Optimal. Leaf size=215 \[ -\frac {\sqrt [4]{2-3 x^2}}{16 x^2}-\frac {15 \tan ^{-1}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32\ 2^{3/4}}-\frac {3 \tan ^{-1}\left (1+\sqrt [4]{4-6 x^2}\right )}{32 \sqrt [4]{2}}+\frac {3 \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{32 \sqrt [4]{2}}-\frac {15 \tanh ^{-1}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32\ 2^{3/4}}+\frac {3 \log \left (\sqrt {2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right )}{64 \sqrt [4]{2}}-\frac {3 \log \left (\sqrt {2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right )}{64 \sqrt [4]{2}} \]
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Rubi [A]
time = 0.16, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps
used = 24, number of rules used = 14, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {454, 272, 44,
65, 218, 212, 209, 455, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {15 \text {ArcTan}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32\ 2^{3/4}}-\frac {3 \text {ArcTan}\left (\sqrt [4]{4-6 x^2}+1\right )}{32 \sqrt [4]{2}}+\frac {3 \text {ArcTan}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{32 \sqrt [4]{2}}-\frac {\sqrt [4]{2-3 x^2}}{16 x^2}+\frac {3 \log \left (\sqrt {2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2}\right )}{64 \sqrt [4]{2}}-\frac {3 \log \left (\sqrt {2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2}\right )}{64 \sqrt [4]{2}}-\frac {15 \tanh ^{-1}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32\ 2^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 209
Rule 210
Rule 212
Rule 217
Rule 218
Rule 272
Rule 454
Rule 455
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx &=\int \left (\frac {1}{4 x^3 \left (2-3 x^2\right )^{3/4}}+\frac {3}{16 x \left (2-3 x^2\right )^{3/4}}-\frac {9 x}{16 \left (2-3 x^2\right )^{3/4} \left (-4+3 x^2\right )}\right ) \, dx\\ &=\frac {3}{16} \int \frac {1}{x \left (2-3 x^2\right )^{3/4}} \, dx+\frac {1}{4} \int \frac {1}{x^3 \left (2-3 x^2\right )^{3/4}} \, dx-\frac {9}{16} \int \frac {x}{\left (2-3 x^2\right )^{3/4} \left (-4+3 x^2\right )} \, dx\\ &=\frac {3}{32} \text {Subst}\left (\int \frac {1}{(2-3 x)^{3/4} x} \, dx,x,x^2\right )+\frac {1}{8} \text {Subst}\left (\int \frac {1}{(2-3 x)^{3/4} x^2} \, dx,x,x^2\right )-\frac {9}{32} \text {Subst}\left (\int \frac {1}{(2-3 x)^{3/4} (-4+3 x)} \, dx,x,x^2\right )\\ &=-\frac {\sqrt [4]{2-3 x^2}}{16 x^2}-\frac {1}{8} \text {Subst}\left (\int \frac {1}{\frac {2}{3}-\frac {x^4}{3}} \, dx,x,\sqrt [4]{2-3 x^2}\right )+\frac {9}{64} \text {Subst}\left (\int \frac {1}{(2-3 x)^{3/4} x} \, dx,x,x^2\right )+\frac {3}{8} \text {Subst}\left (\int \frac {1}{-2-x^4} \, dx,x,\sqrt [4]{2-3 x^2}\right )\\ &=-\frac {\sqrt [4]{2-3 x^2}}{16 x^2}-\frac {3}{16} \text {Subst}\left (\int \frac {1}{\frac {2}{3}-\frac {x^4}{3}} \, dx,x,\sqrt [4]{2-3 x^2}\right )-\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {2}-x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{16 \sqrt {2}}-\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {2}+x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{16 \sqrt {2}}+\frac {3 \text {Subst}\left (\int \frac {\sqrt {2}-x^2}{-2-x^4} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{16 \sqrt {2}}+\frac {3 \text {Subst}\left (\int \frac {\sqrt {2}+x^2}{-2-x^4} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{16 \sqrt {2}}\\ &=-\frac {\sqrt [4]{2-3 x^2}}{16 x^2}-\frac {3 \tan ^{-1}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{16\ 2^{3/4}}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{16\ 2^{3/4}}-\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {2}-2^{3/4} x+x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{32 \sqrt {2}}-\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {2}+2^{3/4} x+x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{32 \sqrt {2}}-\frac {9 \text {Subst}\left (\int \frac {1}{\sqrt {2}-x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{32 \sqrt {2}}-\frac {9 \text {Subst}\left (\int \frac {1}{\sqrt {2}+x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{32 \sqrt {2}}+\frac {3 \text {Subst}\left (\int \frac {2^{3/4}+2 x}{-\sqrt {2}-2^{3/4} x-x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{64 \sqrt [4]{2}}+\frac {3 \text {Subst}\left (\int \frac {2^{3/4}-2 x}{-\sqrt {2}+2^{3/4} x-x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{64 \sqrt [4]{2}}\\ &=-\frac {\sqrt [4]{2-3 x^2}}{16 x^2}-\frac {15 \tan ^{-1}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32\ 2^{3/4}}-\frac {15 \tanh ^{-1}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32\ 2^{3/4}}+\frac {3 \log \left (\sqrt {2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right )}{64 \sqrt [4]{2}}-\frac {3 \log \left (\sqrt {2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right )}{64 \sqrt [4]{2}}-\frac {3 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt [4]{4-6 x^2}\right )}{32 \sqrt [4]{2}}+\frac {3 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt [4]{4-6 x^2}\right )}{32 \sqrt [4]{2}}\\ &=-\frac {\sqrt [4]{2-3 x^2}}{16 x^2}-\frac {15 \tan ^{-1}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32\ 2^{3/4}}-\frac {3 \tan ^{-1}\left (1+\sqrt [4]{4-6 x^2}\right )}{32 \sqrt [4]{2}}+\frac {3 \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{32 \sqrt [4]{2}}-\frac {15 \tanh ^{-1}\left (\frac {\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{32\ 2^{3/4}}+\frac {3 \log \left (\sqrt {2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right )}{64 \sqrt [4]{2}}-\frac {3 \log \left (\sqrt {2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right )}{64 \sqrt [4]{2}}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 155, normalized size = 0.72 \begin {gather*} -\frac {4 \sqrt [4]{2-3 x^2}+15 \sqrt [4]{2} x^2 \tan ^{-1}\left (\sqrt [4]{1-\frac {3 x^2}{2}}\right )-3\ 2^{3/4} x^2 \tan ^{-1}\left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+15 \sqrt [4]{2} x^2 \tanh ^{-1}\left (\sqrt [4]{1-\frac {3 x^2}{2}}\right )+3\ 2^{3/4} x^2 \tanh ^{-1}\left (\frac {2 \sqrt [4]{4-6 x^2}}{2+\sqrt {4-6 x^2}}\right )}{64 x^2} \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 = 25.01, size = 577, normalized size = 2.68
method | result | size |
trager | \(\text {Expression too large to display}\) | \(577\) |
risch | \(\text {Expression too large to display}\) | \(1542\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 352 vs.
\(2 (158) = 316\).
time = 1.64, size = 352, normalized size = 1.64 \begin {gather*} \frac {12 \cdot 8^{\frac {3}{4}} \sqrt {2} x^{2} \arctan \left (\frac {1}{4} \cdot 8^{\frac {1}{4}} \sqrt {2} \sqrt {8^{\frac {3}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 4 \, \sqrt {2} + 4 \, \sqrt {-3 \, x^{2} + 2}} - \frac {1}{2} \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} - 1\right ) + 12 \cdot 8^{\frac {3}{4}} \sqrt {2} x^{2} \arctan \left (\frac {1}{16} \cdot 8^{\frac {1}{4}} \sqrt {2} \sqrt {-16 \cdot 8^{\frac {3}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 64 \, \sqrt {2} + 64 \, \sqrt {-3 \, x^{2} + 2}} - \frac {1}{2} \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 1\right ) - 3 \cdot 8^{\frac {3}{4}} \sqrt {2} x^{2} \log \left (16 \cdot 8^{\frac {3}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 64 \, \sqrt {2} + 64 \, \sqrt {-3 \, x^{2} + 2}\right ) + 3 \cdot 8^{\frac {3}{4}} \sqrt {2} x^{2} \log \left (-16 \cdot 8^{\frac {3}{4}} \sqrt {2} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 64 \, \sqrt {2} + 64 \, \sqrt {-3 \, x^{2} + 2}\right ) + 60 \cdot 8^{\frac {3}{4}} x^{2} \arctan \left (\frac {1}{2} \cdot 8^{\frac {1}{4}} \sqrt {\sqrt {2} + \sqrt {-3 \, x^{2} + 2}} - \frac {1}{2} \cdot 8^{\frac {1}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) - 15 \cdot 8^{\frac {3}{4}} x^{2} \log \left (8^{\frac {3}{4}} + 4 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) + 15 \cdot 8^{\frac {3}{4}} x^{2} \log \left (-8^{\frac {3}{4}} + 4 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) - 32 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}}{512 \, x^{2}} \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}{3 x^{5} \left (2 - 3 x^{2}\right )^{\frac {3}{4}} - 4 x^{3} \left (2 - 3 x^{2}\right )^{\frac {3}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.84, size = 192, normalized size = 0.89 \begin {gather*} -\frac {3}{64} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {3}{64} \cdot 2^{\frac {3}{4}} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {3}{128} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + \frac {3}{128} \cdot 2^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) - \frac {15}{64} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) - \frac {15}{128} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) + \frac {15}{128} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} - {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) - \frac {{\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}}{16 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.56, size = 107, normalized size = 0.50 \begin {gather*} -\frac {{\left (2-3\,x^2\right )}^{1/4}}{16\,x^2}-\frac {15\,2^{1/4}\,\mathrm {atan}\left (\frac {2^{3/4}\,{\left (2-3\,x^2\right )}^{1/4}}{2}\right )}{64}+\frac {2^{1/4}\,\mathrm {atan}\left (\frac {2^{3/4}\,{\left (2-3\,x^2\right )}^{1/4}\,1{}\mathrm {i}}{2}\right )\,15{}\mathrm {i}}{64}+2^{3/4}\,\mathrm {atan}\left (2^{1/4}\,{\left (2-3\,x^2\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {3}{64}-\frac {3}{64}{}\mathrm {i}\right )+\frac {{\left (-1\right )}^{1/4}\,2^{1/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,2^{3/4}\,{\left (2-3\,x^2\right )}^{1/4}}{2}\right )\,3{}\mathrm {i}}{32} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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