Optimal. Leaf size=15 \[ \frac {2}{b \sqrt {\csc (a+b x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, 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 30
Rule 2621
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*}
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Mathematica [A] time = 0.02, size = 15, normalized size = 1.00 \[ \frac {2}{b \sqrt {\csc (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 13, normalized size = 0.87 \[ \frac {2 \, \sqrt {\sin \left (b x + a\right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 13, normalized size = 0.87 \[ \frac {2 \, \sqrt {\sin \left (b x + a\right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 14, normalized size = 0.93 \[ \frac {2}{b \sqrt {\csc \left (b x +a \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 13, normalized size = 0.87 \[ \frac {2 \, \sqrt {\sin \left (b x + a\right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.30, size = 15, normalized size = 1.00 \[ \frac {2}{b\,\sqrt {\frac {1}{\sin \left (a+b\,x\right )}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos {\left (a + b x \right )} \sqrt {\csc {\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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