Optimal. Leaf size=97 \[ -\frac {\left (1-x^2\right )^{2/3}}{6 x^2}+\frac {\tan ^{-1}\left (\frac {1+\sqrt [3]{2-2 x^2}}{\sqrt {3}}\right )}{6\ 2^{2/3} \sqrt {3}}-\frac {\log \left (3+x^2\right )}{36\ 2^{2/3}}+\frac {\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{12\ 2^{2/3}} \]
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Rubi [A]
time = 0.06, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {457, 105, 12,
57, 631, 210, 31} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt [3]{2-2 x^2}+1}{\sqrt {3}}\right )}{6\ 2^{2/3} \sqrt {3}}-\frac {\left (1-x^2\right )^{2/3}}{6 x^2}-\frac {\log \left (x^2+3\right )}{36\ 2^{2/3}}+\frac {\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{12\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 57
Rule 105
Rule 210
Rule 457
Rule 631
Rubi steps
\begin {align*} \int \frac {1}{x^3 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} x^2 (3+x)} \, dx,x,x^2\right )\\ &=-\frac {\left (1-x^2\right )^{2/3}}{6 x^2}-\frac {1}{6} \text {Subst}\left (\int -\frac {1}{3 \sqrt [3]{1-x} (3+x)} \, dx,x,x^2\right )\\ &=-\frac {\left (1-x^2\right )^{2/3}}{6 x^2}+\frac {1}{18} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} (3+x)} \, dx,x,x^2\right )\\ &=-\frac {\left (1-x^2\right )^{2/3}}{6 x^2}-\frac {\log \left (3+x^2\right )}{36\ 2^{2/3}}+\frac {1}{12} \text {Subst}\left (\int \frac {1}{2 \sqrt [3]{2}+2^{2/3} x+x^2} \, dx,x,\sqrt [3]{1-x^2}\right )-\frac {\text {Subst}\left (\int \frac {1}{2^{2/3}-x} \, dx,x,\sqrt [3]{1-x^2}\right )}{12\ 2^{2/3}}\\ &=-\frac {\left (1-x^2\right )^{2/3}}{6 x^2}-\frac {\log \left (3+x^2\right )}{36\ 2^{2/3}}+\frac {\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{12\ 2^{2/3}}-\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\sqrt [3]{2-2 x^2}\right )}{6\ 2^{2/3}}\\ &=-\frac {\left (1-x^2\right )^{2/3}}{6 x^2}+\frac {\tan ^{-1}\left (\frac {1+\sqrt [3]{2-2 x^2}}{\sqrt {3}}\right )}{6\ 2^{2/3} \sqrt {3}}-\frac {\log \left (3+x^2\right )}{36\ 2^{2/3}}+\frac {\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{12\ 2^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 108, normalized size = 1.11 \begin {gather*} \frac {1}{72} \left (-\frac {12 \left (1-x^2\right )^{2/3}}{x^2}+2 \sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {1+\sqrt [3]{2-2 x^2}}{\sqrt {3}}\right )+2 \sqrt [3]{2} \log \left (-2+\sqrt [3]{2-2 x^2}\right )-\sqrt [3]{2} \log \left (4+2 \sqrt [3]{2-2 x^2}+\left (2-2 x^2\right )^{2/3}\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 = 13.13, size = 755, normalized size = 7.78
method | result | size |
risch | \(\text {Expression too large to display}\) | \(755\) |
trager | \(\text {Expression too large to display}\) | \(1108\) |
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 = 0.69, size = 115, normalized size = 1.19 \begin {gather*} \frac {4 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{2} \arctan \left (\frac {1}{6} \cdot 4^{\frac {1}{6}} {\left (4^{\frac {1}{3}} \sqrt {3} + 2 \, \sqrt {3} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right )}\right ) - 4^{\frac {2}{3}} x^{2} \log \left (4^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {2}{3}}\right ) + 2 \cdot 4^{\frac {2}{3}} x^{2} \log \left (-4^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) - 24 \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{144 \, 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}{x^{3} \sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.59, size = 100, normalized size = 1.03 \begin {gather*} \frac {1}{72} \cdot 4^{\frac {2}{3}} \sqrt {3} \arctan \left (\frac {1}{12} \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (4^{\frac {1}{3}} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right )}\right ) - \frac {1}{144} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + {\left (-x^{2} + 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{72} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {1}{3}} - {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right ) - \frac {{\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{6 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.49, size = 120, normalized size = 1.24 \begin {gather*} \frac {2^{1/3}\,\ln \left (\frac {{\left (1-x^2\right )}^{1/3}}{36}-\frac {2^{2/3}}{36}\right )}{36}-\frac {{\left (1-x^2\right )}^{2/3}}{6\,x^2}+\frac {2^{1/3}\,\ln \left (\frac {{\left (1-x^2\right )}^{1/3}}{36}-\frac {2^{2/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{144}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{72}-\frac {2^{1/3}\,\ln \left (\frac {{\left (1-x^2\right )}^{1/3}}{36}-\frac {2^{2/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{144}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{72} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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