Optimal. Leaf size=24 \[ \frac {4 \cos (a+b x)}{b}-\frac {4 \tanh ^{-1}(\cos (a+b x))}{b} \]
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Rubi [A] time = 0.04, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4288, 2592, 321, 206} \[ \frac {4 \cos (a+b x)}{b}-\frac {4 \tanh ^{-1}(\cos (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 206
Rule 321
Rule 2592
Rule 4288
Rubi steps
\begin {align*} \int \csc ^3(a+b x) \sin ^2(2 a+2 b x) \, dx &=4 \int \cos (a+b x) \cot (a+b x) \, dx\\ &=-\frac {4 \operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\cos (a+b x)\right )}{b}\\ &=\frac {4 \cos (a+b x)}{b}-\frac {4 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {4 \tanh ^{-1}(\cos (a+b x))}{b}+\frac {4 \cos (a+b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 1.83 \[ 4 \left (\frac {\cos (a+b x)}{b}+\frac {\log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )}{b}-\frac {\log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )}{b}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 38, normalized size = 1.58 \[ \frac {2 \, {\left (2 \, \cos \left (b x + a\right ) - \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right )\right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.57, size = 57, normalized size = 2.38 \[ -\frac {2 \, {\left (\frac {4}{\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1} - \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )\right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.57, size = 34, normalized size = 1.42 \[ \frac {4 \cos \left (b x +a \right )}{b}+\frac {4 \ln \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.99, size = 92, normalized size = 3.83 \[ \frac {2 \, {\left (2 \, \cos \left (b x + a\right ) - \log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \relax (a) + \cos \relax (a)^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \relax (a) + \sin \relax (a)^{2}\right ) + \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \relax (a) + \cos \relax (a)^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \relax (a) + \sin \relax (a)^{2}\right )\right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 22, normalized size = 0.92 \[ \frac {4\,\cos \left (a+b\,x\right )-4\,\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )}{b} \]
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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