3.4.18 \(\int \frac {1}{(c-a^2 c x^2) \cosh ^{-1}(a x)^2} \, dx\) [318]

Optimal. Leaf size=66 \[ \frac {1}{a c \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}+\frac {a \text {Int}\left (\frac {x}{(-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)},x\right )}{c} \]

[Out]

1/a/c/arccosh(a*x)/(a*x-1)^(1/2)/(a*x+1)^(1/2)+a*Unintegrable(x/(a*x-1)^(3/2)/(a*x+1)^(3/2)/arccosh(a*x),x)/c

________________________________________________________________________________________

Rubi [A]
time = 0.16, 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 {1}{\left (c-a^2 c x^2\right ) \cosh ^{-1}(a x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

1/(a*c*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]) + (a*Defer[Int][x/((-1 + a*x)^(3/2)*(1 + a*x)^(3/2)*ArcCosh[
a*x]), x])/c

Rubi steps

\begin {align*} \int \frac {1}{\left (c-a^2 c x^2\right ) \cosh ^{-1}(a x)^2} \, dx &=\frac {1}{a c \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}+\frac {a \int \frac {x}{(-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)} \, dx}{c}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 2.27, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (c-a^2 c x^2\right ) \cosh ^{-1}(a x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{{\mathrm {acosh}\left (a\,x\right )}^2\,\left (c-a^2\,c\,x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________