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

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

[Out]

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

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Rubi [A]  time = 0.0282825, 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 = {} \[ \int \frac{x^2}{\left (a+b \text{sech}\left (c+d x^2\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^2/(a + b*Sech[c + d*x^2])^2,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 48.4199, size = 0, normalized size = 0. \[ \int \frac{x^2}{\left (a+b \text{sech}\left (c+d x^2\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.085, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{ \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^2/(a+b*sech(d*x^2+c))^2,x)

[Out]

int(x^2/(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{{\left (a^{3} d e^{\left (2 \, c\right )} - a b^{2} d e^{\left (2 \, c\right )}\right )} x^{3} e^{\left (2 \, d x^{2}\right )} - 3 \, a b^{2} x +{\left (a^{3} d - a b^{2} d\right )} x^{3} -{\left (3 \, b^{3} x e^{c} - 2 \,{\left (a^{2} b d e^{c} - b^{3} d e^{c}\right )} x^{3}\right )} e^{\left (d x^{2}\right )}}{3 \,{\left (a^{5} d - a^{3} b^{2} d +{\left (a^{5} d e^{\left (2 \, c\right )} - a^{3} b^{2} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x^{2}\right )} + 2 \,{\left (a^{4} b d e^{c} - a^{2} b^{3} d e^{c}\right )} e^{\left (d x^{2}\right )}\right )}} - \int -\frac{a b^{2} +{\left (b^{3} e^{c} - 2 \,{\left (2 \, a^{2} b d e^{c} - b^{3} d e^{c}\right )} x^{2}\right )} e^{\left (d x^{2}\right )}}{a^{5} d - a^{3} b^{2} d +{\left (a^{5} d e^{\left (2 \, c\right )} - a^{3} b^{2} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x^{2}\right )} + 2 \,{\left (a^{4} b d e^{c} - a^{2} b^{3} d e^{c}\right )} e^{\left (d x^{2}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\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**2/(a+b*sech(d*x**2+c))**2,x)

[Out]

Integral(x**2/(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 \frac{x^{2}}{{\left (b \operatorname{sech}\left (d x^{2} + c\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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