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

Optimal. Leaf size=21 \[ \text {Int}\left (\frac {x^2}{\left (a+b \text {sech}\left (c+d x^2\right )\right )^2},x\right ) \]

[Out]

Unintegrable(x^2/(a+b*sech(d*x^2+c))^2,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 = {} \[ \int \frac {x^2}{\left (a+b \text {sech}\left (c+d x^2\right )\right )^2} \, dx \]

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*}

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Mathematica [A]  time = 52.97, size = 0, normalized size = 0.00 \[ \int \frac {x^2}{\left (a+b \text {sech}\left (c+d x^2\right )\right )^2} \, dx \]

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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fricas [A]  time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2}}{b^{2} \operatorname {sech}\left (d x^{2} + c\right )^{2} + 2 \, a b \operatorname {sech}\left (d x^{2} + c\right ) + a^{2}}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (b \operatorname {sech}\left (d x^{2} + c\right ) + a\right )}^{2}}\,{d x} \]

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)

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maple [A]  time = 0.40, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (a +b \,\mathrm {sech}\left (d \,x^{2}+c \right )\right )^{2}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (a^{3} d e^{\left (2 \, c\right )} - a b^{2} d e^{\left (2 \, c\right )}\right )} x^{3} e^{\left (2 \, d x^{2}\right )} - 3 \, a b^{2} x + {\left (a^{3} d - a b^{2} d\right )} x^{3} - {\left (3 \, b^{3} x e^{c} - 2 \, {\left (a^{2} b d e^{c} - b^{3} d e^{c}\right )} x^{3}\right )} e^{\left (d x^{2}\right )}}{3 \, {\left (a^{5} d - a^{3} b^{2} d + {\left (a^{5} d e^{\left (2 \, c\right )} - a^{3} b^{2} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x^{2}\right )} + 2 \, {\left (a^{4} b d e^{c} - a^{2} b^{3} d e^{c}\right )} e^{\left (d x^{2}\right )}\right )}} - \int -\frac {a b^{2} + {\left (b^{3} e^{c} - 2 \, {\left (2 \, a^{2} b d e^{c} - b^{3} d e^{c}\right )} x^{2}\right )} e^{\left (d x^{2}\right )}}{a^{5} d - a^{3} b^{2} d + {\left (a^{5} d e^{\left (2 \, c\right )} - a^{3} b^{2} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x^{2}\right )} + 2 \, {\left (a^{4} b d e^{c} - a^{2} b^{3} d e^{c}\right )} e^{\left (d x^{2}\right )}}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {x^2}{{\left (a+\frac {b}{\mathrm {cosh}\left (d\,x^2+c\right )}\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + b/cosh(c + d*x^2))^2,x)

[Out]

int(x^2/(a + b/cosh(c + d*x^2))^2, x)

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

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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