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

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

[Out]

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

Verification is Not applicable to the result.

[In]

Int[x^2/(a + b*Sech[c + d*x^2]),x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 7.81, size = 0, normalized size = 0.00 \[ \int \frac {x^2}{a+b \text {sech}\left (c+d x^2\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.38, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2}}{b \operatorname {sech}\left (d x^{2} + c\right ) + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.32, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{a +b \,\mathrm {sech}\left (d \,x^{2}+c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + b/cosh(c + d*x^2)),x)

[Out]

int(x^2/(a + b/cosh(c + d*x^2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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