Optimal. Leaf size=169 \[ -\frac{15 x}{64 c^3 \left (a^2 x^2+1\right )}-\frac{x}{32 c^3 \left (a^2 x^2+1\right )^2}+\frac{3 x \tan ^{-1}(a x)^2}{8 c^3 \left (a^2 x^2+1\right )}+\frac{x \tan ^{-1}(a x)^2}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac{3 \tan ^{-1}(a x)}{8 a c^3 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)}{8 a c^3 \left (a^2 x^2+1\right )^2}+\frac{\tan ^{-1}(a x)^3}{8 a c^3}-\frac{15 \tan ^{-1}(a x)}{64 a c^3} \]
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Rubi [A] time = 0.118644, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {4900, 4892, 4930, 199, 205} \[ -\frac{15 x}{64 c^3 \left (a^2 x^2+1\right )}-\frac{x}{32 c^3 \left (a^2 x^2+1\right )^2}+\frac{3 x \tan ^{-1}(a x)^2}{8 c^3 \left (a^2 x^2+1\right )}+\frac{x \tan ^{-1}(a x)^2}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac{3 \tan ^{-1}(a x)}{8 a c^3 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)}{8 a c^3 \left (a^2 x^2+1\right )^2}+\frac{\tan ^{-1}(a x)^3}{8 a c^3}-\frac{15 \tan ^{-1}(a x)}{64 a c^3} \]
Antiderivative was successfully verified.
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Rule 4900
Rule 4892
Rule 4930
Rule 199
Rule 205
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx &=\frac{\tan ^{-1}(a x)}{8 a c^3 \left (1+a^2 x^2\right )^2}+\frac{x \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{1}{8} \int \frac{1}{\left (c+a^2 c x^2\right )^3} \, dx+\frac{3 \int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c}\\ &=-\frac{x}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac{\tan ^{-1}(a x)}{8 a c^3 \left (1+a^2 x^2\right )^2}+\frac{x \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)^2}{8 c^3 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^3}{8 a c^3}-\frac{3 \int \frac{1}{\left (c+a^2 c x^2\right )^2} \, dx}{32 c}-\frac{(3 a) \int \frac{x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c}\\ &=-\frac{x}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 x}{64 c^3 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)}{8 a c^3 \left (1+a^2 x^2\right )^2}+\frac{3 \tan ^{-1}(a x)}{8 a c^3 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)^2}{8 c^3 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^3}{8 a c^3}-\frac{3 \int \frac{1}{c+a^2 c x^2} \, dx}{64 c^2}-\frac{3 \int \frac{1}{\left (c+a^2 c x^2\right )^2} \, dx}{8 c}\\ &=-\frac{x}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac{15 x}{64 c^3 \left (1+a^2 x^2\right )}-\frac{3 \tan ^{-1}(a x)}{64 a c^3}+\frac{\tan ^{-1}(a x)}{8 a c^3 \left (1+a^2 x^2\right )^2}+\frac{3 \tan ^{-1}(a x)}{8 a c^3 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)^2}{8 c^3 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^3}{8 a c^3}-\frac{3 \int \frac{1}{c+a^2 c x^2} \, dx}{16 c^2}\\ &=-\frac{x}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac{15 x}{64 c^3 \left (1+a^2 x^2\right )}-\frac{15 \tan ^{-1}(a x)}{64 a c^3}+\frac{\tan ^{-1}(a x)}{8 a c^3 \left (1+a^2 x^2\right )^2}+\frac{3 \tan ^{-1}(a x)}{8 a c^3 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)^2}{8 c^3 \left (1+a^2 x^2\right )}+\frac{\tan ^{-1}(a x)^3}{8 a c^3}\\ \end{align*}
Mathematica [A] time = 0.0447453, size = 98, normalized size = 0.58 \[ \frac{-a x \left (15 a^2 x^2+17\right )+8 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^3+8 a x \left (3 a^2 x^2+5\right ) \tan ^{-1}(a x)^2+\left (-15 a^4 x^4-6 a^2 x^2+17\right ) \tan ^{-1}(a x)}{64 a c^3 \left (a^2 x^2+1\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 159, normalized size = 0.9 \begin{align*}{\frac{x \left ( \arctan \left ( ax \right ) \right ) ^{2}}{4\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{3\,x \left ( \arctan \left ( ax \right ) \right ) ^{2}}{8\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) }}+{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{3}}{8\,a{c}^{3}}}+{\frac{\arctan \left ( ax \right ) }{8\,a{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{3\,\arctan \left ( ax \right ) }{8\,a{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) }}-{\frac{15\,{a}^{2}{x}^{3}}{64\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}-{\frac{17\,x}{64\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}-{\frac{15\,\arctan \left ( ax \right ) }{64\,a{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.67147, size = 313, normalized size = 1.85 \begin{align*} \frac{1}{8} \,{\left (\frac{3 \, a^{2} x^{3} + 5 \, x}{a^{4} c^{3} x^{4} + 2 \, a^{2} c^{3} x^{2} + c^{3}} + \frac{3 \, \arctan \left (a x\right )}{a c^{3}}\right )} \arctan \left (a x\right )^{2} - \frac{{\left (15 \, a^{3} x^{3} - 8 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} + 17 \, a x + 15 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a^{2}}{64 \,{\left (a^{7} c^{3} x^{4} + 2 \, a^{5} c^{3} x^{2} + a^{3} c^{3}\right )}} + \frac{{\left (3 \, a^{2} x^{2} - 3 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 4\right )} a \arctan \left (a x\right )}{8 \,{\left (a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.15604, size = 261, normalized size = 1.54 \begin{align*} -\frac{15 \, a^{3} x^{3} - 8 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} - 8 \,{\left (3 \, a^{3} x^{3} + 5 \, a x\right )} \arctan \left (a x\right )^{2} + 17 \, a x +{\left (15 \, a^{4} x^{4} + 6 \, a^{2} x^{2} - 17\right )} \arctan \left (a x\right )}{64 \,{\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a 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{\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{\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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