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

Optimal. Leaf size=12 \[ \text{Unintegrable}\left (\frac{1}{\left (a+b \text{sech}^{-1}(c x)\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0067468, 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 = {} \[ \int \frac{1}{\left (a+b \text{sech}^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcSech[c*x])^(-2),x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 8.59317, size = 0, normalized size = 0. \[ \int \frac{1}{\left (a+b \text{sech}^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.3, size = 0, normalized size = 0. \begin{align*} \int \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/(a+b*arcsech(c*x))^2,x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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*log(c) - a*b*c^2)*x^2 - b^2*log(c) - (b^
2*log(c) + b^2*log(x) - a*b)*sqrt(c*x + 1)*sqrt(-c*x + 1) + a*b - (b^2*c^2*x^2 - sqrt(c*x + 1)*sqrt(-c*x + 1)*
b^2 - b^2)*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1) + (b^2*c^2*x^2 - b^2)*log(x)) + integrate((c^4*x^4 - 2*c^2*x^
2 + (3*c^2*x^2 - 1)*(c*x + 1)*(c*x - 1) + (2*c^4*x^4 - 5*c^2*x^2 + 2)*sqrt(c*x + 1)*sqrt(-c*x + 1) + 1)/((b^2*
c^4*log(c) - a*b*c^4)*x^4 - (b^2*log(c) + b^2*log(x) - a*b)*(c*x + 1)*(c*x - 1) - 2*(b^2*c^2*log(c) - a*b*c^2)
*x^2 + b^2*log(c) - 2*((b^2*c^2*log(c) - a*b*c^2)*x^2 - b^2*log(c) + a*b + (b^2*c^2*x^2 - b^2)*log(x))*sqrt(c*
x + 1)*sqrt(-c*x + 1) - a*b - (b^2*c^4*x^4 - 2*b^2*c^2*x^2 - (c*x + 1)*(c*x - 1)*b^2 - 2*(b^2*c^2*x^2 - b^2)*s
qrt(c*x + 1)*sqrt(-c*x + 1) + b^2)*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1) + (b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)
*log(x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Integral((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}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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