Optimal. Leaf size=105 \[ \frac{3}{16 a c^3 \left (a^2 x^2+1\right )}+\frac{1}{16 a c^3 \left (a^2 x^2+1\right )^2}+\frac{3 x \tan ^{-1}(a x)}{8 c^3 \left (a^2 x^2+1\right )}+\frac{x \tan ^{-1}(a x)}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac{3 \tan ^{-1}(a x)^2}{16 a c^3} \]
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Rubi [A] time = 0.0461152, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {4896, 4892, 261} \[ \frac{3}{16 a c^3 \left (a^2 x^2+1\right )}+\frac{1}{16 a c^3 \left (a^2 x^2+1\right )^2}+\frac{3 x \tan ^{-1}(a x)}{8 c^3 \left (a^2 x^2+1\right )}+\frac{x \tan ^{-1}(a x)}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac{3 \tan ^{-1}(a x)^2}{16 a c^3} \]
Antiderivative was successfully verified.
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Rule 4896
Rule 4892
Rule 261
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^3} \, dx &=\frac{1}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac{x \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c}\\ &=\frac{1}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac{x \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^2}{16 a c^3}-\frac{(3 a) \int \frac{x}{\left (c+a^2 c x^2\right )^2} \, dx}{8 c}\\ &=\frac{1}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac{3}{16 a c^3 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^2}{16 a c^3}\\ \end{align*}
Mathematica [A] time = 0.0256976, size = 68, normalized size = 0.65 \[ \frac{3 a^2 x^2+2 a x \left (3 a^2 x^2+5\right ) \tan ^{-1}(a x)+3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2+4}{16 a c^3 \left (a^2 x^2+1\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 96, normalized size = 0.9 \begin{align*}{\frac{1}{16\,a{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{3}{16\,a{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) }}+{\frac{x\arctan \left ( ax \right ) }{4\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{3\,x\arctan \left ( ax \right ) }{8\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) }}+{\frac{3\, \left ( \arctan \left ( ax \right ) \right ) ^{2}}{16\,a{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.6343, size = 174, normalized size = 1.66 \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 ) + \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}{16 \,{\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 = 1.63665, size = 189, normalized size = 1.8 \begin{align*} \frac{3 \, a^{2} x^{2} + 3 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 2 \,{\left (3 \, a^{3} x^{3} + 5 \, a x\right )} \arctan \left (a x\right ) + 4}{16 \,{\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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RecursionError} \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 )}{{\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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