\(\int x^4 \text {arccosh}(a x)^2 \, dx\) [12]

Optimal result

Integrand size = 10, antiderivative size = 132 \[ \int x^4 \text {arccosh}(a x)^2 \, dx=\frac {16 x}{75 a^4}+\frac {8 x^3}{225 a^2}+\frac {2 x^5}{125}-\frac {16 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{75 a^5}-\frac {8 x^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{75 a^3}-\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{25 a}+\frac {1}{5} x^5 \text {arccosh}(a x)^2 \]

[Out]

16/75*x/a^4+8/225*x^3/a^2+2/125*x^5+1/5*x^5*arccosh(a*x)^2-16/75*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^5-
8/75*x^2*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3-2/25*x^4*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5883, 5939, 5915, 8, 30} \[ \int x^4 \text {arccosh}(a x)^2 \, dx=-\frac {16 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{75 a^5}+\frac {16 x}{75 a^4}-\frac {8 x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{75 a^3}+\frac {8 x^3}{225 a^2}+\frac {1}{5} x^5 \text {arccosh}(a x)^2-\frac {2 x^4 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{25 a}+\frac {2 x^5}{125} \]

[In]

Int[x^4*ArcCosh[a*x]^2,x]

[Out]

(16*x)/(75*a^4) + (8*x^3)/(225*a^2) + (2*x^5)/125 - (16*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(75*a^5) -
(8*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(75*a^3) - (2*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])
/(25*a) + (x^5*ArcCosh[a*x]^2)/5

Rule 8

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 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]

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}{5} x^5 \text {arccosh}(a x)^2-\frac {1}{5} (2 a) \int \frac {x^5 \text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = -\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{25 a}+\frac {1}{5} x^5 \text {arccosh}(a x)^2+\frac {2 \int x^4 \, dx}{25}-\frac {8 \int \frac {x^3 \text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{25 a} \\ & = \frac {2 x^5}{125}-\frac {8 x^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{75 a^3}-\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{25 a}+\frac {1}{5} x^5 \text {arccosh}(a x)^2-\frac {16 \int \frac {x \text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{75 a^3}+\frac {8 \int x^2 \, dx}{75 a^2} \\ & = \frac {8 x^3}{225 a^2}+\frac {2 x^5}{125}-\frac {16 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{75 a^5}-\frac {8 x^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{75 a^3}-\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{25 a}+\frac {1}{5} x^5 \text {arccosh}(a x)^2+\frac {16 \int 1 \, dx}{75 a^4} \\ & = \frac {16 x}{75 a^4}+\frac {8 x^3}{225 a^2}+\frac {2 x^5}{125}-\frac {16 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{75 a^5}-\frac {8 x^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{75 a^3}-\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{25 a}+\frac {1}{5} x^5 \text {arccosh}(a x)^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.61 \[ \int x^4 \text {arccosh}(a x)^2 \, dx=\frac {\frac {240 x}{a^4}+\frac {40 x^3}{a^2}+18 x^5-\frac {30 \sqrt {-1+a x} \sqrt {1+a x} \left (8+4 a^2 x^2+3 a^4 x^4\right ) \text {arccosh}(a x)}{a^5}+225 x^5 \text {arccosh}(a x)^2}{1125} \]

[In]

Integrate[x^4*ArcCosh[a*x]^2,x]

[Out]

((240*x)/a^4 + (40*x^3)/a^2 + 18*x^5 - (30*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(8 + 4*a^2*x^2 + 3*a^4*x^4)*ArcCosh[a*
x])/a^5 + 225*x^5*ArcCosh[a*x]^2)/1125

