### 3.23 $$\int \frac{\cosh ^2(a+b x-c x^2)}{x} \, dx$$

Optimal. Leaf size=32 $\frac{1}{2} \text{Unintegrable}\left (\frac{\cosh \left (2 a+2 b x-2 c x^2\right )}{x},x\right )+\frac{\log (x)}{2}$

[Out]

Log[x]/2 + Unintegrable[Cosh[2*a + 2*b*x - 2*c*x^2]/x, x]/2

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Rubi [A]  time = 0.0334658, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0., 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*}

Mathematica [A]  time = 34.5246, size = 0, normalized size = 0. $\int \frac{\cosh ^2\left (a+b x-c x^2\right )}{x} \, dx$

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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Maple [A]  time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \cosh \left ( -c{x}^{2}+bx+a \right ) \right ) ^{2}}{x}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \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 \left (x\right ) \end{align*}

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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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\cosh \left (c x^{2} - b x - a\right )^{2}}{x}, x\right ) \end{align*}

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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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh ^{2}{\left (a + b x - c x^{2} \right )}}{x}\, dx \end{align*}

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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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (-c x^{2} + b x + a\right )^{2}}{x}\,{d x} \end{align*}

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)