Optimal. Leaf size=16 \[ \frac {\cot ^{-1}(\cot (a+b x))^2}{2 b} \]
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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} \[ \frac {\cot ^{-1}(\cot (a+b x))^2}{2 b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2157
Rubi steps
\begin {align*} \int \cot ^{-1}(\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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.63, size = 10, normalized size = 0.62 \[ \frac {1}{2} x^{2} b + x a \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 10, normalized size = 0.62 \[ \frac {1}{2} \, b x^{2} + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 45, normalized size = 2.81 \[ \frac {-\left (\frac {\pi }{2}-\mathrm {arccot}\left (\cot \left (b x +a \right )\right )\right ) \mathrm {arccot}\left (\cot \left (b x +a \right )\right )-\frac {\left (\frac {\pi }{2}-\mathrm {arccot}\left (\cot \left (b x +a \right )\right )\right )^{2}}{2}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 10, normalized size = 0.62 \[ \frac {1}{2} \, b x^{2} + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.64, size = 16, normalized size = 1.00 \[ x\,\mathrm {acot}\left (\mathrm {cot}\left (a+b\,x\right )\right )-\frac {b\,x^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 19, normalized size = 1.19 \[ \begin {cases} \frac {\operatorname {acot}^{2}{\left (\cot {\left (a + b x \right )} \right )}}{2 b} & \text {for}\: b \neq 0 \\x \operatorname {acot}{\left (\cot {\relax (a )} \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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