\(\int x^3 \text {arccosh}(a x)^3 \, dx\) [23]

Optimal result

Integrand size = 10, antiderivative size = 183 \[ \int x^3 \text {arccosh}(a x)^3 \, dx=-\frac {45 x \sqrt {-1+a x} \sqrt {1+a x}}{256 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{128 a}-\frac {45 \text {arccosh}(a x)}{256 a^4}+\frac {9 x^2 \text {arccosh}(a x)}{32 a^2}+\frac {3}{32} x^4 \text {arccosh}(a x)-\frac {9 x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{16 a}-\frac {3 \text {arccosh}(a x)^3}{32 a^4}+\frac {1}{4} x^4 \text {arccosh}(a x)^3 \]

[Out]

-45/256*arccosh(a*x)/a^4+9/32*x^2*arccosh(a*x)/a^2+3/32*x^4*arccosh(a*x)-3/32*arccosh(a*x)^3/a^4+1/4*x^4*arcco
sh(a*x)^3-45/256*x*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3-3/128*x^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a-9/32*x*arccosh(a*x)
^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3-3/16*x^3*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5883, 5939, 5893, 92, 54, 102, 12} \[ \int x^3 \text {arccosh}(a x)^3 \, dx=-\frac {3 \text {arccosh}(a x)^3}{32 a^4}-\frac {45 \text {arccosh}(a x)}{256 a^4}-\frac {9 x \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{32 a^3}-\frac {45 x \sqrt {a x-1} \sqrt {a x+1}}{256 a^3}+\frac {9 x^2 \text {arccosh}(a x)}{32 a^2}+\frac {1}{4} x^4 \text {arccosh}(a x)^3+\frac {3}{32} x^4 \text {arccosh}(a x)-\frac {3 x^3 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{16 a}-\frac {3 x^3 \sqrt {a x-1} \sqrt {a x+1}}{128 a} \]

[In]

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

[Out]

(-45*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(256*a^3) - (3*x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(128*a) - (45*ArcCosh[a*
x])/(256*a^4) + (9*x^2*ArcCosh[a*x])/(32*a^2) + (3*x^4*ArcCosh[a*x])/32 - (9*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Ar
cCosh[a*x]^2)/(32*a^3) - (3*x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(16*a) - (3*ArcCosh[a*x]^3)/(32*a
^4) + (x^4*ArcCosh[a*x]^3)/4

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 102

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

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}{4} x^4 \text {arccosh}(a x)^3-\frac {1}{4} (3 a) \int \frac {x^4 \text {arccosh}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = -\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{16 a}+\frac {1}{4} x^4 \text {arccosh}(a x)^3+\frac {3}{8} \int x^3 \text {arccosh}(a x) \, dx-\frac {9 \int \frac {x^2 \text {arccosh}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{16 a} \\ & = \frac {3}{32} x^4 \text {arccosh}(a x)-\frac {9 x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{16 a}+\frac {1}{4} x^4 \text {arccosh}(a x)^3-\frac {9 \int \frac {\text {arccosh}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{32 a^3}+\frac {9 \int x \text {arccosh}(a x) \, dx}{16 a^2}-\frac {1}{32} (3 a) \int \frac {x^4}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = -\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{128 a}+\frac {9 x^2 \text {arccosh}(a x)}{32 a^2}+\frac {3}{32} x^4 \text {arccosh}(a x)-\frac {9 x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{16 a}-\frac {3 \text {arccosh}(a x)^3}{32 a^4}+\frac {1}{4} x^4 \text {arccosh}(a x)^3-\frac {3 \int \frac {3 x^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{128 a}-\frac {9 \int \frac {x^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{32 a} \\ & = -\frac {9 x \sqrt {-1+a x} \sqrt {1+a x}}{64 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{128 a}+\frac {9 x^2 \text {arccosh}(a x)}{32 a^2}+\frac {3}{32} x^4 \text {arccosh}(a x)-\frac {9 x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{16 a}-\frac {3 \text {arccosh}(a x)^3}{32 a^4}+\frac {1}{4} x^4 \text {arccosh}(a x)^3-\frac {9 \int \frac {1}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{64 a^3}-\frac {9 \int \frac {x^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{128 a} \\ & = -\frac {45 x \sqrt {-1+a x} \sqrt {1+a x}}{256 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{128 a}-\frac {9 \text {arccosh}(a x)}{64 a^4}+\frac {9 x^2 \text {arccosh}(a x)}{32 a^2}+\frac {3}{32} x^4 \text {arccosh}(a x)-\frac {9 x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{16 a}-\frac {3 \text {arccosh}(a x)^3}{32 a^4}+\frac {1}{4} x^4 \text {arccosh}(a x)^3-\frac {9 \int \frac {1}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{256 a^3} \\ & = -\frac {45 x \sqrt {-1+a x} \sqrt {1+a x}}{256 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{128 a}-\frac {45 \text {arccosh}(a x)}{256 a^4}+\frac {9 x^2 \text {arccosh}(a x)}{32 a^2}+\frac {3}{32} x^4 \text {arccosh}(a x)-\frac {9 x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{32 a^3}-\frac {3 x^3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{16 a}-\frac {3 \text {arccosh}(a x)^3}{32 a^4}+\frac {1}{4} x^4 \text {arccosh}(a x)^3 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.78 \[ \int x^3 \text {arccosh}(a x)^3 \, dx=\frac {-3 a x \sqrt {-1+a x} \sqrt {1+a x} \left (15+2 a^2 x^2\right )+24 a^2 x^2 \left (3+a^2 x^2\right ) \text {arccosh}(a x)-24 a x \sqrt {-1+a x} \sqrt {1+a x} \left (3+2 a^2 x^2\right ) \text {arccosh}(a x)^2+8 \left (-3+8 a^4 x^4\right ) \text {arccosh}(a x)^3-45 \log \left (a x+\sqrt {-1+a x} \sqrt {1+a x}\right )}{256 a^4} \]

