Optimal. Leaf size=17 \[ \frac{2}{3 b \csc ^{\frac{3}{2}}(a+b x)} \]
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Rubi [A] time = 0.0247542, antiderivative size = 17, 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} \[ \frac{2}{3 b \csc ^{\frac{3}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 2621
Rule 30
Rubi steps
\begin{align*} \int \frac{\cos (a+b x)}{\sqrt{\csc (a+b x)}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x^{5/2}} \, dx,x,\csc (a+b x)\right )}{b}\\ &=\frac{2}{3 b \csc ^{\frac{3}{2}}(a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0244207, size = 17, normalized size = 1. \[ \frac{2}{3 b \csc ^{\frac{3}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 14, normalized size = 0.8 \begin{align*}{\frac{2}{3\,b} \left ( \csc \left ( bx+a \right ) \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.943326, size = 18, normalized size = 1.06 \begin{align*} \frac{2 \, \sin \left (b x + a\right )^{\frac{3}{2}}}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.09656, size = 68, normalized size = 4. \begin{align*} -\frac{2 \,{\left (\cos \left (b x + a\right )^{2} - 1\right )}}{3 \, b \sqrt{\sin \left (b x + a\right )}} \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 \frac{\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 [B] time = 1.31045, size = 43, normalized size = 2.53 \begin{align*} \frac{2 \, \mathrm{sgn}\left (\sin \left (b x + a\right )\right )^{3} \sin \left (b x + a\right )^{\frac{3}{2}}}{3 \, b \mathrm{sgn}\left (\mathrm{sgn}\left (\sin \left (b x + a\right )\right )\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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