Optimal. Leaf size=28 \[ \frac{\tanh ^{-1}(\sin (a+b x))}{2 b}-\frac{\sin (a+b x)}{2 b} \]
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Rubi [A] time = 0.0372884, antiderivative size = 28, normalized size of antiderivative = 1., 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 = {4288, 2592, 321, 206} \[ \frac{\tanh ^{-1}(\sin (a+b x))}{2 b}-\frac{\sin (a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 4288
Rule 2592
Rule 321
Rule 206
Rubi steps
\begin{align*} \int \csc (2 a+2 b x) \sin ^3(a+b x) \, dx &=\frac{1}{2} \int \sin (a+b x) \tan (a+b x) \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{1-x^2} \, dx,x,\sin (a+b x)\right )}{2 b}\\ &=-\frac{\sin (a+b x)}{2 b}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sin (a+b x)\right )}{2 b}\\ &=\frac{\tanh ^{-1}(\sin (a+b x))}{2 b}-\frac{\sin (a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0120244, size = 27, normalized size = 0.96 \[ \frac{1}{2} \left (\frac{\tanh ^{-1}(\sin (a+b x))}{b}-\frac{\sin (a+b x)}{b}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 32, normalized size = 1.1 \begin{align*} -{\frac{\sin \left ( bx+a \right ) }{2\,b}}+{\frac{\ln \left ( \sec \left ( bx+a \right ) +\tan \left ( bx+a \right ) \right ) }{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.75747, size = 167, normalized size = 5.96 \begin{align*} -\frac{\log \left (\frac{\cos \left (b x + 2 \, a\right )^{2} + \cos \left (a\right )^{2} - 2 \, \cos \left (a\right ) \sin \left (b x + 2 \, a\right ) + \sin \left (b x + 2 \, a\right )^{2} + 2 \, \cos \left (b x + 2 \, a\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}}{\cos \left (b x + 2 \, a\right )^{2} + \cos \left (a\right )^{2} + 2 \, \cos \left (a\right ) \sin \left (b x + 2 \, a\right ) + \sin \left (b x + 2 \, a\right )^{2} - 2 \, \cos \left (b x + 2 \, a\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}}\right ) + 2 \, \sin \left (b x + a\right )}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.502638, size = 99, normalized size = 3.54 \begin{align*} \frac{\log \left (\sin \left (b x + a\right ) + 1\right ) - \log \left (-\sin \left (b x + a\right ) + 1\right ) - 2 \, \sin \left (b x + a\right )}{4 \, 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 = 1.61057, size = 834, normalized size = 29.79 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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