\(\int \frac {\cosh (\frac {1}{4}+x+x^2)}{x^2} \, dx\) [15]

Optimal result

Integrand size = 13, antiderivative size = 13 \[ \int \frac {\cosh \left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=-\frac {\cosh \left (\frac {1}{4}+x+x^2\right )}{x}+\frac {1}{2} \sqrt {\pi } \text {erf}\left (\frac {1}{2} (-1-2 x)\right )+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (1+2 x)\right )+\text {Int}\left (\frac {\sinh \left (\frac {1}{4}+x+x^2\right )}{x},x\right ) \]

[Out]

-cosh(1/4+x+x^2)/x-1/2*erf(1/2+x)*Pi^(1/2)+1/2*erfi(1/2+x)*Pi^(1/2)+Unintegrable(sinh(1/4+x+x^2)/x,x)

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 13, 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 = {} \[ \int \frac {\cosh \left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=\int \frac {\cosh \left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx \]

[In]

Int[Cosh[1/4 + x + x^2]/x^2,x]

[Out]

-(Cosh[1/4 + x + x^2]/x) + (Sqrt[Pi]*Erf[(-1 - 2*x)/2])/2 + (Sqrt[Pi]*Erfi[(1 + 2*x)/2])/2 + Defer[Int][Sinh[1
/4 + x + x^2]/x, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {\cosh \left (\frac {1}{4}+x+x^2\right )}{x}+2 \int \sinh \left (\frac {1}{4}+x+x^2\right ) \, dx+\int \frac {\sinh \left (\frac {1}{4}+x+x^2\right )}{x} \, dx \\ & = -\frac {\cosh \left (\frac {1}{4}+x+x^2\right )}{x}-\int e^{-\frac {1}{4}-x-x^2} \, dx+\int e^{\frac {1}{4}+x+x^2} \, dx+\int \frac {\sinh \left (\frac {1}{4}+x+x^2\right )}{x} \, dx \\ & = -\frac {\cosh \left (\frac {1}{4}+x+x^2\right )}{x}-\int e^{-\frac {1}{4} (-1-2 x)^2} \, dx+\int e^{\frac {1}{4} (1+2 x)^2} \, dx+\int \frac {\sinh \left (\frac {1}{4}+x+x^2\right )}{x} \, dx \\ & = -\frac {\cosh \left (\frac {1}{4}+x+x^2\right )}{x}+\frac {1}{2} \sqrt {\pi } \text {erf}\left (\frac {1}{2} (-1-2 x)\right )+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (1+2 x)\right )+\int \frac {\sinh \left (\frac {1}{4}+x+x^2\right )}{x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 6.32 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \frac {\cosh \left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=\int \frac {\cosh \left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx \]

[In]

Integrate[Cosh[1/4 + x + x^2]/x^2,x]

[Out]

Integrate[Cosh[1/4 + x + x^2]/x^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh \left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=\int { \frac {\cosh \left (x^{2} + x + \frac {1}{4}\right )}{x^{2}} \,d x } \]

[In]

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

[Out]

integral(cosh(x^2 + x + 1/4)/x^2, x)

Sympy [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {\cosh \left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=\int \frac {\cosh {\left (x^{2} + x + \frac {1}{4} \right )}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.48 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh \left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=\int { \frac {\cosh \left (x^{2} + x + \frac {1}{4}\right )}{x^{2}} \,d x } \]

[In]

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

[Out]

integrate(cosh(x^2 + x + 1/4)/x^2, x)

Giac [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh \left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=\int { \frac {\cosh \left (x^{2} + x + \frac {1}{4}\right )}{x^{2}} \,d x } \]

[In]

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

[Out]

integrate(cosh(x^2 + x + 1/4)/x^2, x)

Mupad [N/A]

Not integrable

Time = 1.57 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh \left (\frac {1}{4}+x+x^2\right )}{x^2} \, dx=\int \frac {\mathrm {cosh}\left (x^2+x+\frac {1}{4}\right )}{x^2} \,d x \]

[In]

int(cosh(x + x^2 + 1/4)/x^2,x)

[Out]

int(cosh(x + x^2 + 1/4)/x^2, x)