Optimal. Leaf size=51 \[ \frac{a \cos (c+d x)}{d}+\frac{a \sin (c+d x) \cos (c+d x)}{2 d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a x}{2} \]
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Rubi [A] time = 0.0637439, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {2838, 2592, 321, 206, 2635, 8} \[ \frac{a \cos (c+d x)}{d}+\frac{a \sin (c+d x) \cos (c+d x)}{2 d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a x}{2} \]
Antiderivative was successfully verified.
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Rule 2838
Rule 2592
Rule 321
Rule 206
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cos ^2(c+d x) \, dx+a \int \cos (c+d x) \cot (c+d x) \, dx\\ &=\frac{a \cos (c+d x) \sin (c+d x)}{2 d}+\frac{1}{2} a \int 1 \, dx-\frac{a \operatorname{Subst}\left (\int \frac{x^2}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac{a x}{2}+\frac{a \cos (c+d x)}{d}+\frac{a \cos (c+d x) \sin (c+d x)}{2 d}-\frac{a \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac{a x}{2}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a \cos (c+d x)}{d}+\frac{a \cos (c+d x) \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0717706, size = 74, normalized size = 1.45 \[ \frac{a (c+d x)}{2 d}+\frac{a \sin (2 (c+d x))}{4 d}+\frac{a \cos (c+d x)}{d}+\frac{a \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{d}-\frac{a \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 63, normalized size = 1.2 \begin{align*}{\frac{\cos \left ( dx+c \right ) a\sin \left ( dx+c \right ) }{2\,d}}+{\frac{ax}{2}}+{\frac{ca}{2\,d}}+{\frac{\cos \left ( dx+c \right ) a}{d}}+{\frac{a\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.23225, size = 77, normalized size = 1.51 \begin{align*} \frac{{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a + 2 \, a{\left (2 \, \cos \left (d x + c\right ) - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71267, size = 174, normalized size = 3.41 \begin{align*} \frac{a d x + a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 2 \, a \cos \left (d x + c\right ) - a \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + a \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{2 \, d} \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*} a \left (\int \cos ^{2}{\left (c + d x \right )} \csc{\left (c + d x \right )}\, dx + \int \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} \csc{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23311, size = 117, normalized size = 2.29 \begin{align*} \frac{{\left (d x + c\right )} a + 2 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - \frac{2 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 2 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, a\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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