Optimal. Leaf size=49 \[ \frac {\sqrt {a x^6} \tan ^{-1}(x)}{2 x^3}-\frac {\sqrt {a x^6} \tanh ^{-1}(x)}{2 x^3}+\frac {1}{2} \tan ^{-1}(x)+\frac {1}{2} \tanh ^{-1}(x) \]
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Rubi [A] time = 0.02, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {212, 206, 203, 15, 1584, 298} \[ \frac {\sqrt {a x^6} \tan ^{-1}(x)}{2 x^3}-\frac {\sqrt {a x^6} \tanh ^{-1}(x)}{2 x^3}+\frac {1}{2} \tan ^{-1}(x)+\frac {1}{2} \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 15
Rule 203
Rule 206
Rule 212
Rule 298
Rule 1584
Rubi steps
\begin {align*} \int \left (\frac {1}{1-x^4}-\frac {\sqrt {a x^6}}{x-x^5}\right ) \, dx &=\int \frac {1}{1-x^4} \, dx-\int \frac {\sqrt {a x^6}}{x-x^5} \, dx\\ &=\frac {1}{2} \int \frac {1}{1-x^2} \, dx+\frac {1}{2} \int \frac {1}{1+x^2} \, dx-\frac {\sqrt {a x^6} \int \frac {x^3}{x-x^5} \, dx}{x^3}\\ &=\frac {1}{2} \tan ^{-1}(x)+\frac {1}{2} \tanh ^{-1}(x)-\frac {\sqrt {a x^6} \int \frac {x^2}{1-x^4} \, dx}{x^3}\\ &=\frac {1}{2} \tan ^{-1}(x)+\frac {1}{2} \tanh ^{-1}(x)-\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 {1}{2} \tan ^{-1}(x)+\frac {\sqrt {a x^6} \tan ^{-1}(x)}{2 x^3}+\frac {1}{2} \tanh ^{-1}(x)-\frac {\sqrt {a x^6} \tanh ^{-1}(x)}{2 x^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 29, normalized size = 0.59 \[ \frac {1}{2} \left (\frac {\sqrt {a x^6} \left (\tan ^{-1}(x)-\tanh ^{-1}(x)\right )}{x^3}+\tan ^{-1}(x)+\tanh ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 256, normalized size = 5.22 \[ \left [\frac {x^{3} \sqrt {-\frac {{\left (a + 1\right )} x^{3} + 2 \, \sqrt {a x^{6}}}{x^{3}}} \log \left (\frac {{\left (a - 1\right )} x^{4} - {\left (a - 1\right )} x^{2} - 2 \, {\left (x^{3} - \sqrt {a x^{6}}\right )} \sqrt {-\frac {{\left (a + 1\right )} x^{3} + 2 \, \sqrt {a x^{6}}}{x^{3}}}}{x^{4} + x^{2}}\right ) + x^{3} \log \left (x + 1\right ) - x^{3} \log \left (x - 1\right ) - \sqrt {a x^{6}} {\left (\log \left (x + 1\right ) - \log \left (x - 1\right )\right )}}{4 \, x^{3}}, \frac {2 \, x^{3} \sqrt {\frac {{\left (a + 1\right )} x^{3} + 2 \, \sqrt {a x^{6}}}{x^{3}}} \arctan \left (-\frac {{\left (x^{3} - \sqrt {a x^{6}}\right )} \sqrt {\frac {{\left (a + 1\right )} x^{3} + 2 \, \sqrt {a x^{6}}}{x^{3}}}}{{\left (a - 1\right )} x^{2}}\right ) + x^{3} \log \left (x + 1\right ) - x^{3} \log \left (x - 1\right ) - \sqrt {a x^{6}} {\left (\log \left (x + 1\right ) - \log \left (x - 1\right )\right )}}{4 \, x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 48, normalized size = 0.98 \[ \frac {1}{4} \, {\left (2 \, \arctan \relax (x) \mathrm {sgn}\relax (x) - \log \left ({\left | x + 1 \right |}\right ) \mathrm {sgn}\relax (x) + \log \left ({\left | x - 1 \right |}\right ) \mathrm {sgn}\relax (x)\right )} \sqrt {a} + \frac {1}{2} \, \arctan \relax (x) + \frac {1}{4} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{4} \, \log \left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 37, normalized size = 0.76 \[ \frac {\arctanh \relax (x )}{2}+\frac {\arctan \relax (x )}{2}+\frac {\sqrt {a \,x^{6}}\, \left (2 \arctan \relax (x )+\ln \left (x -1\right )-\ln \left (x +1\right )\right )}{4 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.04, size = 42, normalized size = 0.86 \[ \frac {1}{2} \, \sqrt {a} \arctan \relax (x) - \frac {1}{4} \, \sqrt {a} \log \left (x + 1\right ) + \frac {1}{4} \, \sqrt {a} \log \left (x - 1\right ) + \frac {1}{2} \, \arctan \relax (x) + \frac {1}{4} \, \log \left (x + 1\right ) - \frac {1}{4} \, \log \left (x - 1\right ) \]
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 \[ \int -\frac {1}{x^4-1}-\frac {\sqrt {a\,x^6}}{x-x^5} \,d x \]
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 \[ - \int \frac {x}{x^{5} - x}\, dx - \int \left (- \frac {\sqrt {a x^{6}}}{x^{5} - x}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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