Optimal. Leaf size=38 \[ \frac{\sin ^3(a+b x)}{3 b}-\frac{2 \sin (a+b x)}{b}-\frac{\csc (a+b x)}{b} \]
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Rubi [A] time = 0.0344497, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2590, 270} \[ \frac{\sin ^3(a+b x)}{3 b}-\frac{2 \sin (a+b x)}{b}-\frac{\csc (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2590
Rule 270
Rubi steps
\begin{align*} \int \cos ^3(a+b x) \cot ^2(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^2}{x^2} \, dx,x,-\sin (a+b x)\right )}{b}\\ &=-\frac{\operatorname{Subst}\left (\int \left (-2+\frac{1}{x^2}+x^2\right ) \, dx,x,-\sin (a+b x)\right )}{b}\\ &=-\frac{\csc (a+b x)}{b}-\frac{2 \sin (a+b x)}{b}+\frac{\sin ^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0165735, size = 38, normalized size = 1. \[ \frac{\sin ^3(a+b x)}{3 b}-\frac{2 \sin (a+b x)}{b}-\frac{\csc (a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 52, normalized size = 1.4 \begin{align*}{\frac{1}{b} \left ( -{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{6}}{\sin \left ( bx+a \right ) }}- \left ({\frac{8}{3}}+ \left ( \cos \left ( bx+a \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}}{3}} \right ) \sin \left ( bx+a \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.990273, size = 43, normalized size = 1.13 \begin{align*} \frac{\sin \left (b x + a\right )^{3} - \frac{3}{\sin \left (b x + a\right )} - 6 \, \sin \left (b x + a\right )}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78223, size = 84, normalized size = 2.21 \begin{align*} \frac{\cos \left (b x + a\right )^{4} + 4 \, \cos \left (b x + a\right )^{2} - 8}{3 \, b \sin \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.675, size = 61, normalized size = 1.61 \begin{align*} \begin{cases} - \frac{8 \sin ^{3}{\left (a + b x \right )}}{3 b} - \frac{4 \sin{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{b} - \frac{\cos ^{4}{\left (a + b x \right )}}{b \sin{\left (a + b x \right )}} & \text{for}\: b \neq 0 \\\frac{x \cos ^{5}{\left (a \right )}}{\sin ^{2}{\left (a \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15322, size = 43, normalized size = 1.13 \begin{align*} \frac{\sin \left (b x + a\right )^{3} - \frac{3}{\sin \left (b x + a\right )} - 6 \, \sin \left (b x + a\right )}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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