Optimal. Leaf size=23 \[ \frac{1}{3} x^3 \tan ^{-1}(\tan (a+b x))-\frac{b x^4}{12} \]
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Rubi [A] time = 0.008892, 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} \[ \frac{1}{3} x^3 \tan ^{-1}(\tan (a+b x))-\frac{b x^4}{12} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 30
Rubi steps
\begin{align*} \int x^2 \tan ^{-1}(\tan (a+b x)) \, dx &=\frac{1}{3} x^3 \tan ^{-1}(\tan (a+b x))-\frac{1}{3} b \int x^3 \, dx\\ &=-\frac{b x^4}{12}+\frac{1}{3} x^3 \tan ^{-1}(\tan (a+b x))\\ \end{align*}
Mathematica [A] time = 0.0162567, size = 20, normalized size = 0.87 \[ -\frac{1}{12} x^3 \left (b x-4 \tan ^{-1}(\tan (a+b x))\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 20, normalized size = 0.9 \begin{align*} -{\frac{b{x}^{4}}{12}}+{\frac{{x}^{3}\arctan \left ( \tan \left ( bx+a \right ) \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.989099, size = 109, normalized size = 4.74 \begin{align*} \frac{\frac{4 \,{\left ({\left (b x + a\right )}^{3} - 3 \,{\left (b x + a\right )}^{2} a + 3 \,{\left (b x + a\right )} a^{2}\right )} \arctan \left (\tan \left (b x + a\right )\right )}{b^{2}} - \frac{{\left (b x + a\right )}^{4} - 4 \,{\left (b x + a\right )}^{3} a + 6 \,{\left (b x + a\right )}^{2} a^{2}}{b^{2}}}{12 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70805, size = 31, normalized size = 1.35 \begin{align*} \frac{1}{4} \, b x^{4} + \frac{1}{3} \, a x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.775372, size = 109, normalized size = 4.74 \begin{align*} \begin{cases} \frac{x^{2} \left (\operatorname{atan}{\left (\tan{\left (a + b x \right )} \right )} + \pi \left \lfloor{\frac{a + b x - \frac{\pi }{2}}{\pi }}\right \rfloor \right )^{2}}{2 b} - \frac{x \left (\operatorname{atan}{\left (\tan{\left (a + b x \right )} \right )} + \pi \left \lfloor{\frac{a + b x - \frac{\pi }{2}}{\pi }}\right \rfloor \right )^{3}}{3 b^{2}} + \frac{\left (\operatorname{atan}{\left (\tan{\left (a + b x \right )} \right )} + \pi \left \lfloor{\frac{a + b x - \frac{\pi }{2}}{\pi }}\right \rfloor \right )^{4}}{12 b^{3}} & \text{for}\: b \neq 0 \\\frac{x^{3} \left (\operatorname{atan}{\left (\tan{\left (a \right )} \right )} + \pi \left \lfloor{\frac{a - \frac{\pi }{2}}{\pi }}\right \rfloor \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 = 1.15182, size = 36, normalized size = 1.57 \begin{align*} \frac{1}{4} \, b x^{4} - \frac{1}{3} \, \pi x^{3} \left \lfloor \frac{a}{\pi } + \frac{1}{2} \right \rfloor + \frac{1}{3} \, a x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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