Optimal. Leaf size=35 \[ -\frac {x}{\sqrt {1-x^2}}+\frac {x^3}{3 \left (1-x^2\right )^{3/2}}+\sin ^{-1}(x) \]
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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 = {288, 216} \[ \frac {x^3}{3 \left (1-x^2\right )^{3/2}}-\frac {x}{\sqrt {1-x^2}}+\sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 216
Rule 288
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 = 26, normalized size = 0.74 \[ \frac {x \left (4 x^2-3\right )}{3 \left (1-x^2\right )^{3/2}}+\sin ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 63, normalized size = 1.80 \[ -\frac {6 \, {\left (x^{4} - 2 \, x^{2} + 1\right )} \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) - {\left (4 \, x^{3} - 3 \, x\right )} \sqrt {-x^{2} + 1}}{3 \, {\left (x^{4} - 2 \, x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.91, size = 29, normalized size = 0.83 \[ \frac {{\left (4 \, x^{2} - 3\right )} \sqrt {-x^{2} + 1} x}{3 \, {\left (x^{2} - 1\right )}^{2}} + \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 30, normalized size = 0.86 \[ \frac {x^{3}}{3 \left (-x^{2}+1\right )^{\frac {3}{2}}}-\frac {x}{\sqrt {-x^{2}+1}}+\arcsin \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 44, normalized size = 1.26 \[ \frac {1}{3} \, x {\left (\frac {3 \, x^{2}}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {2}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}\right )} - \frac {x}{3 \, \sqrt {-x^{2} + 1}} + \arcsin \relax (x) \]
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 \[ \mathrm {asin}\relax (x)+\frac {3\,\sqrt {1-x^2}}{4\,\left (x-1\right )}+\frac {3\,\sqrt {1-x^2}}{4\,\left (x+1\right )}-\sqrt {1-x^2}\,\left (\frac {1}{12\,\left (x-1\right )}-\frac {1}{12\,{\left (x-1\right )}^2}\right )-\sqrt {1-x^2}\,\left (\frac {1}{12\,\left (x+1\right )}+\frac {1}{12\,{\left (x+1\right )}^2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.14, size = 105, normalized size = 3.00 \[ \frac {3 x^{4} \operatorname {asin}{\relax (x )}}{3 x^{4} - 6 x^{2} + 3} + \frac {4 x^{3} \sqrt {1 - x^{2}}}{3 x^{4} - 6 x^{2} + 3} - \frac {6 x^{2} \operatorname {asin}{\relax (x )}}{3 x^{4} - 6 x^{2} + 3} - \frac {3 x \sqrt {1 - x^{2}}}{3 x^{4} - 6 x^{2} + 3} + \frac {3 \operatorname {asin}{\relax (x )}}{3 x^{4} - 6 x^{2} + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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