Optimal. Leaf size=28 \[ \frac {\tanh ^{-1}(\sin (a+b x))}{4 b}-\frac {\csc (a+b x)}{4 b} \]
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Rubi [A] time = 0.04, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4287, 2621, 321, 207} \[ \frac {\tanh ^{-1}(\sin (a+b x))}{4 b}-\frac {\csc (a+b x)}{4 b} \]
Antiderivative was successfully verified.
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Rule 207
Rule 321
Rule 2621
Rule 4287
Rubi steps
\begin {align*} \int \cos (a+b x) \csc ^2(2 a+2 b x) \, dx &=\frac {1}{4} \int \csc ^2(a+b x) \sec (a+b x) \, dx\\ &=-\frac {\operatorname {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{4 b}\\ &=-\frac {\csc (a+b x)}{4 b}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{4 b}\\ &=\frac {\tanh ^{-1}(\sin (a+b x))}{4 b}-\frac {\csc (a+b x)}{4 b}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 29, normalized size = 1.04 \[ -\frac {\csc (a+b x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\sin ^2(a+b x)\right )}{4 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 50, normalized size = 1.79 \[ \frac {\log \left (\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - \log \left (-\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 2}{8 \, b \sin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 38, normalized size = 1.36 \[ -\frac {\frac {2}{\sin \left (b x + a\right )} - \log \left ({\left | \sin \left (b x + a\right ) + 1 \right |}\right ) + \log \left ({\left | \sin \left (b x + a\right ) - 1 \right |}\right )}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.86, size = 34, normalized size = 1.21 \[ -\frac {1}{4 b \sin \left (b x +a \right )}+\frac {\ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 233, normalized size = 8.32 \[ -\frac {{\left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \log \left (\frac {\cos \left (b x + 2 \, a\right )^{2} + \cos \relax (a)^{2} - 2 \, \cos \relax (a) \sin \left (b x + 2 \, a\right ) + \sin \left (b x + 2 \, a\right )^{2} + 2 \, \cos \left (b x + 2 \, a\right ) \sin \relax (a) + \sin \relax (a)^{2}}{\cos \left (b x + 2 \, a\right )^{2} + \cos \relax (a)^{2} + 2 \, \cos \relax (a) \sin \left (b x + 2 \, a\right ) + \sin \left (b x + 2 \, a\right )^{2} - 2 \, \cos \left (b x + 2 \, a\right ) \sin \relax (a) + \sin \relax (a)^{2}}\right ) + 4 \, \cos \left (b x + a\right ) \sin \left (2 \, b x + 2 \, a\right ) - 4 \, \cos \left (2 \, b x + 2 \, a\right ) \sin \left (b x + a\right ) + 4 \, \sin \left (b x + a\right )}{8 \, {\left (b \cos \left (2 \, b x + 2 \, a\right )^{2} + b \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, b \cos \left (2 \, b x + 2 \, a\right ) + b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 26, normalized size = 0.93 \[ \frac {\mathrm {atanh}\left (\sin \left (a+b\,x\right )\right )}{4\,b}-\frac {1}{4\,b\,\sin \left (a+b\,x\right )} \]
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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