Integrand size = 19, antiderivative size = 33 \[ \int \cos ^3(a+b x) \sqrt {\csc (a+b x)} \, dx=-\frac {2}{5 b \csc ^{\frac {5}{2}}(a+b x)}+\frac {2}{b \sqrt {\csc (a+b x)}} \]
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Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2701, 14} \[ \int \cos ^3(a+b x) \sqrt {\csc (a+b x)} \, dx=\frac {2}{b \sqrt {\csc (a+b x)}}-\frac {2}{5 b \csc ^{\frac {5}{2}}(a+b x)} \]
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Rule 14
Rule 2701
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {-1+x^2}{x^{7/2}} \, dx,x,\csc (a+b x)\right )}{b} \\ & = -\frac {\text {Subst}\left (\int \left (-\frac {1}{x^{7/2}}+\frac {1}{x^{3/2}}\right ) \, dx,x,\csc (a+b x)\right )}{b} \\ & = -\frac {2}{5 b \csc ^{\frac {5}{2}}(a+b x)}+\frac {2}{b \sqrt {\csc (a+b x)}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \cos ^3(a+b x) \sqrt {\csc (a+b x)} \, dx=\frac {9+\cos (2 (a+b x))}{5 b \sqrt {\csc (a+b x)}} \]
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Time = 0.74 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79
method | result | size |
default | \(\frac {-\frac {2 \left (\sin ^{\frac {5}{2}}\left (b x +a \right )\right )}{5}+2 \left (\sqrt {\sin }\left (b x +a \right )\right )}{b}\) | \(26\) |
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Time = 0.32 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.70 \[ \int \cos ^3(a+b x) \sqrt {\csc (a+b x)} \, dx=\frac {2 \, {\left (\cos \left (b x + a\right )^{2} + 4\right )} \sqrt {\sin \left (b x + a\right )}}{5 \, b} \]
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Timed out. \[ \int \cos ^3(a+b x) \sqrt {\csc (a+b x)} \, dx=\text {Timed out} \]
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Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \cos ^3(a+b x) \sqrt {\csc (a+b x)} \, dx=\frac {2 \, {\left (\frac {5}{\sin \left (b x + a\right )^{2}} - 1\right )} \sin \left (b x + a\right )^{\frac {5}{2}}}{5 \, b} \]
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Time = 0.31 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.73 \[ \int \cos ^3(a+b x) \sqrt {\csc (a+b x)} \, dx=-\frac {2 \, {\left (\sin \left (b x + a\right )^{\frac {5}{2}} - 5 \, \sqrt {\sin \left (b x + a\right )}\right )}}{5 \, b} \]
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Timed out. \[ \int \cos ^3(a+b x) \sqrt {\csc (a+b x)} \, dx=\int {\cos \left (a+b\,x\right )}^3\,\sqrt {\frac {1}{\sin \left (a+b\,x\right )}} \,d x \]
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