\(\int \frac {x^3 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx\) [130]

Optimal result

Integrand size = 22, antiderivative size = 110 \[ \int \frac {x^3 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\frac {2 x \sqrt {1-a x}}{3 a^3 \sqrt {-1+a x}}+\frac {x^3 \sqrt {1-a x}}{9 a \sqrt {-1+a x}}-\frac {2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)}{3 a^2} \] Output:

2/3*x*(-a*x+1)^(1/2)/a^3/(a*x-1)^(1/2)+1/9*x^3*(-a*x+1)^(1/2)/a/(a*x-1)^(1 
/2)-2/3*(-a^2*x^2+1)^(1/2)*arccosh(a*x)/a^4-1/3*x^2*(-a^2*x^2+1)^(1/2)*arc 
cosh(a*x)/a^2
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.67 \[ \int \frac {x^3 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=-\frac {a x \sqrt {-1+a x} \sqrt {1+a x} \left (6+a^2 x^2\right )-3 \left (-2+a^2 x^2+a^4 x^4\right ) \text {arccosh}(a x)}{9 a^4 \sqrt {1-a^2 x^2}} \] Input:

Integrate[(x^3*ArcCosh[a*x])/Sqrt[1 - a^2*x^2],x]
 

Output:

-1/9*(a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(6 + a^2*x^2) - 3*(-2 + a^2*x^2 + a 
^4*x^4)*ArcCosh[a*x])/(a^4*Sqrt[1 - a^2*x^2])
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6353, 15, 6329, 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.

\(\displaystyle \int \frac {x^3 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx\)

\(\Big \downarrow \) 6353

\(\displaystyle \frac {2 \int \frac {x \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {\sqrt {a x-1} \int x^2dx}{3 a \sqrt {1-a x}}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)}{3 a^2}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {2 \int \frac {x \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arccosh}(a x)}{3 a^2}-\frac {x^3 \sqrt {a x-1}}{9 a \sqrt {1-a x}}\)

\(\Big \downarrow \) 6329

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

\(\Big \downarrow \) 24

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

Input:

Int[(x^3*ArcCosh[a*x])/Sqrt[1 - a^2*x^2],x]
 

Output:

-1/9*(x^3*Sqrt[-1 + a*x])/(a*Sqrt[1 - a*x]) - (x^2*Sqrt[1 - a^2*x^2]*ArcCo 
sh[a*x])/(3*a^2) + (2*(-((x*Sqrt[-1 + a*x])/(a*Sqrt[1 - a*x])) - (Sqrt[1 - 
 a^2*x^2]*ArcCosh[a*x])/a^2))/(3*a^2)
 

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 6329

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

rule 6353

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

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.31

method	result	size
orering	\(\frac {\left (5 a^{4} x^{4}+12 a^{2} x^{2}-24\right ) \operatorname {arccosh}\left (a x \right )}{9 a^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {\left (a^{2} x^{2}+6\right ) \left (a x -1\right ) \left (a x +1\right ) \left (\frac {3 x^{2} \operatorname {arccosh}\left (a x \right )}{\sqrt {-a^{2} x^{2}+1}}+\frac {x^{3} a}{\sqrt {a x -1}\, \sqrt {a x +1}\, \sqrt {-a^{2} x^{2}+1}}+\frac {x^{4} \operatorname {arccosh}\left (a x \right ) a^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{9 x^{2} a^{4}}\)	\(144\)
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (4 a^{4} x^{4}-5 a^{2} x^{2}+4 a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}-3 \sqrt {a x -1}\, \sqrt {a x +1}\, a x +1\right ) \left (-1+3 \,\operatorname {arccosh}\left (a x \right )\right )}{72 a^{4} \left (a^{2} x^{2}-1\right )}-\frac {3 \sqrt {-a^{2} x^{2}+1}\, \left (\sqrt {a x -1}\, \sqrt {a x +1}\, a x +a^{2} x^{2}-1\right ) \left (-1+\operatorname {arccosh}\left (a x \right )\right )}{8 a^{4} \left (a^{2} x^{2}-1\right )}-\frac {3 \sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2}-\sqrt {a x -1}\, \sqrt {a x +1}\, a x -1\right ) \left (1+\operatorname {arccosh}\left (a x \right )\right )}{8 a^{4} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (4 a^{4} x^{4}-5 a^{2} x^{2}-4 a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}+3 \sqrt {a x -1}\, \sqrt {a x +1}\, a x +1\right ) \left (1+3 \,\operatorname {arccosh}\left (a x \right )\right )}{72 a^{4} \left (a^{2} x^{2}-1\right )}\)	\(311\)

