3.27 \(\int \frac {\sinh ^2(\frac {1}{4}+x+x^2)}{x} \, dx\)

Optimal. Leaf size=31 \[ \frac {1}{2} \text {Int}\left (\frac {\cosh \left (2 x^2+2 x+\frac {1}{2}\right )}{x},x\right )-\frac {\log (x)}{2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sinh ^2\left (\frac {1}{4}+x+x^2\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sinh[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*} \int \frac {\sinh ^2\left (\frac {1}{4}+x+x^2\right )}{x} \, dx &=\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 [A]  time = 20.68, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^2\left (\frac {1}{4}+x+x^2\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sinh[1/4 + x + x^2]^2/x,x]

[Out]

Integrate[Sinh[1/4 + x + x^2]^2/x, x]

________________________________________________________________________________________

fricas [A]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sinh \left (x^{2} + x + \frac {1}{4}\right )^{2}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sinh(x^2 + x + 1/4)^2/x, x)

________________________________________________________________________________________

giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh \left (x^{2} + x + \frac {1}{4}\right )^{2}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sinh(x^2 + x + 1/4)^2/x, x)

________________________________________________________________________________________

maple [A]  time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^{2}\left (\frac {1}{4}+x +x^{2}\right )}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{4} \, \int \frac {e^{\left (2 \, x^{2} + 2 \, x + \frac {1}{2}\right )}}{x}\,{d x} + \frac {1}{4} \, \int \frac {e^{\left (-2 \, x^{2} - 2 \, x - \frac {1}{2}\right )}}{x}\,{d x} - \frac {1}{2} \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(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)

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\mathrm {sinh}\left (x^2+x+\frac {1}{4}\right )}^2}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x + x^2 + 1/4)^2/x,x)

[Out]

int(sinh(x + x^2 + 1/4)^2/x, x)

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^{2}{\left (x^{2} + x + \frac {1}{4} \right )}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________