Optimal. Leaf size=40 \[ -\frac{a^2}{d (1-\cos (c+d x))}-\frac{a^2 \log (1-\cos (c+d x))}{d} \]
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Rubi [A] time = 0.0496429, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3879, 43} \[ -\frac{a^2}{d (1-\cos (c+d x))}-\frac{a^2 \log (1-\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3879
Rule 43
Rubi steps
\begin{align*} \int \cot ^3(c+d x) (a+a \sec (c+d x))^2 \, dx &=-\frac{a^4 \operatorname{Subst}\left (\int \frac{x}{(a-a x)^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a^4 \operatorname{Subst}\left (\int \left (\frac{1}{a^2 (-1+x)^2}+\frac{1}{a^2 (-1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a^2}{d (1-\cos (c+d x))}-\frac{a^2 \log (1-\cos (c+d x))}{d}\\ \end{align*}
Mathematica [A] time = 0.0662764, size = 56, normalized size = 1.4 \[ \frac{a^2 \csc ^2\left (\frac{1}{2} (c+d x)\right ) \left (-2 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+2 \cos (c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-1\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 51, normalized size = 1.3 \begin{align*} -{\frac{{a}^{2}}{d \left ( -1+\sec \left ( dx+c \right ) \right ) }}-{\frac{{a}^{2}\ln \left ( -1+\sec \left ( dx+c \right ) \right ) }{d}}+{\frac{{a}^{2}\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 = 1.12142, size = 46, normalized size = 1.15 \begin{align*} -\frac{a^{2} \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac{a^{2}}{\cos \left (d x + c\right ) - 1}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.19576, size = 113, normalized size = 2.82 \begin{align*} \frac{a^{2} -{\left (a^{2} \cos \left (d x + c\right ) - a^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{d \cos \left (d x + c\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^{2} \left (\int 2 \cot ^{3}{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int \cot ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \cot ^{3}{\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 = 1.47374, size = 150, normalized size = 3.75 \begin{align*} -\frac{2 \, a^{2} \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 2 \, a^{2} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - \frac{{\left (a^{2} + \frac{2 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}}{\cos \left (d x + c\right ) - 1}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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