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

Optimal. Leaf size=20 \[ \text{Unintegrable}\left (\frac{1}{x \left (a+b \text{sech}\left (c+d x^2\right )\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0264937, 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 = {} \[ \int \frac{1}{x \left (a+b \text{sech}\left (c+d x^2\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 98.2405, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (a+b \text{sech}\left (c+d x^2\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.115, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \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(1/x/(a+b*sech(d*x^2+c))^2,x)

[Out]

int(1/x/(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{b^{3} e^{\left (d x^{2} + c\right )} + a b^{2}}{{\left (a^{5} d e^{\left (2 \, c\right )} - a^{3} b^{2} d e^{\left (2 \, c\right )}\right )} x^{2} e^{\left (2 \, d x^{2}\right )} + 2 \,{\left (a^{4} b d e^{c} - a^{2} b^{3} d e^{c}\right )} x^{2} e^{\left (d x^{2}\right )} +{\left (a^{5} d - a^{3} b^{2} d\right )} x^{2}} + \frac{\log \left (x\right )}{a^{2}} - \int \frac{2 \,{\left (a b^{2} +{\left (b^{3} e^{c} +{\left (2 \, a^{2} b d e^{c} - b^{3} d e^{c}\right )} x^{2}\right )} e^{\left (d x^{2}\right )}\right )}}{{\left (a^{5} d e^{\left (2 \, c\right )} - a^{3} b^{2} d e^{\left (2 \, c\right )}\right )} x^{3} e^{\left (2 \, d x^{2}\right )} + 2 \,{\left (a^{4} b d e^{c} - a^{2} b^{3} d e^{c}\right )} x^{3} e^{\left (d x^{2}\right )} +{\left (a^{5} d - a^{3} b^{2} d\right )} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b^{2} x \operatorname{sech}\left (d x^{2} + c\right )^{2} + 2 \, a b x \operatorname{sech}\left (d x^{2} + c\right ) + a^{2} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \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(1/x/(a+b*sech(d*x**2+c))**2,x)

[Out]

Integral(1/(x*(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{1}{{\left (b \operatorname{sech}\left (d x^{2} + c\right ) + a\right )}^{2} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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