Integrand size = 13, antiderivative size = 31 \[ \int \frac {1}{x^4 \sqrt {-1+x^3}} \, dx=\frac {\sqrt {-1+x^3}}{3 x^3}+\frac {1}{3} \arctan \left (\sqrt {-1+x^3}\right ) \]
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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, 44, 65, 209} \[ \int \frac {1}{x^4 \sqrt {-1+x^3}} \, dx=\frac {1}{3} \arctan \left (\sqrt {x^3-1}\right )+\frac {\sqrt {x^3-1}}{3 x^3} \]
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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^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 = 28, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^4 \sqrt {-1+x^3}} \, dx=\frac {1}{3} \left (\frac {\sqrt {-1+x^3}}{x^3}+\arctan \left (\sqrt {-1+x^3}\right )\right ) \]
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Time = 1.98 (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 | \(\frac {\arctan \left (\sqrt {x^{3}-1}\right ) x^{3}+\sqrt {x^{3}-1}}{3 x^{3}}\) | \(26\) |
trager | \(\frac {\sqrt {x^{3}-1}}{3 x^{3}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \sqrt {x^{3}-1}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{3}}\right )}{6}\) | \(57\) |
meijerg | \(-\frac {\sqrt {-\operatorname {signum}\left (x^{3}-1\right )}\, \left (-\frac {\sqrt {\pi }\, \left (-4 x^{3}+8\right )}{8 x^{3}}+\frac {\sqrt {\pi }\, \sqrt {-x^{3}+1}}{x^{3}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{3}+1}}{2}\right )-\frac {\left (1-2 \ln \left (2\right )+3 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{2}+\frac {\sqrt {\pi }}{x^{3}}\right )}{3 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x^{3}-1\right )}}\) | \(100\) |
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Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x^4 \sqrt {-1+x^3}} \, dx=\frac {x^{3} \arctan \left (\sqrt {x^{3} - 1}\right ) + \sqrt {x^{3} - 1}}{3 \, x^{3}} \]
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Result contains complex when optimal does not.
Time = 0.95 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.58 \[ \int \frac {1}{x^4 \sqrt {-1+x^3}} \, dx=\begin {cases} \frac {i \operatorname {acosh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{3} - \frac {i}{3 x^{\frac {3}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} + \frac {i}{3 x^{\frac {9}{2}} \sqrt {-1 + \frac {1}{x^{3}}}} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\- \frac {\operatorname {asin}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{3} + \frac {\sqrt {1 - \frac {1}{x^{3}}}}{3 x^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^4 \sqrt {-1+x^3}} \, dx=\frac {\sqrt {x^{3} - 1}}{3 \, x^{3}} + \frac {1}{3} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^4 \sqrt {-1+x^3}} \, dx=\frac {\sqrt {x^{3} - 1}}{3 \, x^{3}} + \frac {1}{3} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \]
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Time = 5.33 (sec) , antiderivative size = 177, normalized size of antiderivative = 5.71 \[ \int \frac {1}{x^4 \sqrt {-1+x^3}} \, dx=\frac {\sqrt {x^3-1}}{3\,x^3}-\frac {\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
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