Integrand size = 33, antiderivative size = 70 \[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-2+3 \cos (c+d x)}} \, dx=-\frac {\sqrt {5} \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {-2+3 \cos (c+d x)}}{\sqrt {\cos (c+d x)}}\right )|\frac {1}{5}\right ) \sqrt {-1+\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{d} \]
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Time = 0.11 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {3073} \[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-2+3 \cos (c+d x)}} \, dx=-\frac {\sqrt {5} \cot (c+d x) \sqrt {\sec (c+d x)-1} \sqrt {\sec (c+d x)+1} E\left (\arcsin \left (\frac {\sqrt {3 \cos (c+d x)-2}}{\sqrt {\cos (c+d x)}}\right )|\frac {1}{5}\right )}{d} \]
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Rule 3073
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {5} \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {-2+3 \cos (c+d x)}}{\sqrt {\cos (c+d x)}}\right )|\frac {1}{5}\right ) \sqrt {-1+\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{d} \\ \end{align*}
\[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-2+3 \cos (c+d x)}} \, dx=\int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-2+3 \cos (c+d x)}} \, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. \(461\) vs. \(2(61)=122\).
Time = 9.83 (sec) , antiderivative size = 462, normalized size of antiderivative = 6.60
method | result | size |
parts | \(\frac {\left (-2 \sqrt {-\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}\, \sqrt {-5 \left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}\, F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right )-\sqrt {-\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}\, \sqrt {-5 \left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}\, E\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right )+5 \left (\csc ^{3}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{3}-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ) \sqrt {-\frac {5 \left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}-1}{\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}}\, \left (\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}-1\right )}{d \left (5 \left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}-1\right ) \left (\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1\right ) {\left (-\frac {\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}-1}{\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}\right )}^{\frac {3}{2}}}-\frac {2 \left (1+\cos \left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right )}{d \sqrt {-2+3 \cos \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )}}\) | \(462\) |
default | \(-\frac {E\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right )+4 \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right ) \left (\cos ^{2}\left (d x +c \right )\right )+2 E\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right )+8 \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right ) \cos \left (d x +c \right )+\sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, E\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right )+4 \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right )+3 \cos \left (d x +c \right ) \sin \left (d x +c \right )-2 \sin \left (d x +c \right )}{d \left (1+\cos \left (d x +c \right )\right ) \sqrt {-2+3 \cos \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )}}\) | \(463\) |
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\[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-2+3 \cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right ) + 1}{\sqrt {3 \, \cos \left (d x + c\right ) - 2} \cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-2+3 \cos (c+d x)}} \, dx=\int \frac {\cos {\left (c + d x \right )} + 1}{\sqrt {3 \cos {\left (c + d x \right )} - 2} \cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]
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\[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-2+3 \cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right ) + 1}{\sqrt {3 \, \cos \left (d x + c\right ) - 2} \cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-2+3 \cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right ) + 1}{\sqrt {3 \, \cos \left (d x + c\right ) - 2} \cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {-2+3 \cos (c+d x)}} \, dx=\int \frac {\cos \left (c+d\,x\right )+1}{{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {3\,\cos \left (c+d\,x\right )-2}} \,d x \]
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