Optimal. Leaf size=14 \[ \frac {\tanh ^{-1}(\sin (a+b x))}{2 b} \]
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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 = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4288, 3770} \[ \frac {\tanh ^{-1}(\sin (a+b x))}{2 b} \]
Antiderivative was successfully verified.
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Rule 3770
Rule 4288
Rubi steps
\begin {align*} \int \csc (2 a+2 b x) \sin (a+b x) \, dx &=\frac {1}{2} \int \sec (a+b x) \, dx\\ &=\frac {\tanh ^{-1}(\sin (a+b x))}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 14, normalized size = 1.00 \[ \frac {\tanh ^{-1}(\sin (a+b x))}{2 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 28, normalized size = 2.00 \[ \frac {\log \left (\sin \left (b x + a\right ) + 1\right ) - \log \left (-\sin \left (b x + a\right ) + 1\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.80, size = 174, normalized size = 12.43 \[ \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, b x + 2 \, a\right ) \tan \left (\frac {1}{2} \, a\right )^{3} + 3 \, \tan \left (\frac {1}{2} \, b x + 2 \, a\right ) \tan \left (\frac {1}{2} \, a\right )^{2} - \tan \left (\frac {1}{2} \, a\right )^{3} - 3 \, \tan \left (\frac {1}{2} \, b x + 2 \, a\right ) \tan \left (\frac {1}{2} \, a\right ) + 3 \, \tan \left (\frac {1}{2} \, a\right )^{2} - \tan \left (\frac {1}{2} \, b x + 2 \, a\right ) + 3 \, \tan \left (\frac {1}{2} \, a\right ) - 1 \right |}\right ) - \log \left ({\left | \tan \left (\frac {1}{2} \, b x + 2 \, a\right ) \tan \left (\frac {1}{2} \, a\right )^{3} - 3 \, \tan \left (\frac {1}{2} \, b x + 2 \, a\right ) \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{3} - 3 \, \tan \left (\frac {1}{2} \, b x + 2 \, a\right ) \tan \left (\frac {1}{2} \, a\right ) + 3 \, \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, b x + 2 \, a\right ) - 3 \, \tan \left (\frac {1}{2} \, a\right ) - 1 \right |}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.68, size = 20, normalized size = 1.43 \[ \frac {\ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 115, normalized size = 8.21 \[ -\frac {\log \left (\frac {\cos \left (b x + 2 \, a\right )^{2} + \cos \relax (a)^{2} - 2 \, \cos \relax (a) \sin \left (b x + 2 \, a\right ) + \sin \left (b x + 2 \, a\right )^{2} + 2 \, \cos \left (b x + 2 \, a\right ) \sin \relax (a) + \sin \relax (a)^{2}}{\cos \left (b x + 2 \, a\right )^{2} + \cos \relax (a)^{2} + 2 \, \cos \relax (a) \sin \left (b x + 2 \, a\right ) + \sin \left (b x + 2 \, a\right )^{2} - 2 \, \cos \left (b x + 2 \, a\right ) \sin \relax (a) + \sin \relax (a)^{2}}\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 12, normalized size = 0.86 \[ \frac {\mathrm {atanh}\left (\sin \left (a+b\,x\right )\right )}{2\,b} \]
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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