Integrand size = 21, antiderivative size = 21 \[ \int \frac {\cosh ^2\left (a+b x+c x^2\right )}{d+e x} \, dx=\frac {\log (d+e x)}{2 e}+\frac {1}{2} \text {Int}\left (\frac {\cosh \left (2 a+2 b x+2 c x^2\right )}{d+e x},x\right ) \]
[Out]
Not integrable
Time = 0.03 (sec) , antiderivative size = 21, 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 = {} \[ \int \frac {\cosh ^2\left (a+b x+c x^2\right )}{d+e x} \, dx=\int \frac {\cosh ^2\left (a+b x+c x^2\right )}{d+e x} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \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*}
Not integrable
Time = 2.49 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\cosh ^2\left (a+b x+c x^2\right )}{d+e x} \, dx=\int \frac {\cosh ^2\left (a+b x+c x^2\right )}{d+e x} \, dx \]
[In]
[Out]
Not integrable
Time = 0.05 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00
\[\int \frac {\cosh \left (c \,x^{2}+b x +a \right )^{2}}{e x +d}d x\]
[In]
[Out]
Not integrable
Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\cosh ^2\left (a+b x+c x^2\right )}{d+e x} \, dx=\int { \frac {\cosh \left (c x^{2} + b x + a\right )^{2}}{e x + d} \,d x } \]
[In]
[Out]
Not integrable
Time = 41.99 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {\cosh ^2\left (a+b x+c x^2\right )}{d+e x} \, dx=\int \frac {\cosh ^{2}{\left (a + b x + c x^{2} \right )}}{d + e x}\, dx \]
[In]
[Out]
Not integrable
Time = 0.36 (sec) , antiderivative size = 72, normalized size of antiderivative = 3.43 \[ \int \frac {\cosh ^2\left (a+b x+c x^2\right )}{d+e x} \, dx=\int { \frac {\cosh \left (c x^{2} + b x + a\right )^{2}}{e x + d} \,d x } \]
[In]
[Out]
Not integrable
Time = 0.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\cosh ^2\left (a+b x+c x^2\right )}{d+e x} \, dx=\int { \frac {\cosh \left (c x^{2} + b x + a\right )^{2}}{e x + d} \,d x } \]
[In]
[Out]
Not integrable
Time = 1.61 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\cosh ^2\left (a+b x+c x^2\right )}{d+e x} \, dx=\int \frac {{\mathrm {cosh}\left (c\,x^2+b\,x+a\right )}^2}{d+e\,x} \,d x \]
[In]
[Out]