Optimal. Leaf size=55 \[ -\frac{3 \tanh ^{-1}(\cos (a+b x))}{8 b}-\frac{\cot ^3(a+b x) \csc (a+b x)}{4 b}+\frac{3 \cot (a+b x) \csc (a+b x)}{8 b} \]
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Rubi [A] time = 0.0421048, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2611, 3770} \[ -\frac{3 \tanh ^{-1}(\cos (a+b x))}{8 b}-\frac{\cot ^3(a+b x) \csc (a+b x)}{4 b}+\frac{3 \cot (a+b x) \csc (a+b x)}{8 b} \]
Antiderivative was successfully verified.
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Rule 2611
Rule 3770
Rubi steps
\begin{align*} \int \cot ^4(a+b x) \csc (a+b x) \, dx &=-\frac{\cot ^3(a+b x) \csc (a+b x)}{4 b}-\frac{3}{4} \int \cot ^2(a+b x) \csc (a+b x) \, dx\\ &=\frac{3 \cot (a+b x) \csc (a+b x)}{8 b}-\frac{\cot ^3(a+b x) \csc (a+b x)}{4 b}+\frac{3}{8} \int \csc (a+b x) \, dx\\ &=-\frac{3 \tanh ^{-1}(\cos (a+b x))}{8 b}+\frac{3 \cot (a+b x) \csc (a+b x)}{8 b}-\frac{\cot ^3(a+b x) \csc (a+b x)}{4 b}\\ \end{align*}
Mathematica [B] time = 0.031234, size = 113, normalized size = 2.05 \[ -\frac{\csc ^4\left (\frac{1}{2} (a+b x)\right )}{64 b}+\frac{5 \csc ^2\left (\frac{1}{2} (a+b x)\right )}{32 b}+\frac{\sec ^4\left (\frac{1}{2} (a+b x)\right )}{64 b}-\frac{5 \sec ^2\left (\frac{1}{2} (a+b x)\right )}{32 b}+\frac{3 \log \left (\sin \left (\frac{1}{2} (a+b x)\right )\right )}{8 b}-\frac{3 \log \left (\cos \left (\frac{1}{2} (a+b x)\right )\right )}{8 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 89, normalized size = 1.6 \begin{align*} -{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{5}}{4\,b \left ( \sin \left ( bx+a \right ) \right ) ^{4}}}+{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{5}}{8\,b \left ( \sin \left ( bx+a \right ) \right ) ^{2}}}+{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{3}}{8\,b}}+{\frac{3\,\cos \left ( bx+a \right ) }{8\,b}}+{\frac{3\,\ln \left ( \csc \left ( bx+a \right ) -\cot \left ( bx+a \right ) \right ) }{8\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.981819, size = 96, normalized size = 1.75 \begin{align*} -\frac{\frac{2 \,{\left (5 \, \cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )\right )}}{\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1} + 3 \, \log \left (\cos \left (b x + a\right ) + 1\right ) - 3 \, \log \left (\cos \left (b x + a\right ) - 1\right )}{16 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.14132, size = 315, normalized size = 5.73 \begin{align*} -\frac{10 \, \cos \left (b x + a\right )^{3} + 3 \,{\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \log \left (\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2}\right ) - 3 \,{\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2}\right ) - 6 \, \cos \left (b x + a\right )}{16 \,{\left (b \cos \left (b x + a\right )^{4} - 2 \, 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 = 4.94042, size = 92, normalized size = 1.67 \begin{align*} \begin{cases} \frac{3 \log{\left (\tan{\left (\frac{a}{2} + \frac{b x}{2} \right )} \right )}}{8 b} + \frac{\tan ^{4}{\left (\frac{a}{2} + \frac{b x}{2} \right )}}{64 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 )}} - \frac{1}{64 b \tan ^{4}{\left (\frac{a}{2} + \frac{b x}{2} \right )}} & \text{for}\: b \neq 0 \\\frac{x \cos ^{4}{\left (a \right )}}{\sin ^{5}{\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 = 1.18685, size = 188, normalized size = 3.42 \begin{align*} -\frac{\frac{{\left (\frac{8 \,{\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac{18 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 1\right )}{\left (\cos \left (b x + a\right ) + 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) - 1\right )}^{2}} - \frac{8 \,{\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac{{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} - 12 \, \log \left (\frac{{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{64 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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