Optimal. Leaf size=27 \[ \frac {\log (\sin (a+b x))}{b}-\frac {\sin ^2(a+b x)}{2 b} \]
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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2670, 14}
\begin {gather*} \frac {\log (\sin (a+b x))}{b}-\frac {\sin ^2(a+b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2670
Rubi steps
\begin {align*} \int \cos ^2(a+b x) \cot (a+b x) \, dx &=\frac {\text {Subst}\left (\int \frac {1-x^2}{x} \, dx,x,-\sin (a+b x)\right )}{b}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{x}-x\right ) \, dx,x,-\sin (a+b x)\right )}{b}\\ &=\frac {\log (\sin (a+b x))}{b}-\frac {\sin ^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 27, normalized size = 1.00 \begin {gather*} \frac {\log (\sin (a+b x))}{b}-\frac {\sin ^2(a+b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 23, normalized size = 0.85
method | result | size |
derivativedivides | \(\frac {\frac {\left (\cos ^{2}\left (b x +a \right )\right )}{2}+\ln \left (\sin \left (b x +a \right )\right )}{b}\) | \(23\) |
default | \(\frac {\frac {\left (\cos ^{2}\left (b x +a \right )\right )}{2}+\ln \left (\sin \left (b x +a \right )\right )}{b}\) | \(23\) |
risch | \(-i x +\frac {{\mathrm e}^{2 i \left (b x +a \right )}}{8 b}+\frac {{\mathrm e}^{-2 i \left (b x +a \right )}}{8 b}-\frac {2 i a}{b}+\frac {\ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}{b}\) | \(57\) |
norman | \(-\frac {2 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b \left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{2}}+\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}-\frac {\ln \left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 25, normalized size = 0.93 \begin {gather*} -\frac {\sin \left (b x + a\right )^{2} - \log \left (\sin \left (b x + a\right )^{2}\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 25, normalized size = 0.93 \begin {gather*} \frac {\cos \left (b x + a\right )^{2} + 2 \, \log \left (\frac {1}{2} \, \sin \left (b x + a\right )\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 369 vs.
\(2 (20) = 40\).
time = 0.67, size = 369, normalized size = 13.67 \begin {gather*} \begin {cases} - \frac {\log {\left (\tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 1 \right )} \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )}}{b \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 2 b \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + b} - \frac {2 \log {\left (\tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 1 \right )} \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )}}{b \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 2 b \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + b} - \frac {\log {\left (\tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 1 \right )}}{b \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 2 b \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + b} + \frac {\log {\left (\tan {\left (\frac {a}{2} + \frac {b x}{2} \right )} \right )} \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )}}{b \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 2 b \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + b} + \frac {2 \log {\left (\tan {\left (\frac {a}{2} + \frac {b x}{2} \right )} \right )} \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )}}{b \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 2 b \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + b} + \frac {\log {\left (\tan {\left (\frac {a}{2} + \frac {b x}{2} \right )} \right )}}{b \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 2 b \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + b} - \frac {2 \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )}}{b \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + 2 b \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )} + b} & \text {for}\: b \neq 0 \\\frac {x \cos ^{3}{\left (a \right )}}{\sin {\left (a \right )}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.00, size = 25, normalized size = 0.93 \begin {gather*} -\frac {\sin \left (b x + a\right )^{2} - \log \left (\sin \left (b x + a\right )^{2}\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.41, size = 35, normalized size = 1.30 \begin {gather*} \frac {\frac {{\cos \left (a+b\,x\right )}^2}{2}-\frac {\ln \left ({\mathrm {tan}\left (a+b\,x\right )}^2+1\right )}{2}+\ln \left (\mathrm {tan}\left (a+b\,x\right )\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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