Input:

int(x^3*arccosh(a*x)/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/9*(5*a^4*x^4+12*a^2*x^2-24)/a^4*arccosh(a*x)/(-a^2*x^2+1)^(1/2)-1/9/x^2* 
(a^2*x^2+6)/a^4*(a*x-1)*(a*x+1)*(3*x^2*arccosh(a*x)/(-a^2*x^2+1)^(1/2)+x^3 
*a/(a*x-1)^(1/2)/(a*x+1)^(1/2)/(-a^2*x^2+1)^(1/2)+x^4*arccosh(a*x)/(-a^2*x 
^2+1)^(3/2)*a^2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.92 \[ \int \frac {x^3 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=-\frac {3 \, {\left (a^{4} x^{4} + a^{2} x^{2} - 2\right )} \sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - {\left (a^{3} x^{3} + 6 \, a x\right )} \sqrt {a^{2} x^{2} - 1} \sqrt {-a^{2} x^{2} + 1}}{9 \, {\left (a^{6} x^{2} - a^{4}\right )}} \] Input:

integrate(x^3*arccosh(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
 

Output:

-1/9*(3*(a^4*x^4 + a^2*x^2 - 2)*sqrt(-a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 
- 1)) - (a^3*x^3 + 6*a*x)*sqrt(a^2*x^2 - 1)*sqrt(-a^2*x^2 + 1))/(a^6*x^2 - 
 a^4)
 

Sympy [F]

\[ \int \frac {x^3 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^{3} \operatorname {acosh}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \] Input:

integrate(x**3*acosh(a*x)/(-a**2*x**2+1)**(1/2),x)
 

Output:

Integral(x**3*acosh(a*x)/sqrt(-(a*x - 1)*(a*x + 1)), x)
 

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.14 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.56 \[ \int \frac {x^3 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\frac {1}{9} \, a {\left (\frac {i \, x^{3}}{a^{2}} + \frac {6 i \, x}{a^{4}}\right )} - \frac {1}{3} \, {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{4}}\right )} \operatorname {arcosh}\left (a x\right ) \] Input:

integrate(x^3*arccosh(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
 

Output:

1/9*a*(I*x^3/a^2 + 6*I*x/a^4) - 1/3*(sqrt(-a^2*x^2 + 1)*x^2/a^2 + 2*sqrt(- 
a^2*x^2 + 1)/a^4)*arccosh(a*x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^3*arccosh(a*x)/(-a^2*x^2+1)^(1/2),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. \[ \int \frac {x^3 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^3\,\mathrm {acosh}\left (a\,x\right )}{\sqrt {1-a^2\,x^2}} \,d x \] Input:

int((x^3*acosh(a*x))/(1 - a^2*x^2)^(1/2),x)
 

Output:

int((x^3*acosh(a*x))/(1 - a^2*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^3 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathit {acosh} \left (a x \right ) x^{3}}{\sqrt {-a^{2} x^{2}+1}}d x \] Input:

int(x^3*acosh(a*x)/(-a^2*x^2+1)^(1/2),x)
 

Output:

int((acosh(a*x)*x**3)/sqrt( - a**2*x**2 + 1),x)