Optimal. Leaf size=34 \[ \frac{\tanh ^{-1}(\cos (a+b x))}{2 b}-\frac{\cot (a+b x) \csc (a+b x)}{2 b} \]
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Rubi [A] time = 0.0226236, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2611, 3770} \[ \frac{\tanh ^{-1}(\cos (a+b x))}{2 b}-\frac{\cot (a+b x) \csc (a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 2611
Rule 3770
Rubi steps
\begin{align*} \int \cot ^2(a+b x) \csc (a+b x) \, dx &=-\frac{\cot (a+b x) \csc (a+b x)}{2 b}-\frac{1}{2} \int \csc (a+b x) \, dx\\ &=\frac{\tanh ^{-1}(\cos (a+b x))}{2 b}-\frac{\cot (a+b x) \csc (a+b x)}{2 b}\\ \end{align*}
Mathematica [B] time = 0.0268574, size = 75, normalized size = 2.21 \[ -\frac{\csc ^2\left (\frac{1}{2} (a+b x)\right )}{8 b}+\frac{\sec ^2\left (\frac{1}{2} (a+b x)\right )}{8 b}-\frac{\log \left (\sin \left (\frac{1}{2} (a+b x)\right )\right )}{2 b}+\frac{\log \left (\cos \left (\frac{1}{2} (a+b x)\right )\right )}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 55, normalized size = 1.6 \begin{align*} -{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{3}}{2\,b \left ( \sin \left ( bx+a \right ) \right ) ^{2}}}-{\frac{\cos \left ( bx+a \right ) }{2\,b}}-{\frac{\ln \left ( \csc \left ( bx+a \right ) -\cot \left ( bx+a \right ) \right ) }{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.981926, size = 62, normalized size = 1.82 \begin{align*} \frac{\frac{2 \, \cos \left (b x + a\right )}{\cos \left (b x + a\right )^{2} - 1} + \log \left (\cos \left (b x + a\right ) + 1\right ) - \log \left (\cos \left (b x + a\right ) - 1\right )}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.57544, size = 200, normalized size = 5.88 \begin{align*} \frac{{\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2}\right ) -{\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2}\right ) + 2 \, \cos \left (b x + a\right )}{4 \,{\left (b \cos \left (b x + a\right )^{2} - b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.6412, size = 58, normalized size = 1.71 \begin{align*} \begin{cases} - \frac{\log{\left (\tan{\left (\frac{a}{2} + \frac{b x}{2} \right )} \right )}}{2 b} + \frac{\tan ^{2}{\left (\frac{a}{2} + \frac{b x}{2} \right )}}{8 b} - \frac{1}{8 b \tan ^{2}{\left (\frac{a}{2} + \frac{b x}{2} \right )}} & \text{for}\: b \neq 0 \\\frac{x \cos ^{2}{\left (a \right )}}{\sin ^{3}{\left (a \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17846, size = 126, normalized size = 3.71 \begin{align*} \frac{\frac{{\left (\frac{2 \,{\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + 1\right )}{\left (\cos \left (b x + a\right ) + 1\right )}}{\cos \left (b x + a\right ) - 1} - \frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 2 \, \log \left (\frac{{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{8 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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