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

Optimal. Leaf size=84 \[ -\frac{\sqrt{c x-1} \text{Unintegrable}\left (\frac{1}{x^2 \left (a+b \cosh ^{-1}(c x)\right )},x\right )}{b c \sqrt{1-c x}}-\frac{\sqrt{c x-1}}{b c x \sqrt{1-c x} \left (a+b \cosh ^{-1}(c x)\right )} \]

[Out]

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

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Rubi [A]  time = 0.527555, 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 = {} \[ \int \frac{1}{x \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 4.60314, size = 0, normalized size = 0. \[ \int \frac{1}{x \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.231, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{2}}{\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

-(c^3*x^3 + (c^2*x^2 - 1)*sqrt(c*x + 1)*sqrt(c*x - 1) - c*x)/(((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((c^5*x^5 - c^3*x^3 + (c^3*x^
3 - 2*c*x)*(c*x + 1)*(c*x - 1) + (2*c^4*x^4 - 3*c^2*x^2 + 1)*sqrt(c*x + 1)*sqrt(c*x - 1))/(((c*x + 1)^(3/2)*(c
*x - 1)*b^2*c^3*x^4 + 2*(b^2*c^4*x^5 - b^2*c^2*x^3)*(c*x + 1)*sqrt(c*x - 1) + (b^2*c^5*x^6 - 2*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)) + ((c*x + 1)^(3/2)*(c*x - 1)*a*
b*c^3*x^4 + 2*(a*b*c^4*x^5 - a*b*c^2*x^3)*(c*x + 1)*sqrt(c*x - 1) + (a*b*c^5*x^6 - 2*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^{2} x^{3} - a^{2} x +{\left (b^{2} c^{2} x^{3} - b^{2} x\right )} \operatorname{arcosh}\left (c x\right )^{2} + 2 \,{\left (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/(a+b*arccosh(c*x))^2/(-c^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

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

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-c^{2} x^{2} + 1}{\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/(a+b*arccosh(c*x))^2/(-c^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

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