\(\int \frac {\cosh ^2(\frac {1}{4}+x+x^2)}{x} \, dx\) [27]

Optimal result

Integrand size = 15, antiderivative size = 15 \[ \int \frac {\cosh ^2\left (\frac {1}{4}+x+x^2\right )}{x} \, dx=\frac {\log (x)}{2}+\frac {1}{2} \text {Int}\left (\frac {\cosh \left (\frac {1}{2}+2 x+2 x^2\right )}{x},x\right ) \]

[Out]

1/2*ln(x)+1/2*Unintegrable(cosh(1/2+2*x+2*x^2)/x,x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 15, 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 = {} \[ \int \frac {\cosh ^2\left (\frac {1}{4}+x+x^2\right )}{x} \, dx=\int \frac {\cosh ^2\left (\frac {1}{4}+x+x^2\right )}{x} \, dx \]

[In]

Int[Cosh[1/4 + x + x^2]^2/x,x]

[Out]

Log[x]/2 + Defer[Int][Cosh[1/2 + 2*x + 2*x^2]/x, x]/2

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2 x}+\frac {\cosh \left (\frac {1}{2}+2 x+2 x^2\right )}{2 x}\right ) \, dx \\ & = \frac {\log (x)}{2}+\frac {1}{2} \int \frac {\cosh \left (\frac {1}{2}+2 x+2 x^2\right )}{x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 6.50 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {\cosh ^2\left (\frac {1}{4}+x+x^2\right )}{x} \, dx=\int \frac {\cosh ^2\left (\frac {1}{4}+x+x^2\right )}{x} \, dx \]

[In]

Integrate[Cosh[1/4 + x + x^2]^2/x,x]

[Out]

Integrate[Cosh[1/4 + x + x^2]^2/x, x]

Maple [N/A] (verified)

Not integrable

Time = 0.07 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87

\[\int \frac {\cosh \left (\frac {1}{4}+x +x^{2}\right )^{2}}{x}d x\]

[In]

int(cosh(1/4+x+x^2)^2/x,x)

[Out]

int(cosh(1/4+x+x^2)^2/x,x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh ^2\left (\frac {1}{4}+x+x^2\right )}{x} \, dx=\int { \frac {\cosh \left (x^{2} + x + \frac {1}{4}\right )^{2}}{x} \,d x } \]

[In]

integrate(cosh(1/4+x+x^2)^2/x,x, algorithm="fricas")

[Out]

integral(cosh(x^2 + x + 1/4)^2/x, x)

Sympy [N/A]

Not integrable

Time = 0.77 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {\cosh ^2\left (\frac {1}{4}+x+x^2\right )}{x} \, dx=\int \frac {\cosh ^{2}{\left (x^{2} + x + \frac {1}{4} \right )}}{x}\, dx \]

[In]

integrate(cosh(1/4+x+x**2)**2/x,x)

[Out]

Integral(cosh(x**2 + x + 1/4)**2/x, x)

Maxima [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 43, normalized size of antiderivative = 2.87 \[ \int \frac {\cosh ^2\left (\frac {1}{4}+x+x^2\right )}{x} \, dx=\int { \frac {\cosh \left (x^{2} + x + \frac {1}{4}\right )^{2}}{x} \,d x } \]

[In]

integrate(cosh(1/4+x+x^2)^2/x,x, algorithm="maxima")

[Out]

1/4*integrate(e^(2*x^2 + 2*x + 1/2)/x, x) + 1/4*integrate(e^(-2*x^2 - 2*x - 1/2)/x, x) + 1/2*log(x)

Giac [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh ^2\left (\frac {1}{4}+x+x^2\right )}{x} \, dx=\int { \frac {\cosh \left (x^{2} + x + \frac {1}{4}\right )^{2}}{x} \,d x } \]

[In]

integrate(cosh(1/4+x+x^2)^2/x,x, algorithm="giac")

[Out]

integrate(cosh(x^2 + x + 1/4)^2/x, x)

Mupad [N/A]

Not integrable

Time = 1.56 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh ^2\left (\frac {1}{4}+x+x^2\right )}{x} \, dx=\int \frac {{\mathrm {cosh}\left (x^2+x+\frac {1}{4}\right )}^2}{x} \,d x \]

[In]

int(cosh(x + x^2 + 1/4)^2/x,x)

[Out]

int(cosh(x + x^2 + 1/4)^2/x, x)