Optimal. Leaf size=133 \[ -\frac {3 x}{8 a c^2 \left (a^2 x^2+1\right )}-\frac {\tan ^{-1}(a x)^3}{2 a^2 c^2 \left (a^2 x^2+1\right )}+\frac {3 x \tan ^{-1}(a x)^2}{4 a c^2 \left (a^2 x^2+1\right )}+\frac {3 \tan ^{-1}(a x)}{4 a^2 c^2 \left (a^2 x^2+1\right )}+\frac {\tan ^{-1}(a x)^3}{4 a^2 c^2}-\frac {3 \tan ^{-1}(a x)}{8 a^2 c^2} \]
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Rubi [A] time = 0.12, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4930, 4892, 199, 205} \[ -\frac {3 x}{8 a c^2 \left (a^2 x^2+1\right )}-\frac {\tan ^{-1}(a x)^3}{2 a^2 c^2 \left (a^2 x^2+1\right )}+\frac {3 x \tan ^{-1}(a x)^2}{4 a c^2 \left (a^2 x^2+1\right )}+\frac {3 \tan ^{-1}(a x)}{4 a^2 c^2 \left (a^2 x^2+1\right )}+\frac {\tan ^{-1}(a x)^3}{4 a^2 c^2}-\frac {3 \tan ^{-1}(a x)}{8 a^2 c^2} \]
Antiderivative was successfully verified.
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Rule 199
Rule 205
Rule 4892
Rule 4930
Rubi steps
\begin {align*} \int \frac {x \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx &=-\frac {\tan ^{-1}(a x)^3}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {3 \int \frac {\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{2 a}\\ &=\frac {3 x \tan ^{-1}(a x)^2}{4 a c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^3}{4 a^2 c^2}-\frac {\tan ^{-1}(a x)^3}{2 a^2 c^2 \left (1+a^2 x^2\right )}-\frac {3}{2} \int \frac {x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx\\ &=\frac {3 \tan ^{-1}(a x)}{4 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {3 x \tan ^{-1}(a x)^2}{4 a c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^3}{4 a^2 c^2}-\frac {\tan ^{-1}(a x)^3}{2 a^2 c^2 \left (1+a^2 x^2\right )}-\frac {3 \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{4 a}\\ &=-\frac {3 x}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {3 \tan ^{-1}(a x)}{4 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {3 x \tan ^{-1}(a x)^2}{4 a c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^3}{4 a^2 c^2}-\frac {\tan ^{-1}(a x)^3}{2 a^2 c^2 \left (1+a^2 x^2\right )}-\frac {3 \int \frac {1}{c+a^2 c x^2} \, dx}{8 a c}\\ &=-\frac {3 x}{8 a c^2 \left (1+a^2 x^2\right )}-\frac {3 \tan ^{-1}(a x)}{8 a^2 c^2}+\frac {3 \tan ^{-1}(a x)}{4 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {3 x \tan ^{-1}(a x)^2}{4 a c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^3}{4 a^2 c^2}-\frac {\tan ^{-1}(a x)^3}{2 a^2 c^2 \left (1+a^2 x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 68, normalized size = 0.51 \[ \frac {2 \left (a^2 x^2-1\right ) \tan ^{-1}(a x)^3+\left (3-3 a^2 x^2\right ) \tan ^{-1}(a x)-3 a x+6 a x \tan ^{-1}(a x)^2}{8 a^2 c^2 \left (a^2 x^2+1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 69, normalized size = 0.52 \[ \frac {6 \, a x \arctan \left (a x\right )^{2} + 2 \, {\left (a^{2} x^{2} - 1\right )} \arctan \left (a x\right )^{3} - 3 \, a x - 3 \, {\left (a^{2} x^{2} - 1\right )} \arctan \left (a x\right )}{8 \, {\left (a^{4} c^{2} x^{2} + a^{2} c^{2}\right )}} \]
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 = 0.05, size = 122, normalized size = 0.92 \[ -\frac {3 x}{8 a \,c^{2} \left (a^{2} x^{2}+1\right )}-\frac {3 \arctan \left (a x \right )}{8 a^{2} c^{2}}+\frac {3 \arctan \left (a x \right )}{4 a^{2} c^{2} \left (a^{2} x^{2}+1\right )}+\frac {3 x \arctan \left (a x \right )^{2}}{4 a \,c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )^{3}}{4 a^{2} c^{2}}-\frac {\arctan \left (a x \right )^{3}}{2 a^{2} c^{2} \left (a^{2} x^{2}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 174, normalized size = 1.31 \[ \frac {3 \, {\left (\frac {x}{a^{2} c x^{2} + c} + \frac {\arctan \left (a x\right )}{a c}\right )} \arctan \left (a x\right )^{2}}{4 \, a c} + \frac {\frac {{\left (2 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} - 3 \, a x - 3 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a^{2}}{a^{5} c x^{2} + a^{3} c} - \frac {6 \, {\left ({\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} - 1\right )} a \arctan \left (a x\right )}{a^{4} c x^{2} + a^{2} c}}{8 \, a c} - \frac {\arctan \left (a x\right )^{3}}{2 \, {\left (a^{2} c x^{2} + c\right )} a^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 114, normalized size = 0.86 \[ {\mathrm {atan}\left (a\,x\right )}^3\,\left (\frac {1}{4\,a^2\,c^2}-\frac {1}{2\,a^4\,c^2\,\left (\frac {1}{a^2}+x^2\right )}\right )-\frac {3\,x}{2\,\left (4\,a^3\,c^2\,x^2+4\,a\,c^2\right )}-\frac {3\,\mathrm {atan}\left (a\,x\right )}{8\,a^2\,c^2}+\frac {3\,\mathrm {atan}\left (a\,x\right )}{4\,a^4\,c^2\,\left (\frac {1}{a^2}+x^2\right )}+\frac {3\,x\,{\mathrm {atan}\left (a\,x\right )}^2}{4\,a^3\,c^2\,\left (\frac {1}{a^2}+x^2\right )} \]
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 \[ \frac {\int \frac {x \operatorname {atan}^{3}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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