Integrand size = 7, antiderivative size = 16 \[ \int \cot ^{-1}(\cot (a+b x)) \, dx=\frac {\cot ^{-1}(\cot (a+b x))^2}{2 b} \]
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Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2188, 30} \[ \int \cot ^{-1}(\cot (a+b x)) \, dx=\frac {\cot ^{-1}(\cot (a+b x))^2}{2 b} \]
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Rule 30
Rule 2188
Rubi steps \begin{align*} \text {integral}& = \frac {\text {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*}
Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \cot ^{-1}(\cot (a+b x)) \, dx=-\frac {b x^2}{2}+x \cot ^{-1}(\cot (a+b x)) \]
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Time = 0.65 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06
method | result | size |
parallelrisch | \(-\frac {x^{2} b}{2}+x \,\operatorname {arccot}\left (\cot \left (b x +a \right )\right )\) | \(17\) |
parts | \(x \,\operatorname {arccot}\left (\cot \left (b x +a \right )\right )+\frac {-\frac {\left (b x +a \right )^{2}}{2}+\left (b x +a \right ) a}{b}\) | \(32\) |
derivativedivides | \(\frac {-\left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (b x +a \right )\right )\right ) \operatorname {arccot}\left (\cot \left (b x +a \right )\right )-\frac {\left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (b x +a \right )\right )\right )^{2}}{2}}{b}\) | \(45\) |
default | \(\frac {-\left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (b x +a \right )\right )\right ) \operatorname {arccot}\left (\cot \left (b x +a \right )\right )-\frac {\left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (b x +a \right )\right )\right )^{2}}{2}}{b}\) | \(45\) |
risch | \(-i x \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )-\frac {\pi x \operatorname {csgn}\left (i {\mathrm e}^{i \left (b x +a \right )}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{2 i \left (b x +a \right )}\right )}{4}+\frac {\pi x \,\operatorname {csgn}\left (i {\mathrm e}^{i \left (b x +a \right )}\right ) \operatorname {csgn}\left (i {\mathrm e}^{2 i \left (b x +a \right )}\right )^{2}}{2}-\frac {\pi x \operatorname {csgn}\left (i {\mathrm e}^{2 i \left (b x +a \right )}\right )^{3}}{4}-\frac {\pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 i \left (b x +a \right )}\right ) \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 i \left (b x +a \right )}-1}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 i \left (b x +a \right )}}{{\mathrm e}^{2 i \left (b x +a \right )}-1}\right ) x}{4}+\frac {\pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 i \left (b x +a \right )}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 i \left (b x +a \right )}}{{\mathrm e}^{2 i \left (b x +a \right )}-1}\right )^{2} x}{4}+\frac {\pi \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 i \left (b x +a \right )}-1}\right )^{2} x}{2}-\frac {\pi \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 i \left (b x +a \right )}-1}\right )^{3} x}{2}+\frac {\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 i \left (b x +a \right )}-1}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 i \left (b x +a \right )}}{{\mathrm e}^{2 i \left (b x +a \right )}-1}\right )^{2} x}{4}-\frac {\pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 i \left (b x +a \right )}}{{\mathrm e}^{2 i \left (b x +a \right )}-1}\right )^{3} x}{4}-\frac {x^{2} b}{2}\) | \(337\) |
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Time = 0.21 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.62 \[ \int \cot ^{-1}(\cot (a+b x)) \, dx=\frac {1}{2} x^{2} b + x a \]
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Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \cot ^{-1}(\cot (a+b x)) \, dx=\begin {cases} \frac {\operatorname {acot}^{2}{\left (\cot {\left (a + b x \right )} \right )}}{2 b} & \text {for}\: b \neq 0 \\x \operatorname {acot}{\left (\cot {\left (a \right )} \right )} & \text {otherwise} \end {cases} \]
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Time = 0.17 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.62 \[ \int \cot ^{-1}(\cot (a+b x)) \, dx=\frac {1}{2} \, b x^{2} + a x \]
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Time = 0.27 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.62 \[ \int \cot ^{-1}(\cot (a+b x)) \, dx=\frac {1}{2} \, b x^{2} + a x \]
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Time = 0.80 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \cot ^{-1}(\cot (a+b x)) \, dx=x\,\mathrm {acot}\left (\mathrm {cot}\left (a+b\,x\right )\right )-\frac {b\,x^2}{2} \]
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