\(\int \frac {\text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx\) [139]

Optimal result

Integrand size = 19, antiderivative size = 32 \[ \int \frac {\text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\frac {\sqrt {-1+a x} \text {arccosh}(a x)^2}{2 a \sqrt {1-a x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {5892} \[ \int \frac {\text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\frac {\sqrt {a x-1} \text {arccosh}(a x)^2}{2 a \sqrt {1-a x}} \]

[In]

Int[ArcCosh[a*x]/Sqrt[1 - a^2*x^2],x]

[Out]

(Sqrt[-1 + a*x]*ArcCosh[a*x]^2)/(2*a*Sqrt[1 - 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 {\sqrt {-1+a x} \text {arccosh}(a x)^2}{2 a \sqrt {1-a x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.41 \[ \int \frac {\text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\frac {\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{2 a \sqrt {1-a^2 x^2}} \]

[In]

Integrate[ArcCosh[a*x]/Sqrt[1 - a^2*x^2],x]

[Out]

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

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.59

method	result	size
default	\(-\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )^{2}}{2 \left (a^{2} x^{2}-1\right ) a}\)	\(51\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(acosh(a*x)/(1 - a^2*x^2)^(1/2),x)

[Out]

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