Optimal. Leaf size=45 \[ \frac {\sqrt {x^6} \tan ^{-1}(x)}{2 x^3}-\frac {\sqrt {x^6} \tanh ^{-1}(x)}{2 x^3}+\frac {1}{2} \tan ^{-1}(x)+\frac {1}{2} \tanh ^{-1}(x) \]
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Rubi [A] time = 0.13, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {6729, 1584, 6725, 212, 206, 203, 15, 298} \[ \frac {\sqrt {x^6} \tan ^{-1}(x)}{2 x^3}-\frac {\sqrt {x^6} \tanh ^{-1}(x)}{2 x^3}+\frac {1}{2} \tan ^{-1}(x)+\frac {1}{2} \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 15
Rule 203
Rule 206
Rule 212
Rule 298
Rule 1584
Rule 6725
Rule 6729
Rubi steps
\begin {align*} \int \frac {x}{x+\sqrt {x^6}} \, dx &=\int \frac {x \left (x-\sqrt {x^6}\right )}{x^2-x^6} \, dx\\ &=\int \frac {x-\sqrt {x^6}}{x \left (1-x^4\right )} \, dx\\ &=\int \left (\frac {1}{1-x^4}+\frac {\sqrt {x^6}}{x \left (-1+x^4\right )}\right ) \, dx\\ &=\int \frac {1}{1-x^4} \, dx+\int \frac {\sqrt {x^6}}{x \left (-1+x^4\right )} \, dx\\ &=\frac {1}{2} \int \frac {1}{1-x^2} \, dx+\frac {1}{2} \int \frac {1}{1+x^2} \, dx+\frac {\sqrt {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 {x^6} \int \frac {1}{1-x^2} \, dx}{2 x^3}+\frac {\sqrt {x^6} \int \frac {1}{1+x^2} \, dx}{2 x^3}\\ &=\frac {1}{2} \tan ^{-1}(x)+\frac {\sqrt {x^6} \tan ^{-1}(x)}{2 x^3}+\frac {1}{2} \tanh ^{-1}(x)-\frac {\sqrt {x^6} \tanh ^{-1}(x)}{2 x^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 27, normalized size = 0.60 \[ \frac {1}{2} \left (\frac {\sqrt {x^6} \left (\tan ^{-1}(x)-\tanh ^{-1}(x)\right )}{x^3}+\tan ^{-1}(x)+\tanh ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 2, normalized size = 0.04 \[ \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 12, normalized size = 0.27 \[ \frac {\arctan \left (x \sqrt {\mathrm {sgn}\relax (x)}\right )}{\sqrt {\mathrm {sgn}\relax (x)}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 27, normalized size = 0.60 \[ \frac {\arctan \left (\sqrt {\frac {\sqrt {x^{6}}}{x^{3}}}\, x \right )}{\sqrt {\frac {\sqrt {x^{6}}}{x^{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 2, normalized size = 0.04 \[ \arctan \relax (x) \]
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 \[ \int \frac {x}{x+\sqrt {x^6}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 2, normalized size = 0.04 \[ \operatorname {atan}{\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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