Integrand size = 33, antiderivative size = 95 \[ \int \frac {1+\cos (c+d x)}{\sqrt {-2-3 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\left .\arcsin \left (\frac {\sqrt {-2-3 \cos (c+d x)}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right )\right |5\right ) \sqrt {-1-\sec (c+d x)} \sqrt {1-\sec (c+d x)}}{d} \]
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Time = 0.22 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {3074, 3073} \[ \int \frac {1+\cos (c+d x)}{\sqrt {-2-3 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) \sqrt {-\sec (c+d x)-1} \sqrt {1-\sec (c+d x)} E\left (\left .\arcsin \left (\frac {\sqrt {-3 \cos (c+d x)-2}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right )\right |5\right )}{d} \]
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Rule 3073
Rule 3074
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {-\cos (c+d x)} \int \frac {1+\cos (c+d x)}{\sqrt {-2-3 \cos (c+d x)} (-\cos (c+d x))^{3/2}} \, dx}{\sqrt {\cos (c+d x)}} \\ & = \frac {\sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\left .\arcsin \left (\frac {\sqrt {-2-3 \cos (c+d x)}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right )\right |5\right ) \sqrt {-1-\sec (c+d x)} \sqrt {1-\sec (c+d x)}}{d} \\ \end{align*}
\[ \int \frac {1+\cos (c+d x)}{\sqrt {-2-3 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {1+\cos (c+d x)}{\sqrt {-2-3 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. \(315\) vs. \(2(83)=166\).
Time = 12.19 (sec) , antiderivative size = 316, normalized size of antiderivative = 3.33
method | result | size |
default | \(-\frac {\left (-\sqrt {5}\, \sqrt {10}\, E\left (\frac {\left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}}{5}, \sqrt {5}\right ) \sqrt {2}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right )-2 \sqrt {5}\, \sqrt {10}\, E\left (\frac {\left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}}{5}, \sqrt {5}\right ) \sqrt {2}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right )-\sqrt {2}\, E\left (\frac {\left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}}{5}, \sqrt {5}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {10}\, \sqrt {5}+30 \cos \left (d x +c \right ) \sin \left (d x +c \right )+20 \sin \left (d x +c \right )\right ) \sqrt {-2-3 \cos \left (d x +c \right )}}{10 d \left (2+3 \cos \left (d x +c \right )\right ) \sqrt {\cos \left (d x +c \right )}\, \left (1+\cos \left (d x +c \right )\right )}\) | \(316\) |
parts | \(-\frac {\left (1+\cos \left (d x +c \right )\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-2-3 \cos \left (d x +c \right )}\, \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, F\left (\frac {\left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}}{5}, \sqrt {5}\right ) \sqrt {5}}{5 d \left (2+3 \cos \left (d x +c \right )\right ) \sqrt {\cos \left (d x +c \right )}}+\frac {\left (\sqrt {5}\, \sqrt {-5 \left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+25}\, \sqrt {-\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}\, E\left (\frac {\left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}}{5}, \sqrt {5}\right )+2 \sqrt {5}\, \sqrt {-5 \left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+25}\, \sqrt {-\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}\, F\left (\frac {\left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}}{5}, \sqrt {5}\right )+5 \left (\csc ^{3}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{3}-25 \csc \left (d x +c \right )+25 \cot \left (d x +c \right )\right ) \sqrt {\frac {\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}-5}{\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}}\, \left (\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}-1\right )}{5 d \left (\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}-5\right ) \left (\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1\right ) {\left (-\frac {\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}-1}{\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}\right )}^{\frac {3}{2}}}\) | \(503\) |
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\[ \int \frac {1+\cos (c+d x)}{\sqrt {-2-3 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {\cos \left (d x + c\right ) + 1}{\sqrt {-3 \, \cos \left (d x + c\right ) - 2} \cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1+\cos (c+d x)}{\sqrt {-2-3 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\cos {\left (c + d x \right )} + 1}{\sqrt {- 3 \cos {\left (c + d x \right )} - 2} \cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]
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\[ \int \frac {1+\cos (c+d x)}{\sqrt {-2-3 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {\cos \left (d x + c\right ) + 1}{\sqrt {-3 \, \cos \left (d x + c\right ) - 2} \cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1+\cos (c+d x)}{\sqrt {-2-3 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {\cos \left (d x + c\right ) + 1}{\sqrt {-3 \, \cos \left (d x + c\right ) - 2} \cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1+\cos (c+d x)}{\sqrt {-2-3 \cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\cos \left (c+d\,x\right )+1}{{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {-3\,\cos \left (c+d\,x\right )-2}} \,d x \]
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