3.326 \(\int \frac{\sqrt{1-c^2 x^2}}{x^3 (a+b \cosh ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=30 \[ \text{Unintegrable}\left (\frac{\sqrt{1-c^2 x^2}}{x^3 \left (a+b \cosh ^{-1}(c x)\right )^2},x\right ) \]

[Out]

Unintegrable[Sqrt[1 - c^2*x^2]/(x^3*(a + b*ArcCosh[c*x])^2), x]

________________________________________________________________________________________

Rubi [A]  time = 0.460794, 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 = {} \[ \int \frac{\sqrt{1-c^2 x^2}}{x^3 \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sqrt[1 - c^2*x^2]/(x^3*(a + b*ArcCosh[c*x])^2),x]

[Out]

(Sqrt[1 - c^2*x^2]*Defer[Int][(Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(x^3*(a + b*ArcCosh[c*x])^2), x])/(Sqrt[-1 + c*x]
*Sqrt[1 + c*x])

Rubi steps

\begin{align*} \int \frac{\sqrt{1-c^2 x^2}}{x^3 \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{\sqrt{-1+c x} \sqrt{1+c x}}{x^3 \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [A]  time = 154., size = 0, normalized size = 0. \[ \int \frac{\sqrt{1-c^2 x^2}}{x^3 \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sqrt[1 - c^2*x^2]/(x^3*(a + b*ArcCosh[c*x])^2),x]

[Out]

Integrate[Sqrt[1 - c^2*x^2]/(x^3*(a + b*ArcCosh[c*x])^2), x]

________________________________________________________________________________________

Maple [A]  time = 0.328, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{2}}\sqrt{-{c}^{2}{x}^{2}+1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*x^2+1)^(1/2)/x^3/(a+b*arccosh(c*x))^2,x)

[Out]

int((-c^2*x^2+1)^(1/2)/x^3/(a+b*arccosh(c*x))^2,x)

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left ({\left (c^{2} x^{2} - 1\right )}{\left (c x + 1\right )} \sqrt{c x - 1} +{\left (c^{3} x^{3} - c x\right )} \sqrt{c x + 1}\right )} \sqrt{-c x + 1}}{a b c^{3} x^{5} + \sqrt{c x + 1} \sqrt{c x - 1} a b c^{2} x^{4} - a b c x^{3} +{\left (b^{2} c^{3} x^{5} + \sqrt{c x + 1} \sqrt{c x - 1} b^{2} c^{2} x^{4} - b^{2} c x^{3}\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )} - \int \frac{{\left ({\left (c^{3} x^{3} - 4 \, c x\right )}{\left (c x + 1\right )}^{\frac{3}{2}}{\left (c x - 1\right )} +{\left (2 \, c^{4} x^{4} - 7 \, c^{2} x^{2} + 3\right )}{\left (c x + 1\right )} \sqrt{c x - 1} +{\left (c^{5} x^{5} - 3 \, c^{3} x^{3} + 2 \, c x\right )} \sqrt{c x + 1}\right )} \sqrt{-c x + 1}}{a b c^{5} x^{8} +{\left (c x + 1\right )}{\left (c x - 1\right )} a b c^{3} x^{6} - 2 \, a b c^{3} x^{6} + a b c x^{4} + 2 \,{\left (a b c^{4} x^{7} - a b c^{2} x^{5}\right )} \sqrt{c x + 1} \sqrt{c x - 1} +{\left (b^{2} c^{5} x^{8} +{\left (c x + 1\right )}{\left (c x - 1\right )} b^{2} c^{3} x^{6} - 2 \, b^{2} c^{3} x^{6} + b^{2} c x^{4} + 2 \,{\left (b^{2} c^{4} x^{7} - b^{2} c^{2} x^{5}\right )} \sqrt{c x + 1} \sqrt{c x - 1}\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*x^2+1)^(1/2)/x^3/(a+b*arccosh(c*x))^2,x, algorithm="maxima")

[Out]

-((c^2*x^2 - 1)*(c*x + 1)*sqrt(c*x - 1) + (c^3*x^3 - c*x)*sqrt(c*x + 1))*sqrt(-c*x + 1)/(a*b*c^3*x^5 + sqrt(c*
x + 1)*sqrt(c*x - 1)*a*b*c^2*x^4 - a*b*c*x^3 + (b^2*c^3*x^5 + sqrt(c*x + 1)*sqrt(c*x - 1)*b^2*c^2*x^4 - b^2*c*
x^3)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))) - integrate(((c^3*x^3 - 4*c*x)*(c*x + 1)^(3/2)*(c*x - 1) + (2*c^4
*x^4 - 7*c^2*x^2 + 3)*(c*x + 1)*sqrt(c*x - 1) + (c^5*x^5 - 3*c^3*x^3 + 2*c*x)*sqrt(c*x + 1))*sqrt(-c*x + 1)/(a
*b*c^5*x^8 + (c*x + 1)*(c*x - 1)*a*b*c^3*x^6 - 2*a*b*c^3*x^6 + a*b*c*x^4 + 2*(a*b*c^4*x^7 - a*b*c^2*x^5)*sqrt(
c*x + 1)*sqrt(c*x - 1) + (b^2*c^5*x^8 + (c*x + 1)*(c*x - 1)*b^2*c^3*x^6 - 2*b^2*c^3*x^6 + b^2*c*x^4 + 2*(b^2*c
^4*x^7 - b^2*c^2*x^5)*sqrt(c*x + 1)*sqrt(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))), x)

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-c^{2} x^{2} + 1}}{b^{2} x^{3} \operatorname{arcosh}\left (c x\right )^{2} + 2 \, a b x^{3} \operatorname{arcosh}\left (c x\right ) + a^{2} x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*x^2+1)^(1/2)/x^3/(a+b*arccosh(c*x))^2,x, algorithm="fricas")

[Out]

integral(sqrt(-c^2*x^2 + 1)/(b^2*x^3*arccosh(c*x)^2 + 2*a*b*x^3*arccosh(c*x) + a^2*x^3), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*x**2+1)**(1/2)/x**3/(a+b*acosh(c*x))**2,x)

[Out]

Integral(sqrt(-(c*x - 1)*(c*x + 1))/(x**3*(a + b*acosh(c*x))**2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*x^2+1)^(1/2)/x^3/(a+b*arccosh(c*x))^2,x, algorithm="giac")

[Out]

integrate(sqrt(-c^2*x^2 + 1)/((b*arccosh(c*x) + a)^2*x^3), x)