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

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

[Out]

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

Verification is Not applicable to the result.

[In]

Int[(a + b*Sech[c + d*x^2])^2/x^2,x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 9.85, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \text {sech}\left (c+d x^2\right )\right )^2}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {sech}\left (d x^{2} + c\right )^{2} + 2 \, a b \operatorname {sech}\left (d x^{2} + c\right ) + a^{2}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/cosh(c + d*x^2))^2/x^2,x)

[Out]

int((a + b/cosh(c + d*x^2))^2/x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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