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

Optimal. Leaf size=156 \[ -\frac{2 \sqrt{1-c x} \text{Unintegrable}\left (\frac{c^2 x^2-1}{x^3 \left (a+b \cosh ^{-1}(c x)\right )},x\right )}{b c \sqrt{c x-1}}-\frac{2 c \sqrt{1-c x} \text{Unintegrable}\left (\frac{c^2 x^2-1}{x \left (a+b \cosh ^{-1}(c x)\right )},x\right )}{b \sqrt{c x-1}}-\frac{\sqrt{c x-1} \sqrt{c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c x^2 \left (a+b \cosh ^{-1}(c x)\right )} \]

[Out]

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

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Rubi [A]  time = 0.661832, 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 = {} \[ \int \frac{\left (1-c^2 x^2\right )^{3/2}}{x^2 \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 60.6377, size = 0, normalized size = 0. \[ \int \frac{\left (1-c^2 x^2\right )^{3/2}}{x^2 \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.555, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{2}} \left ( -{c}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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