Integrand size = 25, antiderivative size = 25 \[ \int \frac {\cos ^{\frac {4}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\text {Int}\left (\frac {\cos ^{\frac {4}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}},x\right ) \]
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Not integrable
Time = 0.06 (sec) , antiderivative size = 25, 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 = {} \[ \int \frac {\cos ^{\frac {4}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\int \frac {\cos ^{\frac {4}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos ^{\frac {4}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx \\ \end{align*}
Not integrable
Time = 33.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\cos ^{\frac {4}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\int \frac {\cos ^{\frac {4}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx \]
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Not integrable
Time = 0.58 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84
\[\int \frac {\cos ^{\frac {4}{3}}\left (d x +c \right )}{\sqrt {a +\cos \left (d x +c \right ) b}}d x\]
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Not integrable
Time = 0.77 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {\cos ^{\frac {4}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )^{\frac {4}{3}}}{\sqrt {b \cos \left (d x + c\right ) + a}} \,d x } \]
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Not integrable
Time = 102.89 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {\cos ^{\frac {4}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\int \frac {\cos ^{\frac {4}{3}}{\left (c + d x \right )}}{\sqrt {a + b \cos {\left (c + d x \right )}}}\, dx \]
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Not integrable
Time = 0.79 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {\cos ^{\frac {4}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )^{\frac {4}{3}}}{\sqrt {b \cos \left (d x + c\right ) + a}} \,d x } \]
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Not integrable
Time = 20.09 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {\cos ^{\frac {4}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )^{\frac {4}{3}}}{\sqrt {b \cos \left (d x + c\right ) + a}} \,d x } \]
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Not integrable
Time = 15.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {\cos ^{\frac {4}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^{4/3}}{\sqrt {a+b\,\cos \left (c+d\,x\right )}} \,d x \]
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