Optimal. Leaf size=16 \[ -\frac {\cot ^{-1}(\tan (a+b x))^2}{2 b} \]
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Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, 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} \[ -\frac {\cot ^{-1}(\tan (a+b x))^2}{2 b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2157
Rubi steps
\begin {align*} \int \cot ^{-1}(\tan (a+b x)) \, dx &=-\frac {\operatorname {Subst}\left (\int x \, dx,x,\cot ^{-1}(\tan (a+b x))\right )}{b}\\ &=-\frac {\cot ^{-1}(\tan (a+b x))^2}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 1.12 \[ x \cot ^{-1}(\tan (a+b x))+\frac {b x^2}{2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 15, normalized size = 0.94 \[ -\frac {1}{2} \, b x^{2} + \frac {1}{2} \, {\left (\pi - 2 \, a\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 30, normalized size = 1.88 \[ -\frac {1}{2} \, b x^{2} + \pi x \left \lfloor \frac {b x + a}{\pi } + \frac {1}{2} \right \rfloor + \frac {1}{2} \, \pi x - a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 20, normalized size = 1.25 \[ \frac {\pi x}{2}-\frac {\arctan \left (\tan \left (b x +a \right )\right )^{2}}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 17, normalized size = 1.06 \[ \frac {1}{2} \, \pi x - \frac {{\left (b x + a\right )}^{2}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 21, normalized size = 1.31 \[ \frac {\Pi \,x}{2}+\frac {b\,x^2}{2}-x\,\mathrm {atan}\left (\mathrm {tan}\left (a+b\,x\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.14, size = 48, normalized size = 3.00 \[ \frac {\pi x}{2} - \begin {cases} \frac {\left (\operatorname {atan}{\left (\tan {\left (a + b x \right )} \right )} + \pi \left \lfloor {\frac {a + b x - \frac {\pi }{2}}{\pi }}\right \rfloor \right )^{2}}{2 b} & \text {for}\: b \neq 0 \\x \left (\operatorname {atan}{\left (\tan {\relax (a )} \right )} + \pi \left \lfloor {\frac {a - \frac {\pi }{2}}{\pi }}\right \rfloor \right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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