Optimal. Leaf size=55 \[ \frac {\tanh ^{-1}(\cos (a+b x))}{8 b}-\frac {\cot (a+b x) \csc ^3(a+b x)}{4 b}+\frac {\cot (a+b x) \csc (a+b x)}{8 b} \]
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Rubi [A] time = 0.05, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2611, 3768, 3770} \[ \frac {\tanh ^{-1}(\cos (a+b x))}{8 b}-\frac {\cot (a+b x) \csc ^3(a+b x)}{4 b}+\frac {\cot (a+b x) \csc (a+b x)}{8 b} \]
Antiderivative was successfully verified.
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Rule 2611
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \cot ^2(a+b x) \csc ^3(a+b x) \, dx &=-\frac {\cot (a+b x) \csc ^3(a+b x)}{4 b}-\frac {1}{4} \int \csc ^3(a+b x) \, dx\\ &=\frac {\cot (a+b x) \csc (a+b x)}{8 b}-\frac {\cot (a+b x) \csc ^3(a+b x)}{4 b}-\frac {1}{8} \int \csc (a+b x) \, dx\\ &=\frac {\tanh ^{-1}(\cos (a+b x))}{8 b}+\frac {\cot (a+b x) \csc (a+b x)}{8 b}-\frac {\cot (a+b x) \csc ^3(a+b x)}{4 b}\\ \end {align*}
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Mathematica [B] time = 0.03, size = 113, normalized size = 2.05 \[ -\frac {\csc ^4\left (\frac {1}{2} (a+b x)\right )}{64 b}+\frac {\csc ^2\left (\frac {1}{2} (a+b x)\right )}{32 b}+\frac {\sec ^4\left (\frac {1}{2} (a+b x)\right )}{64 b}-\frac {\sec ^2\left (\frac {1}{2} (a+b x)\right )}{32 b}-\frac {\log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )}{8 b}+\frac {\log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )}{8 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 111, normalized size = 2.02 \[ -\frac {2 \, \cos \left (b x + a\right )^{3} - {\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + {\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + 2 \, \cos \left (b x + a\right )}{16 \, {\left (b \cos \left (b x + a\right )^{4} - 2 \, b \cos \left (b x + a\right )^{2} + b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 98, normalized size = 1.78 \[ \frac {\frac {{\left (\frac {2 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (b x + a\right ) + 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) - 1\right )}^{2}} + \frac {{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} - 4 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{64 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 76, normalized size = 1.38 \[ -\frac {\cos ^{3}\left (b x +a \right )}{4 b \sin \left (b x +a \right )^{4}}-\frac {\cos ^{3}\left (b x +a \right )}{8 b \sin \left (b x +a \right )^{2}}-\frac {\cos \left (b x +a \right )}{8 b}-\frac {\ln \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{8 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 65, normalized size = 1.18 \[ -\frac {\frac {2 \, {\left (\cos \left (b x + a\right )^{3} + \cos \left (b x + a\right )\right )}}{\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1} - \log \left (\cos \left (b x + a\right ) + 1\right ) + \log \left (\cos \left (b x + a\right ) - 1\right )}{16 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.47, size = 48, normalized size = 0.87 \[ \frac {{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^4}{64\,b}-\frac {1}{64\,b\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^4}-\frac {\ln \left (\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )\right )}{8\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.95, size = 58, normalized size = 1.05 \[ \begin {cases} - \frac {\log {\left (\tan {\left (\frac {a}{2} + \frac {b x}{2} \right )} \right )}}{8 b} + \frac {\tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )}}{64 b} - \frac {1}{64 b \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )}} & \text {for}\: b \neq 0 \\\frac {x \cos ^{2}{\relax (a )}}{\sin ^{5}{\relax (a )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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