Optimal. Leaf size=39 \[ -\frac {1}{2} \log \left (x^2+1\right )-\frac {\tan ^{-1}(x)^2}{2 x^2}+\log (x)-\frac {1}{2} \tan ^{-1}(x)^2-\frac {\tan ^{-1}(x)}{x} \]
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Rubi [A] time = 0.07, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {4852, 4918, 266, 36, 29, 31, 4884} \[ -\frac {1}{2} \log \left (x^2+1\right )-\frac {\tan ^{-1}(x)^2}{2 x^2}+\log (x)-\frac {1}{2} \tan ^{-1}(x)^2-\frac {\tan ^{-1}(x)}{x} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 4852
Rule 4884
Rule 4918
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(x)^2}{x^3} \, dx &=-\frac {\tan ^{-1}(x)^2}{2 x^2}+\int \frac {\tan ^{-1}(x)}{x^2 \left (1+x^2\right )} \, dx\\ &=-\frac {\tan ^{-1}(x)^2}{2 x^2}+\int \frac {\tan ^{-1}(x)}{x^2} \, dx-\int \frac {\tan ^{-1}(x)}{1+x^2} \, dx\\ &=-\frac {\tan ^{-1}(x)}{x}-\frac {1}{2} \tan ^{-1}(x)^2-\frac {\tan ^{-1}(x)^2}{2 x^2}+\int \frac {1}{x \left (1+x^2\right )} \, dx\\ &=-\frac {\tan ^{-1}(x)}{x}-\frac {1}{2} \tan ^{-1}(x)^2-\frac {\tan ^{-1}(x)^2}{2 x^2}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x (1+x)} \, dx,x,x^2\right )\\ &=-\frac {\tan ^{-1}(x)}{x}-\frac {1}{2} \tan ^{-1}(x)^2-\frac {\tan ^{-1}(x)^2}{2 x^2}+\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}-\frac {1}{2} \tan ^{-1}(x)^2-\frac {\tan ^{-1}(x)^2}{2 x^2}+\log (x)-\frac {1}{2} \log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 38, normalized size = 0.97 \[ -\frac {1}{2} \log \left (x^2+1\right )+\frac {\left (-x^2-1\right ) \tan ^{-1}(x)^2}{2 x^2}+\log (x)-\frac {\tan ^{-1}(x)}{x} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{-1}(x)^2}{x^3} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.04, size = 38, normalized size = 0.97 \[ -\frac {{\left (x^{2} + 1\right )} \arctan \relax (x)^{2} + x^{2} \log \left (x^{2} + 1\right ) - 2 \, x^{2} \log \relax (x) + 2 \, x \arctan \relax (x)}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arctan \relax (x)^{2}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 34, normalized size = 0.87
method | result | size |
default | \(-\frac {\arctan \relax (x )}{x}-\frac {\arctan \relax (x )^{2}}{2}-\frac {\arctan \relax (x )^{2}}{2 x^{2}}+\ln \relax (x )-\frac {\ln \left (x^{2}+1\right )}{2}\) | \(34\) |
risch | \(\frac {\left (x^{2}+1\right ) \ln \left (i x +1\right )^{2}}{8 x^{2}}-\frac {\left (x^{2} \ln \left (-i x +1\right )-2 i x +\ln \left (-i x +1\right )\right ) \ln \left (i x +1\right )}{4 x^{2}}+\frac {x^{2} \ln \left (-i x +1\right )^{2}-4 i \ln \left (-i x +1\right ) x +8 x^{2} \ln \relax (x )-4 \ln \left (x^{2}+1\right ) x^{2}+\ln \left (-i x +1\right )^{2}}{8 x^{2}}\) | \(113\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 36, normalized size = 0.92 \[ -{\left (\frac {1}{x} + \arctan \relax (x)\right )} \arctan \relax (x) + \frac {1}{2} \, \arctan \relax (x)^{2} - \frac {\arctan \relax (x)^{2}}{2 \, x^{2}} - \frac {1}{2} \, \log \left (x^{2} + 1\right ) + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 31, normalized size = 0.79 \[ \ln \relax (x)-\frac {\ln \left (x^2+1\right )}{2}-\frac {\mathrm {atan}\relax (x)}{x}-{\mathrm {atan}\relax (x)}^2\,\left (\frac {1}{2\,x^2}+\frac {1}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.53, size = 32, normalized size = 0.82 \[ \log {\relax (x )} - \frac {\log {\left (x^{2} + 1 \right )}}{2} - \frac {\operatorname {atan}^{2}{\relax (x )}}{2} - \frac {\operatorname {atan}{\relax (x )}}{x} - \frac {\operatorname {atan}^{2}{\relax (x )}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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