Integrand size = 20, antiderivative size = 37 \[ \int \frac {1+4 x^3}{x \sqrt {-1+x^6}} \, dx=\frac {2}{3} \arctan \left (x^3+\sqrt {-1+x^6}\right )+\frac {4}{3} \log \left (x^3+\sqrt {-1+x^6}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.89, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1489, 858, 223, 212, 272, 65, 209} \[ \int \frac {1+4 x^3}{x \sqrt {-1+x^6}} \, dx=\frac {1}{3} \arctan \left (\sqrt {x^6-1}\right )+\frac {4}{3} \text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right ) \]
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Rule 65
Rule 209
Rule 212
Rule 223
Rule 272
Rule 858
Rule 1489
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {1+4 x}{x \sqrt {-1+x^2}} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \frac {1}{x \sqrt {-1+x^2}} \, dx,x,x^3\right )+\frac {4}{3} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^3\right ) \\ & = \frac {1}{6} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )+\frac {4}{3} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right ) \\ & = \frac {4}{3} \text {arctanh}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right ) \\ & = \frac {1}{3} \arctan \left (\sqrt {-1+x^6}\right )+\frac {4}{3} \text {arctanh}\left (\frac {x^3}{\sqrt {-1+x^6}}\right ) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.11 \[ \int \frac {1+4 x^3}{x \sqrt {-1+x^6}} \, dx=-\frac {2}{3} \arctan \left (x^3-\sqrt {-1+x^6}\right )-\frac {4}{3} \log \left (-x^3+\sqrt {-1+x^6}\right ) \]
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Time = 0.94 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.70
method | result | size |
pseudoelliptic | \(\frac {4 \ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{3}-\frac {\arctan \left (\frac {1}{\sqrt {x^{6}-1}}\right )}{3}\) | \(26\) |
trager | \(\frac {4 \ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{3}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{3}\) | \(43\) |
meijerg | \(\frac {4 \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \arcsin \left (x^{3}\right )}{3 \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}+\frac {\sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )+\left (-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }\right )}{6 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}\) | \(86\) |
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Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.89 \[ \int \frac {1+4 x^3}{x \sqrt {-1+x^6}} \, dx=\frac {2}{3} \, \arctan \left (-x^{3} + \sqrt {x^{6} - 1}\right ) - \frac {4}{3} \, \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) \]
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Time = 3.60 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.97 \[ \int \frac {1+4 x^3}{x \sqrt {-1+x^6}} \, dx=\frac {\begin {cases} \operatorname {acos}{\left (\frac {1}{x^{3}} \right )} & \text {for}\: x^{3} > -1 \wedge x^{3} < 1 \end {cases}}{3} + \frac {4 \log {\left (2 x^{3} + 2 \sqrt {x^{6} - 1} \right )}}{3} \]
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Time = 0.29 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.16 \[ \int \frac {1+4 x^3}{x \sqrt {-1+x^6}} \, dx=\frac {1}{3} \, \arctan \left (\sqrt {x^{6} - 1}\right ) + \frac {2}{3} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} + 1\right ) - \frac {2}{3} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} - 1\right ) \]
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\[ \int \frac {1+4 x^3}{x \sqrt {-1+x^6}} \, dx=\int { \frac {4 \, x^{3} + 1}{\sqrt {x^{6} - 1} x} \,d x } \]
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Time = 5.37 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.68 \[ \int \frac {1+4 x^3}{x \sqrt {-1+x^6}} \, dx=\frac {4\,\ln \left (\sqrt {x^6-1}+x^3\right )}{3}+\frac {\mathrm {atan}\left (\sqrt {x^6-1}\right )}{3} \]
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