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

Optimal. Leaf size=116 \[ \text{Unintegrable}\left (\frac{1}{x \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )},x\right )-\frac{\sqrt{c x-1} \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{b \sqrt{1-c x}}+\frac{\sqrt{c x-1} \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{b \sqrt{1-c x}} \]

[Out]

-((Sqrt[-1 + c*x]*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/(b*Sqrt[1 - c*x])) + (Sqrt[-1 + c*x]*Sinh[a/
b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(b*Sqrt[1 - c*x]) + Unintegrable[1/(x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos
h[c*x])), x]

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Rubi [A]  time = 1.07703, 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 = {} \[ \int \frac{\sqrt{1-c^2 x^2}}{x \left (a+b \cosh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.20046, size = 0, normalized size = 0. \[ \int \frac{\sqrt{1-c^2 x^2}}{x \left (a+b \cosh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.234, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( a+b{\rm arccosh} \left (cx\right ) \right ) }\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/(a+b*arccosh(c*x)),x)

[Out]

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

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Maxima [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 )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-c^2*x^2 + 1)/((b*arccosh(c*x) + a)*x), 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}}{b x \operatorname{arcosh}\left (c x\right ) + a x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 \left (a + b \operatorname{acosh}{\left (c x \right )}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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