Integrand size = 13, antiderivative size = 53 \[ \int x^{20} \sqrt {-1+x^6} \, dx=\frac {\sqrt {-1+x^6} \left (-15 x^3-10 x^9-8 x^{15}+48 x^{21}\right )}{1152}-\frac {5}{384} \log \left (x^3+\sqrt {-1+x^6}\right ) \]
Time = 0.14 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.98 \[ \int x^{20} \sqrt {-1+x^6} \, dx=\frac {x^3 \sqrt {-1+x^6} \left (-15-10 x^6-8 x^{12}+48 x^{18}\right )}{1152}-\frac {5}{384} \log \left (x^3+\sqrt {-1+x^6}\right ) \]
(x^3*Sqrt[-1 + x^6]*(-15 - 10*x^6 - 8*x^12 + 48*x^18))/1152 - (5*Log[x^3 + Sqrt[-1 + x^6]])/384
Time = 0.20 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.92, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {807, 248, 262, 262, 262, 224, 219}
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.
\(\displaystyle \int x^{20} \sqrt {x^6-1} \, dx\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {1}{3} \int x^{18} \sqrt {x^6-1}dx^3\) |
\(\Big \downarrow \) 248 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{8} x^{21} \sqrt {x^6-1}-\frac {1}{8} \int \frac {x^{18}}{\sqrt {x^6-1}}dx^3\right )\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{8} \left (-\frac {5}{6} \int \frac {x^{12}}{\sqrt {x^6-1}}dx^3-\frac {1}{6} \sqrt {x^6-1} x^{15}\right )+\frac {1}{8} \sqrt {x^6-1} x^{21}\right )\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{8} \left (-\frac {5}{6} \left (\frac {3}{4} \int \frac {x^6}{\sqrt {x^6-1}}dx^3+\frac {1}{4} \sqrt {x^6-1} x^9\right )-\frac {1}{6} \sqrt {x^6-1} x^{15}\right )+\frac {1}{8} \sqrt {x^6-1} x^{21}\right )\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{8} \left (-\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {x^6-1}}dx^3+\frac {1}{2} \sqrt {x^6-1} x^3\right )+\frac {1}{4} \sqrt {x^6-1} x^9\right )-\frac {1}{6} \sqrt {x^6-1} x^{15}\right )+\frac {1}{8} \sqrt {x^6-1} x^{21}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{8} \left (-\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{1-x^6}d\frac {x^3}{\sqrt {x^6-1}}+\frac {1}{2} \sqrt {x^6-1} x^3\right )+\frac {1}{4} \sqrt {x^6-1} x^9\right )-\frac {1}{6} \sqrt {x^6-1} x^{15}\right )+\frac {1}{8} \sqrt {x^6-1} x^{21}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{8} \left (-\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )+\frac {1}{2} \sqrt {x^6-1} x^3\right )+\frac {1}{4} \sqrt {x^6-1} x^9\right )-\frac {1}{6} \sqrt {x^6-1} x^{15}\right )+\frac {1}{8} \sqrt {x^6-1} x^{21}\right )\) |
((x^21*Sqrt[-1 + x^6])/8 + (-1/6*(x^15*Sqrt[-1 + x^6]) - (5*((x^9*Sqrt[-1 + x^6])/4 + (3*((x^3*Sqrt[-1 + x^6])/2 + ArcTanh[x^3/Sqrt[-1 + x^6]]/2))/4 ))/6)/8)/3
3.7.76.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 2*p + 1))), x] + Simp[2*a*(p/(m + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && GtQ[ p, 0] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Time = 1.13 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.85
method | result | size |
trager | \(\frac {x^{3} \left (48 x^{18}-8 x^{12}-10 x^{6}-15\right ) \sqrt {x^{6}-1}}{1152}-\frac {5 \ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{384}\) | \(45\) |
pseudoelliptic | \(\frac {\sqrt {x^{6}-1}\, \left (48 x^{21}-8 x^{15}-10 x^{9}-15 x^{3}\right )}{1152}-\frac {5 \ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{384}\) | \(46\) |
risch | \(\frac {x^{3} \left (48 x^{18}-8 x^{12}-10 x^{6}-15\right ) \sqrt {x^{6}-1}}{1152}-\frac {5 \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \arcsin \left (x^{3}\right )}{384 \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}\) | \(55\) |
meijerg | \(-\frac {i \sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, \left (-\frac {i \sqrt {\pi }\, x^{3} \left (-336 x^{18}+56 x^{12}+70 x^{6}+105\right ) \sqrt {-x^{6}+1}}{672}+\frac {5 i \sqrt {\pi }\, \arcsin \left (x^{3}\right )}{32}\right )}{12 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}}\) | \(71\) |
Time = 0.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.89 \[ \int x^{20} \sqrt {-1+x^6} \, dx=\frac {1}{1152} \, {\left (48 \, x^{21} - 8 \, x^{15} - 10 \, x^{9} - 15 \, x^{3}\right )} \sqrt {x^{6} - 1} + \frac {5}{384} \, \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) \]
Result contains complex when optimal does not.
