Optimal. Leaf size=23 \[ \frac{1}{2} x^2 \tan ^{-1}(\cot (a+b x))+\frac{b x^3}{6} \]
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Rubi [A] time = 0.0071134, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5173, 30} \[ \frac{1}{2} x^2 \tan ^{-1}(\cot (a+b x))+\frac{b x^3}{6} \]
Antiderivative was successfully verified.
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Rule 5173
Rule 30
Rubi steps
\begin{align*} \int x \tan ^{-1}(\cot (a+b x)) \, dx &=\frac{1}{2} x^2 \tan ^{-1}(\cot (a+b x))+\frac{1}{2} b \int x^2 \, dx\\ &=\frac{b x^3}{6}+\frac{1}{2} x^2 \tan ^{-1}(\cot (a+b x))\\ \end{align*}
Mathematica [A] time = 0.0142949, size = 20, normalized size = 0.87 \[ \frac{1}{6} x^2 \left (3 \tan ^{-1}(\cot (a+b x))+b x\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.057, size = 54, normalized size = 2.4 \begin{align*}{\frac{\pi \,{x}^{2}}{4}}-{\frac{{x}^{2}{\rm arccot} \left (\cot \left ( bx+a \right ) \right )}{2}}-{\frac{1}{2\,{b}^{2}} \left ( -{\frac{ \left ( bx+a \right ) ^{3}}{3}}+ \left ( bx+a \right ) ^{2}a-{a}^{2} \left ( bx+a \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.991631, size = 23, normalized size = 1. \begin{align*} -\frac{1}{3} \, b x^{3} + \frac{1}{4} \,{\left (\pi - 2 \, a\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.04841, size = 45, normalized size = 1.96 \begin{align*} -\frac{1}{3} \, b x^{3} + \frac{1}{4} \,{\left (\pi - 2 \, a\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.381749, size = 65, normalized size = 2.83 \begin{align*} \begin{cases} \frac{b x^{3}}{6} - \frac{x^{2} \operatorname{acot}{\left (\cot{\left (a + b x \right )} \right )}}{2} + \frac{\pi x \operatorname{acot}{\left (\cot{\left (a + b x \right )} \right )}}{2 b} - \frac{\pi \operatorname{acot}^{2}{\left (\cot{\left (a + b x \right )} \right )}}{4 b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{2} \left (- \operatorname{acot}{\left (\cot{\left (a \right )} \right )} + \frac{\pi }{2}\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 = 1.11006, size = 26, normalized size = 1.13 \begin{align*} -\frac{1}{3} \, b x^{3} + \frac{1}{4} \, \pi x^{2} - \frac{1}{2} \, a x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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