Integrand size = 13, antiderivative size = 63 \[ \int \frac {1}{x^{10} \sqrt {-1+x^3}} \, dx=\frac {\sqrt {-1+x^3}}{9 x^9}+\frac {5 \sqrt {-1+x^3}}{36 x^6}+\frac {5 \sqrt {-1+x^3}}{24 x^3}+\frac {5}{24} \arctan \left (\sqrt {-1+x^3}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 44, 65, 209} \[ \int \frac {1}{x^{10} \sqrt {-1+x^3}} \, dx=\frac {5}{24} \arctan \left (\sqrt {x^3-1}\right )+\frac {5 \sqrt {x^3-1}}{24 x^3}+\frac {\sqrt {x^3-1}}{9 x^9}+\frac {5 \sqrt {x^3-1}}{36 x^6} \]
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Rule 44
Rule 65
Rule 209
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^4} \, dx,x,x^3\right ) \\ & = \frac {\sqrt {-1+x^3}}{9 x^9}+\frac {5}{18} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^3} \, dx,x,x^3\right ) \\ & = \frac {\sqrt {-1+x^3}}{9 x^9}+\frac {5 \sqrt {-1+x^3}}{36 x^6}+\frac {5}{24} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^2} \, dx,x,x^3\right ) \\ & = \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}{48} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^3\right ) \\ & = \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} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^3}\right ) \\ & = \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} \tan ^{-1}\left (\sqrt {-1+x^3}\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.68 \[ \int \frac {1}{x^{10} \sqrt {-1+x^3}} \, dx=\frac {\sqrt {-1+x^3} \left (8+10 x^3+15 x^6\right )}{72 x^9}+\frac {5}{24} \arctan \left (\sqrt {-1+x^3}\right ) \]
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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 | \(-\frac {\sqrt {-\operatorname {signum}\left (x^{3}-1\right )}\, \left (\frac {\sqrt {\pi }}{3 x^{9}}+\frac {\sqrt {\pi }}{4 x^{6}}+\frac {3 \sqrt {\pi }}{8 x^{3}}-\frac {5 \left (\frac {37}{30}-2 \ln \left (2\right )+3 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{16}-\frac {\sqrt {\pi }\, \left (-148 x^{9}+144 x^{6}+96 x^{3}+128\right )}{384 x^{9}}+\frac {\sqrt {\pi }\, \left (240 x^{6}+160 x^{3}+128\right ) \sqrt {-x^{3}+1}}{384 x^{9}}+\frac {5 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{3}+1}}{2}\right )}{8}\right )}{3 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x^{3}-1\right )}}\) | \(141\) |
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Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.62 \[ \int \frac {1}{x^{10} \sqrt {-1+x^3}} \, dx=\frac {15 \, x^{9} \arctan \left (\sqrt {x^{3} - 1}\right ) + {\left (15 \, x^{6} + 10 \, x^{3} + 8\right )} \sqrt {x^{3} - 1}}{72 \, x^{9}} \]
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Result contains complex when optimal does not.
Time = 6.62 (sec) , antiderivative size = 182, normalized size of antiderivative = 2.89 \[ \int \frac {1}{x^{10} \sqrt {-1+x^3}} \, dx=\begin {cases} \frac {5 i \operatorname {acosh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{24} - \frac {5 i}{24 x^{\frac {3}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} + \frac {5 i}{72 x^{\frac {9}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} + \frac {i}{36 x^{\frac {15}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} + \frac {i}{9 x^{\frac {21}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\- \frac {5 \operatorname {asin}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{24} + \frac {5}{24 x^{\frac {3}{2}} \sqrt {1 - \frac {1}{x^{3}}}} - \frac {5}{72 x^{\frac {9}{2}} \sqrt {1 - \frac {1}{x^{3}}}} - \frac {1}{36 x^{\frac {15}{2}} \sqrt {1 - \frac {1}{x^{3}}}} - \frac {1}{9 x^{\frac {21}{2}} \sqrt {1 - \frac {1}{x^{3}}}} & \text {otherwise} \end {cases} \]
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Time = 0.33 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x^{10} \sqrt {-1+x^3}} \, dx=\frac {15 \, {\left (x^{3} - 1\right )}^{\frac {5}{2}} + 40 \, {\left (x^{3} - 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {x^{3} - 1}}{72 \, {\left ({\left (x^{3} - 1\right )}^{3} + 3 \, x^{3} + 3 \, {\left (x^{3} - 1\right )}^{2} - 2\right )}} + \frac {5}{24} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \]
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Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.70 \[ \int \frac {1}{x^{10} \sqrt {-1+x^3}} \, dx=\frac {15 \, {\left (x^{3} - 1\right )}^{\frac {5}{2}} + 40 \, {\left (x^{3} - 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {x^{3} - 1}}{72 \, x^{9}} + \frac {5}{24} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 201, normalized size of antiderivative = 3.19 \[ \int \frac {1}{x^{10} \sqrt {-1+x^3}} \, dx=\frac {5\,\sqrt {x^3-1}}{24\,x^3}+\frac {5\,\sqrt {x^3-1}}{36\,x^6}+\frac {\sqrt {x^3-1}}{9\,x^9}-\frac {5\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{8\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
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