\(\int x \text {arccosh}(a x)^3 \, dx\) [25]

Optimal result

Integrand size = 8, antiderivative size = 107 \[ \int x \text {arccosh}(a x)^3 \, dx=-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x}}{8 a}-\frac {3 \text {arccosh}(a x)}{8 a^2}+\frac {3}{4} x^2 \text {arccosh}(a x)-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{4 a}-\frac {\text {arccosh}(a x)^3}{4 a^2}+\frac {1}{2} x^2 \text {arccosh}(a x)^3 \]

[Out]

-3/8*arccosh(a*x)/a^2+3/4*x^2*arccosh(a*x)-1/4*arccosh(a*x)^3/a^2+1/2*x^2*arccosh(a*x)^3-3/8*x*(a*x-1)^(1/2)*(
a*x+1)^(1/2)/a-3/4*x*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5883, 5939, 5893, 92, 54} \[ \int x \text {arccosh}(a x)^3 \, dx=-\frac {\text {arccosh}(a x)^3}{4 a^2}-\frac {3 \text {arccosh}(a x)}{8 a^2}+\frac {1}{2} x^2 \text {arccosh}(a x)^3+\frac {3}{4} x^2 \text {arccosh}(a x)-\frac {3 x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{4 a}-\frac {3 x \sqrt {a x-1} \sqrt {a x+1}}{8 a} \]

[In]

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

[Out]

(-3*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(8*a) - (3*ArcCosh[a*x])/(8*a^2) + (3*x^2*ArcCosh[a*x])/4 - (3*x*Sqrt[-1 +
a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(4*a) - ArcCosh[a*x]^3/(4*a^2) + (x^2*ArcCosh[a*x]^3)/2

Rule 54

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[b*(x/a)]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rule 92

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a + b*x
)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 5883

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcC
osh[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5893

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*Arc
Cosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && NeQ[n
, -1]

Rule 5939

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_
))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1
*e2*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m + 2*p + 1))), Int[(f*x)^(m - 2)*(d1 + e1*x)^p*(d2 + e2*x)
^p*(a + b*ArcCosh[c*x])^n, x], x] - Dist[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 +
e2*x)^p/(-1 + c*x)^p], Int[(f*x)^(m - 1)*(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, f, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IG
tQ[m, 1] && NeQ[m + 2*p + 1, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \text {arccosh}(a x)^3-\frac {1}{2} (3 a) \int \frac {x^2 \text {arccosh}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = -\frac {3 x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{4 a}+\frac {1}{2} x^2 \text {arccosh}(a x)^3+\frac {3}{2} \int x \text {arccosh}(a x) \, dx-\frac {3 \int \frac {\text {arccosh}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{4 a} \\ & = \frac {3}{4} x^2 \text {arccosh}(a x)-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{4 a}-\frac {\text {arccosh}(a x)^3}{4 a^2}+\frac {1}{2} x^2 \text {arccosh}(a x)^3-\frac {1}{4} (3 a) \int \frac {x^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = -\frac {3 x \sqrt {-1+a x} \sqrt {1+a x}}{8 a}+\frac {3}{4} x^2 \text {arccosh}(a x)-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{4 a}-\frac {\text {arccosh}(a x)^3}{4 a^2}+\frac {1}{2} x^2 \text {arccosh}(a x)^3-\frac {3 \int \frac {1}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{8 a} \\ & = -\frac {3 x \sqrt {-1+a x} \sqrt {1+a x}}{8 a}-\frac {3 \text {arccosh}(a x)}{8 a^2}+\frac {3}{4} x^2 \text {arccosh}(a x)-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{4 a}-\frac {\text {arccosh}(a x)^3}{4 a^2}+\frac {1}{2} x^2 \text {arccosh}(a x)^3 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.06 \[ \int x \text {arccosh}(a x)^3 \, dx=\frac {6 a^2 x^2 \text {arccosh}(a x)-6 a x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2+\left (-2+4 a^2 x^2\right ) \text {arccosh}(a x)^3-3 \left (a x \sqrt {-1+a x} \sqrt {1+a x}+\log \left (a x+\sqrt {-1+a x} \sqrt {1+a x}\right )\right )}{8 a^2} \]

[In]

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

[Out]

(6*a^2*x^2*ArcCosh[a*x] - 6*a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2 + (-2 + 4*a^2*x^2)*ArcCosh[a*x]^3
- 3*(a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x] + Log[a*x + Sqrt[-1 + a*x]*Sqrt[1 + a*x]]))/(8*a^2)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.82

method	result	size
derivativedivides	\(\frac {\frac {a^{2} x^{2} \operatorname {arccosh}\left (a x \right )^{3}}{2}-\frac {3 a x \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{4}-\frac {\operatorname {arccosh}\left (a x \right )^{3}}{4}+\frac {3 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )}{4}-\frac {3 \sqrt {a x -1}\, \sqrt {a x +1}\, a x}{8}-\frac {3 \,\operatorname {arccosh}\left (a x \right )}{8}}{a^{2}}\)	\(88\)
default	\(\frac {\frac {a^{2} x^{2} \operatorname {arccosh}\left (a x \right )^{3}}{2}-\frac {3 a x \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{4}-\frac {\operatorname {arccosh}\left (a x \right )^{3}}{4}+\frac {3 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )}{4}-\frac {3 \sqrt {a x -1}\, \sqrt {a x +1}\, a x}{8}-\frac {3 \,\operatorname {arccosh}\left (a x \right )}{8}}{a^{2}}\)	\(88\)

[In]

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

[Out]

1/a^2*(1/2*a^2*x^2*arccosh(a*x)^3-3/4*a*x*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-1/4*arccosh(a*x)^3+3/4*a^
2*x^2*arccosh(a*x)-3/8*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a*x-3/8*arccosh(a*x))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.05 \[ \int x \text {arccosh}(a x)^3 \, dx=-\frac {6 \, \sqrt {a^{2} x^{2} - 1} a x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} - 2 \, {\left (2 \, a^{2} x^{2} - 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{3} + 3 \, \sqrt {a^{2} x^{2} - 1} a x - 3 \, {\left (2 \, a^{2} x^{2} - 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{8 \, a^{2}} \]

[In]

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

[Out]

-1/8*(6*sqrt(a^2*x^2 - 1)*a*x*log(a*x + sqrt(a^2*x^2 - 1))^2 - 2*(2*a^2*x^2 - 1)*log(a*x + sqrt(a^2*x^2 - 1))^
3 + 3*sqrt(a^2*x^2 - 1)*a*x - 3*(2*a^2*x^2 - 1)*log(a*x + sqrt(a^2*x^2 - 1)))/a^2

Sympy [F]

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

[In]

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

[Out]

Integral(x*acosh(a*x)**3, x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate(x*arccosh(a*x)^3, x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(x*acosh(a*x)^3, x)