Optimal. Leaf size=35 \[ -\frac{2 a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc (c+d x)}{d}+a^2 (-x) \]
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Rubi [A] time = 0.0721307, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3886, 3473, 8, 2606, 3767} \[ -\frac{2 a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc (c+d x)}{d}+a^2 (-x) \]
Antiderivative was successfully verified.
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Rule 3886
Rule 3473
Rule 8
Rule 2606
Rule 3767
Rubi steps
\begin{align*} \int \cot ^2(c+d x) (a+a \sec (c+d x))^2 \, dx &=\int \left (a^2 \cot ^2(c+d x)+2 a^2 \cot (c+d x) \csc (c+d x)+a^2 \csc ^2(c+d x)\right ) \, dx\\ &=a^2 \int \cot ^2(c+d x) \, dx+a^2 \int \csc ^2(c+d x) \, dx+\left (2 a^2\right ) \int \cot (c+d x) \csc (c+d x) \, dx\\ &=-\frac{a^2 \cot (c+d x)}{d}-a^2 \int 1 \, dx-\frac{a^2 \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}-\frac{\left (2 a^2\right ) \operatorname{Subst}(\int 1 \, dx,x,\csc (c+d x))}{d}\\ &=-a^2 x-\frac{2 a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc (c+d x)}{d}\\ \end{align*}
Mathematica [C] time = 0.0382712, size = 46, normalized size = 1.31 \[ -\frac{2 a^2 \cot \left (\frac{c}{2}+\frac{d x}{2}\right ) \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-\tan ^2\left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 50, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( -\cot \left ( dx+c \right ) -dx-c \right ) -2\,{\frac{{a}^{2}}{\sin \left ( dx+c \right ) }}-{a}^{2}\cot \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.66891, size = 65, normalized size = 1.86 \begin{align*} -\frac{{\left (d x + c + \frac{1}{\tan \left (d x + c\right )}\right )} a^{2} + \frac{2 \, a^{2}}{\sin \left (d x + c\right )} + \frac{a^{2}}{\tan \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.898445, size = 96, normalized size = 2.74 \begin{align*} -\frac{a^{2} d x \sin \left (d x + c\right ) + 2 \, a^{2} \cos \left (d x + c\right ) + 2 \, a^{2}}{d \sin \left (d x + c\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*} a^{2} \left (\int 2 \cot ^{2}{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int \cot ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \cot ^{2}{\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 = 1.36719, size = 42, normalized size = 1.2 \begin{align*} -\frac{{\left (d x + c\right )} a^{2} + \frac{2 \, a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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