Optimal. Leaf size=15 \[ \frac{2}{b \sqrt{\csc (a+b x)}} \]
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Rubi [A] time = 0.0246537, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2621, 30} \[ \frac{2}{b \sqrt{\csc (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2621
Rule 30
Rubi steps
\begin{align*} \int \cos (a+b x) \sqrt{\csc (a+b x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x^{3/2}} \, dx,x,\csc (a+b x)\right )}{b}\\ &=\frac{2}{b \sqrt{\csc (a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.0205176, size = 15, normalized size = 1. \[ \frac{2}{b \sqrt{\csc (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 14, normalized size = 0.9 \begin{align*} 2\,{\frac{1}{b\sqrt{\csc \left ( bx+a \right ) }}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.938317, size = 18, normalized size = 1.2 \begin{align*} \frac{2 \, \sqrt{\sin \left (b x + a\right )}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.10671, size = 32, normalized size = 2.13 \begin{align*} \frac{2 \, \sqrt{\sin \left (b x + a\right )}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos{\left (a + b x \right )} \sqrt{\csc{\left (a + b x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14225, size = 27, normalized size = 1.8 \begin{align*} \frac{2 \, \mathrm{sgn}\left (\sin \left (b x + a\right )\right ) \sqrt{\sin \left (b x + a\right )}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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