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

Optimal. Leaf size=66 \[ \text {Int}\left (\frac {1}{x^2 \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )},x\right )-\frac {c \sqrt {c x-1} \log \left (a+b \cosh ^{-1}(c x)\right )}{b \sqrt {1-c x}} \]

[Out]

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

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Rubi [A]  time = 0.93, 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 = {} \[ \int \frac {\sqrt {1-c^2 x^2}}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(c*Sqrt[1 - c^2*x^2]*Log[a + b*ArcCosh[c*x]])/(b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (Sqrt[1 - c^2*x^2]*Defer[Int]
[1/(x^2*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^2 \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^2 \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 {c^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {1}{x^2 \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^2 \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 {1}{\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 {c \sqrt {1-c^2 x^2} \log \left (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^2 \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*}

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Mathematica [A]  time = 1.09, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {1-c^2 x^2}}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} x^{2} + 1}}{b x^{2} \operatorname {arcosh}\left (c x\right ) + a x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-c^{2} x^{2} + 1}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.37, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-c^{2} x^{2}+1}}{x^{2} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-c^{2} x^{2} + 1}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {1-c^2\,x^2}}{x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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