Integrand size = 24, antiderivative size = 191 \[ \int \frac {1}{x^3 \left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=-\frac {\left (-1+3 x^2\right )^{3/4}}{4 x^2}+\frac {3}{4} \arctan \left (\sqrt [4]{-1+3 x^2}\right )+\frac {9 \arctan \left (1-\sqrt {2} \sqrt [4]{-1+3 x^2}\right )}{8 \sqrt {2}}-\frac {9 \arctan \left (1+\sqrt {2} \sqrt [4]{-1+3 x^2}\right )}{8 \sqrt {2}}-\frac {3}{4} \text {arctanh}\left (\sqrt [4]{-1+3 x^2}\right )-\frac {9 \log \left (1-\sqrt {2} \sqrt [4]{-1+3 x^2}+\sqrt {-1+3 x^2}\right )}{16 \sqrt {2}}+\frac {9 \log \left (1+\sqrt {2} \sqrt [4]{-1+3 x^2}+\sqrt {-1+3 x^2}\right )}{16 \sqrt {2}} \]
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Time = 0.11 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {457, 105, 162, 65, 303, 1176, 631, 210, 1179, 642, 304, 209, 212} \[ \int \frac {1}{x^3 \left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\frac {3}{4} \arctan \left (\sqrt [4]{3 x^2-1}\right )+\frac {9 \arctan \left (1-\sqrt {2} \sqrt [4]{3 x^2-1}\right )}{8 \sqrt {2}}-\frac {9 \arctan \left (\sqrt {2} \sqrt [4]{3 x^2-1}+1\right )}{8 \sqrt {2}}-\frac {3}{4} \text {arctanh}\left (\sqrt [4]{3 x^2-1}\right )-\frac {\left (3 x^2-1\right )^{3/4}}{4 x^2}-\frac {9 \log \left (\sqrt {3 x^2-1}-\sqrt {2} \sqrt [4]{3 x^2-1}+1\right )}{16 \sqrt {2}}+\frac {9 \log \left (\sqrt {3 x^2-1}+\sqrt {2} \sqrt [4]{3 x^2-1}+1\right )}{16 \sqrt {2}} \]
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Rule 65
Rule 105
Rule 162
Rule 209
Rule 210
Rule 212
Rule 303
Rule 304
Rule 457
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 (-2+3 x) \sqrt [4]{-1+3 x}} \, dx,x,x^2\right ) \\ & = -\frac {\left (-1+3 x^2\right )^{3/4}}{4 x^2}-\frac {1}{4} \text {Subst}\left (\int \frac {-\frac {9}{2}+\frac {9 x}{4}}{x (-2+3 x) \sqrt [4]{-1+3 x}} \, dx,x,x^2\right ) \\ & = -\frac {\left (-1+3 x^2\right )^{3/4}}{4 x^2}-\frac {9}{16} \text {Subst}\left (\int \frac {1}{x \sqrt [4]{-1+3 x}} \, dx,x,x^2\right )+\frac {9}{8} \text {Subst}\left (\int \frac {1}{(-2+3 x) \sqrt [4]{-1+3 x}} \, dx,x,x^2\right ) \\ & = -\frac {\left (-1+3 x^2\right )^{3/4}}{4 x^2}-\frac {3}{4} \text {Subst}\left (\int \frac {x^2}{\frac {1}{3}+\frac {x^4}{3}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )+\frac {3}{2} \text {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt [4]{-1+3 x^2}\right ) \\ & = -\frac {\left (-1+3 x^2\right )^{3/4}}{4 x^2}+\frac {3}{8} \text {Subst}\left (\int \frac {1-x^2}{\frac {1}{3}+\frac {x^4}{3}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )-\frac {3}{8} \text {Subst}\left (\int \frac {1+x^2}{\frac {1}{3}+\frac {x^4}{3}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )-\frac {3}{4} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )+\frac {3}{4} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right ) \\ & = -\frac {\left (-1+3 x^2\right )^{3/4}}{4 x^2}+\frac {3}{4} \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-\frac {3}{4} \tanh ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-\frac {9}{16} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )-\frac {9}{16} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )-\frac {9 \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{16 \sqrt {2}}-\frac {9 \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{16 \sqrt {2}} \\ & = -\frac {\left (-1+3 x^2\right )^{3/4}}{4 x^2}+\frac {3}{4} \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-\frac {3}{4} \tanh ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-\frac {9 \log \left (1-\sqrt {2} \sqrt [4]{-1+3 x^2}+\sqrt {-1+3 x^2}\right )}{16 \sqrt {2}}+\frac {9 \log \left (1+\sqrt {2} \sqrt [4]{-1+3 x^2}+\sqrt {-1+3 x^2}\right )}{16 \sqrt {2}}-\frac {9 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt [4]{-1+3 x^2}\right )}{8 \sqrt {2}}+\frac {9 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt [4]{-1+3 x^2}\right )}{8 \sqrt {2}} \\ & = -\frac {\left (-1+3 x^2\right )^{3/4}}{4 x^2}+\frac {3}{4} \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )+\frac {9 \tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{-1+3 x^2}\right )}{8 \sqrt {2}}-\frac {9 \tan ^{-1}\left (1+\sqrt {2} \sqrt [4]{-1+3 x^2}\right )}{8 \sqrt {2}}-\frac {3}{4} \tanh ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-\frac {9 \log \left (1-\sqrt {2} \sqrt [4]{-1+3 x^2}+\sqrt {-1+3 x^2}\right )}{16 \sqrt {2}}+\frac {9 \log \left (1+\sqrt {2} \sqrt [4]{-1+3 x^2}+\sqrt {-1+3 x^2}\right )}{16 \sqrt {2}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.66 \[ \int \frac {1}{x^3 \left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\frac {1}{16} \left (-\frac {4 \left (-1+3 x^2\right )^{3/4}}{x^2}+12 \arctan \left (\sqrt [4]{-1+3 x^2}\right )-9 \sqrt {2} \arctan \left (\frac {-1+\sqrt {-1+3 x^2}}{\sqrt {2} \sqrt [4]{-1+3 x^2}}\right )-12 \text {arctanh}\left (\sqrt [4]{-1+3 x^2}\right )+9 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-1+3 x^2}}{1+\sqrt {-1+3 x^2}}\right )\right ) \]
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Time = 7.82 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.92
method | result | size |
pseudoelliptic | \(\frac {-18 \arctan \left (1+\left (3 x^{2}-1\right )^{\frac {1}{4}} \sqrt {2}\right ) \sqrt {2}\, x^{2}-18 \arctan \left (-1+\left (3 x^{2}-1\right )^{\frac {1}{4}} \sqrt {2}\right ) \sqrt {2}\, x^{2}-9 \ln \left (\frac {1-\left (3 x^{2}-1\right )^{\frac {1}{4}} \sqrt {2}+\sqrt {3 x^{2}-1}}{1+\left (3 x^{2}-1\right )^{\frac {1}{4}} \sqrt {2}+\sqrt {3 x^{2}-1}}\right ) \sqrt {2}\, x^{2}-8 \left (3 x^{2}-1\right )^{\frac {3}{4}}+12 \ln \left (-1+\left (3 x^{2}-1\right )^{\frac {1}{4}}\right ) x^{2}-12 \ln \left (1+\left (3 x^{2}-1\right )^{\frac {1}{4}}\right ) x^{2}+24 \arctan \left (\left (3 x^{2}-1\right )^{\frac {1}{4}}\right ) x^{2}}{32 x^{2}}\) | \(176\) |
trager | \(-\frac {\left (3 x^{2}-1\right )^{\frac {3}{4}}}{4 x^{2}}+\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \sqrt {3 x^{2}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \left (3 x^{2}-1\right )^{\frac {1}{4}}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}-2 \left (3 x^{2}-1\right )^{\frac {3}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{2}}\right )}{16}+\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \left (3 x^{2}-1\right )^{\frac {1}{4}}-2 \sqrt {3 x^{2}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )+2 \left (3 x^{2}-1\right )^{\frac {3}{4}}}{x^{2}}\right )}{16}+\frac {3 \ln \left (-\frac {2 \left (3 x^{2}-1\right )^{\frac {3}{4}}-2 \sqrt {3 x^{2}-1}-3 x^{2}+2 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{3 x^{2}-2}\right )}{8}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \sqrt {3 x^{2}-1}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+2 \left (3 x^{2}-1\right )^{\frac {3}{4}}-2 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{3 x^{2}-2}\right )}{8}\) | \(315\) |
