Optimal. Leaf size=35 \[ \frac {x^3}{3 \left (1-x^2\right )^{3/2}}-\frac {x}{\sqrt {1-x^2}}+\sin ^{-1}(x) \]
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Rubi [A]
time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {294, 222}
\begin {gather*} -\frac {x}{\sqrt {1-x^2}}+\frac {x^3}{3 \left (1-x^2\right )^{3/2}}+\sin ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 294
Rubi steps
\begin {align*} \int \frac {x^4}{\left (1-x^2\right )^{5/2}} \, dx &=\frac {x^3}{3 \left (1-x^2\right )^{3/2}}-\int \frac {x^2}{\left (1-x^2\right )^{3/2}} \, dx\\ &=\frac {x^3}{3 \left (1-x^2\right )^{3/2}}-\frac {x}{\sqrt {1-x^2}}+\int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {x^3}{3 \left (1-x^2\right )^{3/2}}-\frac {x}{\sqrt {1-x^2}}+\sin ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 44, normalized size = 1.26 \begin {gather*} \frac {x \left (-3+4 x^2\right )}{3 \left (1-x^2\right )^{3/2}}+2 \tan ^{-1}\left (\frac {x}{-1+\sqrt {1-x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 2.89, size = 55, normalized size = 1.57 \begin {gather*} \frac {-x \sqrt {1-x^2}-2 x^2 \text {ArcSin}\left [x\right ]+\frac {4 x^3 \sqrt {1-x^2}}{3}+x^4 \text {ArcSin}\left [x\right ]+\text {ArcSin}\left [x\right ]}{1-2 x^2+x^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.00, size = 30, normalized size = 0.86
method | result | size |
default | \(\frac {x^{3}}{3 \left (-x^{2}+1\right )^{\frac {3}{2}}}+\arcsin \left (x \right )-\frac {x}{\sqrt {-x^{2}+1}}\) | \(30\) |
risch | \(-\frac {x \left (4 x^{2}-3\right )}{3 \left (x^{2}-1\right ) \sqrt {-x^{2}+1}}+\arcsin \left (x \right )\) | \(30\) |
meijerg | \(-\frac {2 i \left (-\frac {i \sqrt {\pi }\, x \left (-20 x^{2}+15\right )}{10 \left (-x^{2}+1\right )^{\frac {3}{2}}}+\frac {3 i \sqrt {\pi }\, \arcsin \left (x \right )}{2}\right )}{3 \sqrt {\pi }}\) | \(39\) |
trager | \(\frac {\left (4 x^{2}-3\right ) x \sqrt {-x^{2}+1}}{3 \left (x^{2}-1\right )^{2}}+\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+1}+x \right )\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.37, size = 44, normalized size = 1.26 \begin {gather*} \frac {1}{3} \, x {\left (\frac {3 \, x^{2}}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {2}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}\right )} - \frac {x}{3 \, \sqrt {-x^{2} + 1}} + \arcsin \left (x\right ) \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. 63 vs.
\(2 (29) = 58\).
time = 0.31, size = 63, normalized size = 1.80 \begin {gather*} -\frac {6 \, {\left (x^{4} - 2 \, x^{2} + 1\right )} \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) - {\left (4 \, x^{3} - 3 \, x\right )} \sqrt {-x^{2} + 1}}{3 \, {\left (x^{4} - 2 \, x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 105 vs.
\(2 (26) = 52\)
time = 0.46, size = 105, normalized size = 3.00 \begin {gather*} \frac {3 x^{4} \operatorname {asin}{\left (x \right )}}{3 x^{4} - 6 x^{2} + 3} + \frac {4 x^{3} \sqrt {1 - x^{2}}}{3 x^{4} - 6 x^{2} + 3} - \frac {6 x^{2} \operatorname {asin}{\left (x \right )}}{3 x^{4} - 6 x^{2} + 3} - \frac {3 x \sqrt {1 - x^{2}}}{3 x^{4} - 6 x^{2} + 3} + \frac {3 \operatorname {asin}{\left (x \right )}}{3 x^{4} - 6 x^{2} + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 35, normalized size = 1.00 \begin {gather*} \frac {2 \left (\frac {2}{3} x x-\frac 1{2}\right ) x \sqrt {-x^{2}+1}}{\left (-x^{2}+1\right )^{2}}+\arcsin x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.00, size = 91, normalized size = 2.60 \begin {gather*} \mathrm {asin}\left (x\right )+\frac {3\,\sqrt {1-x^2}}{4\,\left (x-1\right )}+\frac {3\,\sqrt {1-x^2}}{4\,\left (x+1\right )}-\sqrt {1-x^2}\,\left (\frac {1}{12\,\left (x-1\right )}-\frac {1}{12\,{\left (x-1\right )}^2}\right )-\sqrt {1-x^2}\,\left (\frac {1}{12\,\left (x+1\right )}+\frac {1}{12\,{\left (x+1\right )}^2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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