Integrand size = 17, antiderivative size = 17 \[ \int \frac {\sinh ^2\left (a+b x+c x^2\right )}{x} \, dx=-\frac {\log (x)}{2}+\frac {1}{2} \text {Int}\left (\frac {\cosh \left (2 a+2 b x+2 c x^2\right )}{x},x\right ) \]
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Not integrable
Time = 0.02 (sec) , antiderivative size = 17, 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 = {} \[ \int \frac {\sinh ^2\left (a+b x+c x^2\right )}{x} \, dx=\int \frac {\sinh ^2\left (a+b x+c x^2\right )}{x} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{2 x}+\frac {\cosh \left (2 a+2 b x+2 c x^2\right )}{2 x}\right ) \, dx \\ & = -\frac {\log (x)}{2}+\frac {1}{2} \int \frac {\cosh \left (2 a+2 b x+2 c x^2\right )}{x} \, dx \\ \end{align*}
Not integrable
Time = 12.59 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {\sinh ^2\left (a+b x+c x^2\right )}{x} \, dx=\int \frac {\sinh ^2\left (a+b x+c x^2\right )}{x} \, dx \]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00
\[\int \frac {\sinh \left (c \,x^{2}+b x +a \right )^{2}}{x}d x\]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {\sinh ^2\left (a+b x+c x^2\right )}{x} \, dx=\int { \frac {\sinh \left (c x^{2} + b x + a\right )^{2}}{x} \,d x } \]
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Not integrable
Time = 0.76 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {\sinh ^2\left (a+b x+c x^2\right )}{x} \, dx=\int \frac {\sinh ^{2}{\left (a + b x + c x^{2} \right )}}{x}\, dx \]
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Not integrable
Time = 0.35 (sec) , antiderivative size = 51, normalized size of antiderivative = 3.00 \[ \int \frac {\sinh ^2\left (a+b x+c x^2\right )}{x} \, dx=\int { \frac {\sinh \left (c x^{2} + b x + a\right )^{2}}{x} \,d x } \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {\sinh ^2\left (a+b x+c x^2\right )}{x} \, dx=\int { \frac {\sinh \left (c x^{2} + b x + a\right )^{2}}{x} \,d x } \]
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Not integrable
Time = 1.11 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {\sinh ^2\left (a+b x+c x^2\right )}{x} \, dx=\int \frac {{\mathrm {sinh}\left (c\,x^2+b\,x+a\right )}^2}{x} \,d x \]
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