Optimal. Leaf size=37 \[ -\frac {\sqrt {a x^6} \tan ^{-1}(x)}{2 x^3}+\frac {\sqrt {a x^6} \tanh ^{-1}(x)}{2 x^3} \]
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Rubi [A]
time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {15, 304, 209,
212} \begin {gather*} \frac {\sqrt {a x^6} \tanh ^{-1}(x)}{2 x^3}-\frac {\sqrt {a x^6} \text {ArcTan}(x)}{2 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 209
Rule 212
Rule 304
Rubi steps
\begin {align*} \int \frac {\sqrt {a x^6}}{x \left (1-x^4\right )} \, dx &=\frac {\sqrt {a x^6} \int \frac {x^2}{1-x^4} \, dx}{x^3}\\ &=\frac {\sqrt {a x^6} \int \frac {1}{1-x^2} \, dx}{2 x^3}-\frac {\sqrt {a x^6} \int \frac {1}{1+x^2} \, dx}{2 x^3}\\ &=-\frac {\sqrt {a x^6} \tan ^{-1}(x)}{2 x^3}+\frac {\sqrt {a x^6} \tanh ^{-1}(x)}{2 x^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 33, normalized size = 0.89 \begin {gather*} -\frac {\sqrt {a x^6} \left (2 \tan ^{-1}(x)+\log (1-x)-\log (1+x)\right )}{4 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 28, normalized size = 0.76
method | result | size |
default | \(-\frac {\sqrt {a \,x^{6}}\, \left (\ln \left (-1+x \right )-\ln \left (1+x \right )+2 \arctan \left (x \right )\right )}{4 x^{3}}\) | \(28\) |
meijerg | \(-\frac {\sqrt {a \,x^{6}}\, \left (\ln \left (1-\left (x^{4}\right )^{\frac {1}{4}}\right )-\ln \left (1+\left (x^{4}\right )^{\frac {1}{4}}\right )+2 \arctan \left (\left (x^{4}\right )^{\frac {1}{4}}\right )\right )}{4 \left (x^{4}\right )^{\frac {3}{4}}}\) | \(44\) |
risch | \(-\frac {i \sqrt {a \,x^{6}}\, \ln \left (x +i\right )}{4 x^{3}}+\frac {i \sqrt {a \,x^{6}}\, \ln \left (x -i\right )}{4 x^{3}}+\frac {\sqrt {a \,x^{6}}\, \ln \left (1+x \right )}{4 x^{3}}-\frac {\sqrt {a \,x^{6}}\, \ln \left (-1+x \right )}{4 x^{3}}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 26, normalized size = 0.70 \begin {gather*} -\frac {1}{2} \, \sqrt {a} \arctan \left (x\right ) + \frac {1}{4} \, \sqrt {a} \log \left (x + 1\right ) - \frac {1}{4} \, \sqrt {a} \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 29, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {a x^{6}} {\left (2 \, \arctan \left (x\right ) - \log \left (\frac {x + 1}{x - 1}\right )\right )}}{4 \, x^{3}} \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 {\sqrt {a x^{6}}}{x^{5} - x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.82, size = 29, normalized size = 0.78 \begin {gather*} -\frac {1}{4} \, {\left (2 \, \arctan \left (x\right ) \mathrm {sgn}\left (x\right ) - \log \left ({\left | x + 1 \right |}\right ) \mathrm {sgn}\left (x\right ) + \log \left ({\left | x - 1 \right |}\right ) \mathrm {sgn}\left (x\right )\right )} \sqrt {a} \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 = -0.03 \begin {gather*} -\int \frac {\sqrt {a\,x^6}}{x\,\left (x^4-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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