Optimal. Leaf size=14 \[ \frac {\log (\sin (a+b x))}{2 b} \]
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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 = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4372, 3556}
\begin {gather*} \frac {\log (\sin (a+b x))}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 4372
Rubi steps
\begin {align*} \int \cos ^2(a+b x) \csc (2 a+2 b x) \, dx &=\frac {1}{2} \int \cot (a+b x) \, dx\\ &=\frac {\log (\sin (a+b x))}{2 b}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 22, normalized size = 1.57 \begin {gather*} \frac {\log (\cos (a+b x))+\log (\tan (a+b x))}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 13, normalized size = 0.93
method | result | size |
default | \(\frac {\ln \left (\sin \left (x b +a \right )\right )}{2 b}\) | \(13\) |
risch | \(-\frac {i x}{2}-\frac {i a}{b}+\frac {\ln \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )}{2 b}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 82 vs.
\(2 (12) = 24\).
time = 0.27, size = 82, normalized size = 5.86 \begin {gather*} \frac {\log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}\right ) + \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}\right )}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.82, size = 14, normalized size = 1.00 \begin {gather*} \frac {\log \left (\frac {1}{2} \, \sin \left (b x + a\right )\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 13, normalized size = 0.93 \begin {gather*} \frac {\log \left ({\left | \sin \left (b x + a\right ) \right |}\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.16, size = 14, normalized size = 1.00 \begin {gather*} \frac {\ln \left ({\sin \left (a+b\,x\right )}^2\right )}{4\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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