Optimal. Leaf size=72 \[ \frac {1}{192} \sqrt [4]{-x^2+x^4} \left (-7 x-4 x^3+32 x^5\right )+\frac {7}{128} \text {ArcTan}\left (\frac {x}{\sqrt [4]{-x^2+x^4}}\right )-\frac {7}{128} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x^2+x^4}}\right ) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(153\) vs. \(2(72)=144\).
time = 0.11, antiderivative size = 153, normalized size of antiderivative = 2.12, number of steps
used = 9, number of rules used = 8, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {2046, 2049,
2057, 335, 338, 304, 209, 212} \begin {gather*} \frac {7 \left (x^2-1\right )^{3/4} x^{3/2} \text {ArcTan}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{128 \left (x^4-x^2\right )^{3/4}}-\frac {7}{192} \sqrt [4]{x^4-x^2} x-\frac {7 \left (x^2-1\right )^{3/4} x^{3/2} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{128 \left (x^4-x^2\right )^{3/4}}+\frac {1}{6} \sqrt [4]{x^4-x^2} x^5-\frac {1}{48} \sqrt [4]{x^4-x^2} x^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 304
Rule 335
Rule 338
Rule 2046
Rule 2049
Rule 2057
Rubi steps
\begin {align*} \int x^4 \sqrt [4]{-x^2+x^4} \, dx &=\frac {1}{6} x^5 \sqrt [4]{-x^2+x^4}-\frac {1}{12} \int \frac {x^6}{\left (-x^2+x^4\right )^{3/4}} \, dx\\ &=-\frac {1}{48} x^3 \sqrt [4]{-x^2+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x^2+x^4}-\frac {7}{96} \int \frac {x^4}{\left (-x^2+x^4\right )^{3/4}} \, dx\\ &=-\frac {7}{192} x \sqrt [4]{-x^2+x^4}-\frac {1}{48} x^3 \sqrt [4]{-x^2+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x^2+x^4}-\frac {7}{128} \int \frac {x^2}{\left (-x^2+x^4\right )^{3/4}} \, dx\\ &=-\frac {7}{192} x \sqrt [4]{-x^2+x^4}-\frac {1}{48} x^3 \sqrt [4]{-x^2+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x^2+x^4}-\frac {\left (7 x^{3/2} \left (-1+x^2\right )^{3/4}\right ) \int \frac {\sqrt {x}}{\left (-1+x^2\right )^{3/4}} \, dx}{128 \left (-x^2+x^4\right )^{3/4}}\\ &=-\frac {7}{192} x \sqrt [4]{-x^2+x^4}-\frac {1}{48} x^3 \sqrt [4]{-x^2+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x^2+x^4}-\frac {\left (7 x^{3/2} \left (-1+x^2\right )^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{64 \left (-x^2+x^4\right )^{3/4}}\\ &=-\frac {7}{192} x \sqrt [4]{-x^2+x^4}-\frac {1}{48} x^3 \sqrt [4]{-x^2+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x^2+x^4}-\frac {\left (7 x^{3/2} \left (-1+x^2\right )^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{64 \left (-x^2+x^4\right )^{3/4}}\\ &=-\frac {7}{192} x \sqrt [4]{-x^2+x^4}-\frac {1}{48} x^3 \sqrt [4]{-x^2+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x^2+x^4}-\frac {\left (7 x^{3/2} \left (-1+x^2\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{128 \left (-x^2+x^4\right )^{3/4}}+\frac {\left (7 x^{3/2} \left (-1+x^2\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{128 \left (-x^2+x^4\right )^{3/4}}\\ &=-\frac {7}{192} x \sqrt [4]{-x^2+x^4}-\frac {1}{48} x^3 \sqrt [4]{-x^2+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x^2+x^4}+\frac {7 x^{3/2} \left (-1+x^2\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{128 \left (-x^2+x^4\right )^{3/4}}-\frac {7 x^{3/2} \left (-1+x^2\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{128 \left (-x^2+x^4\right )^{3/4}}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 96, normalized size = 1.33 \begin {gather*} \frac {x^{3/2} \left (-1+x^2\right )^{3/4} \left (2 x^{3/2} \sqrt [4]{-1+x^2} \left (-7-4 x^2+32 x^4\right )+21 \text {ArcTan}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )-21 \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )\right )}{384 \left (x^2 \left (-1+x^2\right )\right )^{3/4}} \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 = 4.05, size = 33, normalized size = 0.46
method | result | size |
