Optimal. Leaf size=140 \[ -\frac{x^4}{32 c^3 \left (a^2 x^2+1\right )^2}+\frac{3}{32 a^4 c^3 \left (a^2 x^2+1\right )}+\frac{x^4 \tan ^{-1}(a x)^2}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac{x^3 \tan ^{-1}(a x)}{8 a c^3 \left (a^2 x^2+1\right )^2}+\frac{3 x \tan ^{-1}(a x)}{16 a^3 c^3 \left (a^2 x^2+1\right )}-\frac{3 \tan ^{-1}(a x)^2}{32 a^4 c^3} \]
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Rubi [A] time = 0.185193, antiderivative size = 140, normalized size of antiderivative = 1., 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 = {4944, 4938, 4934, 4884} \[ -\frac{x^4}{32 c^3 \left (a^2 x^2+1\right )^2}+\frac{3}{32 a^4 c^3 \left (a^2 x^2+1\right )}+\frac{x^4 \tan ^{-1}(a x)^2}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac{x^3 \tan ^{-1}(a x)}{8 a c^3 \left (a^2 x^2+1\right )^2}+\frac{3 x \tan ^{-1}(a x)}{16 a^3 c^3 \left (a^2 x^2+1\right )}-\frac{3 \tan ^{-1}(a x)^2}{32 a^4 c^3} \]
Antiderivative was successfully verified.
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Rule 4944
Rule 4938
Rule 4934
Rule 4884
Rubi steps
\begin{align*} \int \frac{x^3 \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx &=\frac{x^4 \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{1}{2} a \int \frac{x^4 \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^3} \, dx\\ &=-\frac{x^4}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac{x^3 \tan ^{-1}(a x)}{8 a c^3 \left (1+a^2 x^2\right )^2}+\frac{x^4 \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 \int \frac{x^2 \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{8 a c}\\ &=-\frac{x^4}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac{3}{32 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{x^3 \tan ^{-1}(a x)}{8 a c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)}{16 a^3 c^3 \left (1+a^2 x^2\right )}+\frac{x^4 \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 \int \frac{\tan ^{-1}(a x)}{c+a^2 c x^2} \, dx}{16 a^3 c^2}\\ &=-\frac{x^4}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac{3}{32 a^4 c^3 \left (1+a^2 x^2\right )}+\frac{x^3 \tan ^{-1}(a x)}{8 a c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)}{16 a^3 c^3 \left (1+a^2 x^2\right )}-\frac{3 \tan ^{-1}(a x)^2}{32 a^4 c^3}+\frac{x^4 \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}\\ \end{align*}
Mathematica [A] time = 0.0856415, size = 74, normalized size = 0.53 \[ \frac{5 a^2 x^2+2 a x \left (5 a^2 x^2+3\right ) \tan ^{-1}(a x)+\left (5 a^4 x^4-6 a^2 x^2-3\right ) \tan ^{-1}(a x)^2+4}{32 a^4 c^3 \left (a^2 x^2+1\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 154, normalized size = 1.1 \begin{align*}{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{2}}{4\,{c}^{3}{a}^{4} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}-{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{2}}{2\,{c}^{3}{a}^{4} \left ({a}^{2}{x}^{2}+1 \right ) }}+{\frac{5\,{x}^{3}\arctan \left ( ax \right ) }{16\,a{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{3\,\arctan \left ( ax \right ) x}{16\,{c}^{3}{a}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{5\, \left ( \arctan \left ( ax \right ) \right ) ^{2}}{32\,{c}^{3}{a}^{4}}}-{\frac{1}{32\,{c}^{3}{a}^{4} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{5}{32\,{c}^{3}{a}^{4} \left ({a}^{2}{x}^{2}+1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.58772, size = 250, normalized size = 1.79 \begin{align*} \frac{1}{16} \, a{\left (\frac{5 \, a^{2} x^{3} + 3 \, x}{a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}} + \frac{5 \, \arctan \left (a x\right )}{a^{5} c^{3}}\right )} \arctan \left (a x\right ) + \frac{{\left (5 \, a^{2} x^{2} - 5 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 4\right )} a^{2}}{32 \,{\left (a^{10} c^{3} x^{4} + 2 \, a^{8} c^{3} x^{2} + a^{6} c^{3}\right )}} - \frac{{\left (2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2}}{4 \,{\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.18483, size = 192, normalized size = 1.37 \begin{align*} \frac{5 \, a^{2} x^{2} +{\left (5 \, a^{4} x^{4} - 6 \, a^{2} x^{2} - 3\right )} \arctan \left (a x\right )^{2} + 2 \,{\left (5 \, a^{3} x^{3} + 3 \, a x\right )} \arctan \left (a x\right ) + 4}{32 \,{\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x^{3} \operatorname{atan}^{2}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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