Optimal. Leaf size=135 \[ \frac {\tan ^{-1}(a x)^4}{8 a^3 c^2}+\frac {3 \tan ^{-1}(a x)^2}{8 a^3 c^2}-\frac {x \tan ^{-1}(a x)^3}{2 a^2 c^2 \left (a^2 x^2+1\right )}+\frac {3 x \tan ^{-1}(a x)}{4 a^2 c^2 \left (a^2 x^2+1\right )}+\frac {3}{8 a^3 c^2 \left (a^2 x^2+1\right )}-\frac {3 \tan ^{-1}(a x)^2}{4 a^3 c^2 \left (a^2 x^2+1\right )} \]
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Rubi [A] time = 0.14, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4936, 4930, 4892, 261} \[ \frac {3}{8 a^3 c^2 \left (a^2 x^2+1\right )}-\frac {x \tan ^{-1}(a x)^3}{2 a^2 c^2 \left (a^2 x^2+1\right )}-\frac {3 \tan ^{-1}(a x)^2}{4 a^3 c^2 \left (a^2 x^2+1\right )}+\frac {3 x \tan ^{-1}(a x)}{4 a^2 c^2 \left (a^2 x^2+1\right )}+\frac {\tan ^{-1}(a x)^4}{8 a^3 c^2}+\frac {3 \tan ^{-1}(a x)^2}{8 a^3 c^2} \]
Antiderivative was successfully verified.
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Rule 261
Rule 4892
Rule 4930
Rule 4936
Rubi steps
\begin {align*} \int \frac {x^2 \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx &=-\frac {x \tan ^{-1}(a x)^3}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^4}{8 a^3 c^2}+\frac {3 \int \frac {x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{2 a}\\ &=-\frac {3 \tan ^{-1}(a x)^2}{4 a^3 c^2 \left (1+a^2 x^2\right )}-\frac {x \tan ^{-1}(a x)^3}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^4}{8 a^3 c^2}+\frac {3 \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{2 a^2}\\ &=\frac {3 x \tan ^{-1}(a x)}{4 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {3 \tan ^{-1}(a x)^2}{8 a^3 c^2}-\frac {3 \tan ^{-1}(a x)^2}{4 a^3 c^2 \left (1+a^2 x^2\right )}-\frac {x \tan ^{-1}(a x)^3}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^4}{8 a^3 c^2}-\frac {3 \int \frac {x}{\left (c+a^2 c x^2\right )^2} \, dx}{4 a}\\ &=\frac {3}{8 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {3 x \tan ^{-1}(a x)}{4 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {3 \tan ^{-1}(a x)^2}{8 a^3 c^2}-\frac {3 \tan ^{-1}(a x)^2}{4 a^3 c^2 \left (1+a^2 x^2\right )}-\frac {x \tan ^{-1}(a x)^3}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^4}{8 a^3 c^2}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 74, normalized size = 0.55 \[ \frac {\left (a^2 x^2+1\right ) \tan ^{-1}(a x)^4+3 \left (a^2 x^2-1\right ) \tan ^{-1}(a x)^2-4 a x \tan ^{-1}(a x)^3+6 a x \tan ^{-1}(a x)+3}{8 a^3 c^2 \left (a^2 x^2+1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 76, normalized size = 0.56 \[ -\frac {4 \, a x \arctan \left (a x\right )^{3} - {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{4} - 6 \, a x \arctan \left (a x\right ) - 3 \, {\left (a^{2} x^{2} - 1\right )} \arctan \left (a x\right )^{2} - 3}{8 \, {\left (a^{5} c^{2} x^{2} + a^{3} c^{2}\right )}} \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 124, normalized size = 0.92 \[ \frac {3}{8 a^{3} c^{2} \left (a^{2} x^{2}+1\right )}+\frac {3 x \arctan \left (a x \right )}{4 a^{2} c^{2} \left (a^{2} x^{2}+1\right )}+\frac {3 \arctan \left (a x \right )^{2}}{8 a^{3} c^{2}}-\frac {3 \arctan \left (a x \right )^{2}}{4 a^{3} c^{2} \left (a^{2} x^{2}+1\right )}-\frac {x \arctan \left (a x \right )^{3}}{2 a^{2} c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )^{4}}{8 a^{3} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 218, normalized size = 1.61 \[ -\frac {1}{2} \, {\left (\frac {x}{a^{4} c^{2} x^{2} + a^{2} c^{2}} - \frac {\arctan \left (a x\right )}{a^{3} c^{2}}\right )} \arctan \left (a x\right )^{3} - \frac {3 \, {\left ({\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 1\right )} a \arctan \left (a x\right )^{2}}{4 \, {\left (a^{6} c^{2} x^{2} + a^{4} c^{2}\right )}} - \frac {1}{8} \, {\left (\frac {{\left ({\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{4} + 3 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} - 3\right )} a^{2}}{a^{8} c^{2} x^{2} + a^{6} c^{2}} - \frac {2 \, {\left (2 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} + 3 \, a x + 3 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a \arctan \left (a x\right )}{a^{7} c^{2} x^{2} + a^{5} c^{2}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 119, normalized size = 0.88 \[ \frac {3}{2\,a^2\,\left (4\,a^3\,c^2\,x^2+4\,a\,c^2\right )}+{\mathrm {atan}\left (a\,x\right )}^2\,\left (\frac {3}{8\,a^3\,c^2}-\frac {3}{4\,a^5\,c^2\,\left (\frac {1}{a^2}+x^2\right )}\right )+\frac {{\mathrm {atan}\left (a\,x\right )}^4}{8\,a^3\,c^2}+\frac {3\,x\,\mathrm {atan}\left (a\,x\right )}{4\,a^4\,c^2\,\left (\frac {1}{a^2}+x^2\right )}-\frac {x\,{\mathrm {atan}\left (a\,x\right )}^3}{2\,a^4\,c^2\,\left (\frac {1}{a^2}+x^2\right )} \]
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 \[ \frac {\int \frac {x^{2} \operatorname {atan}^{3}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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