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

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

[Out]

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

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Rubi [A]  time = 0.549333, 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 = {} \[ \int \frac{1}{x \left (1-c^2 x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 21.8446, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (1-c^2 x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.369, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \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(1/x/(-c^2*x^2+1)^(3/2)/(a+b*arccosh(c*x))^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(((c*x + 1)*sqrt(c*x - 1)*b^2*c^2*x^2 + (b^2*c^3*x^3 - b^2*c*x)*sqrt(c*x +
1))*sqrt(-c*x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + ((c*x + 1)*sqrt(c*x - 1)*a*b*c^2*x^2 + (a*b*c^3*x
^3 - a*b*c*x)*sqrt(c*x + 1))*sqrt(-c*x + 1)) + integrate((3*c^5*x^5 - 3*c^3*x^3 + (3*c^3*x^3 - 2*c*x)*(c*x + 1
)*(c*x - 1) + (6*c^4*x^4 - 5*c^2*x^2 + 1)*sqrt(c*x + 1)*sqrt(c*x - 1))/(((b^2*c^5*x^6 - b^2*c^3*x^4)*(c*x + 1)
^(3/2)*(c*x - 1) + 2*(b^2*c^6*x^7 - 2*b^2*c^4*x^5 + b^2*c^2*x^3)*(c*x + 1)*sqrt(c*x - 1) + (b^2*c^7*x^8 - 3*b^
2*c^5*x^6 + 3*b^2*c^3*x^4 - b^2*c*x^2)*sqrt(c*x + 1))*sqrt(-c*x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) +
((a*b*c^5*x^6 - a*b*c^3*x^4)*(c*x + 1)^(3/2)*(c*x - 1) + 2*(a*b*c^6*x^7 - 2*a*b*c^4*x^5 + a*b*c^2*x^3)*(c*x +
1)*sqrt(c*x - 1) + (a*b*c^7*x^8 - 3*a*b*c^5*x^6 + 3*a*b*c^3*x^4 - a*b*c*x^2)*sqrt(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{\sqrt{-c^{2} x^{2} + 1}}{a^{2} c^{4} x^{5} - 2 \, a^{2} c^{2} x^{3} + a^{2} x +{\left (b^{2} c^{4} x^{5} - 2 \, b^{2} c^{2} x^{3} + b^{2} x\right )} \operatorname{arcosh}\left (c x\right )^{2} + 2 \,{\left (a b c^{4} x^{5} - 2 \, a b c^{2} x^{3} + a b x\right )} \operatorname{arcosh}\left (c x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(-c^2*x^2 + 1)/(a^2*c^4*x^5 - 2*a^2*c^2*x^3 + a^2*x + (b^2*c^4*x^5 - 2*b^2*c^2*x^3 + b^2*x)*arcco
sh(c*x)^2 + 2*(a*b*c^4*x^5 - 2*a*b*c^2*x^3 + a*b*x)*arccosh(c*x)), 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(1/x/(-c**2*x**2+1)**(3/2)/(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{1}{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (b \operatorname{arcosh}\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/(-c^2*x^2+1)^(3/2)/(a+b*arccosh(c*x))^2,x, algorithm="giac")

[Out]

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