Optimal. Leaf size=23 \[ \frac{1}{2} x^2 \tan ^{-1}(\tan (a+b x))-\frac{b x^3}{6} \]
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Rubi [A] time = 0.0070475, 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 = {5171, 30} \[ \frac{1}{2} x^2 \tan ^{-1}(\tan (a+b x))-\frac{b x^3}{6} \]
Antiderivative was successfully verified.
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Rule 5171
Rule 30
Rubi steps
\begin{align*} \int x \tan ^{-1}(\tan (a+b x)) \, dx &=\frac{1}{2} x^2 \tan ^{-1}(\tan (a+b x))-\frac{1}{2} b \int x^2 \, dx\\ &=-\frac{b x^3}{6}+\frac{1}{2} x^2 \tan ^{-1}(\tan (a+b x))\\ \end{align*}
Mathematica [A] time = 0.0148217, size = 20, normalized size = 0.87 \[ -\frac{1}{6} x^2 \left (b x-3 \tan ^{-1}(\tan (a+b x))\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 20, normalized size = 0.9 \begin{align*} -{\frac{b{x}^{3}}{6}}+{\frac{{x}^{2}\arctan \left ( \tan \left ( bx+a \right ) \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.00108, size = 77, normalized size = 3.35 \begin{align*} \frac{\frac{3 \,{\left ({\left (b x + a\right )}^{2} - 2 \,{\left (b x + a\right )} a\right )} \arctan \left (\tan \left (b x + a\right )\right )}{b} - \frac{{\left (b x + a\right )}^{3} - 3 \,{\left (b x + a\right )}^{2} a}{b}}{6 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74936, size = 31, normalized size = 1.35 \begin{align*} \frac{1}{3} \, b x^{3} + \frac{1}{2} \, a x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.241458, size = 32, normalized size = 1.39 \begin{align*} - \frac{b x^{3}}{6} + \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11709, size = 36, normalized size = 1.57 \begin{align*} \frac{1}{3} \, b x^{3} - \frac{1}{2} \, \pi x^{2} \left \lfloor \frac{a}{\pi } + \frac{1}{2} \right \rfloor + \frac{1}{2} \, a x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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