\(\int \text {arccosh}(a x)^2 \, dx\) [16]

Optimal result

Integrand size = 6, antiderivative size = 39 \[ \int \text {arccosh}(a x)^2 \, dx=2 x-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{a}+x \text {arccosh}(a x)^2 \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5879, 5915, 8} \[ \int \text {arccosh}(a x)^2 \, dx=x \text {arccosh}(a x)^2-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{a}+2 x \]

[In]

Int[ArcCosh[a*x]^2,x]

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 5879

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[x*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 5915

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sy
mbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Dist[b*
(n/(2*c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Int[(1 + c*x)^(p + 1/2)*(-
1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, p}, x] && EqQ[e1, c
*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int \text {arccosh}(a x)^2 \, dx=2 x-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{a}+x \text {arccosh}(a x)^2 \]

[In]

Integrate[ArcCosh[a*x]^2,x]

[Out]

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

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\(\frac {a x \operatorname {arccosh}\left (a x \right )^{2}-2 \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )+2 a x}{a}\)	\(39\)
default	\(\frac {a x \operatorname {arccosh}\left (a x \right )^{2}-2 \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )+2 a x}{a}\)	\(39\)

[In]

int(arccosh(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.51 \[ \int \text {arccosh}(a x)^2 \, dx=\frac {a x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} + 2 \, a x - 2 \, \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{a} \]

[In]

integrate(arccosh(a*x)^2,x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

integrate(acosh(a*x)**2,x)

[Out]

Integral(acosh(a*x)**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.82 \[ \int \text {arccosh}(a x)^2 \, dx=x \operatorname {arcosh}\left (a x\right )^{2} + 2 \, x - \frac {2 \, \sqrt {a^{2} x^{2} - 1} \operatorname {arcosh}\left (a x\right )}{a} \]

[In]

integrate(arccosh(a*x)^2,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.59 \[ \int \text {arccosh}(a x)^2 \, dx=x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} + 2 \, a {\left (\frac {x}{a} - \frac {\sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{a^{2}}\right )} \]

[In]

integrate(arccosh(a*x)^2,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

int(acosh(a*x)^2,x)

[Out]

int(acosh(a*x)^2, x)