Integrand size = 18, antiderivative size = 18 \[ \int \frac {1}{x (a+a \sin (e+f x))^{3/2}} \, dx=\text {Int}\left (\frac {1}{x (a+a \sin (e+f x))^{3/2}},x\right ) \]
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Not integrable
Time = 0.06 (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 = {} \[ \int \frac {1}{x (a+a \sin (e+f x))^{3/2}} \, dx=\int \frac {1}{x (a+a \sin (e+f x))^{3/2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x (a+a \sin (e+f x))^{3/2}} \, dx \\ \end{align*}
Not integrable
Time = 24.71 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x (a+a \sin (e+f x))^{3/2}} \, dx=\int \frac {1}{x (a+a \sin (e+f x))^{3/2}} \, dx \]
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Not integrable
Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89
\[\int \frac {1}{x \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}d x\]
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Not integrable
Time = 0.31 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.78 \[ \int \frac {1}{x (a+a \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} x} \,d x } \]
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Not integrable
Time = 2.41 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x (a+a \sin (e+f x))^{3/2}} \, dx=\int \frac {1}{x \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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Not integrable
Time = 0.71 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x (a+a \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} x} \,d x } \]
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Not integrable
Time = 83.43 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x (a+a \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} x} \,d x } \]
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Not integrable
Time = 1.17 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x (a+a \sin (e+f x))^{3/2}} \, dx=\int \frac {1}{x\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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