Optimal. Leaf size=81 \[ -\frac{1}{6} a^3 c^2 x^2-\frac{4}{3} a c^2 \log \left (a^2 x^2+1\right )+\frac{1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)+2 a^2 c^2 x \tan ^{-1}(a x)+a c^2 \log (x)-\frac{c^2 \tan ^{-1}(a x)}{x} \]
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Rubi [A] time = 0.116758, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {4948, 4846, 260, 4852, 266, 36, 29, 31, 43} \[ -\frac{1}{6} a^3 c^2 x^2-\frac{4}{3} a c^2 \log \left (a^2 x^2+1\right )+\frac{1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)+2 a^2 c^2 x \tan ^{-1}(a x)+a c^2 \log (x)-\frac{c^2 \tan ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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Rule 4948
Rule 4846
Rule 260
Rule 4852
Rule 266
Rule 36
Rule 29
Rule 31
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)}{x^2} \, dx &=\int \left (2 a^2 c^2 \tan ^{-1}(a x)+\frac{c^2 \tan ^{-1}(a x)}{x^2}+a^4 c^2 x^2 \tan ^{-1}(a x)\right ) \, dx\\ &=c^2 \int \frac{\tan ^{-1}(a x)}{x^2} \, dx+\left (2 a^2 c^2\right ) \int \tan ^{-1}(a x) \, dx+\left (a^4 c^2\right ) \int x^2 \tan ^{-1}(a x) \, dx\\ &=-\frac{c^2 \tan ^{-1}(a x)}{x}+2 a^2 c^2 x \tan ^{-1}(a x)+\frac{1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)+\left (a c^2\right ) \int \frac{1}{x \left (1+a^2 x^2\right )} \, dx-\left (2 a^3 c^2\right ) \int \frac{x}{1+a^2 x^2} \, dx-\frac{1}{3} \left (a^5 c^2\right ) \int \frac{x^3}{1+a^2 x^2} \, dx\\ &=-\frac{c^2 \tan ^{-1}(a x)}{x}+2 a^2 c^2 x \tan ^{-1}(a x)+\frac{1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)-a c^2 \log \left (1+a^2 x^2\right )+\frac{1}{2} \left (a c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )-\frac{1}{6} \left (a^5 c^2\right ) \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )\\ &=-\frac{c^2 \tan ^{-1}(a x)}{x}+2 a^2 c^2 x \tan ^{-1}(a x)+\frac{1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)-a c^2 \log \left (1+a^2 x^2\right )+\frac{1}{2} \left (a c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )-\frac{1}{2} \left (a^3 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+a^2 x} \, dx,x,x^2\right )-\frac{1}{6} \left (a^5 c^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^2}-\frac{1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{6} a^3 c^2 x^2-\frac{c^2 \tan ^{-1}(a x)}{x}+2 a^2 c^2 x \tan ^{-1}(a x)+\frac{1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)+a c^2 \log (x)-\frac{4}{3} a c^2 \log \left (1+a^2 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0513391, size = 62, normalized size = 0.77 \[ \frac{c^2 \left (2 \left (a^4 x^4+6 a^2 x^2-3\right ) \tan ^{-1}(a x)-a x \left (a^2 x^2+8 \log \left (a^2 x^2+1\right )-6 \log (x)\right )\right )}{6 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 78, normalized size = 1. \begin{align*}{\frac{{a}^{4}{c}^{2}{x}^{3}\arctan \left ( ax \right ) }{3}}+2\,{a}^{2}{c}^{2}x\arctan \left ( ax \right ) -{\frac{{c}^{2}\arctan \left ( ax \right ) }{x}}-{\frac{{c}^{2}{x}^{2}{a}^{3}}{6}}-{\frac{4\,a{c}^{2}\ln \left ({a}^{2}{x}^{2}+1 \right ) }{3}}+a{c}^{2}\ln \left ( ax \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.968203, size = 96, normalized size = 1.19 \begin{align*} -\frac{1}{6} \,{\left (a^{2} c^{2} x^{2} + 8 \, c^{2} \log \left (a^{2} x^{2} + 1\right ) - 6 \, c^{2} \log \left (x\right )\right )} a + \frac{1}{3} \,{\left (a^{4} c^{2} x^{3} + 6 \, a^{2} c^{2} x - \frac{3 \, c^{2}}{x}\right )} \arctan \left (a x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70552, size = 167, normalized size = 2.06 \begin{align*} -\frac{a^{3} c^{2} x^{3} + 8 \, a c^{2} x \log \left (a^{2} x^{2} + 1\right ) - 6 \, a c^{2} x \log \left (x\right ) - 2 \,{\left (a^{4} c^{2} x^{4} + 6 \, a^{2} c^{2} x^{2} - 3 \, c^{2}\right )} \arctan \left (a x\right )}{6 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.85078, size = 82, normalized size = 1.01 \begin{align*} \begin{cases} \frac{a^{4} c^{2} x^{3} \operatorname{atan}{\left (a x \right )}}{3} - \frac{a^{3} c^{2} x^{2}}{6} + 2 a^{2} c^{2} x \operatorname{atan}{\left (a x \right )} + a c^{2} \log{\left (x \right )} - \frac{4 a c^{2} \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{3} - \frac{c^{2} \operatorname{atan}{\left (a x \right )}}{x} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10871, size = 97, normalized size = 1.2 \begin{align*} -\frac{1}{6} \, a^{3} c^{2} x^{2} - \frac{4}{3} \, a c^{2} \log \left (a^{2} x^{2} + 1\right ) + \frac{1}{2} \, a c^{2} \log \left (x^{2}\right ) + \frac{1}{3} \,{\left (a^{4} c^{2} x^{3} + 6 \, a^{2} c^{2} x - \frac{3 \, c^{2}}{x}\right )} \arctan \left (a x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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