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 = 0.0430228, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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 4288
Rule 2592
Rule 321
Rule 206
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*}
Mathematica [A] time = 0.0256678, 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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Maple [A] time = 0.029, size = 34, normalized size = 1.4 \begin{align*} 4\,{\frac{\cos \left ( bx+a \right ) }{b}}+4\,{\frac{\ln \left ( \csc \left ( bx+a \right ) -\cot \left ( bx+a \right ) \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.05241, size = 124, normalized size = 5.17 \begin{align*} \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 \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 )\right )}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.503847, size = 112, normalized size = 4.67 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.24985, size = 77, normalized size = 3.21 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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