Optimal. Leaf size=173 \[ -\frac {1}{2} \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )+\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{-1+3 x^2}\right )}{2 \sqrt {2}}-\frac {\tan ^{-1}\left (1+\sqrt {2} \sqrt [4]{-1+3 x^2}\right )}{2 \sqrt {2}}-\frac {1}{2} \tanh ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )+\frac {\log \left (1-\sqrt {2} \sqrt [4]{-1+3 x^2}+\sqrt {-1+3 x^2}\right )}{4 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} \sqrt [4]{-1+3 x^2}+\sqrt {-1+3 x^2}\right )}{4 \sqrt {2}} \]
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Rubi [A]
time = 0.09, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 12, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {457, 88, 65,
217, 1179, 642, 1176, 631, 210, 218, 212, 209} \begin {gather*} -\frac {1}{2} \text {ArcTan}\left (\sqrt [4]{3 x^2-1}\right )+\frac {\text {ArcTan}\left (1-\sqrt {2} \sqrt [4]{3 x^2-1}\right )}{2 \sqrt {2}}-\frac {\text {ArcTan}\left (\sqrt {2} \sqrt [4]{3 x^2-1}+1\right )}{2 \sqrt {2}}+\frac {\log \left (\sqrt {3 x^2-1}-\sqrt {2} \sqrt [4]{3 x^2-1}+1\right )}{4 \sqrt {2}}-\frac {\log \left (\sqrt {3 x^2-1}+\sqrt {2} \sqrt [4]{3 x^2-1}+1\right )}{4 \sqrt {2}}-\frac {1}{2} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 88
Rule 209
Rule 210
Rule 212
Rule 217
Rule 218
Rule 457
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {1}{x \left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x (-2+3 x) (-1+3 x)^{3/4}} \, dx,x,x^2\right )\\ &=-\left (\frac {1}{4} \text {Subst}\left (\int \frac {1}{x (-1+3 x)^{3/4}} \, dx,x,x^2\right )\right )+\frac {3}{4} \text {Subst}\left (\int \frac {1}{(-2+3 x) (-1+3 x)^{3/4}} \, dx,x,x^2\right )\\ &=-\left (\frac {1}{3} \text {Subst}\left (\int \frac {1}{\frac {1}{3}+\frac {x^4}{3}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )\right )+\text {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt [4]{-1+3 x^2}\right )\\ &=-\left (\frac {1}{6} \text {Subst}\left (\int \frac {1-x^2}{\frac {1}{3}+\frac {x^4}{3}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )\right )-\frac {1}{6} \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 {1}{2} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )\\ &=-\frac {1}{2} \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-\frac {1}{2} \tanh ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )+\frac {\text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{4 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{4 \sqrt {2}}\\ &=-\frac {1}{2} \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-\frac {1}{2} \tanh ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )+\frac {\log \left (1-\sqrt {2} \sqrt [4]{-1+3 x^2}+\sqrt {-1+3 x^2}\right )}{4 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} \sqrt [4]{-1+3 x^2}+\sqrt {-1+3 x^2}\right )}{4 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt [4]{-1+3 x^2}\right )}{2 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt [4]{-1+3 x^2}\right )}{2 \sqrt {2}}\\ &=-\frac {1}{2} \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )+\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{-1+3 x^2}\right )}{2 \sqrt {2}}-\frac {\tan ^{-1}\left (1+\sqrt {2} \sqrt [4]{-1+3 x^2}\right )}{2 \sqrt {2}}-\frac {1}{2} \tanh ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )+\frac {\log \left (1-\sqrt {2} \sqrt [4]{-1+3 x^2}+\sqrt {-1+3 x^2}\right )}{4 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} \sqrt [4]{-1+3 x^2}+\sqrt {-1+3 x^2}\right )}{4 \sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 111, normalized size = 0.64 \begin {gather*} \frac {1}{4} \left (-2 \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-\sqrt {2} \tan ^{-1}\left (\frac {-1+\sqrt {-1+3 x^2}}{\sqrt {2} \sqrt [4]{-1+3 x^2}}\right )-2 \tanh ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-1+3 x^2}}{1+\sqrt {-1+3 x^2}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 3.68, size = 302, normalized size = 1.75
method | result | size |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {3 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \left (3 x^{2}-1\right )^{\frac {1}{4}}-2 \sqrt {3 x^{2}-1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )+2 \left (3 x^{2}-1\right )^{\frac {3}{4}}}{x^{2}}\right )}{4}-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {2 \sqrt {3 x^{2}-1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \left (3 x^{2}-1\right )^{\frac {1}{4}}-3 \RootOf \left (\textit {\_Z}^{4}+1\right ) x^{2}+2 \left (3 x^{2}-1\right )^{\frac {3}{4}}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )}{x^{2}}\right )}{4}-\frac {\ln \left (-\frac {2 \left (3 x^{2}-1\right )^{\frac {3}{4}}+2 \sqrt {3 x^{2}-1}+3 x^{2}+2 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{3 x^{2}-2}\right )}{4}-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \sqrt {3 x^{2}-1}-3 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+2 \left (3 x^{2}-1\right )^{\frac {3}{4}}-2 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{3 x^{2}-2}\right )}{4}\) | \(302\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.46, size = 215, normalized size = 1.24 \begin {gather*} \frac {1}{2} \, \sqrt {2} \arctan \left (\sqrt {2} \sqrt {\sqrt {2} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + \sqrt {3 \, x^{2} - 1} + 1} - \sqrt {2} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1\right ) + \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-4 \, \sqrt {2} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {3 \, x^{2} - 1} + 4} - \sqrt {2} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {1}{8} \, \sqrt {2} \log \left (4 \, \sqrt {2} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {3 \, x^{2} - 1} + 4\right ) + \frac {1}{8} \, \sqrt {2} \log \left (-4 \, \sqrt {2} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {3 \, x^{2} - 1} + 4\right ) - \frac {1}{2} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{4} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{4} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (3 x^{2} - 2\right ) \left (3 x^{2} - 1\right )^{\frac {3}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.06, size = 155, normalized size = 0.90 \begin {gather*} -\frac {1}{4} \, \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 {1}{4} \, \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 {1}{8} \, \sqrt {2} \log \left (\sqrt {2} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + \sqrt {3 \, x^{2} - 1} + 1\right ) + \frac {1}{8} \, \sqrt {2} \log \left (-\sqrt {2} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + \sqrt {3 \, x^{2} - 1} + 1\right ) - \frac {1}{2} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{4} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{4} \, \log \left ({\left | {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.51, size = 77, normalized size = 0.45 \begin {gather*} -\frac {\mathrm {atan}\left ({\left (3\,x^2-1\right )}^{1/4}\right )}{2}+\frac {\mathrm {atan}\left ({\left (3\,x^2-1\right )}^{1/4}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (3\,x^2-1\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (3\,x^2-1\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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