3.9 \(\int x^4 (a+b \text{sech}(c+d x^2))^2 \, dx\)

Optimal. Leaf size=20 \[ \text{Unintegrable}\left (x^4 \left (a+b \text{sech}\left (c+d x^2\right )\right )^2,x\right ) \]

[Out]

Unintegrable[x^4*(a + b*Sech[c + d*x^2])^2, x]

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Rubi [A]  time = 0.0246905, 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 = {} \[ \int x^4 \left (a+b \text{sech}\left (c+d x^2\right )\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^4*(a + b*Sech[c + d*x^2])^2,x]

[Out]

Defer[Int][x^4*(a + b*Sech[c + d*x^2])^2, x]

Rubi steps

\begin{align*} \int x^4 \left (a+b \text{sech}\left (c+d x^2\right )\right )^2 \, dx &=\int x^4 \left (a+b \text{sech}\left (c+d x^2\right )\right )^2 \, dx\\ \end{align*}

Mathematica [A]  time = 8.12017, size = 0, normalized size = 0. \[ \int x^4 \left (a+b \text{sech}\left (c+d x^2\right )\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x^4*(a + b*Sech[c + d*x^2])^2,x]

[Out]

Integrate[x^4*(a + b*Sech[c + d*x^2])^2, x]

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Maple [A]  time = 0.07, size = 0, normalized size = 0. \begin{align*} \int{x}^{4} \left ( a+b{\rm sech} \left (d{x}^{2}+c\right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*sech(d*x^2+c))^2,x)

[Out]

int(x^4*(a+b*sech(d*x^2+c))^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{5} \, a^{2} x^{5} - \frac{b^{2} x^{3}}{d e^{\left (2 \, d x^{2} + 2 \, c\right )} + d} + \int \frac{4 \, a b d x^{4} e^{\left (d x^{2} + c\right )} + 3 \, b^{2} x^{2}}{d e^{\left (2 \, d x^{2} + 2 \, c\right )} + d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*sech(d*x^2+c))^2,x, algorithm="maxima")

[Out]

1/5*a^2*x^5 - b^2*x^3/(d*e^(2*d*x^2 + 2*c) + d) + integrate((4*a*b*d*x^4*e^(d*x^2 + c) + 3*b^2*x^2)/(d*e^(2*d*
x^2 + 2*c) + d), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{2} x^{4} \operatorname{sech}\left (d x^{2} + c\right )^{2} + 2 \, a b x^{4} \operatorname{sech}\left (d x^{2} + c\right ) + a^{2} x^{4}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*sech(d*x^2+c))^2,x, algorithm="fricas")

[Out]

integral(b^2*x^4*sech(d*x^2 + c)^2 + 2*a*b*x^4*sech(d*x^2 + c) + a^2*x^4, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \left (a + b \operatorname{sech}{\left (c + d x^{2} \right )}\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a+b*sech(d*x**2+c))**2,x)

[Out]

Integral(x**4*(a + b*sech(c + d*x**2))**2, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{sech}\left (d x^{2} + c\right ) + a\right )}^{2} x^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*sech(d*x^2+c))^2,x, algorithm="giac")

[Out]

integrate((b*sech(d*x^2 + c) + a)^2*x^4, x)