Optimal. Leaf size=19 \[ \log (x) \left (\tan ^{-1}(\cot (a+b x))+b x\right )-b x \]
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Rubi [A] time = 0.03, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2158, 29} \[ \log (x) \left (\tan ^{-1}(\cot (a+b x))+b x\right )-b x \]
Antiderivative was successfully verified.
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Rule 29
Rule 2158
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(\cot (a+b x))}{x} \, dx &=-b x-\left (-b x-\tan ^{-1}(\cot (a+b x))\right ) \int \frac {1}{x} \, dx\\ &=-b x+\left (b x+\tan ^{-1}(\cot (a+b x))\right ) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 19, normalized size = 1.00 \[ \log (x) \left (\tan ^{-1}(\cot (a+b x))+b x\right )-b x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 14, normalized size = 0.74 \[ -b x + \frac {1}{2} \, {\left (\pi - 2 \, a\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 15, normalized size = 0.79 \[ -b x + \frac {1}{2} \, {\left (\pi - 2 \, a\right )} \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.39, size = 35, normalized size = 1.84 \[ \frac {\pi \ln \relax (x )}{2}-b x -a \ln \relax (x )-\ln \relax (x ) \left (\mathrm {arccot}\left (\cot \left (b x +a \right )\right )-b x -a \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 14, normalized size = 0.74 \[ -b x + \frac {1}{2} \, {\left (\pi - 2 \, a\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {\frac {\Pi }{2}-\mathrm {acot}\left (\mathrm {cot}\left (a+b\,x\right )\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.57, size = 27, normalized size = 1.42 \[ b x \log {\relax (x )} - b x - \log {\relax (x )} \operatorname {acot}{\left (\cot {\left (a + b x \right )} \right )} + \frac {\pi \log {\relax (x )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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