3.4.24 \(\int \frac {\sqrt {1-c^2 x^2}}{x (a+b \cosh ^{-1}(c x))^2} \, dx\) [324]

Optimal. Leaf size=182 \[ -\frac {\sqrt {-1+c x} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c x \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\sqrt {1-c x} \text {Chi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b^2 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{b^2 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \text {Int}\left (\frac {1}{x^2 \left (a+b \cosh ^{-1}(c x)\right )},x\right )}{b c \sqrt {-1+c x}} \]

[Out]

cosh(a/b)*Shi((a+b*arccosh(c*x))/b)*(-c*x+1)^(1/2)/b^2/(c*x-1)^(1/2)-Chi((a+b*arccosh(c*x))/b)*sinh(a/b)*(-c*x
+1)^(1/2)/b^2/(c*x-1)^(1/2)-(c*x-1)^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/b/c/x/(a+b*arccosh(c*x))+(-c*x+1)^(
1/2)*Unintegrable(1/x^2/(a+b*arccosh(c*x)),x)/b/c/(c*x-1)^(1/2)

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Rubi [A]
time = 0.17, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt {1-c^2 x^2}}{x \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(b*c*x*(a + b*ArcCosh[c*x]))) - (Sqrt[1 - c*x]*CoshIntegral
[(a + b*ArcCosh[c*x])/b]*Sinh[a/b])/(b^2*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[a/b]*SinhIntegral[(a + b*ArcCos
h[c*x])/b])/(b^2*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Defer[Int][1/(x^2*(a + b*ArcCosh[c*x])), x])/(b*c*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 )^2} \, 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 )^2} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x) \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c x \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\sqrt {1-c^2 x^2} \int \frac {1}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c \sqrt {1-c^2 x^2}\right ) \int \frac {1}{a+b \cosh ^{-1}(c x)} \, dx}{b \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x) \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c x \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \cosh ^{-1}(c x)\right )}{b^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \int \frac {1}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{b c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x) \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c x \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\sqrt {1-c^2 x^2} \int \frac {1}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (\sqrt {1-c^2 x^2} \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \cosh ^{-1}(c x)\right )}{b^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (\sqrt {1-c^2 x^2} \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \cosh ^{-1}(c x)\right )}{b^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x) \sqrt {1+c x} \sqrt {1-c^2 x^2}}{b c x \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\sqrt {1-c^2 x^2} \text {Chi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{b^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \int \frac {1}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{b c \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]
time = 25.46, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1-c^2 x^2}}{x \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {-c^{2} x^{2}+1}}{x \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}\, dx\]

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))^2,x)

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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))^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{x \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \end {gather*}

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))**2,x)

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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))^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {1-c^2\,x^2}}{x\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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