Optimal. Leaf size=49 \[ -\frac{\log \left (a^2 x^2+1\right )}{2 a^3 c}-\frac{\tan ^{-1}(a x)^2}{2 a^3 c}+\frac{x \tan ^{-1}(a x)}{a^2 c} \]
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Rubi [A] time = 0.072521, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4916, 4846, 260, 4884} \[ -\frac{\log \left (a^2 x^2+1\right )}{2 a^3 c}-\frac{\tan ^{-1}(a x)^2}{2 a^3 c}+\frac{x \tan ^{-1}(a x)}{a^2 c} \]
Antiderivative was successfully verified.
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Rule 4916
Rule 4846
Rule 260
Rule 4884
Rubi steps
\begin{align*} \int \frac{x^2 \tan ^{-1}(a x)}{c+a^2 c x^2} \, dx &=-\frac{\int \frac{\tan ^{-1}(a x)}{c+a^2 c x^2} \, dx}{a^2}+\frac{\int \tan ^{-1}(a x) \, dx}{a^2 c}\\ &=\frac{x \tan ^{-1}(a x)}{a^2 c}-\frac{\tan ^{-1}(a x)^2}{2 a^3 c}-\frac{\int \frac{x}{1+a^2 x^2} \, dx}{a c}\\ &=\frac{x \tan ^{-1}(a x)}{a^2 c}-\frac{\tan ^{-1}(a x)^2}{2 a^3 c}-\frac{\log \left (1+a^2 x^2\right )}{2 a^3 c}\\ \end{align*}
Mathematica [A] time = 0.0278609, size = 49, normalized size = 1. \[ -\frac{\log \left (a^2 x^2+1\right )}{2 a^3 c}-\frac{\tan ^{-1}(a x)^2}{2 a^3 c}+\frac{x \tan ^{-1}(a x)}{a^2 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 46, normalized size = 0.9 \begin{align*}{\frac{x\arctan \left ( ax \right ) }{{a}^{2}c}}-{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{2}}{2\,{a}^{3}c}}-{\frac{\ln \left ({a}^{2}{x}^{2}+1 \right ) }{2\,{a}^{3}c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.67045, size = 73, normalized size = 1.49 \begin{align*}{\left (\frac{x}{a^{2} c} - \frac{\arctan \left (a x\right )}{a^{3} c}\right )} \arctan \left (a x\right ) + \frac{\arctan \left (a x\right )^{2} - \log \left (a^{2} x^{2} + 1\right )}{2 \, a^{3} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70957, size = 92, normalized size = 1.88 \begin{align*} \frac{2 \, a x \arctan \left (a x\right ) - \arctan \left (a x\right )^{2} - \log \left (a^{2} x^{2} + 1\right )}{2 \, a^{3} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.41703, size = 75, normalized size = 1.53 \begin{align*} \begin{cases} \frac{x \operatorname{atan}{\left (a x \right )}}{a^{2} c} - \frac{\log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{2 a^{3} c} - \frac{\operatorname{atan}^{2}{\left (a x \right )}}{2 a^{3} c} & \text{for}\: c \neq 0 \\\tilde{\infty } \left (\frac{x^{3} \operatorname{atan}{\left (a x \right )}}{3} - \frac{x^{2}}{6 a} + \frac{\log{\left (a^{2} x^{2} + 1 \right )}}{6 a^{3}}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15817, size = 50, normalized size = 1.02 \begin{align*} \frac{2 \, a x \arctan \left (a x\right ) - \arctan \left (a x\right )^{2} - \log \left (a^{2} x^{2} + 1\right )}{2 \, a^{3} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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