Optimal. Leaf size=48 \[ -\frac{a^2 \cot (c+d x)}{d}+a^2 (-x)-\frac{2 a b \csc (c+d x)}{d}-\frac{b^2 \cot (c+d x)}{d} \]
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Rubi [A] time = 0.0746202, antiderivative size = 48, 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} \[ -\frac{a^2 \cot (c+d x)}{d}+a^2 (-x)-\frac{2 a b \csc (c+d x)}{d}-\frac{b^2 \cot (c+d x)}{d} \]
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+b \sec (c+d x))^2 \, dx &=\int \left (a^2 \cot ^2(c+d x)+2 a b \cot (c+d x) \csc (c+d x)+b^2 \csc ^2(c+d x)\right ) \, dx\\ &=a^2 \int \cot ^2(c+d x) \, dx+(2 a b) \int \cot (c+d x) \csc (c+d x) \, dx+b^2 \int \csc ^2(c+d x) \, dx\\ &=-\frac{a^2 \cot (c+d x)}{d}-a^2 \int 1 \, dx-\frac{(2 a b) \operatorname{Subst}(\int 1 \, dx,x,\csc (c+d x))}{d}-\frac{b^2 \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}\\ &=-a^2 x-\frac{a^2 \cot (c+d x)}{d}-\frac{b^2 \cot (c+d x)}{d}-\frac{2 a b \csc (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.366417, size = 39, normalized size = 0.81 \[ -\frac{\left (a^2+b^2\right ) \cot (c+d x)+a (a (c+d x)+2 b \csc (c+d x))}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 49, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( -\cot \left ( dx+c \right ) -dx-c \right ) -2\,{\frac{ab}{\sin \left ( dx+c \right ) }}-{b}^{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 = 1.46353, size = 63, normalized size = 1.31 \begin{align*} -\frac{{\left (d x + c + \frac{1}{\tan \left (d x + c\right )}\right )} a^{2} + \frac{2 \, a b}{\sin \left (d x + c\right )} + \frac{b^{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 = 0.852383, size = 104, normalized size = 2.17 \begin{align*} -\frac{a^{2} d x \sin \left (d x + c\right ) + 2 \, a b +{\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )}{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*} \int \left (a + b \sec{\left (c + d x \right )}\right )^{2} \cot ^{2}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36982, size = 108, normalized size = 2.25 \begin{align*} -\frac{2 \,{\left (d x + c\right )} a^{2} - a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{a^{2} + 2 \, a b + b^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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