Integrand size = 19, antiderivative size = 92 \[ \int \frac {\cos ^4(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\frac {4 \cos (a+b x)}{15 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {2 \cos ^3(a+b x)}{9 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {8 \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{15 b} \]
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Time = 0.15 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2708, 3856, 2719} \[ \int \frac {\cos ^4(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\frac {2 \cos ^3(a+b x)}{9 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{15 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {8 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{15 b} \]
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Rule 2708
Rule 2719
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {2 \cos ^3(a+b x)}{9 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {2}{3} \int \frac {\cos ^2(a+b x)}{\sqrt {\csc (a+b x)}} \, dx \\ & = \frac {4 \cos (a+b x)}{15 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {2 \cos ^3(a+b x)}{9 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {4}{15} \int \frac {1}{\sqrt {\csc (a+b x)}} \, dx \\ & = \frac {4 \cos (a+b x)}{15 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {2 \cos ^3(a+b x)}{9 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {1}{15} \left (4 \sqrt {\csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \sqrt {\sin (a+b x)} \, dx \\ & = \frac {4 \cos (a+b x)}{15 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {2 \cos ^3(a+b x)}{9 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {8 \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{15 b} \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.68 \[ \int \frac {\cos ^4(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\frac {39 \cos (a+b x)+5 \cos (3 (a+b x))-\frac {48 E\left (\left .\frac {1}{4} (-2 a+\pi -2 b x)\right |2\right )}{\sin ^{\frac {3}{2}}(a+b x)}}{90 b \csc ^{\frac {3}{2}}(a+b x)} \]
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Time = 1.13 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.65
method | result | size |
default | \(\frac {-\frac {2 \left (\cos ^{6}\left (b x +a \right )\right )}{9}-\frac {8 \sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, E\left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )}{15}+\frac {4 \sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, F\left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )}{15}-\frac {2 \left (\cos ^{4}\left (b x +a \right )\right )}{45}+\frac {4 \left (\cos ^{2}\left (b x +a \right )\right )}{15}}{\cos \left (b x +a \right ) \sqrt {\sin \left (b x +a \right )}\, b}\) | \(152\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.03 \[ \int \frac {\cos ^4(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\frac {2 \, {\left (6 \, \sqrt {2 i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + 6 \, \sqrt {-2 i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) - \frac {5 \, \cos \left (b x + a\right )^{5} + \cos \left (b x + a\right )^{3} - 6 \, \cos \left (b x + a\right )}{\sqrt {\sin \left (b x + a\right )}}\right )}}{45 \, b} \]
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\[ \int \frac {\cos ^4(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\int \frac {\cos ^{4}{\left (a + b x \right )}}{\sqrt {\csc {\left (a + b x \right )}}}\, dx \]
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\[ \int \frac {\cos ^4(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\int { \frac {\cos \left (b x + a\right )^{4}}{\sqrt {\csc \left (b x + a\right )}} \,d x } \]
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\[ \int \frac {\cos ^4(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\int { \frac {\cos \left (b x + a\right )^{4}}{\sqrt {\csc \left (b x + a\right )}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^4(a+b x)}{\sqrt {\csc (a+b x)}} \, dx=\int \frac {{\cos \left (a+b\,x\right )}^4}{\sqrt {\frac {1}{\sin \left (a+b\,x\right )}}} \,d x \]
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