Optimal. Leaf size=11 \[ \frac {\log (\sin (a+b x))}{b} \]
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Rubi [A] time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3475} \[ \frac {\log (\sin (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 3475
Rubi steps
\begin {align*} \int \cot (a+b x) \, dx &=\frac {\log (\sin (a+b x))}{b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 19, normalized size = 1.73 \[ \frac {\log (\tan (a+b x))+\log (\cos (a+b x))}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 27, normalized size = 2.45 \[ \frac {\log \left (\frac {\tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} + 1}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.34, size = 56, normalized size = 5.09 \[ \frac {\log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right ) - 2 \, \log \left ({\left | -\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1 \right |}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 29, normalized size = 2.64 \[ -\frac {\ln \left (\tan ^{2}\left (b x +a \right )+1\right )}{2 b}+\frac {\ln \left (\tan \left (b x +a \right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 11, normalized size = 1.00 \[ \frac {\log \left (\sin \left (b x + a\right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 28, normalized size = 2.55 \[ \frac {\ln \left (\mathrm {tan}\left (a+b\,x\right )\right )}{b}-\frac {\ln \left ({\mathrm {tan}\left (a+b\,x\right )}^2+1\right )}{2\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 29, normalized size = 2.64 \[ \begin {cases} - \frac {\log {\left (\tan ^{2}{\left (a + b x \right )} + 1 \right )}}{2 b} + \frac {\log {\left (\tan {\left (a + b x \right )} \right )}}{b} & \text {for}\: b \neq 0 \\\frac {x}{\tan {\relax (a )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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