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

Optimal. Leaf size=16 \[ \text{Unintegrable}\left (\frac{1}{x \left (a+b \text{sech}^{-1}(c x)\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0253484, 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 = {} \[ \int \frac{1}{x \left (a+b \text{sech}^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x*(a + b*ArcSech[c*x])^2),x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 4.99292, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (a+b \text{sech}^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.286, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( a+b{\rm arcsech} \left (cx\right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{c^{2} x^{3} +{\left (c^{2} x^{3} - x\right )} \sqrt{c x + 1} \sqrt{-c x + 1} - x}{{\left (b^{2} c^{2} x^{2} - b^{2}\right )} x \log \left (x\right ) -{\left (b^{2} x \log \left (x\right ) +{\left (b^{2} \log \left (c\right ) - a b\right )} x\right )} \sqrt{c x + 1} \sqrt{-c x + 1} +{\left ({\left (b^{2} c^{2} \log \left (c\right ) - a b c^{2}\right )} x^{2} - b^{2} \log \left (c\right ) + a b\right )} x +{\left (\sqrt{c x + 1} \sqrt{-c x + 1} b^{2} x -{\left (b^{2} c^{2} x^{2} - b^{2}\right )} x\right )} \log \left (\sqrt{c x + 1} \sqrt{-c x + 1} + 1\right )} + \int -\frac{2 \,{\left (c x + 1\right )}{\left (c x - 1\right )} c^{2} x^{2} +{\left (c^{4} x^{4} - 2 \, c^{2} x^{2}\right )} \sqrt{c x + 1} \sqrt{-c x + 1}}{{\left (b^{2} x \log \left (x\right ) +{\left (b^{2} \log \left (c\right ) - a b\right )} x\right )}{\left (c x + 1\right )}{\left (c x - 1\right )} -{\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} x \log \left (x\right ) + 2 \,{\left ({\left (b^{2} c^{2} x^{2} - b^{2}\right )} x \log \left (x\right ) +{\left ({\left (b^{2} c^{2} \log \left (c\right ) - a b c^{2}\right )} x^{2} - b^{2} \log \left (c\right ) + a b\right )} x\right )} \sqrt{c x + 1} \sqrt{-c x + 1} -{\left ({\left (b^{2} c^{4} \log \left (c\right ) - a b c^{4}\right )} x^{4} - 2 \,{\left (b^{2} c^{2} \log \left (c\right ) - a b c^{2}\right )} x^{2} + b^{2} \log \left (c\right ) - a b\right )} x -{\left ({\left (c x + 1\right )}{\left (c x - 1\right )} b^{2} x + 2 \,{\left (b^{2} c^{2} x^{2} - b^{2}\right )} \sqrt{c x + 1} \sqrt{-c x + 1} x -{\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} x\right )} \log \left (\sqrt{c x + 1} \sqrt{-c x + 1} + 1\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(c^2*x^3 + (c^2*x^3 - x)*sqrt(c*x + 1)*sqrt(-c*x + 1) - x)/((b^2*c^2*x^2 - b^2)*x*log(x) - (b^2*x*log(x) + (b
^2*log(c) - a*b)*x)*sqrt(c*x + 1)*sqrt(-c*x + 1) + ((b^2*c^2*log(c) - a*b*c^2)*x^2 - b^2*log(c) + a*b)*x + (sq
rt(c*x + 1)*sqrt(-c*x + 1)*b^2*x - (b^2*c^2*x^2 - b^2)*x)*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1)) + integrate(-
(2*(c*x + 1)*(c*x - 1)*c^2*x^2 + (c^4*x^4 - 2*c^2*x^2)*sqrt(c*x + 1)*sqrt(-c*x + 1))/((b^2*x*log(x) + (b^2*log
(c) - a*b)*x)*(c*x + 1)*(c*x - 1) - (b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*x*log(x) + 2*((b^2*c^2*x^2 - b^2)*x*lo
g(x) + ((b^2*c^2*log(c) - a*b*c^2)*x^2 - b^2*log(c) + a*b)*x)*sqrt(c*x + 1)*sqrt(-c*x + 1) - ((b^2*c^4*log(c)
- a*b*c^4)*x^4 - 2*(b^2*c^2*log(c) - a*b*c^2)*x^2 + b^2*log(c) - a*b)*x - ((c*x + 1)*(c*x - 1)*b^2*x + 2*(b^2*
c^2*x^2 - b^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*x - (b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*x)*log(sqrt(c*x + 1)*sqrt(
-c*x + 1) + 1)), 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{arsech}\left (c x\right )^{2} + 2 \, a b x \operatorname{arsech}\left (c x\right ) + a^{2} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/(b^2*x*arcsech(c*x)^2 + 2*a*b*x*arcsech(c*x) + 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{asech}{\left (c x \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \operatorname{arsech}\left (c x\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*arcsech(c*x))^2,x, algorithm="giac")

[Out]

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