Integrand size = 15, antiderivative size = 53 \[ \int \frac {1}{x^7 \sqrt {-1-x^3}} \, dx=\frac {\sqrt {-1-x^3}}{6 x^6}-\frac {\sqrt {-1-x^3}}{4 x^3}+\frac {1}{4} \arctan \left (\sqrt {-1-x^3}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {272, 44, 65, 210} \[ \int \frac {1}{x^7 \sqrt {-1-x^3}} \, dx=\frac {1}{4} \arctan \left (\sqrt {-x^3-1}\right )-\frac {\sqrt {-x^3-1}}{4 x^3}+\frac {\sqrt {-x^3-1}}{6 x^6} \]
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Rule 44
Rule 65
Rule 210
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt {-1-x} x^3} \, dx,x,x^3\right ) \\ & = \frac {\sqrt {-1-x^3}}{6 x^6}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt {-1-x} x^2} \, dx,x,x^3\right ) \\ & = \frac {\sqrt {-1-x^3}}{6 x^6}-\frac {\sqrt {-1-x^3}}{4 x^3}+\frac {1}{8} \text {Subst}\left (\int \frac {1}{\sqrt {-1-x} x} \, dx,x,x^3\right ) \\ & = \frac {\sqrt {-1-x^3}}{6 x^6}-\frac {\sqrt {-1-x^3}}{4 x^3}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {-1-x^3}\right ) \\ & = \frac {\sqrt {-1-x^3}}{6 x^6}-\frac {\sqrt {-1-x^3}}{4 x^3}+\frac {1}{4} \tan ^{-1}\left (\sqrt {-1-x^3}\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.79 \[ \int \frac {1}{x^7 \sqrt {-1-x^3}} \, dx=\frac {\left (2-3 x^3\right ) \sqrt {-1-x^3}}{12 x^6}+\frac {1}{4} \arctan \left (\sqrt {-1-x^3}\right ) \]
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Time = 4.14 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.72
method | result | size |
risch | \(\frac {3 x^{6}+x^{3}-2}{12 x^{6} \sqrt {-x^{3}-1}}+\frac {\arctan \left (\sqrt {-x^{3}-1}\right )}{4}\) | \(38\) |
default | \(\frac {\arctan \left (\sqrt {-x^{3}-1}\right )}{4}+\frac {\sqrt {-x^{3}-1}}{6 x^{6}}-\frac {\sqrt {-x^{3}-1}}{4 x^{3}}\) | \(42\) |
elliptic | \(\frac {\arctan \left (\sqrt {-x^{3}-1}\right )}{4}+\frac {\sqrt {-x^{3}-1}}{6 x^{6}}-\frac {\sqrt {-x^{3}-1}}{4 x^{3}}\) | \(42\) |
pseudoelliptic | \(\frac {3 \arctan \left (\sqrt {-x^{3}-1}\right ) x^{6}-3 x^{3} \sqrt {-x^{3}-1}+2 \sqrt {-x^{3}-1}}{12 x^{6}}\) | \(47\) |
trager | \(-\frac {\left (3 x^{3}-2\right ) \sqrt {-x^{3}-1}}{12 x^{6}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {x^{3} \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )-2 \sqrt {-x^{3}-1}}{x^{3}}\right )}{8}\) | \(66\) |
meijerg | \(-\frac {i \left (-\frac {\sqrt {\pi }}{2 x^{6}}+\frac {\sqrt {\pi }}{2 x^{3}}+\frac {3 \left (\frac {7}{6}-2 \ln \left (2\right )+3 \ln \left (x \right )\right ) \sqrt {\pi }}{8}+\frac {\sqrt {\pi }\, \left (-7 x^{6}-8 x^{3}+8\right )}{16 x^{6}}-\frac {\sqrt {\pi }\, \left (-12 x^{3}+8\right ) \sqrt {x^{3}+1}}{16 x^{6}}-\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{3}+1}}{2}\right )}{4}\right )}{3 \sqrt {\pi }}\) | \(98\) |
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Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^7 \sqrt {-1-x^3}} \, dx=\frac {3 \, x^{6} \arctan \left (\sqrt {-x^{3} - 1}\right ) - {\left (3 \, x^{3} - 2\right )} \sqrt {-x^{3} - 1}}{12 \, x^{6}} \]
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Result contains complex when optimal does not.
Time = 2.15 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.25 \[ \int \frac {1}{x^7 \sqrt {-1-x^3}} \, dx=\frac {i \operatorname {asinh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{4} - \frac {i}{4 x^{\frac {3}{2}} \sqrt {1 + \frac {1}{x^{3}}}} - \frac {i}{12 x^{\frac {9}{2}} \sqrt {1 + \frac {1}{x^{3}}}} + \frac {i}{6 x^{\frac {15}{2}} \sqrt {1 + \frac {1}{x^{3}}}} \]
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Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^7 \sqrt {-1-x^3}} \, dx=-\frac {3 \, {\left (-x^{3} - 1\right )}^{\frac {3}{2}} + 5 \, \sqrt {-x^{3} - 1}}{12 \, {\left (2 \, x^{3} - {\left (x^{3} + 1\right )}^{2} + 1\right )}} + \frac {1}{4} \, \arctan \left (\sqrt {-x^{3} - 1}\right ) \]
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Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x^7 \sqrt {-1-x^3}} \, dx=\frac {3 \, {\left (-x^{3} - 1\right )}^{\frac {3}{2}} + 5 \, \sqrt {-x^{3} - 1}}{12 \, x^{6}} + \frac {1}{4} \, \arctan \left (\sqrt {-x^{3} - 1}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 209, normalized size of antiderivative = 3.94 \[ \int \frac {1}{x^7 \sqrt {-1-x^3}} \, dx=\frac {\sqrt {-x^3-1}}{6\,x^6}-\frac {\sqrt {-x^3-1}}{4\,x^3}-\frac {3\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {x^3+1}\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{4\,\sqrt {-x^3-1}\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
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