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85

method	result	size
derivativedivides	\(\frac {\frac {a^{5} x^{5} \operatorname {arccosh}\left (a x \right )^{2}}{5}-\frac {16 \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )}{75}-\frac {2 a^{4} x^{4} \operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}}{25}-\frac {8 a^{2} x^{2} \operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}}{75}+\frac {16 a x}{75}+\frac {2 a^{5} x^{5}}{125}+\frac {8 a^{3} x^{3}}{225}}{a^{5}}\)	\(112\)
default	\(\frac {\frac {a^{5} x^{5} \operatorname {arccosh}\left (a x \right )^{2}}{5}-\frac {16 \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )}{75}-\frac {2 a^{4} x^{4} \operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}}{25}-\frac {8 a^{2} x^{2} \operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}}{75}+\frac {16 a x}{75}+\frac {2 a^{5} x^{5}}{125}+\frac {8 a^{3} x^{3}}{225}}{a^{5}}\)	\(112\)

[In]

int(x^4*arccosh(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

1/a^5*(1/5*a^5*x^5*arccosh(a*x)^2-16/75*(a*x-1)^(1/2)*(a*x+1)^(1/2)*arccosh(a*x)-2/25*a^4*x^4*arccosh(a*x)*(a*
x-1)^(1/2)*(a*x+1)^(1/2)-8/75*a^2*x^2*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)+16/75*a*x+2/125*a^5*x^5+8/225*a
^3*x^3)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.75 \[ \int x^4 \text {arccosh}(a x)^2 \, dx=\frac {225 \, a^{5} x^{5} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} + 18 \, a^{5} x^{5} + 40 \, a^{3} x^{3} - 30 \, {\left (3 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 8\right )} \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) + 240 \, a x}{1125 \, a^{5}} \]

[In]

integrate(x^4*arccosh(a*x)^2,x, algorithm="fricas")

[Out]

1/1125*(225*a^5*x^5*log(a*x + sqrt(a^2*x^2 - 1))^2 + 18*a^5*x^5 + 40*a^3*x^3 - 30*(3*a^4*x^4 + 4*a^2*x^2 + 8)*
sqrt(a^2*x^2 - 1)*log(a*x + sqrt(a^2*x^2 - 1)) + 240*a*x)/a^5

Sympy [F]

\[ \int x^4 \text {arccosh}(a x)^2 \, dx=\int x^{4} \operatorname {acosh}^{2}{\left (a x \right )}\, dx \]

[In]

integrate(x**4*acosh(a*x)**2,x)

[Out]

Integral(x**4*acosh(a*x)**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.75 \[ \int x^4 \text {arccosh}(a x)^2 \, dx=\frac {1}{5} \, x^{5} \operatorname {arcosh}\left (a x\right )^{2} - \frac {2}{75} \, {\left (\frac {3 \, \sqrt {a^{2} x^{2} - 1} x^{4}}{a^{2}} + \frac {4 \, \sqrt {a^{2} x^{2} - 1} x^{2}}{a^{4}} + \frac {8 \, \sqrt {a^{2} x^{2} - 1}}{a^{6}}\right )} a \operatorname {arcosh}\left (a x\right ) + \frac {2 \, {\left (9 \, a^{4} x^{5} + 20 \, a^{2} x^{3} + 120 \, x\right )}}{1125 \, a^{4}} \]

[In]

integrate(x^4*arccosh(a*x)^2,x, algorithm="maxima")

[Out]

1/5*x^5*arccosh(a*x)^2 - 2/75*(3*sqrt(a^2*x^2 - 1)*x^4/a^2 + 4*sqrt(a^2*x^2 - 1)*x^2/a^4 + 8*sqrt(a^2*x^2 - 1)
/a^6)*a*arccosh(a*x) + 2/1125*(9*a^4*x^5 + 20*a^2*x^3 + 120*x)/a^4

Giac [F(-2)]

Exception generated. \[ \int x^4 \text {arccosh}(a x)^2 \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^4*arccosh(a*x)^2,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. \[ \int x^4 \text {arccosh}(a x)^2 \, dx=\int x^4\,{\mathrm {acosh}\left (a\,x\right )}^2 \,d x \]

[In]

int(x^4*acosh(a*x)^2,x)

[Out]

int(x^4*acosh(a*x)^2, x)