Integrand size = 13, antiderivative size = 31 \[ \int \frac {\sqrt {-1+x^3}}{x^4} \, dx=-\frac {\sqrt {-1+x^3}}{3 x^3}+\frac {1}{3} \arctan \left (\sqrt {-1+x^3}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 43, 65, 209} \[ \int \frac {\sqrt {-1+x^3}}{x^4} \, dx=\frac {1}{3} \arctan \left (\sqrt {x^3-1}\right )-\frac {\sqrt {x^3-1}}{3 x^3} \]
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Rule 43
Rule 65
Rule 209
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {\sqrt {-1+x}}{x^2} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt {-1+x^3}}{3 x^3}+\frac {1}{6} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt {-1+x^3}}{3 x^3}+\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^3}\right ) \\ & = -\frac {\sqrt {-1+x^3}}{3 x^3}+\frac {1}{3} \arctan \left (\sqrt {-1+x^3}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-1+x^3}}{x^4} \, dx=-\frac {\sqrt {-1+x^3}}{3 x^3}+\frac {1}{3} \arctan \left (\sqrt {-1+x^3}\right ) \]
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Time = 1.96 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77
method | result | size |
default | \(-\frac {\sqrt {x^{3}-1}}{3 x^{3}}+\frac {\arctan \left (\sqrt {x^{3}-1}\right )}{3}\) | \(24\) |
risch | \(-\frac {\sqrt {x^{3}-1}}{3 x^{3}}+\frac {\arctan \left (\sqrt {x^{3}-1}\right )}{3}\) | \(24\) |
elliptic | \(-\frac {\sqrt {x^{3}-1}}{3 x^{3}}+\frac {\arctan \left (\sqrt {x^{3}-1}\right )}{3}\) | \(24\) |
pseudoelliptic | \(\frac {\arctan \left (\sqrt {x^{3}-1}\right ) x^{3}-\sqrt {x^{3}-1}}{3 x^{3}}\) | \(28\) |
trager | \(-\frac {\sqrt {x^{3}-1}}{3 x^{3}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \sqrt {x^{3}-1}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{3}}\right )}{6}\) | \(56\) |
meijerg | \(\frac {\sqrt {\operatorname {signum}\left (x^{3}-1\right )}\, \left (\frac {\sqrt {\pi }\, \left (-4 x^{3}+8\right )}{4 x^{3}}-\frac {2 \sqrt {\pi }\, \sqrt {-x^{3}+1}}{x^{3}}+2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{3}+1}}{2}\right )-\left (-2 \ln \left (2\right )-1+3 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }-\frac {2 \sqrt {\pi }}{x^{3}}\right )}{6 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{3}-1\right )}}\) | \(103\) |
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Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {-1+x^3}}{x^4} \, dx=\frac {x^{3} \arctan \left (\sqrt {x^{3} - 1}\right ) - \sqrt {x^{3} - 1}}{3 \, x^{3}} \]
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Result contains complex when optimal does not.
Time = 0.82 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.65 \[ \int \frac {\sqrt {-1+x^3}}{x^4} \, dx=\begin {cases} \frac {i \operatorname {acosh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{3} + \frac {i}{3 x^{\frac {3}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} - \frac {i}{3 x^{\frac {9}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\- \frac {\operatorname {asin}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{3} - \frac {\sqrt {1 - \frac {1}{x^{3}}}}{3 x^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {-1+x^3}}{x^4} \, dx=-\frac {\sqrt {x^{3} - 1}}{3 \, x^{3}} + \frac {1}{3} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {-1+x^3}}{x^4} \, dx=-\frac {\sqrt {x^{3} - 1}}{3 \, x^{3}} + \frac {1}{3} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \]
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Time = 5.27 (sec) , antiderivative size = 177, normalized size of antiderivative = 5.71 \[ \int \frac {\sqrt {-1+x^3}}{x^4} \, dx=-\frac {\sqrt {x^3-1}}{3\,x^3}-\frac {\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
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