Optimal. Leaf size=32 \[ \frac{1}{2} \text{Unintegrable}\left (\frac{\cosh \left (2 a+2 b x+2 c x^2\right )}{x},x\right )-\frac{\log (x)}{2} \]
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Rubi [A] time = 0.031242, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sinh ^2\left (a+b x+c x^2\right )}{x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\sinh ^2\left (a+b x+c x^2\right )}{x} \, dx &=\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*}
Mathematica [A] time = 59.4538, size = 0, normalized size = 0. \[ \int \frac{\sinh ^2\left (a+b x+c x^2\right )}{x} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.054, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \sinh \left ( c{x}^{2}+bx+a \right ) \right ) ^{2}}{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{4} \, \int \frac{e^{\left (2 \, c x^{2} + 2 \, b x + 2 \, a\right )}}{x}\,{d x} + \frac{1}{4} \, \int \frac{e^{\left (-2 \, c x^{2} - 2 \, b x - 2 \, a\right )}}{x}\,{d x} - \frac{1}{2} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sinh \left (c x^{2} + b x + a\right )^{2}}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh ^{2}{\left (a + b x + c x^{2} \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh \left (c x^{2} + b x + a\right )^{2}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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