Optimal. Leaf size=58 \[ \frac{\left (3 a^2 c-d\right ) \log \left (a^2 x^2+1\right )}{6 a^3}+c x \cot ^{-1}(a x)+\frac{d x^2}{6 a}+\frac{1}{3} d x^3 \cot ^{-1}(a x) \]
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Rubi [A] time = 0.061042, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4913, 1593, 444, 43} \[ \frac{\left (3 a^2 c-d\right ) \log \left (a^2 x^2+1\right )}{6 a^3}+c x \cot ^{-1}(a x)+\frac{d x^2}{6 a}+\frac{1}{3} d x^3 \cot ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 4913
Rule 1593
Rule 444
Rule 43
Rubi steps
\begin{align*} \int \left (c+d x^2\right ) \cot ^{-1}(a x) \, dx &=c x \cot ^{-1}(a x)+\frac{1}{3} d x^3 \cot ^{-1}(a x)+a \int \frac{c x+\frac{d x^3}{3}}{1+a^2 x^2} \, dx\\ &=c x \cot ^{-1}(a x)+\frac{1}{3} d x^3 \cot ^{-1}(a x)+a \int \frac{x \left (c+\frac{d x^2}{3}\right )}{1+a^2 x^2} \, dx\\ &=c x \cot ^{-1}(a x)+\frac{1}{3} d x^3 \cot ^{-1}(a x)+\frac{1}{2} a \operatorname{Subst}\left (\int \frac{c+\frac{d x}{3}}{1+a^2 x} \, dx,x,x^2\right )\\ &=c x \cot ^{-1}(a x)+\frac{1}{3} d x^3 \cot ^{-1}(a x)+\frac{1}{2} a \operatorname{Subst}\left (\int \left (\frac{d}{3 a^2}+\frac{3 a^2 c-d}{3 a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{d x^2}{6 a}+c x \cot ^{-1}(a x)+\frac{1}{3} d x^3 \cot ^{-1}(a x)+\frac{\left (3 a^2 c-d\right ) \log \left (1+a^2 x^2\right )}{6 a^3}\\ \end{align*}
Mathematica [A] time = 0.0089964, size = 67, normalized size = 1.16 \[ \frac{c \log \left (a^2 x^2+1\right )}{2 a}-\frac{d \log \left (a^2 x^2+1\right )}{6 a^3}+c x \cot ^{-1}(a x)+\frac{d x^2}{6 a}+\frac{1}{3} d x^3 \cot ^{-1}(a x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 60, normalized size = 1. \begin{align*}{\frac{d{x}^{3}{\rm arccot} \left (ax\right )}{3}}+cx{\rm arccot} \left (ax\right )+{\frac{d{x}^{2}}{6\,a}}+{\frac{\ln \left ({a}^{2}{x}^{2}+1 \right ) c}{2\,a}}-{\frac{\ln \left ({a}^{2}{x}^{2}+1 \right ) d}{6\,{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.953438, size = 72, normalized size = 1.24 \begin{align*} \frac{1}{6} \, a{\left (\frac{d x^{2}}{a^{2}} + \frac{{\left (3 \, a^{2} c - d\right )} \log \left (a^{2} x^{2} + 1\right )}{a^{4}}\right )} + \frac{1}{3} \,{\left (d x^{3} + 3 \, c x\right )} \operatorname{arccot}\left (a x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.14745, size = 127, normalized size = 2.19 \begin{align*} \frac{a^{2} d x^{2} + 2 \,{\left (a^{3} d x^{3} + 3 \, a^{3} c x\right )} \operatorname{arccot}\left (a x\right ) +{\left (3 \, a^{2} c - d\right )} \log \left (a^{2} x^{2} + 1\right )}{6 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.836268, size = 73, normalized size = 1.26 \begin{align*} \begin{cases} c x \operatorname{acot}{\left (a x \right )} + \frac{d x^{3} \operatorname{acot}{\left (a x \right )}}{3} + \frac{c \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{2 a} + \frac{d x^{2}}{6 a} - \frac{d \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{6 a^{3}} & \text{for}\: a \neq 0 \\\frac{\pi \left (c x + \frac{d x^{3}}{3}\right )}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10448, size = 74, normalized size = 1.28 \begin{align*} \frac{d x^{2}}{6 \, a} + \frac{1}{3} \,{\left (d x^{3} + 3 \, c x\right )} \arctan \left (\frac{1}{a x}\right ) + \frac{{\left (3 \, a^{2} c - d\right )} \log \left (a^{2} x^{2} + 1\right )}{6 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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