Optimal. Leaf size=29 \[ \frac{\sin (a+b x)}{b}-\frac{\sin (a-c) \tanh ^{-1}(\cos (b x+c))}{b} \]
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Rubi [A] time = 0.0155587, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {4578, 2637, 3770} \[ \frac{\sin (a+b x)}{b}-\frac{\sin (a-c) \tanh ^{-1}(\cos (b x+c))}{b} \]
Antiderivative was successfully verified.
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Rule 4578
Rule 2637
Rule 3770
Rubi steps
\begin{align*} \int \cot (c+b x) \sin (a+b x) \, dx &=\sin (a-c) \int \csc (c+b x) \, dx+\int \cos (a+b x) \, dx\\ &=-\frac{\tanh ^{-1}(\cos (c+b x)) \sin (a-c)}{b}+\frac{\sin (a+b x)}{b}\\ \end{align*}
Mathematica [C] time = 0.0525759, size = 93, normalized size = 3.21 \[ -\frac{2 i \sin (a-c) \tan ^{-1}\left (\frac{(\cos (c)-i \sin (c)) \left (\cos (c) \cos \left (\frac{b x}{2}\right )-\sin (c) \sin \left (\frac{b x}{2}\right )\right )}{\sin (c) \cos \left (\frac{b x}{2}\right )+i \cos (c) \cos \left (\frac{b x}{2}\right )}\right )}{b}+\frac{\sin (a) \cos (b x)}{b}+\frac{\cos (a) \sin (b x)}{b} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.085, size = 95, normalized size = 3.3 \begin{align*}{\frac{-{\frac{i}{2}}{{\rm e}^{i \left ( bx+a \right ) }}}{b}}+{\frac{{\frac{i}{2}}{{\rm e}^{-i \left ( bx+a \right ) }}}{b}}+{\frac{\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}-{{\rm e}^{i \left ( a-c \right ) }} \right ) \sin \left ( a-c \right ) }{b}}-{\frac{\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}+{{\rm e}^{i \left ( a-c \right ) }} \right ) \sin \left ( a-c \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.16384, size = 142, normalized size = 4.9 \begin{align*} \frac{\log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right ) \sin \left (-a + c\right ) - \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right ) \sin \left (-a + c\right ) + 2 \, \sin \left (b x + a\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.53109, size = 531, normalized size = 18.31 \begin{align*} \frac{\frac{\sqrt{2} \log \left (\frac{2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) + \frac{2 \, \sqrt{2}{\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \cos \left (b x + a\right ) - \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right )\right )}}{\sqrt{\cos \left (-2 \, a + 2 \, c\right ) + 1}} - \cos \left (-2 \, a + 2 \, c\right ) + 3}{2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - \cos \left (-2 \, a + 2 \, c\right ) - 1}\right ) \sin \left (-2 \, a + 2 \, c\right )}{\sqrt{\cos \left (-2 \, a + 2 \, c\right ) + 1}} + 4 \, \sin \left (b x + a\right )}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin{\left (a + b x \right )} \cot{\left (b x + c \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19333, size = 305, normalized size = 10.52 \begin{align*} -\frac{2 \,{\left (\frac{{\left (\tan \left (\frac{1}{2} \, a\right )^{2} \tan \left (\frac{1}{2} \, c\right )^{2} - \tan \left (\frac{1}{2} \, a\right ) \tan \left (\frac{1}{2} \, c\right )^{3} + \tan \left (\frac{1}{2} \, a\right ) \tan \left (\frac{1}{2} \, c\right ) - \tan \left (\frac{1}{2} \, c\right )^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, b x\right ) \tan \left (\frac{1}{2} \, c\right ) - 1 \right |}\right )}{\tan \left (\frac{1}{2} \, a\right )^{2} \tan \left (\frac{1}{2} \, c\right )^{3} + \tan \left (\frac{1}{2} \, a\right )^{2} \tan \left (\frac{1}{2} \, c\right ) + \tan \left (\frac{1}{2} \, c\right )^{3} + \tan \left (\frac{1}{2} \, c\right )} - \frac{{\left (\tan \left (\frac{1}{2} \, a\right )^{2} \tan \left (\frac{1}{2} \, c\right ) - \tan \left (\frac{1}{2} \, a\right ) \tan \left (\frac{1}{2} \, c\right )^{2} + \tan \left (\frac{1}{2} \, a\right ) - \tan \left (\frac{1}{2} \, c\right )\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, b x\right ) + \tan \left (\frac{1}{2} \, c\right ) \right |}\right )}{\tan \left (\frac{1}{2} \, a\right )^{2} \tan \left (\frac{1}{2} \, c\right )^{2} + \tan \left (\frac{1}{2} \, a\right )^{2} + \tan \left (\frac{1}{2} \, c\right )^{2} + 1} + \frac{\tan \left (\frac{1}{2} \, b x\right ) \tan \left (\frac{1}{2} \, a\right )^{2} - \tan \left (\frac{1}{2} \, b x\right ) - 2 \, \tan \left (\frac{1}{2} \, a\right )}{{\left (\tan \left (\frac{1}{2} \, b x\right )^{2} + 1\right )}{\left (\tan \left (\frac{1}{2} \, a\right )^{2} + 1\right )}}\right )}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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