meijerg | \(\frac {2 \mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{4}} x^{\frac {11}{2}} \hypergeom \left (\left [-\frac {1}{4}, \frac {11}{4}\right ], \left [\frac {15}{4}\right ], x^{2}\right )}{11 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{4}}}\) | \(33\) |
trager | \(\frac {x \left (32 x^{4}-4 x^{2}-7\right ) \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{192}+\frac {7 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}-x^{2}}\, x +2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{4}-x^{2}\right )^{\frac {3}{4}}-2 x^{2} \left (x^{4}-x^{2}\right )^{\frac {1}{4}}-x \RootOf \left (\textit {\_Z}^{2}+1\right )}{x}\right )}{256}+\frac {7 \ln \left (\frac {2 \left (x^{4}-x^{2}\right )^{\frac {3}{4}}-2 \sqrt {x^{4}-x^{2}}\, x +2 x^{2} \left (x^{4}-x^{2}\right )^{\frac {1}{4}}-2 x^{3}+x}{x}\right )}{256}\) | \(168\) |
risch | \(\frac {x \left (32 x^{4}-4 x^{2}-7\right ) \left (x^{2} \left (x^{2}-1\right )\right )^{\frac {1}{4}}}{192}+\frac {\left (\frac {7 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{8}-3 x^{6}+3 x^{4}-x^{2}\right )^{\frac {1}{4}} x^{4}+2 x^{6}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{8}-3 x^{6}+3 x^{4}-x^{2}\right )^{\frac {3}{4}}+4 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{8}-3 x^{6}+3 x^{4}-x^{2}\right )^{\frac {1}{4}} x^{2}-2 \sqrt {x^{8}-3 x^{6}+3 x^{4}-x^{2}}\, x^{2}-5 x^{4}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{8}-3 x^{6}+3 x^{4}-x^{2}\right )^{\frac {1}{4}}+2 \sqrt {x^{8}-3 x^{6}+3 x^{4}-x^{2}}+4 x^{2}-1}{\left (-1+x \right )^{2} \left (1+x \right )^{2}}\right )}{256}-\frac {7 \ln \left (\frac {2 x^{6}+2 \left (x^{8}-3 x^{6}+3 x^{4}-x^{2}\right )^{\frac {1}{4}} x^{4}+2 \sqrt {x^{8}-3 x^{6}+3 x^{4}-x^{2}}\, x^{2}-5 x^{4}+2 \left (x^{8}-3 x^{6}+3 x^{4}-x^{2}\right )^{\frac {3}{4}}-4 \left (x^{8}-3 x^{6}+3 x^{4}-x^{2}\right )^{\frac {1}{4}} x^{2}-2 \sqrt {x^{8}-3 x^{6}+3 x^{4}-x^{2}}+4 x^{2}+2 \left (x^{8}-3 x^{6}+3 x^{4}-x^{2}\right )^{\frac {1}{4}}-1}{\left (-1+x \right )^{2} \left (1+x \right )^{2}}\right )}{256}\right ) \left (x^{2} \left (x^{2}-1\right )\right )^{\frac {1}{4}} \left (x^{2} \left (x^{2}-1\right )^{3}\right )^{\frac {1}{4}}}{x \left (x^{2}-1\right )}\) | \(451\) |
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. 123 vs.
\(2 (60) = 120\).
time = 0.88, size = 123, normalized size = 1.71 \begin {gather*} \frac {1}{192} \, {\left (32 \, x^{5} - 4 \, x^{3} - 7 \, x\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} - \frac {7}{256} \, \arctan \left (\frac {2 \, {\left ({\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} x^{2} + {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}\right )}}{x}\right ) + \frac {7}{256} \, \log \left (-\frac {2 \, x^{3} - 2 \, {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{4} - x^{2}} x - x - 2 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{x}\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 x^{4} \sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 89, normalized size = 1.24 \begin {gather*} \frac {1}{192} \, {\left (7 \, {\left (\frac {1}{x^{2}} - 1\right )}^{2} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 18 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {5}{4}} - 21 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right )} x^{6} + \frac {7}{128} \, \arctan \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {7}{256} \, \log \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {7}{256} \, \log \left (-{\left (-\frac {1}{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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^4\,{\left (x^4-x^2\right )}^{1/4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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