\(\int \text {arccosh}(a x)^3 \, dx\) [26]

Optimal result

Integrand size = 6, antiderivative size = 68 \[ \int \text {arccosh}(a x)^3 \, dx=-\frac {6 \sqrt {-1+a x} \sqrt {1+a x}}{a}+6 x \text {arccosh}(a x)-\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{a}+x \text {arccosh}(a x)^3 \]

[Out]

6*x*arccosh(a*x)+x*arccosh(a*x)^3-6*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a-3*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)
/a

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5879, 5915, 75} \[ \int \text {arccosh}(a x)^3 \, dx=x \text {arccosh}(a x)^3-\frac {3 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{a}+6 x \text {arccosh}(a x)-\frac {6 \sqrt {a x-1} \sqrt {a x+1}}{a} \]

[In]

Int[ArcCosh[a*x]^3,x]

[Out]

(-6*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/a + 6*x*ArcCosh[a*x] - (3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/a + x
*ArcCosh[a*x]^3

Rule 75

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

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)^3-(3 a) \int \frac {x \text {arccosh}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = -\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{a}+x \text {arccosh}(a x)^3+6 \int \text {arccosh}(a x) \, dx \\ & = 6 x \text {arccosh}(a x)-\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{a}+x \text {arccosh}(a x)^3-(6 a) \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = -\frac {6 \sqrt {-1+a x} \sqrt {1+a x}}{a}+6 x \text {arccosh}(a x)-\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{a}+x \text {arccosh}(a x)^3 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00 \[ \int \text {arccosh}(a x)^3 \, dx=-\frac {6 \sqrt {-1+a x} \sqrt {1+a x}}{a}+6 x \text {arccosh}(a x)-\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{a}+x \text {arccosh}(a x)^3 \]

[In]

Integrate[ArcCosh[a*x]^3,x]

[Out]

(-6*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/a + 6*x*ArcCosh[a*x] - (3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/a + x
*ArcCosh[a*x]^3

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.90

method	result	size
derivativedivides	\(\frac {a x \operatorname {arccosh}\left (a x \right )^{3}-3 \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}+6 a x \,\operatorname {arccosh}\left (a x \right )-6 \sqrt {a x -1}\, \sqrt {a x +1}}{a}\)	\(61\)
default	\(\frac {a x \operatorname {arccosh}\left (a x \right )^{3}-3 \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}+6 a x \,\operatorname {arccosh}\left (a x \right )-6 \sqrt {a x -1}\, \sqrt {a x +1}}{a}\)	\(61\)

[In]

int(arccosh(a*x)^3,x,method=_RETURNVERBOSE)

[Out]

1/a*(a*x*arccosh(a*x)^3-3*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+6*a*x*arccosh(a*x)-6*(a*x-1)^(1/2)*(a*x+1
)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.32 \[ \int \text {arccosh}(a x)^3 \, dx=\frac {a x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{3} + 6 \, a x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - 3 \, \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} - 6 \, \sqrt {a^{2} x^{2} - 1}}{a} \]

[In]

integrate(arccosh(a*x)^3,x, algorithm="fricas")

[Out]

(a*x*log(a*x + sqrt(a^2*x^2 - 1))^3 + 6*a*x*log(a*x + sqrt(a^2*x^2 - 1)) - 3*sqrt(a^2*x^2 - 1)*log(a*x + sqrt(
a^2*x^2 - 1))^2 - 6*sqrt(a^2*x^2 - 1))/a

Sympy [F]

\[ \int \text {arccosh}(a x)^3 \, dx=\int \operatorname {acosh}^{3}{\left (a x \right )}\, dx \]

[In]

integrate(acosh(a*x)**3,x)

[Out]

Integral(acosh(a*x)**3, x)

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.84 \[ \int \text {arccosh}(a x)^3 \, dx=x \operatorname {arcosh}\left (a x\right )^{3} - \frac {3 \, \sqrt {a^{2} x^{2} - 1} \operatorname {arcosh}\left (a x\right )^{2}}{a} + \frac {6 \, {\left (a x \operatorname {arcosh}\left (a x\right ) - \sqrt {a^{2} x^{2} - 1}\right )}}{a} \]

[In]

integrate(arccosh(a*x)^3,x, algorithm="maxima")

[Out]

x*arccosh(a*x)^3 - 3*sqrt(a^2*x^2 - 1)*arccosh(a*x)^2/a + 6*(a*x*arccosh(a*x) - sqrt(a^2*x^2 - 1))/a

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.44 \[ \int \text {arccosh}(a x)^3 \, dx=x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{3} - 3 \, a {\left (\frac {\sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2}}{a^{2}} - \frac {2 \, {\left (x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - \frac {\sqrt {a^{2} x^{2} - 1}}{a}\right )}}{a}\right )} \]

[In]

integrate(arccosh(a*x)^3,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \text {arccosh}(a x)^3 \, dx=\int {\mathrm {acosh}\left (a\,x\right )}^3 \,d x \]

[In]

int(acosh(a*x)^3,x)

[Out]

int(acosh(a*x)^3, x)