risch | \(-\frac {\left (3 x^{2}-1\right )^{\frac {3}{4}}}{4 x^{2}}+\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \sqrt {3 x^{2}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \left (3 x^{2}-1\right )^{\frac {1}{4}}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}-2 \left (3 x^{2}-1\right )^{\frac {3}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{2}}\right )}{16}+\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \left (3 x^{2}-1\right )^{\frac {1}{4}}-2 \sqrt {3 x^{2}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )+2 \left (3 x^{2}-1\right )^{\frac {3}{4}}}{x^{2}}\right )}{16}+\frac {3 \ln \left (-\frac {2 \left (3 x^{2}-1\right )^{\frac {3}{4}}-2 \sqrt {3 x^{2}-1}-3 x^{2}+2 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{3 x^{2}-2}\right )}{8}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \sqrt {3 x^{2}-1}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+2 \left (3 x^{2}-1\right )^{\frac {3}{4}}-2 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{3 x^{2}-2}\right )}{8}\) | \(315\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^3 \left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\frac {-\left (9 i - 9\right ) \, \sqrt {2} x^{2} \log \left (\left (i + 1\right ) \, \sqrt {2} + 2 \, {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) + \left (9 i + 9\right ) \, \sqrt {2} x^{2} \log \left (-\left (i - 1\right ) \, \sqrt {2} + 2 \, {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \left (9 i + 9\right ) \, \sqrt {2} x^{2} \log \left (\left (i - 1\right ) \, \sqrt {2} + 2 \, {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) + \left (9 i - 9\right ) \, \sqrt {2} x^{2} \log \left (-\left (i + 1\right ) \, \sqrt {2} + 2 \, {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) + 24 \, x^{2} \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - 12 \, x^{2} \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + 12 \, x^{2} \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1\right ) - 8 \, {\left (3 \, x^{2} - 1\right )}^{\frac {3}{4}}}{32 \, x^{2}} \]
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\[ \int \frac {1}{x^3 \left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\int \frac {1}{x^{3} \cdot \left (3 x^{2} - 2\right ) \sqrt [4]{3 x^{2} - 1}}\, dx \]
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\[ \int \frac {1}{x^3 \left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\int { \frac {1}{{\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} {\left (3 \, x^{2} - 2\right )} x^{3}} \,d x } \]
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Time = 0.38 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^3 \left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=-\frac {9}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {9}{16} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {9}{32} \, \sqrt {2} \log \left (\sqrt {2} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + \sqrt {3 \, x^{2} - 1} + 1\right ) - \frac {9}{32} \, \sqrt {2} \log \left (-\sqrt {2} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + \sqrt {3 \, x^{2} - 1} + 1\right ) - \frac {{\left (3 \, x^{2} - 1\right )}^{\frac {3}{4}}}{4 \, x^{2}} + \frac {3}{4} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \frac {3}{8} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {3}{8} \, \log \left ({\left | {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]
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Time = 0.22 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.43 \[ \int \frac {1}{x^3 \left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\frac {3\,\mathrm {atan}\left ({\left (3\,x^2-1\right )}^{1/4}\right )}{4}+\frac {\mathrm {atan}\left ({\left (3\,x^2-1\right )}^{1/4}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{4}-\frac {{\left (3\,x^2-1\right )}^{3/4}}{4\,x^2}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,{\left (3\,x^2-1\right )}^{1/4}\,1{}\mathrm {i}\right )\,9{}\mathrm {i}}{8}-\frac {{\left (-1\right )}^{3/4}\,\mathrm {atan}\left ({\left (-1\right )}^{3/4}\,{\left (3\,x^2-1\right )}^{1/4}\,1{}\mathrm {i}\right )\,9{}\mathrm {i}}{8} \]
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