18}}+\frac {5\,{\left (x^6-1\right )}^{5/2}}{48\,x^{18}} \] Input:
int(1/(x^19*(x^6 - 1)^(1/2)),x)
Output:
(5*atan((x^6 - 1)^(1/2)))/48 + (11*(x^6 - 1)^(1/2))/(48*x^18) + (5*(x^6 - 1)^(3/2))/(18*x^18) + (5*(x^6 - 1)^(5/2))/(48*x^18)
Time = 0.16 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.65 \[ \int \frac {1}{x^{19} \sqrt {-1+x^6}} \, dx=\frac {15 \mathit {atan} \left (\frac {\sqrt {x^{6}-1}\, x^{6}-2 \sqrt {x^{6}-1}}{2 x^{6}-2}\right ) x^{18}+30 \sqrt {x^{6}-1}\, x^{12}+20 \sqrt {x^{6}-1}\, x^{6}+16 \sqrt {x^{6}-1}}{288 x^{18}} \] Input:
int(1/x^19/(x^6-1)^(1/2),x)
Output:
(15*atan((sqrt(x**6 - 1)*x**6 - 2*sqrt(x**6 - 1))/(2*x**6 - 2))*x**18 + 30 *sqrt(x**6 - 1)*x**12 + 20*sqrt(x**6 - 1)*x**6 + 16*sqrt(x**6 - 1))/(288*x **18)