Optimal. Leaf size=57 \[ -\frac {\sqrt {1-x^2} \tan ^{-1}(x)}{x}-\tanh ^{-1}\left (\sqrt {1-x^2}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1-x^2}}{\sqrt {2}}\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {264, 4976, 446, 83, 63, 206} \[ -\frac {\sqrt {1-x^2} \tan ^{-1}(x)}{x}-\tanh ^{-1}\left (\sqrt {1-x^2}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1-x^2}}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 83
Rule 206
Rule 264
Rule 446
Rule 4976
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(x)}{x^2 \sqrt {1-x^2}} \, dx &=-\frac {\sqrt {1-x^2} \tan ^{-1}(x)}{x}+\int \frac {\sqrt {1-x^2}}{x \left (1+x^2\right )} \, dx\\ &=-\frac {\sqrt {1-x^2} \tan ^{-1}(x)}{x}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {1-x}}{x (1+x)} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1-x^2} \tan ^{-1}(x)}{x}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,x^2\right )-\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} (1+x)} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1-x^2} \tan ^{-1}(x)}{x}+2 \operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1-x^2}\right )-\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x^2}\right )\\ &=-\frac {\sqrt {1-x^2} \tan ^{-1}(x)}{x}-\tanh ^{-1}\left (\sqrt {1-x^2}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1-x^2}}{\sqrt {2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.07, size = 77, normalized size = 1.35 \[ -\frac {\log \left (x^2+1\right )}{\sqrt {2}}+\frac {\log \left (-x^2+2 \sqrt {2-2 x^2}+3\right )}{\sqrt {2}}-\log \left (\sqrt {1-x^2}+1\right )-\frac {\sqrt {1-x^2} \tan ^{-1}(x)}{x}+\log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 81, normalized size = 1.42 \[ \frac {\sqrt {2} x \log \left (\frac {x^{2} - 2 \, \sqrt {2} \sqrt {-x^{2} + 1} - 3}{x^{2} + 1}\right ) - x \log \left (\sqrt {-x^{2} + 1} + 1\right ) + x \log \left (\sqrt {-x^{2} + 1} - 1\right ) - 2 \, \sqrt {-x^{2} + 1} \arctan \relax (x)}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.09, size = 104, normalized size = 1.82 \[ \frac {1}{2} \, {\left (\frac {x}{\sqrt {-x^{2} + 1} - 1} - \frac {\sqrt {-x^{2} + 1} - 1}{x}\right )} \arctan \relax (x) - \frac {1}{2} \, \sqrt {2} \log \left (\frac {\sqrt {2} - \sqrt {-x^{2} + 1}}{\sqrt {2} + \sqrt {-x^{2} + 1}}\right ) - \frac {1}{2} \, \log \left (\sqrt {-x^{2} + 1} + 1\right ) + \frac {1}{2} \, \log \left (-\sqrt {-x^{2} + 1} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {\arctan \relax (x )}{\sqrt {-x^{2}+1}\, x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arctan \relax (x)}{\sqrt {-x^{2} + 1} x^{2}}\,{d x} \]
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 \[ \int \frac {\mathrm {atan}\relax (x)}{x^2\,\sqrt {1-x^2}} \,d x \]
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 \[ \int \frac {\operatorname {atan}{\relax (x )}}{x^{2} \sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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