Integrand size = 16, antiderivative size = 16 \[ \int \frac {a+b \csc \left (c+d x^2\right )}{x} \, dx=a \log (x)+b \text {Int}\left (\frac {\csc \left (c+d x^2\right )}{x},x\right ) \]
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Not integrable
Time = 0.02 (sec) , antiderivative size = 16, 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 = {} \[ \int \frac {a+b \csc \left (c+d x^2\right )}{x} \, dx=\int \frac {a+b \csc \left (c+d x^2\right )}{x} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a}{x}+\frac {b \csc \left (c+d x^2\right )}{x}\right ) \, dx \\ & = a \log (x)+b \int \frac {\csc \left (c+d x^2\right )}{x} \, dx \\ \end{align*}
Not integrable
Time = 3.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {a+b \csc \left (c+d x^2\right )}{x} \, dx=\int \frac {a+b \csc \left (c+d x^2\right )}{x} \, dx \]
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Not integrable
Time = 0.11 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00
\[\int \frac {a +b \csc \left (d \,x^{2}+c \right )}{x}d x\]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {a+b \csc \left (c+d x^2\right )}{x} \, dx=\int { \frac {b \csc \left (d x^{2} + c\right ) + a}{x} \,d x } \]
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Not integrable
Time = 0.62 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {a+b \csc \left (c+d x^2\right )}{x} \, dx=\int \frac {a + b \csc {\left (c + d x^{2} \right )}}{x}\, dx \]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 108, normalized size of antiderivative = 6.75 \[ \int \frac {a+b \csc \left (c+d x^2\right )}{x} \, dx=\int { \frac {b \csc \left (d x^{2} + c\right ) + a}{x} \,d x } \]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {a+b \csc \left (c+d x^2\right )}{x} \, dx=\int { \frac {b \csc \left (d x^{2} + c\right ) + a}{x} \,d x } \]
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Not integrable
Time = 18.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \frac {a+b \csc \left (c+d x^2\right )}{x} \, dx=\int \frac {a+\frac {b}{\sin \left (d\,x^2+c\right )}}{x} \,d x \]
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