Optimal. Leaf size=60 \[ -\frac {\tan ^{-1}(x)}{6 x^3}-\frac {1}{12 x^2}-\frac {1}{6} \log \left (x^2+1\right )-\frac {\left (x^2+1\right )^2 \tan ^{-1}(x)^2}{4 x^4}+\frac {\log (x)}{3}-\frac {\tan ^{-1}(x)}{2 x} \]
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Rubi [A] time = 0.08, antiderivative size = 60, 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 = {4944, 4950, 4852, 266, 44, 36, 29, 31} \[ -\frac {1}{12 x^2}-\frac {1}{6} \log \left (x^2+1\right )-\frac {\left (x^2+1\right )^2 \tan ^{-1}(x)^2}{4 x^4}-\frac {\tan ^{-1}(x)}{6 x^3}+\frac {\log (x)}{3}-\frac {\tan ^{-1}(x)}{2 x} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 44
Rule 266
Rule 4852
Rule 4944
Rule 4950
Rubi steps
\begin {align*} \int \frac {\left (1+x^2\right ) \tan ^{-1}(x)^2}{x^5} \, dx &=-\frac {\left (1+x^2\right )^2 \tan ^{-1}(x)^2}{4 x^4}+\frac {1}{2} \int \frac {\left (1+x^2\right ) \tan ^{-1}(x)}{x^4} \, dx\\ &=-\frac {\left (1+x^2\right )^2 \tan ^{-1}(x)^2}{4 x^4}+\frac {1}{2} \int \frac {\tan ^{-1}(x)}{x^4} \, dx+\frac {1}{2} \int \frac {\tan ^{-1}(x)}{x^2} \, dx\\ &=-\frac {\tan ^{-1}(x)}{6 x^3}-\frac {\tan ^{-1}(x)}{2 x}-\frac {\left (1+x^2\right )^2 \tan ^{-1}(x)^2}{4 x^4}+\frac {1}{6} \int \frac {1}{x^3 \left (1+x^2\right )} \, dx+\frac {1}{2} \int \frac {1}{x \left (1+x^2\right )} \, dx\\ &=-\frac {\tan ^{-1}(x)}{6 x^3}-\frac {\tan ^{-1}(x)}{2 x}-\frac {\left (1+x^2\right )^2 \tan ^{-1}(x)^2}{4 x^4}+\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{x^2 (1+x)} \, dx,x,x^2\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x (1+x)} \, dx,x,x^2\right )\\ &=-\frac {\tan ^{-1}(x)}{6 x^3}-\frac {\tan ^{-1}(x)}{2 x}-\frac {\left (1+x^2\right )^2 \tan ^{-1}(x)^2}{4 x^4}+\frac {1}{12} \operatorname {Subst}\left (\int \left (\frac {1}{x^2}-\frac {1}{x}+\frac {1}{1+x}\right ) \, dx,x,x^2\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,x^2\right )\\ &=-\frac {1}{12 x^2}-\frac {\tan ^{-1}(x)}{6 x^3}-\frac {\tan ^{-1}(x)}{2 x}-\frac {\left (1+x^2\right )^2 \tan ^{-1}(x)^2}{4 x^4}+\frac {\log (x)}{3}-\frac {1}{6} \log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 56, normalized size = 0.93 \[ \frac {-2 \left (3 x^3+x\right ) \tan ^{-1}(x)+x^2 \left (4 x^2 \log (x)-2 x^2 \log \left (x^2+1\right )-1\right )-3 \left (x^2+1\right )^2 \tan ^{-1}(x)^2}{12 x^4} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (1+x^2\right ) \tan ^{-1}(x)^2}{x^5} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.00, size = 54, normalized size = 0.90 \[ -\frac {2 \, x^{4} \log \left (x^{2} + 1\right ) - 4 \, x^{4} \log \relax (x) + 3 \, {\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \relax (x)^{2} + x^{2} + 2 \, {\left (3 \, x^{3} + x\right )} \arctan \relax (x)}{12 \, x^{4}} \]
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 \[ \int \frac {{\left (x^{2} + 1\right )} \arctan \relax (x)^{2}}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 57, normalized size = 0.95
method | result | size |
default | \(-\frac {\arctan \relax (x )^{2}}{2 x^{2}}-\frac {\arctan \relax (x )^{2}}{4 x^{4}}-\frac {\arctan \relax (x )}{6 x^{3}}-\frac {\arctan \relax (x )}{2 x}-\frac {\arctan \relax (x )^{2}}{4}-\frac {\ln \left (x^{2}+1\right )}{6}-\frac {1}{12 x^{2}}+\frac {\ln \relax (x )}{3}\) | \(57\) |
risch | \(\frac {\left (x^{4}+2 x^{2}+1\right ) \ln \left (i x +1\right )^{2}}{16 x^{4}}-\frac {\left (3 x^{4} \ln \left (-i x +1\right )-6 i x^{3}+6 x^{2} \ln \left (-i x +1\right )-2 i x +3 \ln \left (-i x +1\right )\right ) \ln \left (i x +1\right )}{24 x^{4}}+\frac {3 x^{4} \ln \left (-i x +1\right )^{2}+16 x^{4} \ln \relax (x )-8 \ln \left (x^{2}+1\right ) x^{4}-12 i x^{3} \ln \left (-i x +1\right )+6 x^{2} \ln \left (-i x +1\right )^{2}-4 i \ln \left (-i x +1\right ) x -4 x^{2}+3 \ln \left (-i x +1\right )^{2}}{48 x^{4}}\) | \(174\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 71, normalized size = 1.18 \[ -\frac {1}{6} \, {\left (\frac {3 \, x^{2} + 1}{x^{3}} + 3 \, \arctan \relax (x)\right )} \arctan \relax (x) + \frac {3 \, x^{2} \arctan \relax (x)^{2} - 2 \, x^{2} \log \left (x^{2} + 1\right ) + 4 \, x^{2} \log \relax (x) - 1}{12 \, x^{2}} - \frac {{\left (2 \, x^{2} + 1\right )} \arctan \relax (x)^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 51, normalized size = 0.85 \[ \frac {\ln \relax (x)}{3}-\frac {\ln \left (x^2+1\right )}{6}-{\mathrm {atan}\relax (x)}^2\,\left (\frac {\frac {x^2}{2}+\frac {1}{4}}{x^4}+\frac {1}{4}\right )-\frac {1}{12\,x^2}-\frac {\mathrm {atan}\relax (x)\,\left (\frac {x^2}{2}+\frac {1}{6}\right )}{x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.90, size = 61, normalized size = 1.02 \[ \frac {\log {\relax (x )}}{3} - \frac {\log {\left (x^{2} + 1 \right )}}{6} - \frac {\operatorname {atan}^{2}{\relax (x )}}{4} - \frac {\operatorname {atan}{\relax (x )}}{2 x} - \frac {\operatorname {atan}^{2}{\relax (x )}}{2 x^{2}} - \frac {1}{12 x^{2}} - \frac {\operatorname {atan}{\relax (x )}}{6 x^{3}} - \frac {\operatorname {atan}^{2}{\relax (x )}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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