Integrand size = 18, antiderivative size = 64 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^7} \, dx=\frac {\left (1-2 x^3\right ) \sqrt {-1+x^6}}{6 x^6}+\frac {1}{3} \arctan \left (\frac {1+x^3}{\sqrt {-1+x^6}}\right )+\frac {2}{3} \text {arctanh}\left (\frac {1+x^3}{\sqrt {-1+x^6}}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.88, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {1489, 825, 858, 223, 212, 272, 65, 209} \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^7} \, dx=-\frac {1}{6} \arctan \left (\sqrt {x^6-1}\right )+\frac {1}{3} \text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )+\frac {\sqrt {x^6-1} \left (1-2 x^3\right )}{6 x^6} \]
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Rule 65
Rule 209
Rule 212
Rule 223
Rule 272
Rule 825
Rule 858
Rule 1489
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {(-1+x) \sqrt {-1+x^2}}{x^3} \, dx,x,x^3\right ) \\ & = \frac {\left (1-2 x^3\right ) \sqrt {-1+x^6}}{6 x^6}+\frac {1}{12} \text {Subst}\left (\int \frac {-2+4 x}{x \sqrt {-1+x^2}} \, dx,x,x^3\right ) \\ & = \frac {\left (1-2 x^3\right ) \sqrt {-1+x^6}}{6 x^6}-\frac {1}{6} \text {Subst}\left (\int \frac {1}{x \sqrt {-1+x^2}} \, dx,x,x^3\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^3\right ) \\ & = \frac {\left (1-2 x^3\right ) \sqrt {-1+x^6}}{6 x^6}-\frac {1}{12} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right ) \\ & = \frac {\left (1-2 x^3\right ) \sqrt {-1+x^6}}{6 x^6}+\frac {1}{3} \text {arctanh}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )-\frac {1}{6} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right ) \\ & = \frac {\left (1-2 x^3\right ) \sqrt {-1+x^6}}{6 x^6}-\frac {1}{6} \arctan \left (\sqrt {-1+x^6}\right )+\frac {1}{3} \text {arctanh}\left (\frac {x^3}{\sqrt {-1+x^6}}\right ) \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.02 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^7} \, dx=\frac {1}{6} \left (\frac {\left (1-2 x^3\right ) \sqrt {-1+x^6}}{x^6}+2 \arctan \left (\frac {\sqrt {-1+x^6}}{-1+x^3}\right )+4 \text {arctanh}\left (\frac {\sqrt {-1+x^6}}{-1+x^3}\right )\right ) \]
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Time = 1.40 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.86
method | result | size |
pseudoelliptic | \(\frac {2 \ln \left (x^{3}+\sqrt {x^{6}-1}\right ) x^{6}+\arctan \left (\frac {1}{\sqrt {x^{6}-1}}\right ) x^{6}-2 x^{3} \sqrt {x^{6}-1}+\sqrt {x^{6}-1}}{6 x^{6}}\) | \(55\) |
trager | \(-\frac {\left (2 x^{3}-1\right ) \sqrt {x^{6}-1}}{6 x^{6}}-\frac {\ln \left (-x^{3}+\sqrt {x^{6}-1}\right )}{3}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{6}\) | \(66\) |
risch | \(-\frac {2 x^{9}-x^{6}-2 x^{3}+1}{6 x^{6} \sqrt {x^{6}-1}}-\frac {\sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )+\left (-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }\right )}{12 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}+\frac {\sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \arcsin \left (x^{3}\right )}{3 \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}\) | \(115\) |
meijerg | \(-\frac {\sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, \left (\frac {\sqrt {\pi }\, \left (-4 x^{6}+8\right )}{4 x^{6}}-\frac {2 \sqrt {\pi }\, \sqrt {-x^{6}+1}}{x^{6}}+2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )-\left (-2 \ln \left (2\right )-1+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }-\frac {2 \sqrt {\pi }}{x^{6}}\right )}{12 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}}-\frac {i \sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, \left (-\frac {4 i \sqrt {\pi }\, \sqrt {-x^{6}+1}}{x^{3}}-4 i \sqrt {\pi }\, \arcsin \left (x^{3}\right )\right )}{12 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}}\) | \(157\) |
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Time = 0.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^7} \, dx=-\frac {2 \, x^{6} \arctan \left (-x^{3} + \sqrt {x^{6} - 1}\right ) + 2 \, x^{6} \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) + 2 \, x^{6} + \sqrt {x^{6} - 1} {\left (2 \, x^{3} - 1\right )}}{6 \, x^{6}} \]
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Result contains complex when optimal does not.
Time = 3.84 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.36 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^7} \, dx=- \begin {cases} \frac {i \operatorname {acosh}{\left (\frac {1}{x^{3}} \right )}}{6} + \frac {i}{6 x^{3} \sqrt {-1 + \frac {1}{x^{6}}}} - \frac {i}{6 x^{9} \sqrt {-1 + \frac {1}{x^{6}}}} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\- \frac {\operatorname {asin}{\left (\frac {1}{x^{3}} \right )}}{6} - \frac {\sqrt {1 - \frac {1}{x^{6}}}}{6 x^{3}} & \text {otherwise} \end {cases} + \begin {cases} - \frac {x^{3}}{3 \sqrt {x^{6} - 1}} + \frac {\operatorname {acosh}{\left (x^{3} \right )}}{3} + \frac {1}{3 x^{3} \sqrt {x^{6} - 1}} & \text {for}\: \left |{x^{6}}\right | > 1 \\\frac {i x^{3}}{3 \sqrt {1 - x^{6}}} - \frac {i \operatorname {asin}{\left (x^{3} \right )}}{3} - \frac {i}{3 x^{3} \sqrt {1 - x^{6}}} & \text {otherwise} \end {cases} \]
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Time = 0.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.05 \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^7} \, dx=-\frac {\sqrt {x^{6} - 1}}{3 \, x^{3}} + \frac {\sqrt {x^{6} - 1}}{6 \, x^{6}} - \frac {1}{6} \, \arctan \left (\sqrt {x^{6} - 1}\right ) + \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} + 1\right ) - \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} - 1\right ) \]
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\[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^7} \, dx=\int { \frac {\sqrt {x^{6} - 1} {\left (x^{3} - 1\right )}}{x^{7}} \,d x } \]
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Timed out. \[ \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^7} \, dx=\int \frac {\left (x^3-1\right )\,\sqrt {x^6-1}}{x^7} \,d x \]
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