3.1.19 \(\int \frac {\sinh ^2(a+b x+c x^2)}{x} \, dx\) [19]

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

[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 = 0.02, 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 = {} \begin {gather*} \int \frac {\sinh ^2\left (a+b x+c x^2\right )}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Sinh[a + b*x + c*x^2]^2/x,x]

[Out]

-1/2*Log[x] + Defer[Int][Cosh[2*a + 2*b*x + 2*c*x^2]/x, x]/2

Rubi steps

\begin {align*} \int \frac {\sinh ^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 = 40.71, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sinh ^2\left (a+b x+c x^2\right )}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.58, size = 0, normalized size = 0.00 \[\int \frac {\sinh ^{2}\left (c \,x^{2}+b x +a \right )}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sinh ^{2}{\left (a + b x + c x^{2} \right )}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {sinh}\left (c\,x^2+b\,x+a\right )}^2}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a + b*x + c*x^2)^2/x,x)

[Out]

int(sinh(a + b*x + c*x^2)^2/x, x)

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