Optimal. Leaf size=31 \[ -\frac {1}{12 x^3}-\frac {1}{4 x}-\frac {\left (1+x^2\right )^2 \tan ^{-1}(x)}{4 x^4} \]
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Rubi [A]
time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {5064, 14}
\begin {gather*} -\frac {\left (x^2+1\right )^2 \text {ArcTan}(x)}{4 x^4}-\frac {1}{12 x^3}-\frac {1}{4 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 5064
Rubi steps
\begin {align*} \int \frac {\left (1+x^2\right ) \tan ^{-1}(x)}{x^5} \, dx &=-\frac {\left (1+x^2\right )^2 \tan ^{-1}(x)}{4 x^4}+\frac {1}{4} \int \frac {1+x^2}{x^4} \, dx\\ &=-\frac {\left (1+x^2\right )^2 \tan ^{-1}(x)}{4 x^4}+\frac {1}{4} \int \left (\frac {1}{x^4}+\frac {1}{x^2}\right ) \, dx\\ &=-\frac {1}{12 x^3}-\frac {1}{4 x}-\frac {\left (1+x^2\right )^2 \tan ^{-1}(x)}{4 x^4}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.01, size = 59, normalized size = 1.90 \begin {gather*} -\frac {\tan ^{-1}(x)}{4 x^4}-\frac {\tan ^{-1}(x)}{2 x^2}-\frac {\, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-x^2\right )}{12 x^3}-\frac {\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-x^2\right )}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 30, normalized size = 0.97
method | result | size |
default | \(-\frac {\arctan \left (x \right )}{4 x^{4}}-\frac {\arctan \left (x \right )}{2 x^{2}}-\frac {1}{12 x^{3}}-\frac {1}{4 x}-\frac {\arctan \left (x \right )}{4}\) | \(30\) |
meijerg | \(-\frac {1}{12 x^{3}}-\frac {1}{4 x}-\frac {2 \left (-\frac {3 x^{4}}{8}+\frac {3}{8}\right ) \arctan \left (\sqrt {x^{2}}\right )}{3 x^{3} \sqrt {x^{2}}}-\frac {\left (x^{2}+1\right ) \arctan \left (x \right )}{2 x^{2}}\) | \(47\) |
risch | \(\frac {i \left (2 x^{2}+1\right ) \ln \left (i x +1\right )}{8 x^{4}}-\frac {i \left (3 \ln \left (x +i\right ) x^{4}-3 \ln \left (x -i\right ) x^{4}-6 i x^{3}+6 x^{2} \ln \left (-i x +1\right )-2 i x +3 \ln \left (-i x +1\right )\right )}{24 x^{4}}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.91, size = 31, normalized size = 1.00 \begin {gather*} -\frac {3 \, x^{2} + 1}{12 \, x^{3}} - \frac {{\left (2 \, x^{2} + 1\right )} \arctan \left (x\right )}{4 \, x^{4}} - \frac {1}{4} \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.57, size = 26, normalized size = 0.84 \begin {gather*} -\frac {3 \, x^{3} + 3 \, {\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \left (x\right ) + x}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.25, size = 34, normalized size = 1.10 \begin {gather*} - \frac {\operatorname {atan}{\left (x \right )}}{4} - \frac {1}{4 x} - \frac {\operatorname {atan}{\left (x \right )}}{2 x^{2}} - \frac {1}{12 x^{3}} - \frac {\operatorname {atan}{\left (x \right )}}{4 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.65, size = 31, normalized size = 1.00 \begin {gather*} -\frac {3 \, x^{2} + 1}{12 \, x^{3}} - \frac {{\left (2 \, x^{2} + 1\right )} \arctan \left (x\right )}{4 \, x^{4}} - \frac {1}{4} \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.32, size = 30, normalized size = 0.97 \begin {gather*} -\frac {\mathrm {atan}\left (x\right )}{4}-\frac {\frac {x}{12}+\frac {\mathrm {atan}\left (x\right )}{4}+\frac {x^2\,\mathrm {atan}\left (x\right )}{2}+\frac {x^3}{4}}{x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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