3.1.33 \(\int \frac {\sinh ^2(a+b x+c x^2)}{d+e x} \, dx\) [33]

Optimal. Leaf size=44 \[ -\frac {\log (d+e x)}{2 e}+\frac {1}{2} \text {Int}\left (\frac {\cosh \left (2 a+2 b x+2 c x^2\right )}{d+e x},x\right ) \]

[Out]

-1/2*ln(e*x+d)/e+1/2*Unintegrable(cosh(2*c*x^2+2*b*x+2*a)/(e*x+d),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 = {} \begin {gather*} \int \frac {\sinh ^2\left (a+b x+c x^2\right )}{d+e x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\sinh ^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*}

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Mathematica [A]
time = 10.86, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sinh ^2\left (a+b x+c x^2\right )}{d+e x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sinh ^{2}\left (c \,x^{2}+b x +a \right )}{e x +d}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-1/2*e^(-1)*log(x*e + d) + 1/4*integrate(e^(2*c*x^2 + 2*b*x + 2*a)/(x*e + d), x) + 1/4*integrate(e^(-2*c*x^2 -
2*b*x)/(d*e^(2*a) + x*e^(2*a + 1)), 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/(e*x+d),x, algorithm="fricas")

[Out]

integral(sinh(c*x^2 + b*x + a)^2/(x*e + d), 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 )}}{d + e x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sinh(a + b*x + c*x**2)**2/(d + e*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/(e*x+d),x, algorithm="giac")

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\mathrm {sinh}\left (c\,x^2+b\,x+a\right )}^2}{d+e\,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/(d + e*x),x)

[Out]

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

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