Optimal. Leaf size=43 \[ \frac{1}{2} \text{Unintegrable}\left (\frac{\cosh \left (2 a+2 b x+2 c x^2\right )}{d+e x},x\right )+\frac{\log (d+e x)}{2 e} \]
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Rubi [A] time = 0.0472074, 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 = {} \[ \int \frac{\cosh ^2\left (a+b x+c x^2\right )}{d+e x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\cosh ^2\left (a+b x+c x^2\right )}{d+e x} \, dx &=\int \left (\frac{1}{2 (d+e x)}+\frac{\cosh \left (2 a+2 b x+2 c x^2\right )}{2 (d+e x)}\right ) \, dx\\ &=\frac{\log (d+e x)}{2 e}+\frac{1}{2} \int \frac{\cosh \left (2 a+2 b x+2 c x^2\right )}{d+e x} \, dx\\ \end{align*}
Mathematica [A] time = 8.14217, size = 0, normalized size = 0. \[ \int \frac{\cosh ^2\left (a+b x+c x^2\right )}{d+e x} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.116, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \cosh \left ( c{x}^{2}+bx+a \right ) \right ) ^{2}}{ex+d}}\, 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{\log \left (e x + d\right )}{2 \, e} + \frac{1}{4} \, \int \frac{e^{\left (2 \, c x^{2} + 2 \, b x + 2 \, a\right )}}{e x + d}\,{d x} + \frac{1}{4} \, \int \frac{e^{\left (-2 \, c x^{2} - 2 \, b x\right )}}{e x e^{\left (2 \, a\right )} + d e^{\left (2 \, a\right )}}\,{d x} \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{\cosh \left (c x^{2} + b x + a\right )^{2}}{e x + d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\cosh \left (c x^{2} + b x + a\right )^{2}}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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