Optimal. Leaf size=26 \[ \frac{\csc (a+b x)}{b}-\frac{\csc ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.0209366, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {2606} \[ \frac{\csc (a+b x)}{b}-\frac{\csc ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 2606
Rubi steps
\begin{align*} \int \cot ^3(a+b x) \csc (a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\csc (a+b x)\right )}{b}\\ &=\frac{\csc (a+b x)}{b}-\frac{\csc ^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0136318, size = 26, normalized size = 1. \[ \frac{\csc (a+b x)}{b}-\frac{\csc ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 60, normalized size = 2.3 \begin{align*}{\frac{1}{b} \left ( -{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{4}}{3\, \left ( \sin \left ( bx+a \right ) \right ) ^{3}}}+{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{4}}{3\,\sin \left ( bx+a \right ) }}+{\frac{ \left ( 2+ \left ( \cos \left ( bx+a \right ) \right ) ^{2} \right ) \sin \left ( bx+a \right ) }{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.993042, size = 34, normalized size = 1.31 \begin{align*} \frac{3 \, \sin \left (b x + a\right )^{2} - 1}{3 \, b \sin \left (b x + a\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09488, size = 89, normalized size = 3.42 \begin{align*} \frac{3 \, \cos \left (b x + a\right )^{2} - 2}{3 \,{\left (b \cos \left (b x + a\right )^{2} - b\right )} \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 = 1.62873, size = 42, normalized size = 1.62 \begin{align*} \begin{cases} \frac{2}{3 b \sin{\left (a + b x \right )}} - \frac{\cos ^{2}{\left (a + b x \right )}}{3 b \sin ^{3}{\left (a + b x \right )}} & \text{for}\: b \neq 0 \\\frac{x \cos ^{3}{\left (a \right )}}{\sin ^{4}{\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 = 1.12884, size = 34, normalized size = 1.31 \begin{align*} \frac{3 \, \sin \left (b x + a\right )^{2} - 1}{3 \, b \sin \left (b x + a\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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