Optimal. Leaf size=14 \[ -\frac {\tanh ^{-1}(\cos (a+b x))}{2 b} \]
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Rubi [A] time = 0.02, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4287, 3770} \[ -\frac {\tanh ^{-1}(\cos (a+b x))}{2 b} \]
Antiderivative was successfully verified.
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Rule 3770
Rule 4287
Rubi steps
\begin {align*} \int \cos (a+b x) \csc (2 a+2 b x) \, dx &=\frac {1}{2} \int \csc (a+b x) \, dx\\ &=-\frac {\tanh ^{-1}(\cos (a+b x))}{2 b}\\ \end {align*}
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Mathematica [B] time = 0.01, size = 42, normalized size = 3.00 \[ \frac {1}{2} \left (\frac {\log \left (\sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b}-\frac {\log \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.60, size = 30, normalized size = 2.14 \[ -\frac {\log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 16, normalized size = 1.14 \[ \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) \right |}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.55, size = 22, normalized size = 1.57 \[ \frac {\ln \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 84, normalized size = 6.00 \[ -\frac {\log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \relax (a) + \cos \relax (a)^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \relax (a) + \sin \relax (a)^{2}\right ) - \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \relax (a) + \cos \relax (a)^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \relax (a) + \sin \relax (a)^{2}\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 12, normalized size = 0.86 \[ -\frac {\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )}{2\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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