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

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

[Out]

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

Verification is Not applicable to the result.

[In]

Int[1/(x*(a + b*Sech[c + d*x^2])),x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 5.18, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a+b \text {sech}\left (c+d x^2\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b x \operatorname {sech}\left (d x^{2} + c\right ) + a x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a +b \,\mathrm {sech}\left (d \,x^{2}+c \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b/cosh(c + d*x^2))),x)

[Out]

int(1/(x*(a + b/cosh(c + d*x^2))), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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