Optimal. Leaf size=23 \[ \frac{1}{3} x^3 \tan ^{-1}(\cot (a+b x))+\frac{b x^4}{12} \]
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Rubi [A] time = 0.0083937, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2168, 30} \[ \frac{1}{3} x^3 \tan ^{-1}(\cot (a+b x))+\frac{b x^4}{12} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 30
Rubi steps
\begin{align*} \int x^2 \tan ^{-1}(\cot (a+b x)) \, dx &=\frac{1}{3} x^3 \tan ^{-1}(\cot (a+b x))+\frac{1}{3} b \int x^3 \, dx\\ &=\frac{b x^4}{12}+\frac{1}{3} x^3 \tan ^{-1}(\cot (a+b x))\\ \end{align*}
Mathematica [A] time = 0.0151806, size = 20, normalized size = 0.87 \[ \frac{1}{12} x^3 \left (4 \tan ^{-1}(\cot (a+b x))+b x\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.061, size = 65, normalized size = 2.8 \begin{align*}{\frac{\pi \,{x}^{3}}{6}}-{\frac{{x}^{3}{\rm arccot} \left (\cot \left ( bx+a \right ) \right )}{3}}-{\frac{1}{3\,{b}^{3}} \left ( -{\frac{ \left ( bx+a \right ) ^{4}}{4}}+a \left ( bx+a \right ) ^{3}-{\frac{3\,{a}^{2} \left ( bx+a \right ) ^{2}}{2}}+ \left ( bx+a \right ){a}^{3} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.964157, size = 23, normalized size = 1. \begin{align*} -\frac{1}{4} \, b x^{4} + \frac{1}{6} \,{\left (\pi - 2 \, a\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.07415, size = 45, normalized size = 1.96 \begin{align*} -\frac{1}{4} \, b x^{4} + \frac{1}{6} \,{\left (\pi - 2 \, a\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.782186, size = 68, normalized size = 2.96 \begin{align*} \begin{cases} \frac{\pi x^{3}}{6} - \frac{x^{2} \operatorname{acot}^{2}{\left (\cot{\left (a + b x \right )} \right )}}{2 b} + \frac{x \operatorname{acot}^{3}{\left (\cot{\left (a + b x \right )} \right )}}{3 b^{2}} - \frac{\operatorname{acot}^{4}{\left (\cot{\left (a + b x \right )} \right )}}{12 b^{3}} & \text{for}\: b \neq 0 \\\frac{x^{3} \left (- \operatorname{acot}{\left (\cot{\left (a \right )} \right )} + \frac{\pi }{2}\right )}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15063, size = 26, normalized size = 1.13 \begin{align*} -\frac{1}{4} \, b x^{4} + \frac{1}{6} \, \pi x^{3} - \frac{1}{3} \, a x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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