3.19 \(\int \frac {\cosh ^2(a+b x+c x^2)}{x} \, dx\)

Optimal. Leaf size=33 \[ \frac {1}{2} \text {Int}\left (\frac {\cosh \left (2 a+2 b x+2 c x^2\right )}{x},x\right )+\frac {\log (x)}{2} \]

[Out]

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

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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 = {} \[ \int \frac {\cosh ^2\left (a+b x+c x^2\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Cosh[a + b*x + c*x^2]^2/x,x]

[Out]

Log[x]/2 + Defer[Int][Cosh[2*a + 2*b*x + 2*c*x^2]/x, x]/2

Rubi steps

\begin {align*} \int \frac {\cosh ^2\left (a+b x+c x^2\right )}{x} \, dx &=\int \left (\frac {1}{2 x}+\frac {\cosh \left (2 a+2 b x+2 c x^2\right )}{2 x}\right ) \, dx\\ &=\frac {\log (x)}{2}+\frac {1}{2} \int \frac {\cosh \left (2 a+2 b x+2 c x^2\right )}{x} \, dx\\ \end {align*}

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Mathematica [A]  time = 35.25, size = 0, normalized size = 0.00 \[ \int \frac {\cosh ^2\left (a+b x+c x^2\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Cosh[a + b*x + c*x^2]^2/x,x]

[Out]

Integrate[Cosh[a + b*x + c*x^2]^2/x, x]

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fricas [A]  time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cosh \left (c x^{2} + b x + a\right )^{2}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(c*x^2+b*x+a)^2/x,x, algorithm="fricas")

[Out]

integral(cosh(c*x^2 + b*x + a)^2/x, x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (c x^{2} + b x + a\right )^{2}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(c*x^2+b*x+a)^2/x,x, algorithm="giac")

[Out]

integrate(cosh(c*x^2 + b*x + a)^2/x, x)

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maple [A]  time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {\cosh ^{2}\left (c \,x^{2}+b x +a \right )}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(c*x^2+b*x+a)^2/x,x)

[Out]

int(cosh(c*x^2+b*x+a)^2/x,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(c*x^2+b*x+a)^2/x,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x + c*x^2)^2/x,x)

[Out]

int(cosh(a + b*x + c*x^2)^2/x, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh ^{2}{\left (a + b x + c x^{2} \right )}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(c*x**2+b*x+a)**2/x,x)

[Out]

Integral(cosh(a + b*x + c*x**2)**2/x, x)

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