Optimal. Leaf size=21 \[ b x-\log (x) \left (b x-\tan ^{-1}(\tan (a+b x))\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 21, 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} \[ b x-\log (x) \left (b x-\tan ^{-1}(\tan (a+b x))\right ) \]
Antiderivative was successfully verified.
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Rule 29
Rule 2158
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(\tan (a+b x))}{x} \, dx &=b x-\left (b x-\tan ^{-1}(\tan (a+b x))\right ) \int \frac {1}{x} \, dx\\ &=b x-\left (b x-\tan ^{-1}(\tan (a+b x))\right ) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 19, normalized size = 0.90 \[ \log (x) \left (\tan ^{-1}(\tan (a+b x))-b x\right )+b x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 8, normalized size = 0.38 \[ b x + a \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 21, normalized size = 1.00 \[ \ln \relax (x ) \arctan \left (\tan \left (b x +a \right )\right )-\ln \relax (x ) x b +b x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 42, normalized size = 2.00 \[ \frac {b \arctan \left (\tan \left (b x + a\right )\right ) \log \left (b x\right ) + {\left (b x - {\left (b x + a\right )} \log \left (b x\right ) + a \log \left (b x\right ) + a\right )} b}{b} \]
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 \[ \int \frac {\mathrm {atan}\left (\mathrm {tan}\left (a+b\,x\right )\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.93, size = 34, normalized size = 1.62 \[ - b x \log {\relax (x )} + b x + \left (\operatorname {atan}{\left (\tan {\left (a + b x \right )} \right )} + \pi \left \lfloor {\frac {a + b x - \frac {\pi }{2}}{\pi }}\right \rfloor \right ) \log {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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