Optimal. Leaf size=79 \[ \frac {5}{32 \left (x^2+1\right )}-\frac {1}{32 \left (x^2+1\right )^2}+\frac {3 x \tan ^{-1}(x)}{16 \left (x^2+1\right )}+\frac {x^4 \tan ^{-1}(x)^2}{4 \left (x^2+1\right )^2}+\frac {x^3 \tan ^{-1}(x)}{8 \left (x^2+1\right )^2}-\frac {3}{32} \tan ^{-1}(x)^2 \]
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Rubi [A] time = 0.13, antiderivative size = 82, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {4944, 4938, 4934, 4884} \[ -\frac {x^4}{32 \left (x^2+1\right )^2}+\frac {3}{32 \left (x^2+1\right )}+\frac {x^4 \tan ^{-1}(x)^2}{4 \left (x^2+1\right )^2}+\frac {x^3 \tan ^{-1}(x)}{8 \left (x^2+1\right )^2}+\frac {3 x \tan ^{-1}(x)}{16 \left (x^2+1\right )}-\frac {3}{32} \tan ^{-1}(x)^2 \]
Antiderivative was successfully verified.
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Rule 4884
Rule 4934
Rule 4938
Rule 4944
Rubi steps
\begin {align*} \int \frac {x^3 \tan ^{-1}(x)^2}{\left (1+x^2\right )^3} \, dx &=\frac {x^4 \tan ^{-1}(x)^2}{4 \left (1+x^2\right )^2}-\frac {1}{2} \int \frac {x^4 \tan ^{-1}(x)}{\left (1+x^2\right )^3} \, dx\\ &=-\frac {x^4}{32 \left (1+x^2\right )^2}+\frac {x^3 \tan ^{-1}(x)}{8 \left (1+x^2\right )^2}+\frac {x^4 \tan ^{-1}(x)^2}{4 \left (1+x^2\right )^2}-\frac {3}{8} \int \frac {x^2 \tan ^{-1}(x)}{\left (1+x^2\right )^2} \, dx\\ &=-\frac {x^4}{32 \left (1+x^2\right )^2}+\frac {3}{32 \left (1+x^2\right )}+\frac {x^3 \tan ^{-1}(x)}{8 \left (1+x^2\right )^2}+\frac {3 x \tan ^{-1}(x)}{16 \left (1+x^2\right )}+\frac {x^4 \tan ^{-1}(x)^2}{4 \left (1+x^2\right )^2}-\frac {3}{16} \int \frac {\tan ^{-1}(x)}{1+x^2} \, dx\\ &=-\frac {x^4}{32 \left (1+x^2\right )^2}+\frac {3}{32 \left (1+x^2\right )}+\frac {x^3 \tan ^{-1}(x)}{8 \left (1+x^2\right )^2}+\frac {3 x \tan ^{-1}(x)}{16 \left (1+x^2\right )}-\frac {3}{32} \tan ^{-1}(x)^2+\frac {x^4 \tan ^{-1}(x)^2}{4 \left (1+x^2\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 47, normalized size = 0.59 \[ \frac {5 x^2+2 \left (5 x^2+3\right ) x \tan ^{-1}(x)+\left (5 x^4-6 x^2-3\right ) \tan ^{-1}(x)^2+4}{32 \left (x^2+1\right )^2} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^3 \tan ^{-1}(x)^2}{\left (1+x^2\right )^3} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.94, size = 51, normalized size = 0.65 \[ \frac {{\left (5 \, x^{4} - 6 \, x^{2} - 3\right )} \arctan \relax (x)^{2} + 5 \, x^{2} + 2 \, {\left (5 \, x^{3} + 3 \, x\right )} \arctan \relax (x) + 4}{32 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} \]
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 \[ \int \frac {x^{3} \arctan \relax (x)^{2}}{{\left (x^{2} + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.42, size = 78, normalized size = 0.99
method | result | size |
default | \(\frac {\arctan \relax (x )^{2}}{4 \left (x^{2}+1\right )^{2}}-\frac {\arctan \relax (x )^{2}}{2 \left (x^{2}+1\right )}+\frac {5 x^{3} \arctan \relax (x )}{16 \left (x^{2}+1\right )^{2}}+\frac {3 x \arctan \relax (x )}{16 \left (x^{2}+1\right )^{2}}+\frac {5 \arctan \relax (x )^{2}}{32}-\frac {1}{32 \left (x^{2}+1\right )^{2}}+\frac {5}{32 \left (x^{2}+1\right )}\) | \(78\) |
risch | \(-\frac {\left (5 x^{4}-6 x^{2}-3\right ) \ln \left (i x +1\right )^{2}}{128 \left (x^{2}+1\right )^{2}}+\frac {\left (-6 x^{2} \ln \left (-i x +1\right )-3 \ln \left (-i x +1\right )+5 x^{4} \ln \left (-i x +1\right )-10 i x^{3}-6 i x \right ) \ln \left (i x +1\right )}{64 \left (x +i\right )^{2} \left (x -i\right )^{2}}-\frac {5 x^{4} \ln \left (-i x +1\right )^{2}-6 x^{2} \ln \left (-i x +1\right )^{2}-3 \ln \left (-i x +1\right )^{2}-20 i x^{3} \ln \left (-i x +1\right )-12 i \ln \left (-i x +1\right ) x -20 x^{2}-16}{128 \left (x +i\right )^{2} \left (x -i\right )^{2}}\) | \(181\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 94, normalized size = 1.19 \[ \frac {1}{16} \, {\left (\frac {5 \, x^{3} + 3 \, x}{x^{4} + 2 \, x^{2} + 1} + 5 \, \arctan \relax (x)\right )} \arctan \relax (x) - \frac {{\left (2 \, x^{2} + 1\right )} \arctan \relax (x)^{2}}{4 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} - \frac {5 \, {\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \relax (x)^{2} - 5 \, x^{2} - 4}{32 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 56, normalized size = 0.71 \[ -\frac {-5\,x^4\,{\mathrm {atan}\relax (x)}^2+4\,x^4-10\,x^3\,\mathrm {atan}\relax (x)+6\,x^2\,{\mathrm {atan}\relax (x)}^2+3\,x^2-6\,x\,\mathrm {atan}\relax (x)+3\,{\mathrm {atan}\relax (x)}^2}{32\,{\left (x^2+1\right )}^2} \]
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 \[ \int \frac {x^{3} \operatorname {atan}^{2}{\relax (x )}}{\left (x^{2} + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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