[In]

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

[Out]

(-3*a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(15 + 2*a^2*x^2) + 24*a^2*x^2*(3 + a^2*x^2)*ArcCosh[a*x] - 24*a*x*Sqrt[-1
+ a*x]*Sqrt[1 + a*x]*(3 + 2*a^2*x^2)*ArcCosh[a*x]^2 + 8*(-3 + 8*a^4*x^4)*ArcCosh[a*x]^3 - 45*Log[a*x + Sqrt[-
1 + a*x]*Sqrt[1 + a*x]])/(256*a^4)

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.82

method	result	size
derivativedivides	\(\frac {\frac {a^{4} x^{4} \operatorname {arccosh}\left (a x \right )^{3}}{4}-\frac {3 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{16}-\frac {9 a x \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{32}-\frac {3 \operatorname {arccosh}\left (a x \right )^{3}}{32}+\frac {3 a^{4} x^{4} \operatorname {arccosh}\left (a x \right )}{32}-\frac {3 a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}}{128}-\frac {45 \sqrt {a x -1}\, \sqrt {a x +1}\, a x}{256}-\frac {45 \,\operatorname {arccosh}\left (a x \right )}{256}+\frac {9 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )}{32}}{a^{4}}\)	\(150\)
default	\(\frac {\frac {a^{4} x^{4} \operatorname {arccosh}\left (a x \right )^{3}}{4}-\frac {3 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{16}-\frac {9 a x \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{32}-\frac {3 \operatorname {arccosh}\left (a x \right )^{3}}{32}+\frac {3 a^{4} x^{4} \operatorname {arccosh}\left (a x \right )}{32}-\frac {3 a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}}{128}-\frac {45 \sqrt {a x -1}\, \sqrt {a x +1}\, a x}{256}-\frac {45 \,\operatorname {arccosh}\left (a x \right )}{256}+\frac {9 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )}{32}}{a^{4}}\)	\(150\)

[In]

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

[Out]

1/a^4*(1/4*a^4*x^4*arccosh(a*x)^3-3/16*a^3*x^3*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-9/32*a*x*arccosh(a*x
)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-3/32*arccosh(a*x)^3+3/32*a^4*x^4*arccosh(a*x)-3/128*a^3*x^3*(a*x-1)^(1/2)*(a*x
+1)^(1/2)-45/256*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a*x-45/256*arccosh(a*x)+9/32*a^2*x^2*arccosh(a*x))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.78 \[ \int x^3 \text {arccosh}(a x)^3 \, dx=\frac {8 \, {\left (8 \, a^{4} x^{4} - 3\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{3} - 24 \, {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} + 3 \, {\left (8 \, a^{4} x^{4} + 24 \, a^{2} x^{2} - 15\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - 3 \, {\left (2 \, a^{3} x^{3} + 15 \, a x\right )} \sqrt {a^{2} x^{2} - 1}}{256 \, a^{4}} \]

[In]

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

[Out]

1/256*(8*(8*a^4*x^4 - 3)*log(a*x + sqrt(a^2*x^2 - 1))^3 - 24*(2*a^3*x^3 + 3*a*x)*sqrt(a^2*x^2 - 1)*log(a*x + s
qrt(a^2*x^2 - 1))^2 + 3*(8*a^4*x^4 + 24*a^2*x^2 - 15)*log(a*x + sqrt(a^2*x^2 - 1)) - 3*(2*a^3*x^3 + 15*a*x)*sq
rt(a^2*x^2 - 1))/a^4

Sympy [F]

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

[In]

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

[Out]

Integral(x**3*acosh(a*x)**3, x)

Maxima [F]

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

[In]

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

[Out]

1/4*x^4*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^3 - integrate(3/4*(a^3*x^6 + sqrt(a*x + 1)*sqrt(a*x - 1)*a^2*x^
5 - a*x^4)*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(-2)]

Exception generated. \[ \int x^3 \text {arccosh}(a x)^3 \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^3*acosh(a*x)^3, x)