Integrand size = 13, antiderivative size = 63 \[ \int \frac {1}{x^{10} \sqrt {-1+x^3}} \, dx=\frac {\sqrt {-1+x^3}}{9 x^9}+\frac {5 \sqrt {-1+x^3}}{36 x^6}+\frac {5 \sqrt {-1+x^3}}{24 x^3}+\frac {5}{24} \arctan \left (\sqrt {-1+x^3}\right ) \]
5/24*arctan((x^3-1)^(1/2))+1/9*(x^3-1)^(1/2)/x^9+5/36*(x^3-1)^(1/2)/x^6+5/ 24*(x^3-1)^(1/2)/x^3
Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.68 \[ \int \frac {1}{x^{10} \sqrt {-1+x^3}} \, dx=\frac {\sqrt {-1+x^3} \left (8+10 x^3+15 x^6\right )}{72 x^9}+\frac {5}{24} \arctan \left (\sqrt {-1+x^3}\right ) \]
Time = 0.17 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.11, 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.
\(\displaystyle \int \frac {1}{x^{10} \sqrt {x^3-1}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{3} \int \frac {1}{x^{12} \sqrt {x^3-1}}dx^3\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{3} \left (\frac {5}{6} \int \frac {1}{x^9 \sqrt {x^3-1}}dx^3+\frac {\sqrt {x^3-1}}{3 x^9}\right )\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{3} \left (\frac {5}{6} \left (\frac {3}{4} \int \frac {1}{x^6 \sqrt {x^3-1}}dx^3+\frac {\sqrt {x^3-1}}{2 x^6}\right )+\frac {\sqrt {x^3-1}}{3 x^9}\right )\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{3} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{x^3 \sqrt {x^3-1}}dx^3+\frac {\sqrt {x^3-1}}{x^3}\right )+\frac {\sqrt {x^3-1}}{2 x^6}\right )+\frac {\sqrt {x^3-1}}{3 x^9}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{3} \left (\frac {5}{6} \left (\frac {3}{4} \left (\int \frac {1}{x^6+1}d\sqrt {x^3-1}+\frac {\sqrt {x^3-1}}{x^3}\right )+\frac {\sqrt {x^3-1}}{2 x^6}\right )+\frac {\sqrt {x^3-1}}{3 x^9}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{3} \left (\frac {5}{6} \left (\frac {3}{4} \left (\arctan \left (\sqrt {x^3-1}\right )+\frac {\sqrt {x^3-1}}{x^3}\right )+\frac {\sqrt {x^3-1}}{2 x^6}\right )+\frac {\sqrt {x^3-1}}{3 x^9}\right )\) |
(Sqrt[-1 + x^3]/(3*x^9) + (5*(Sqrt[-1 + x^3]/(2*x^6) + (3*(Sqrt[-1 + x^3]/ x^3 + ArcTan[Sqrt[-1 + x^3]]))/4))/6)/3
3.5.81.3.1 Defintions of rubi rules used
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 = 4.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.63
method | result | size |
pseudoelliptic | \(\frac {\left (15 x^{6}+10 x^{3}+8\right ) \sqrt {x^{3}-1}+15 \arctan \left (\sqrt {x^{3}-1}\right ) x^{9}}{72 x^{9}}\) | \(40\) |
risch | \(\frac {15 x^{9}-5 x^{6}-2 x^{3}-8}{72 x^{9} \sqrt {x^{3}-1}}+\frac {5 \arctan \left (\sqrt {x^{3}-1}\right )}{24}\) | \(41\) |
default | \(\frac {5 \arctan \left (\sqrt {x^{3}-1}\right )}{24}+\frac {\sqrt {x^{3}-1}}{9 x^{9}}+\frac {5 \sqrt {x^{3}-1}}{36 x^{6}}+\frac {5 \sqrt {x^{3}-1}}{24 x^{3}}\) | \(48\) |
elliptic | \(\frac {5 \arctan \left (\sqrt {x^{3}-1}\right )}{24}+\frac {\sqrt {x^{3}-1}}{9 x^{9}}+\frac {5 \sqrt {x^{3}-1}}{36 x^{6}}+\frac {5 \sqrt {x^{3}-1}}{24 x^{3}}\) | \(48\) |
trager | \(\frac {\left (15 x^{6}+10 x^{3}+8\right ) \sqrt {x^{3}-1}}{72 x^{9}}-\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {x^{3} \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+2 \sqrt {x^{3}-1}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{3}}\right )}{48}\) | \(68\) |
meijerg | \(-\frac {\sqrt {-\operatorname {signum}\left (x^{3}-1\right )}\, \left (\frac {\sqrt {\pi }}{3 x^{9}}+\frac {\sqrt {\pi }}{4 x^{6}}+\frac {3 \sqrt {\pi }}{8 x^{3}}-\frac {5 \left (\frac {37}{30}-2 \ln \left (2\right )+3 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{16}-\frac {\sqrt {\pi }\, \left (-148 x^{9}+144 x^{6}+96 x^{3}+128\right )}{384 x^{9}}+\frac {\sqrt {\pi }\, \left (240 x^{6}+160 x^{3}+128\right ) \sqrt {-x^{3}+1}}{384 x^{9}}+\frac {5 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{3}+1}}{2}\right )}{8}\right )}{3 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x^{3}-1\right )}}\) | \(141\) |
Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.62 \[ \int \frac {1}{x^{10} \sqrt {-1+x^3}} \, dx=\frac {15 \, x^{9} \arctan \left (\sqrt {x^{3} - 1}\right ) + {\left (15 \, x^{6} + 10 \, x^{3} + 8\right )} \sqrt {x^{3} - 1}}{72 \, x^{9}} \]
Result contains complex when optimal does not.
