Optimal. Leaf size=22 \[ -\log \left (x^2+1\right )+\log (x)+x \tan ^{-1}(x)-\frac {\tan ^{-1}(x)}{x} \]
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Rubi [A] time = 0.03, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {4950, 4852, 266, 36, 29, 31, 4846, 260} \[ -\log \left (x^2+1\right )+\log (x)+x \tan ^{-1}(x)-\frac {\tan ^{-1}(x)}{x} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 260
Rule 266
Rule 4846
Rule 4852
Rule 4950
Rubi steps
\begin {align*} \int \frac {\left (1+x^2\right ) \tan ^{-1}(x)}{x^2} \, dx &=\int \tan ^{-1}(x) \, dx+\int \frac {\tan ^{-1}(x)}{x^2} \, dx\\ &=-\frac {\tan ^{-1}(x)}{x}+x \tan ^{-1}(x)+\int \frac {1}{x \left (1+x^2\right )} \, dx-\int \frac {x}{1+x^2} \, dx\\ &=-\frac {\tan ^{-1}(x)}{x}+x \tan ^{-1}(x)-\frac {1}{2} \log \left (1+x^2\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x (1+x)} \, dx,x,x^2\right )\\ &=-\frac {\tan ^{-1}(x)}{x}+x \tan ^{-1}(x)-\frac {1}{2} \log \left (1+x^2\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,x^2\right )\\ &=-\frac {\tan ^{-1}(x)}{x}+x \tan ^{-1}(x)+\log (x)-\log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 1.00 \[ -\log \left (x^2+1\right )+\log (x)+x \tan ^{-1}(x)-\frac {\tan ^{-1}(x)}{x} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (1+x^2\right ) \tan ^{-1}(x)}{x^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.19, size = 26, normalized size = 1.18 \[ \frac {{\left (x^{2} - 1\right )} \arctan \relax (x) - x \log \left (x^{2} + 1\right ) + x \log \relax (x)}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.99, size = 25, normalized size = 1.14 \[ {\left (x - \frac {1}{x}\right )} \arctan \relax (x) - \log \left (x^{2} + 1\right ) + \frac {1}{2} \, \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 23, normalized size = 1.05
method | result | size |
default | \(-\frac {\arctan \relax (x )}{x}+x \arctan \relax (x )+\ln \relax (x )-\ln \left (x^{2}+1\right )\) | \(23\) |
meijerg | \(-\frac {\arctan \left (\sqrt {x^{2}}\right )}{\sqrt {x^{2}}}-\ln \left (x^{2}+1\right )+\ln \relax (x )+\frac {x^{2} \arctan \left (\sqrt {x^{2}}\right )}{\sqrt {x^{2}}}\) | \(40\) |
risch | \(-\frac {i \left (x^{2}-1\right ) \ln \left (i x +1\right )}{2 x}+\frac {i \left (-2 i \ln \relax (x ) x +2 i \ln \left (x^{2}+1\right ) x +x^{2} \ln \left (-i x +1\right )-\ln \left (-i x +1\right )\right )}{2 x}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 21, normalized size = 0.95 \[ {\left (x - \frac {1}{x}\right )} \arctan \relax (x) - \log \left (x^{2} + 1\right ) + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 22, normalized size = 1.00 \[ \ln \relax (x)-\ln \left (x^2+1\right )-\frac {\mathrm {atan}\relax (x)}{x}+x\,\mathrm {atan}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 19, normalized size = 0.86 \[ x \operatorname {atan}{\relax (x )} + \log {\relax (x )} - \log {\left (x^{2} + 1 \right )} - \frac {\operatorname {atan}{\relax (x )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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