Integrand size = 19, antiderivative size = 35 \[ \int \frac {\cos ^3(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=-\frac {2}{7 b \csc ^{\frac {7}{2}}(a+b x)}+\frac {2}{3 b \csc ^{\frac {3}{2}}(a+b x)} \]
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Time = 0.05 (sec) , antiderivative size = 35, 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} \[ \int \frac {\cos ^3(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\frac {2}{3 b \csc ^{\frac {3}{2}}(a+b x)}-\frac {2}{7 b \csc ^{\frac {7}{2}}(a+b x)} \]
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Rule 14
Rule 2701
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {-1+x^2}{x^{9/2}} \, dx,x,\csc (a+b x)\right )}{b} \\ & = -\frac {\text {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*}
Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \frac {\cos ^3(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\frac {2 \left (-3+7 \csc ^2(a+b x)\right )}{21 b \csc ^{\frac {7}{2}}(a+b x)} \]
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Time = 0.91 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.74
method | result | size |
default | \(\frac {-\frac {2 \left (\sin ^{\frac {7}{2}}\left (b x +a \right )\right )}{7}+\frac {2 \left (\sin ^{\frac {3}{2}}\left (b x +a \right )\right )}{3}}{b}\) | \(26\) |
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Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int \frac {\cos ^3(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=-\frac {2 \, {\left (3 \, \cos \left (b x + a\right )^{4} + \cos \left (b x + a\right )^{2} - 4\right )}}{21 \, b \sqrt {\sin \left (b x + a\right )}} \]
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Timed out. \[ \int \frac {\cos ^3(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\text {Timed out} \]
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Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.71 \[ \int \frac {\cos ^3(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\frac {2 \, {\left (\frac {7}{\sin \left (b x + a\right )^{2}} - 3\right )} \sin \left (b x + a\right )^{\frac {7}{2}}}{21 \, b} \]
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Time = 0.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.74 \[ \int \frac {\cos ^3(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=-\frac {2 \, {\left (3 \, \sin \left (b x + a\right )^{\frac {7}{2}} - 7 \, \sin \left (b x + a\right )^{\frac {3}{2}}\right )}}{21 \, b} \]
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Timed out. \[ \int \frac {\cos ^3(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\int \frac {{\cos \left (a+b\,x\right )}^3}{\sqrt {\frac {1}{\sin \left (a+b\,x\right )}}} \,d x \]
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