3.275 \(\int \frac {\sqrt {1-c^2 x^2}}{x^4 (a+b \cosh ^{-1}(c x))} \, dx\)

Optimal. Leaf size=31 \[ \text {Int}\left (\frac {\sqrt {1-c^2 x^2}}{x^4 \left (a+b \cosh ^{-1}(c x)\right )},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.44, 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 = {} \[ \int \frac {\sqrt {1-c^2 x^2}}{x^4 \left (a+b \cosh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sqrt[1 - c^2*x^2]/(x^4*(a + b*ArcCosh[c*x])),x]

[Out]

(Sqrt[1 - c^2*x^2]*Defer[Int][(Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(x^4*(a + b*ArcCosh[c*x])), x])/(Sqrt[-1 + c*x]*S
qrt[1 + c*x])

Rubi steps

\begin {align*} \int \frac {\sqrt {1-c^2 x^2}}{x^4 \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^4 \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.90, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {1-c^2 x^2}}{x^4 \left (a+b \cosh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sqrt[1 - c^2*x^2]/(x^4*(a + b*ArcCosh[c*x])),x]

[Out]

Integrate[Sqrt[1 - c^2*x^2]/(x^4*(a + b*ArcCosh[c*x])), x]

________________________________________________________________________________________

fricas [A]  time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} x^{2} + 1}}{b x^{4} \operatorname {arcosh}\left (c x\right ) + a x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*x^2+1)^(1/2)/x^4/(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*x^2+1)^(1/2)/x^4/(a+b*arccosh(c*x)),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.84, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-c^{2} x^{2}+1}}{x^{4} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*x^2+1)^(1/2)/x^4/(a+b*arccosh(c*x)),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\sqrt {1-c^2\,x^2}}{x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - c^2*x^2)^(1/2)/(x^4*(a + b*acosh(c*x))),x)

[Out]

int((1 - c^2*x^2)^(1/2)/(x^4*(a + b*acosh(c*x))), x)

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{x^{4} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*x**2+1)**(1/2)/x**4/(a+b*acosh(c*x)),x)

[Out]

Integral(sqrt(-(c*x - 1)*(c*x + 1))/(x**4*(a + b*acosh(c*x))), x)

________________________________________________________________________________________