Optimal. Leaf size=41 \[ -\frac {1}{20 x^4}+\frac {1}{5 x^2}-\frac {\left (1-x^2\right )^{5/2} \sin ^{-1}(x)}{5 x^5}+\frac {\log (x)}{5} \]
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Rubi [A]
time = 0.04, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4771, 272, 45}
\begin {gather*} -\frac {\left (1-x^2\right )^{5/2} \text {ArcSin}(x)}{5 x^5}-\frac {1}{20 x^4}+\frac {1}{5 x^2}+\frac {\log (x)}{5} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 4771
Rubi steps
\begin {align*} \int \frac {\left (1-x^2\right )^{3/2} \sin ^{-1}(x)}{x^6} \, dx &=-\frac {\left (1-x^2\right )^{5/2} \sin ^{-1}(x)}{5 x^5}+\frac {1}{5} \int \frac {\left (1-x^2\right )^2}{x^5} \, dx\\ &=-\frac {\left (1-x^2\right )^{5/2} \sin ^{-1}(x)}{5 x^5}+\frac {1}{10} \text {Subst}\left (\int \frac {(1-x)^2}{x^3} \, dx,x,x^2\right )\\ &=-\frac {\left (1-x^2\right )^{5/2} \sin ^{-1}(x)}{5 x^5}+\frac {1}{10} \text {Subst}\left (\int \left (\frac {1}{x^3}-\frac {2}{x^2}+\frac {1}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{20 x^4}+\frac {1}{5 x^2}-\frac {\left (1-x^2\right )^{5/2} \sin ^{-1}(x)}{5 x^5}+\frac {\log (x)}{5}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 36, normalized size = 0.88 \begin {gather*} -\frac {x-4 x^3+4 \left (1-x^2\right )^{5/2} \sin ^{-1}(x)-4 x^5 \log (x)}{20 x^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.72, size = 201, normalized size = 4.90
method | result | size |
default | \(-\frac {2 i \arcsin \left (x \right )}{5}+\frac {\left (-\sqrt {-x^{2}+1}\, x^{4}+i x^{5}+2 x^{2} \sqrt {-x^{2}+1}-\sqrt {-x^{2}+1}\right ) \left (20 \arcsin \left (x \right ) x^{8}-4 i x^{8}-4 \sqrt {-x^{2}+1}\, x^{7}-40 \arcsin \left (x \right ) x^{6}+i x^{6}+9 \sqrt {-x^{2}+1}\, x^{5}+40 \arcsin \left (x \right ) x^{4}-6 \sqrt {-x^{2}+1}\, x^{3}-20 x^{2} \arcsin \left (x \right )+x \sqrt {-x^{2}+1}+4 \arcsin \left (x \right )\right )}{20 \left (5 x^{8}-10 x^{6}+10 x^{4}-5 x^{2}+1\right ) x^{5}}+\frac {\ln \left (\left (i x +\sqrt {-x^{2}+1}\right )^{2}-1\right )}{5}\) | \(201\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.32, size = 35, normalized size = 0.85 \begin {gather*} -\frac {{\left (-x^{2} + 1\right )}^{\frac {5}{2}} \arcsin \left (x\right )}{5 \, x^{5}} + \frac {4 \, x^{2} - 1}{20 \, x^{4}} + \frac {1}{10} \, \log \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.56, size = 44, normalized size = 1.07 \begin {gather*} \frac {4 \, x^{5} \log \left (x\right ) + 4 \, x^{3} - 4 \, {\left (x^{4} - 2 \, x^{2} + 1\right )} \sqrt {-x^{2} + 1} \arcsin \left (x\right ) - x}{20 \, x^{5}} \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 {\left (- \left (x - 1\right ) \left (x + 1\right )\right )^{\frac {3}{2}} \operatorname {asin}{\left (x \right )}}{x^{6}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 135 vs.
\(2 (31) = 62\).
time = 0.88, size = 135, normalized size = 3.29 \begin {gather*} -\frac {1}{160} \, {\left (\frac {x^{5} {\left (\frac {5 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - \frac {10 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{4}}{x^{4}} - 1\right )}}{{\left (\sqrt {-x^{2} + 1} - 1\right )}^{5}} + \frac {10 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}{x} - \frac {5 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{3}}{x^{3}} + \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{5}}{x^{5}}\right )} \arcsin \left (x\right ) - \frac {3 \, x^{4} - 4 \, x^{2} + 1}{20 \, x^{4}} + \frac {1}{10} \, \log \left (x^{2}\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 = -0.02 \begin {gather*} \int \frac {\mathrm {asin}\left (x\right )\,{\left (1-x^2\right )}^{3/2}}{x^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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