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

Optimal. Leaf size=31 \[ \text {Int}\left (\frac {1}{x^2 \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.51, 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 = {} \[ \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.31, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} x^{2} + 1}}{a c^{2} x^{4} - a x^{2} + {\left (b c^{2} x^{4} - b x^{2}\right )} \operatorname {arcosh}\left (c x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-c^2*x^2 + 1)/(a*c^2*x^4 - a*x^2 + (b*c^2*x^4 - b*x^2)*arccosh(c*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.44, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right ) \sqrt {-c^{2} x^{2}+1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(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 = -0.03 \[ \int \frac {1}{x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {1-c^2\,x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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