Optimal. Leaf size=111 \[ \frac {\tan ^{-1}(a x)^2}{16 a^3 c^3}+\frac {x \tan ^{-1}(a x)}{8 a^2 c^3 \left (a^2 x^2+1\right )}-\frac {x \tan ^{-1}(a x)}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac {1}{16 a^3 c^3 \left (a^2 x^2+1\right )}-\frac {1}{16 a^3 c^3 \left (a^2 x^2+1\right )^2} \]
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Rubi [A] time = 0.07, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4934, 4892, 261} \[ \frac {1}{16 a^3 c^3 \left (a^2 x^2+1\right )}-\frac {1}{16 a^3 c^3 \left (a^2 x^2+1\right )^2}+\frac {x \tan ^{-1}(a x)}{8 a^2 c^3 \left (a^2 x^2+1\right )}-\frac {x \tan ^{-1}(a x)}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac {\tan ^{-1}(a x)^2}{16 a^3 c^3} \]
Antiderivative was successfully verified.
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Rule 261
Rule 4892
Rule 4934
Rubi steps
\begin {align*} \int \frac {x^2 \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^3} \, dx &=-\frac {1}{16 a^3 c^3 \left (1+a^2 x^2\right )^2}-\frac {x \tan ^{-1}(a x)}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {\int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{4 a^2 c}\\ &=-\frac {1}{16 a^3 c^3 \left (1+a^2 x^2\right )^2}-\frac {x \tan ^{-1}(a x)}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \tan ^{-1}(a x)}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^2}{16 a^3 c^3}-\frac {\int \frac {x}{\left (c+a^2 c x^2\right )^2} \, dx}{8 a c}\\ &=-\frac {1}{16 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {1}{16 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {x \tan ^{-1}(a x)}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \tan ^{-1}(a x)}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^2}{16 a^3 c^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 64, normalized size = 0.58 \[ \frac {a^2 x^2+2 a x \left (a^2 x^2-1\right ) \tan ^{-1}(a x)+\left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2}{16 a^3 c^3 \left (a^2 x^2+1\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 83, normalized size = 0.75 \[ \frac {a^{2} x^{2} + {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 2 \, {\left (a^{3} x^{3} - a x\right )} \arctan \left (a x\right )}{16 \, {\left (a^{7} c^{3} x^{4} + 2 \, a^{5} c^{3} x^{2} + a^{3} c^{3}\right )}} \]
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 = 101, normalized size = 0.91 \[ \frac {\arctan \left (a x \right ) x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {x \arctan \left (a x \right )}{8 a^{2} c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )^{2}}{16 a^{3} c^{3}}-\frac {1}{16 a^{3} c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {1}{16 a^{3} c^{3} \left (a^{2} x^{2}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 129, normalized size = 1.16 \[ \frac {1}{8} \, {\left (\frac {a^{2} x^{3} - x}{a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}} + \frac {\arctan \left (a x\right )}{a^{3} c^{3}}\right )} \arctan \left (a x\right ) + \frac {{\left (a^{2} x^{2} - {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2}\right )} a}{16 \, {\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.47, size = 80, normalized size = 0.72 \[ \frac {a^4\,x^4\,{\mathrm {atan}\left (a\,x\right )}^2+2\,a^3\,x^3\,\mathrm {atan}\left (a\,x\right )+2\,a^2\,x^2\,{\mathrm {atan}\left (a\,x\right )}^2+a^2\,x^2-2\,a\,x\,\mathrm {atan}\left (a\,x\right )+{\mathrm {atan}\left (a\,x\right )}^2}{16\,a^3\,c^3\,{\left (a^2\,x^2+1\right )}^2} \]
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 \[ \frac {\int \frac {x^{2} \operatorname {atan}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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