Integrand size = 18, antiderivative size = 18 \[ \int \frac {1}{x \sqrt {a+a \cos (c+d x)}} \, dx=\text {Int}\left (\frac {1}{x \sqrt {a+a \cos (c+d x)}},x\right ) \]
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Not integrable
Time = 0.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{x \sqrt {a+a \cos (c+d x)}} \, dx=\int \frac {1}{x \sqrt {a+a \cos (c+d x)}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x \sqrt {a+a \cos (c+d x)}} \, dx \\ \end{align*}
Not integrable
Time = 1.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x \sqrt {a+a \cos (c+d x)}} \, dx=\int \frac {1}{x \sqrt {a+a \cos (c+d x)}} \, dx \]
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Not integrable
Time = 0.17 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89
\[\int \frac {1}{x \sqrt {a +\cos \left (d x +c \right ) a}}d x\]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.67 \[ \int \frac {1}{x \sqrt {a+a \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {a \cos \left (d x + c\right ) + a} x} \,d x } \]
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Not integrable
Time = 1.86 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x \sqrt {a+a \cos (c+d x)}} \, dx=\int \frac {1}{x \sqrt {a \left (\cos {\left (c + d x \right )} + 1\right )}}\, dx \]
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Not integrable
Time = 0.58 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {a+a \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {a \cos \left (d x + c\right ) + a} x} \,d x } \]
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Not integrable
Time = 0.56 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {a+a \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {a \cos \left (d x + c\right ) + a} x} \,d x } \]
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Not integrable
Time = 14.36 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {a+a \cos (c+d x)}} \, dx=\int \frac {1}{x\,\sqrt {a+a\,\cos \left (c+d\,x\right )}} \,d x \]
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