3.1.29 \(\int \frac {1}{x^2 (a+b \text {sech}(c+d x^2))^2} \, dx\) [29]

Optimal. Leaf size=21 \[ \text {Int}\left (\frac {1}{x^2 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2},x\right ) \]

[Out]

Unintegrable(1/x^2/(a+b*sech(d*x^2+c))^2,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 {1}{x^2 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/(x^2*(a + b*Sech[c + d*x^2])^2),x]

[Out]

Defer[Int][1/(x^2*(a + b*Sech[c + d*x^2])^2), x]

Rubi steps

\begin {align*} \int \frac {1}{x^2 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2} \, dx &=\int \frac {1}{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 = 44.07, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^2 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[1/(x^2*(a + b*Sech[c + d*x^2])^2),x]

[Out]

Integrate[1/(x^2*(a + b*Sech[c + d*x^2])^2), x]

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Maple [A]
time = 1.20, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{2} \left (a +b \,\mathrm {sech}\left (d \,x^{2}+c \right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(a+b*sech(d*x^2+c))^2,x)

[Out]

int(1/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 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*sech(d*x^2+c))^2,x, algorithm="maxima")

[Out]

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

[Out]

integral(1/(b^2*x^2*sech(d*x^2 + c)^2 + 2*a*b*x^2*sech(d*x^2 + c) + a^2*x^2), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{2} \left (a + b \operatorname {sech}{\left (c + d x^{2} \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(a+b*sech(d*x**2+c))**2,x)

[Out]

Integral(1/(x**2*(a + b*sech(c + d*x**2))**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(1/x^2/(a+b*sech(d*x^2+c))^2,x, algorithm="giac")

[Out]

integrate(1/((b*sech(d*x^2 + c) + a)^2*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b/cosh(c + d*x^2))^2),x)

[Out]

int(1/(x^2*(a + b/cosh(c + d*x^2))^2), x)

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