### 3.33 $$\int \frac{\cosh ^2(a+b x+c x^2)}{d+e x} \, dx$$

Optimal. Leaf size=43 $\frac{1}{2} \text{Unintegrable}\left (\frac{\cosh \left (2 a+2 b x+2 c x^2\right )}{d+e x},x\right )+\frac{\log (d+e x)}{2 e}$

[Out]

Log[d + e*x]/(2*e) + Unintegrable[Cosh[2*a + 2*b*x + 2*c*x^2]/(d + e*x), x]/2

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Rubi [A]  time = 0.0472074, 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 = {} $\int \frac{\cosh ^2\left (a+b x+c x^2\right )}{d+e x} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

Int[Cosh[a + b*x + c*x^2]^2/(d + e*x),x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 8.14217, size = 0, normalized size = 0. $\int \frac{\cosh ^2\left (a+b x+c x^2\right )}{d+e x} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

Integrate[Cosh[a + b*x + c*x^2]^2/(d + e*x),x]

[Out]

Integrate[Cosh[a + b*x + c*x^2]^2/(d + e*x), x]

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Maple [A]  time = 0.116, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \cosh \left ( c{x}^{2}+bx+a \right ) \right ) ^{2}}{ex+d}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\log \left (e x + d\right )}{2 \, e} + \frac{1}{4} \, \int \frac{e^{\left (2 \, c x^{2} + 2 \, b x + 2 \, a\right )}}{e x + d}\,{d x} + \frac{1}{4} \, \int \frac{e^{\left (-2 \, c x^{2} - 2 \, b x\right )}}{e x e^{\left (2 \, a\right )} + d e^{\left (2 \, a\right )}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(e*x + d)/e + 1/4*integrate(e^(2*c*x^2 + 2*b*x + 2*a)/(e*x + d), x) + 1/4*integrate(e^(-2*c*x^2 - 2*b*x
)/(e*x*e^(2*a) + d*e^(2*a)), 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}}{e x + d}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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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}}{e x + d}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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