\(\int \frac {1}{\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}} \, dx\) [411]

Optimal result

Integrand size = 24, antiderivative size = 46 \[ \int \frac {1}{\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}} \, dx=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {5892} \[ \int \frac {1}{\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}} \, dx=-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)} \sqrt {c-a^2 c x^2}} \]

[In]

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

[Out]

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

Rule 5892

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*S
imp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])]*(a + b*ArcCosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e,
n}, x] && EqQ[c^2*d + e, 0] && NeQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}} \, dx=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.89

method	result	size
default	\(-\frac {2 \sqrt {a x -1}\, \sqrt {a x +1}}{\sqrt {-c \left (a x -1\right ) \left (a x +1\right )}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\, a}\)	\(41\)

[In]

int(1/(-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.28 \[ \int \frac {1}{\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}} \, dx=\frac {2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} - 1}}{{\left (a^{3} c x^{2} - a c\right )} \sqrt {\log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}} \]

[In]

integrate(1/(-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(3/2),x, algorithm="fricas")

[Out]

2*sqrt(-a^2*c*x^2 + c)*sqrt(a^2*x^2 - 1)/((a^3*c*x^2 - a*c)*sqrt(log(a*x + sqrt(a^2*x^2 - 1))))

Sympy [F]

\[ \int \frac {1}{\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}} \, dx=\int \frac {1}{\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {acosh}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]

[In]

integrate(1/(-a**2*c*x**2+c)**(1/2)/acosh(a*x)**(3/2),x)

[Out]

Integral(1/(sqrt(-c*(a*x - 1)*(a*x + 1))*acosh(a*x)**(3/2)), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}} \, dx=\int { \frac {1}{\sqrt {-a^{2} c x^{2} + c} \operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/(-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(3/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {1}{\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}} \, dx=\int { \frac {1}{\sqrt {-a^{2} c x^{2} + c} \operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/(-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(3/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}} \, dx=\int \frac {1}{{\mathrm {acosh}\left (a\,x\right )}^{3/2}\,\sqrt {c-a^2\,c\,x^2}} \,d x \]

[In]

int(1/(acosh(a*x)^(3/2)*(c - a^2*c*x^2)^(1/2)),x)

[Out]

int(1/(acosh(a*x)^(3/2)*(c - a^2*c*x^2)^(1/2)), x)