Optimal. Leaf size=35 \[ \frac {2}{3 b \csc ^{\frac {3}{2}}(a+b x)}-\frac {2}{7 b \csc ^{\frac {7}{2}}(a+b x)} \]
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Rubi [A] time = 0.03, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2621, 14} \[ \frac {2}{3 b \csc ^{\frac {3}{2}}(a+b x)}-\frac {2}{7 b \csc ^{\frac {7}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2621
Rubi steps
\begin {align*} \int \frac {\cos ^3(a+b x)}{\sqrt {\csc (a+b x)}} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {-1+x^2}{x^{9/2}} \, dx,x,\csc (a+b x)\right )}{b}\\ &=-\frac {\operatorname {Subst}\left (\int \left (-\frac {1}{x^{9/2}}+\frac {1}{x^{5/2}}\right ) \, dx,x,\csc (a+b x)\right )}{b}\\ &=-\frac {2}{7 b \csc ^{\frac {7}{2}}(a+b x)}+\frac {2}{3 b \csc ^{\frac {3}{2}}(a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 29, normalized size = 0.83 \[ \frac {2 \left (7 \csc ^2(a+b x)-3\right )}{21 b \csc ^{\frac {7}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 33, normalized size = 0.94 \[ -\frac {2 \, {\left (3 \, \cos \left (b x + a\right )^{4} + \cos \left (b x + a\right )^{2} - 4\right )}}{21 \, b \sqrt {\sin \left (b x + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.51, size = 25, normalized size = 0.71 \[ \frac {2 \, {\left (\frac {7}{\sin \left (b x + a\right )^{2}} - 3\right )} \sin \left (b x + a\right )^{\frac {7}{2}}}{21 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 26, normalized size = 0.74 \[ \frac {-\frac {2 \left (\sin ^{\frac {7}{2}}\left (b x +a \right )\right )}{7}+\frac {2 \left (\sin ^{\frac {3}{2}}\left (b x +a \right )\right )}{3}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 25, normalized size = 0.71 \[ \frac {2 \, {\left (\frac {7}{\sin \left (b x + a\right )^{2}} - 3\right )} \sin \left (b x + a\right )^{\frac {7}{2}}}{21 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\cos \left (a+b\,x\right )}^3}{\sqrt {\frac {1}{\sin \left (a+b\,x\right )}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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