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

Optimal result

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

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

Mathematica [A] (verified)

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

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

Output:

((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
 

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6298, 6354, 15, 6354, 15, 6330, 24}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^4 \text {arccosh}(a x)^2 \, dx\)

\(\Big \downarrow \) 6298

\(\displaystyle \frac {1}{5} x^5 \text {arccosh}(a x)^2-\frac {2}{5} a \int \frac {x^5 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx\)

\(\Big \downarrow \) 6354

\(\displaystyle \frac {1}{5} x^5 \text {arccosh}(a x)^2-\frac {2}{5} a \left (\frac {4 \int \frac {x^3 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx}{5 a^2}-\frac {\int x^4dx}{5 a}+\frac {x^4 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{5 a^2}\right )\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{5} x^5 \text {arccosh}(a x)^2-\frac {2}{5} a \left (\frac {4 \int \frac {x^3 \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx}{5 a^2}+\frac {x^4 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{5 a^2}-\frac {x^5}{25 a}\right )\)

\(\Big \downarrow \) 6354

\(\displaystyle \frac {1}{5} x^5 \text {arccosh}(a x)^2-\frac {2}{5} a \left (\frac {4 \left (\frac {2 \int \frac {x \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx}{3 a^2}-\frac {\int x^2dx}{3 a}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{3 a^2}\right )}{5 a^2}+\frac {x^4 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{5 a^2}-\frac {x^5}{25 a}\right )\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{5} x^5 \text {arccosh}(a x)^2-\frac {2}{5} a \left (\frac {4 \left (\frac {2 \int \frac {x \text {arccosh}(a x)}{\sqrt {a x-1} \sqrt {a x+1}}dx}{3 a^2}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{3 a^2}-\frac {x^3}{9 a}\right )}{5 a^2}+\frac {x^4 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{5 a^2}-\frac {x^5}{25 a}\right )\)

\(\Big \downarrow \) 6330

\(\displaystyle \frac {1}{5} x^5 \text {arccosh}(a x)^2-\frac {2}{5} a \left (\frac {4 \left (\frac {2 \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{a^2}-\frac {\int 1dx}{a}\right )}{3 a^2}+\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{3 a^2}-\frac {x^3}{9 a}\right )}{5 a^2}+\frac {x^4 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{5 a^2}-\frac {x^5}{25 a}\right )\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {1}{5} x^5 \text {arccosh}(a x)^2-\frac {2}{5} a \left (\frac {x^4 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{5 a^2}+\frac {4 \left (\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{3 a^2}+\frac {2 \left (\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{a^2}-\frac {x}{a}\right )}{3 a^2}-\frac {x^3}{9 a}\right )}{5 a^2}-\frac {x^5}{25 a}\right )\)

Input:

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

Output:

(x^5*ArcCosh[a*x]^2)/5 - (2*a*(-1/25*x^5/a + (x^4*Sqrt[-1 + a*x]*Sqrt[1 + 
a*x]*ArcCosh[a*x])/(5*a^2) + (4*(-1/9*x^3/a + (x^2*Sqrt[-1 + a*x]*Sqrt[1 + 
 a*x]*ArcCosh[a*x])/(3*a^2) + (2*(-(x/a) + (Sqrt[-1 + a*x]*Sqrt[1 + a*x]*A 
rcCosh[a*x])/a^2))/(3*a^2)))/(5*a^2)))/5
 

Defintions of rubi rules used

rule 15

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

rule 24

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

rule 6298

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
 :> Simp[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(d*(m + 1))), x] - Simp[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 6330

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p 
_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + 
e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Simp[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] && E 
qQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]
 

rule 6354

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 
1_.)*(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] + (Simp[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] - Simp[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] && IGtQ[m, 1] && N 
eQ[m + 2*p + 1, 0]
 

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85

method	result	size
derivativedivides	\(\frac {\frac {a^{5} x^{5} \operatorname {arccosh}\left (a x \right )^{2}}{5}-\frac {16 \,\operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}}{75}-\frac {2 a^{4} x^{4} \operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}}{25}-\frac {8 \,\operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}\, a^{2} x^{2}}{75}+\frac {16 a x}{75}+\frac {2 a^{5} x^{5}}{125}+\frac {8 a^{3} x^{3}}{225}}{a^{5}}\)	\(112\)
default	\(\frac {\frac {a^{5} x^{5} \operatorname {arccosh}\left (a x \right )^{2}}{5}-\frac {16 \,\operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}}{75}-\frac {2 a^{4} x^{4} \operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}}{25}-\frac {8 \,\operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}\, a^{2} x^{2}}{75}+\frac {16 a x}{75}+\frac {2 a^{5} x^{5}}{125}+\frac {8 a^{3} x^{3}}{225}}{a^{5}}\)	\(112\)
orering	\(\frac {\left (549 a^{6} x^{6}+200 a^{4} x^{4}+1760 a^{2} x^{2}-2880\right ) \operatorname {arccosh}\left (a x \right )^{2}}{1125 a^{6} x}-\frac {\left (108 a^{6} x^{6}+83 a^{4} x^{4}+740 a^{2} x^{2}-1080\right ) \left (4 x^{3} \operatorname {arccosh}\left (a x \right )^{2}+\frac {2 x^{4} \operatorname {arccosh}\left (a x \right ) a}{\sqrt {a x -1}\, \sqrt {a x +1}}\right )}{1125 x^{4} a^{6}}+\frac {\left (9 a^{4} x^{4}+20 a^{2} x^{2}+120\right ) \left (a x -1\right ) \left (a x +1\right ) \left (12 x^{2} \operatorname {arccosh}\left (a x \right )^{2}+\frac {16 x^{3} \operatorname {arccosh}\left (a x \right ) a}{\sqrt {a x -1}\, \sqrt {a x +1}}+\frac {2 x^{4} a^{2}}{\left (a x -1\right ) \left (a x +1\right )}-\frac {x^{4} \operatorname {arccosh}\left (a x \right ) a^{2}}{\left (a x -1\right )^{\frac {3}{2}} \sqrt {a x +1}}-\frac {x^{4} \operatorname {arccosh}\left (a x \right ) a^{2}}{\sqrt {a x -1}\, \left (a x +1\right )^{\frac {3}{2}}}\right )}{1125 x^{3} a^{6}}\)	\(258\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.75 \[ \int x^4 \text {arccosh}(a x)^2 \, dx=\frac {225 \, a^{5} x^{5} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} + 18 \, a^{5} x^{5} + 40 \, a^{3} x^{3} - 30 \, {\left (3 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 8\right )} \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) + 240 \, a x}{1125 \, a^{5}} \] Input:

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

Output:

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]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.75 \[ \int x^4 \text {arccosh}(a x)^2 \, dx=\frac {1}{5} \, x^{5} \operatorname {arcosh}\left (a x\right )^{2} - \frac {2}{75} \, {\left (\frac {3 \, \sqrt {a^{2} x^{2} - 1} x^{4}}{a^{2}} + \frac {4 \, \sqrt {a^{2} x^{2} - 1} x^{2}}{a^{4}} + \frac {8 \, \sqrt {a^{2} x^{2} - 1}}{a^{6}}\right )} a \operatorname {arcosh}\left (a x\right ) + \frac {2 \, {\left (9 \, a^{4} x^{5} + 20 \, a^{2} x^{3} + 120 \, x\right )}}{1125 \, a^{4}} \] Input:

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

Output:

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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