3.189 \(\int \frac{a+b \text{sech}^{-1}(c x)}{x \sqrt{1-c^4 x^4}} \, dx\)

Optimal. Leaf size=28 \[ \text{Unintegrable}\left (\frac{a+b \text{sech}^{-1}(c x)}{x \sqrt{1-c^4 x^4}},x\right ) \]

[Out]

Unintegrable[(a + b*ArcSech[c*x])/(x*Sqrt[1 - c^4*x^4]), x]

________________________________________________________________________________________

Rubi [A]  time = 0.0910904, 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 = {} \[ \int \frac{a+b \text{sech}^{-1}(c x)}{x \sqrt{1-c^4 x^4}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcSech[c*x])/(x*Sqrt[1 - c^4*x^4]),x]

[Out]

Defer[Int][(a + b*ArcSech[c*x])/(x*Sqrt[1 - c^4*x^4]), x]

Rubi steps

\begin{align*} \int \frac{a+b \text{sech}^{-1}(c x)}{x \sqrt{1-c^4 x^4}} \, dx &=\int \frac{a+b \text{sech}^{-1}(c x)}{x \sqrt{1-c^4 x^4}} \, dx\\ \end{align*}

Mathematica [A]  time = 0.391118, size = 0, normalized size = 0. \[ \int \frac{a+b \text{sech}^{-1}(c x)}{x \sqrt{1-c^4 x^4}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*ArcSech[c*x])/(x*Sqrt[1 - c^4*x^4]),x]

[Out]

Integrate[(a + b*ArcSech[c*x])/(x*Sqrt[1 - c^4*x^4]), x]

________________________________________________________________________________________

Maple [A]  time = 1.691, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\rm arcsech} \left (cx\right )}{x}{\frac{1}{\sqrt{-{c}^{4}{x}^{4}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsech(c*x))/x/(-c^4*x^4+1)^(1/2),x)

[Out]

int((a+b*arcsech(c*x))/x/(-c^4*x^4+1)^(1/2),x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsech(c*x))/x/(-c^4*x^4+1)^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c^{4} x^{4} + 1}{\left (b \operatorname{arsech}\left (c x\right ) + a\right )}}{c^{4} x^{5} - x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsech(c*x))/x/(-c^4*x^4+1)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-c^4*x^4 + 1)*(b*arcsech(c*x) + a)/(c^4*x^5 - x), x)

________________________________________________________________________________________

Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{asech}{\left (c x \right )}}{x \sqrt{- \left (c x - 1\right ) \left (c x + 1\right ) \left (c^{2} x^{2} + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asech(c*x))/x/(-c**4*x**4+1)**(1/2),x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsech(c*x))/x/(-c^4*x^4+1)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arcsech(c*x) + a)/(sqrt(-c^4*x^4 + 1)*x), x)