Optimal. Leaf size=61 \[ \frac{1}{2 a d (\cos (c+d x)+1)}+\frac{\log (1-\cos (c+d x))}{4 a d}+\frac{3 \log (\cos (c+d x)+1)}{4 a d} \]
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Rubi [A] time = 0.0556855, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3879, 88} \[ \frac{1}{2 a d (\cos (c+d x)+1)}+\frac{\log (1-\cos (c+d x))}{4 a d}+\frac{3 \log (\cos (c+d x)+1)}{4 a d} \]
Antiderivative was successfully verified.
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Rule 3879
Rule 88
Rubi steps
\begin{align*} \int \frac{\cot (c+d x)}{a+a \sec (c+d x)} \, dx &=-\frac{a^2 \operatorname{Subst}\left (\int \frac{x^2}{(a-a x) (a+a x)^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a^2 \operatorname{Subst}\left (\int \left (-\frac{1}{4 a^3 (-1+x)}+\frac{1}{2 a^3 (1+x)^2}-\frac{3}{4 a^3 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac{1}{2 a d (1+\cos (c+d x))}+\frac{\log (1-\cos (c+d x))}{4 a d}+\frac{3 \log (1+\cos (c+d x))}{4 a d}\\ \end{align*}
Mathematica [A] time = 0.120026, size = 67, normalized size = 1.1 \[ \frac{\sec (c+d x) \left (2 \cos ^2\left (\frac{1}{2} (c+d x)\right ) \left (\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+3 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+1\right )}{2 a d (\sec (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.067, size = 54, normalized size = 0.9 \begin{align*}{\frac{1}{2\,da \left ( \cos \left ( dx+c \right ) +1 \right ) }}+{\frac{3\,\ln \left ( \cos \left ( dx+c \right ) +1 \right ) }{4\,da}}+{\frac{\ln \left ( -1+\cos \left ( dx+c \right ) \right ) }{4\,da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.25335, size = 63, normalized size = 1.03 \begin{align*} \frac{\frac{3 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a} + \frac{\log \left (\cos \left (d x + c\right ) - 1\right )}{a} + \frac{2}{a \cos \left (d x + c\right ) + a}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.16061, size = 182, normalized size = 2.98 \begin{align*} \frac{3 \,{\left (\cos \left (d x + c\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) +{\left (\cos \left (d x + c\right ) + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 2}{4 \,{\left (a d \cos \left (d x + c\right ) + a d\right )}} \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*} \frac{\int \frac{\cot{\left (c + d x \right )}}{\sec{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.41428, size = 116, normalized size = 1.9 \begin{align*} \frac{\frac{\log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a} - \frac{4 \, \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a} - \frac{\cos \left (d x + c\right ) - 1}{a{\left (\cos \left (d x + c\right ) + 1\right )}}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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