Time = 17.97 (sec) , antiderivative size = 175, normalized size of antiderivative = 3.30 \[ \int x^{20} \sqrt {-1+x^6} \, dx=\begin {cases} \frac {x^{27}}{24 \sqrt {x^{6} - 1}} - \frac {7 x^{21}}{144 \sqrt {x^{6} - 1}} - \frac {x^{15}}{576 \sqrt {x^{6} - 1}} - \frac {5 x^{9}}{1152 \sqrt {x^{6} - 1}} + \frac {5 x^{3}}{384 \sqrt {x^{6} - 1}} - \frac {5 \operatorname {acosh}{\left (x^{3} \right )}}{384} & \text {for}\: \left |{x^{6}}\right | > 1 \\- \frac {i x^{27}}{24 \sqrt {1 - x^{6}}} + \frac {7 i x^{21}}{144 \sqrt {1 - x^{6}}} + \frac {i x^{15}}{576 \sqrt {1 - x^{6}}} + \frac {5 i x^{9}}{1152 \sqrt {1 - x^{6}}} - \frac {5 i x^{3}}{384 \sqrt {1 - x^{6}}} + \frac {5 i \operatorname {asin}{\left (x^{3} \right )}}{384} & \text {otherwise} \end {cases} \]
Piecewise((x**27/(24*sqrt(x**6 - 1)) - 7*x**21/(144*sqrt(x**6 - 1)) - x**1 5/(576*sqrt(x**6 - 1)) - 5*x**9/(1152*sqrt(x**6 - 1)) + 5*x**3/(384*sqrt(x **6 - 1)) - 5*acosh(x**3)/384, Abs(x**6) > 1), (-I*x**27/(24*sqrt(1 - x**6 )) + 7*I*x**21/(144*sqrt(1 - x**6)) + I*x**15/(576*sqrt(1 - x**6)) + 5*I*x **9/(1152*sqrt(1 - x**6)) - 5*I*x**3/(384*sqrt(1 - x**6)) + 5*I*asin(x**3) /384, True))
Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (45) = 90\).
Time = 0.18 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.53 \[ \int x^{20} \sqrt {-1+x^6} \, dx=-\frac {\frac {15 \, \sqrt {x^{6} - 1}}{x^{3}} + \frac {73 \, {\left (x^{6} - 1\right )}^{\frac {3}{2}}}{x^{9}} - \frac {55 \, {\left (x^{6} - 1\right )}^{\frac {5}{2}}}{x^{15}} + \frac {15 \, {\left (x^{6} - 1\right )}^{\frac {7}{2}}}{x^{21}}}{1152 \, {\left (\frac {4 \, {\left (x^{6} - 1\right )}}{x^{6}} - \frac {6 \, {\left (x^{6} - 1\right )}^{2}}{x^{12}} + \frac {4 \, {\left (x^{6} - 1\right )}^{3}}{x^{18}} - \frac {{\left (x^{6} - 1\right )}^{4}}{x^{24}} - 1\right )}} - \frac {5}{768} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} + 1\right ) + \frac {5}{768} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} - 1\right ) \]
-1/1152*(15*sqrt(x^6 - 1)/x^3 + 73*(x^6 - 1)^(3/2)/x^9 - 55*(x^6 - 1)^(5/2 )/x^15 + 15*(x^6 - 1)^(7/2)/x^21)/(4*(x^6 - 1)/x^6 - 6*(x^6 - 1)^2/x^12 + 4*(x^6 - 1)^3/x^18 - (x^6 - 1)^4/x^24 - 1) - 5/768*log(sqrt(x^6 - 1)/x^3 + 1) + 5/768*log(sqrt(x^6 - 1)/x^3 - 1)
Time = 0.30 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.30 \[ \int x^{20} \sqrt {-1+x^6} \, dx=\frac {1}{1152} \, {\left (2 \, {\left (4 \, {\left (6 \, x^{6} - 1\right )} x^{6} - 5\right )} x^{6} - 15\right )} \sqrt {x^{6} - 1} x^{3} - \frac {5 \, {\left (\log \left (\sqrt {-\frac {1}{x^{6}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{x^{6}} + 1} + 1\right )\right )}}{768 \, \mathrm {sgn}\left (x\right )} \]
1/1152*(2*(4*(6*x^6 - 1)*x^6 - 5)*x^6 - 15)*sqrt(x^6 - 1)*x^3 - 5/768*(log (sqrt(-1/x^6 + 1) + 1) - log(-sqrt(-1/x^6 + 1) + 1))/sgn(x)
Timed out. \[ \int x^{20} \sqrt {-1+x^6} \, dx=\int x^{20}\,\sqrt {x^6-1} \,d x \]