Time = 6.62 (sec) , antiderivative size = 182, normalized size of antiderivative = 2.89 \[ \int \frac {1}{x^{10} \sqrt {-1+x^3}} \, dx=\begin {cases} \frac {5 i \operatorname {acosh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{24} - \frac {5 i}{24 x^{\frac {3}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} + \frac {5 i}{72 x^{\frac {9}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} + \frac {i}{36 x^{\frac {15}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} + \frac {i}{9 x^{\frac {21}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\- \frac {5 \operatorname {asin}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{24} + \frac {5}{24 x^{\frac {3}{2}} \sqrt {1 - \frac {1}{x^{3}}}} - \frac {5}{72 x^{\frac {9}{2}} \sqrt {1 - \frac {1}{x^{3}}}} - \frac {1}{36 x^{\frac {15}{2}} \sqrt {1 - \frac {1}{x^{3}}}} - \frac {1}{9 x^{\frac {21}{2}} \sqrt {1 - \frac {1}{x^{3}}}} & \text {otherwise} \end {cases} \]
Piecewise((5*I*acosh(x**(-3/2))/24 - 5*I/(24*x**(3/2)*sqrt(-1 + x**(-3))) + 5*I/(72*x**(9/2)*sqrt(-1 + x**(-3))) + I/(36*x**(15/2)*sqrt(-1 + x**(-3) )) + I/(9*x**(21/2)*sqrt(-1 + x**(-3))), 1/Abs(x**3) > 1), (-5*asin(x**(-3 /2))/24 + 5/(24*x**(3/2)*sqrt(1 - 1/x**3)) - 5/(72*x**(9/2)*sqrt(1 - 1/x** 3)) - 1/(36*x**(15/2)*sqrt(1 - 1/x**3)) - 1/(9*x**(21/2)*sqrt(1 - 1/x**3)) , True))
Time = 0.33 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x^{10} \sqrt {-1+x^3}} \, dx=\frac {15 \, {\left (x^{3} - 1\right )}^{\frac {5}{2}} + 40 \, {\left (x^{3} - 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {x^{3} - 1}}{72 \, {\left ({\left (x^{3} - 1\right )}^{3} + 3 \, x^{3} + 3 \, {\left (x^{3} - 1\right )}^{2} - 2\right )}} + \frac {5}{24} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \]
1/72*(15*(x^3 - 1)^(5/2) + 40*(x^3 - 1)^(3/2) + 33*sqrt(x^3 - 1))/((x^3 - 1)^3 + 3*x^3 + 3*(x^3 - 1)^2 - 2) + 5/24*arctan(sqrt(x^3 - 1))
Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.70 \[ \int \frac {1}{x^{10} \sqrt {-1+x^3}} \, dx=\frac {15 \, {\left (x^{3} - 1\right )}^{\frac {5}{2}} + 40 \, {\left (x^{3} - 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {x^{3} - 1}}{72 \, x^{9}} + \frac {5}{24} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \]
1/72*(15*(x^3 - 1)^(5/2) + 40*(x^3 - 1)^(3/2) + 33*sqrt(x^3 - 1))/x^9 + 5/ 24*arctan(sqrt(x^3 - 1))
Time = 0.04 (sec) , antiderivative size = 201, normalized size of antiderivative = 3.19 \[ \int \frac {1}{x^{10} \sqrt {-1+x^3}} \, dx=\frac {5\,\sqrt {x^3-1}}{24\,x^3}+\frac {5\,\sqrt {x^3-1}}{36\,x^6}+\frac {\sqrt {x^3-1}}{9\,x^9}-\frac {5\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{8\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
(5*(x^3 - 1)^(1/2))/(24*x^3) + (5*(x^3 - 1)^(1/2))/(36*x^6) + (x^3 - 1)^(1 /2)/(9*x^9) - (5*((3^(1/2)*1i)/2 + 3/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^( 1/2)*1i)/2 - 3/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 3/2 ))^(1/2)*(-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticPi((3^(1/2)*1i)/2 + 3/2, asin((-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)))/(8*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) + x^3)^(1/2) )