Optimal. Leaf size=16 \[ \frac{a \log (1-\cos (c+d x))}{d} \]
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Rubi [A] time = 0.0203078, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3879, 31} \[ \frac{a \log (1-\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3879
Rule 31
Rubi steps
\begin{align*} \int \cot (c+d x) (a+a \sec (c+d x)) \, dx &=-\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{a-a x} \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac{a \log (1-\cos (c+d x))}{d}\\ \end{align*}
Mathematica [A] time = 0.024625, size = 29, normalized size = 1.81 \[ \frac{2 a \left (\log \left (\tan \left (\frac{1}{2} (c+d x)\right )\right )+\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 29, normalized size = 1.8 \begin{align*}{\frac{a\ln \left ( -1+\sec \left ( dx+c \right ) \right ) }{d}}-{\frac{a\ln \left ( \sec \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 = 1.14439, size = 19, normalized size = 1.19 \begin{align*} \frac{a \log \left (\cos \left (d x + c\right ) - 1\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.847773, size = 46, normalized size = 2.88 \begin{align*} \frac{a \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{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 \cot{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int \cot{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.41107, size = 78, normalized size = 4.88 \begin{align*} \frac{a \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - a \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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