Optimal. Leaf size=16 \[ \frac{\cot ^{-1}(\cot (a+b x))^2}{2 b} \]
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Rubi [A] time = 0.0128469, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2157, 30} \[ \frac{\cot ^{-1}(\cot (a+b x))^2}{2 b} \]
Antiderivative was successfully verified.
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Rule 2157
Rule 30
Rubi steps
\begin{align*} \int \cot ^{-1}(\cot (a+b x)) \, dx &=\frac{\operatorname{Subst}\left (\int x \, dx,x,\cot ^{-1}(\cot (a+b x))\right )}{b}\\ &=\frac{\cot ^{-1}(\cot (a+b x))^2}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0062584, size = 18, normalized size = 1.12 \[ x \cot ^{-1}(\cot (a+b x))-\frac{b x^2}{2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.049, size = 45, normalized size = 2.8 \begin{align*}{\frac{1}{b} \left ( - \left ({\frac{\pi }{2}}-{\rm arccot} \left (\cot \left ( bx+a \right ) \right ) \right ){\rm arccot} \left (\cot \left ( bx+a \right ) \right )-{\frac{1}{2} \left ({\frac{\pi }{2}}-{\rm arccot} \left (\cot \left ( bx+a \right ) \right ) \right ) ^{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.945519, size = 14, normalized size = 0.88 \begin{align*} \frac{1}{2} \, b x^{2} + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96274, size = 23, normalized size = 1.44 \begin{align*} \frac{1}{2} x^{2} b + x a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.179973, size = 15, normalized size = 0.94 \begin{align*} - \frac{b x^{2}}{2} + x \operatorname{acot}{\left (\cot{\left (a + b x \right )} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1046, size = 14, normalized size = 0.88 \begin{align*} \frac{1}{2} \, b x^{2} + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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