3.1.19 \(\int \frac {x^2}{a+b \text {sech}(c+d x^2)} \, dx\) [19]

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 = 0.02, 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 {x^2}{a+b \text {sech}\left (c+d x^2\right )} \, dx \end {gather*}

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 = 5.10, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2}{a+b \text {sech}\left (c+d x^2\right )} \, dx \end {gather*}

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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Maple [A]
time = 1.35, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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

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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