\(\int \frac {x^4 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx\) [129]

Optimal result

Integrand size = 22, antiderivative size = 145 \[ \int \frac {x^4 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\frac {3 x^2 \sqrt {1-a x}}{16 a^3 \sqrt {-1+a x}}+\frac {x^4 \sqrt {1-a x}}{16 a \sqrt {-1+a x}}-\frac {3 x \sqrt {1-a^2 x^2} \text {arccosh}(a x)}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arccosh}(a x)}{4 a^2}-\frac {3 \sqrt {1-a x} \text {arccosh}(a x)^2}{16 a^5 \sqrt {-1+a x}} \] Output:

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

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.64 \[ \int \frac {x^4 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x) (-16 \cosh (2 \text {arccosh}(a x))-\cosh (4 \text {arccosh}(a x))+4 \text {arccosh}(a x) (6 \text {arccosh}(a x)+8 \sinh (2 \text {arccosh}(a x))+\sinh (4 \text {arccosh}(a x))))}{128 a^5 \sqrt {-((-1+a x) (1+a x))}} \] Input:

Integrate[(x^4*ArcCosh[a*x])/Sqrt[1 - a^2*x^2],x]
 

Output:

(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*(-16*Cosh[2*ArcCosh[a*x]] - Cosh[4*A 
rcCosh[a*x]] + 4*ArcCosh[a*x]*(6*ArcCosh[a*x] + 8*Sinh[2*ArcCosh[a*x]] + S 
inh[4*ArcCosh[a*x]])))/(128*a^5*Sqrt[-((-1 + a*x)*(1 + a*x))])
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6353, 15, 6353, 15, 6307}

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.

\(\displaystyle \int \frac {x^4 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx\)

\(\Big \downarrow \) 6353

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

\(\Big \downarrow \) 15

\(\displaystyle \frac {3 \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arccosh}(a x)}{4 a^2}-\frac {x^4 \sqrt {a x-1}}{16 a \sqrt {1-a x}}\)

\(\Big \downarrow \) 6353

\(\displaystyle \frac {3 \left (\frac {\int \frac {\text {arccosh}(a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {\sqrt {a x-1} \int xdx}{2 a \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arccosh}(a x)}{4 a^2}-\frac {x^4 \sqrt {a x-1}}{16 a \sqrt {1-a x}}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {3 \left (\frac {\int \frac {\text {arccosh}(a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)}{2 a^2}-\frac {x^2 \sqrt {a x-1}}{4 a \sqrt {1-a x}}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arccosh}(a x)}{4 a^2}-\frac {x^4 \sqrt {a x-1}}{16 a \sqrt {1-a x}}\)

\(\Big \downarrow \) 6307

\(\displaystyle -\frac {x^3 \sqrt {1-a^2 x^2} \text {arccosh}(a x)}{4 a^2}+\frac {3 \left (\frac {\sqrt {a x-1} \text {arccosh}(a x)^2}{4 a^3 \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)}{2 a^2}-\frac {x^2 \sqrt {a x-1}}{4 a \sqrt {1-a x}}\right )}{4 a^2}-\frac {x^4 \sqrt {a x-1}}{16 a \sqrt {1-a x}}\)

Input:

Int[(x^4*ArcCosh[a*x])/Sqrt[1 - a^2*x^2],x]
 

Output:

-1/16*(x^4*Sqrt[-1 + a*x])/(a*Sqrt[1 - a*x]) - (x^3*Sqrt[1 - a^2*x^2]*ArcC 
osh[a*x])/(4*a^2) + (3*(-1/4*(x^2*Sqrt[-1 + a*x])/(a*Sqrt[1 - a*x]) - (x*S 
qrt[1 - a^2*x^2]*ArcCosh[a*x])/(2*a^2) + (Sqrt[-1 + a*x]*ArcCosh[a*x]^2)/( 
4*a^3*Sqrt[1 - a*x])))/(4*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 6307

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d 
 + e*x^2])]*(a + b*ArcCosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x 
] && EqQ[c^2*d + e, 0] && NeQ[n, -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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(455\) vs. \(2(119)=238\).

Time = 0.48 (sec) , antiderivative size = 456, normalized size of antiderivative = 3.14

method	result	size
default	\(-\frac {3 \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )^{2}}{16 a^{5} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (8 a^{5} x^{5}-12 a^{3} x^{3}+8 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}+4 a x -8 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (-1+4 \,\operatorname {arccosh}\left (a x \right )\right )}{256 a^{5} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (2 a^{3} x^{3}-2 a x +2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (-1+2 \,\operatorname {arccosh}\left (a x \right )\right )}{16 a^{5} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (-2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+2 a^{3} x^{3}+\sqrt {a x -1}\, \sqrt {a x +1}-2 a x \right ) \left (1+2 \,\operatorname {arccosh}\left (a x \right )\right )}{16 a^{5} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (-8 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}+8 a^{5} x^{5}+8 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-12 a^{3} x^{3}-\sqrt {a x -1}\, \sqrt {a x +1}+4 a x \right ) \left (1+4 \,\operatorname {arccosh}\left (a x \right )\right )}{256 a^{5} \left (a^{2} x^{2}-1\right )}\)	\(456\)

Input:

int(x^4*arccosh(a*x)/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-3/16*(-a^2*x^2+1)^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^5/(a^2*x^2-1)*arcco 
sh(a*x)^2-1/256*(-a^2*x^2+1)^(1/2)*(8*a^5*x^5-12*a^3*x^3+8*(a*x-1)^(1/2)*( 
a*x+1)^(1/2)*a^4*x^4+4*a*x-8*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+(a*x-1)^( 
1/2)*(a*x+1)^(1/2))*(-1+4*arccosh(a*x))/a^5/(a^2*x^2-1)-1/16*(-a^2*x^2+1)^ 
(1/2)*(2*a^3*x^3-2*a*x+2*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-(a*x-1)^(1/2) 
*(a*x+1)^(1/2))*(-1+2*arccosh(a*x))/a^5/(a^2*x^2-1)-1/16*(-a^2*x^2+1)^(1/2 
)*(-2*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+2*a^3*x^3+(a*x-1)^(1/2)*(a*x+1)^ 
(1/2)-2*a*x)*(1+2*arccosh(a*x))/a^5/(a^2*x^2-1)-1/256*(-a^2*x^2+1)^(1/2)*( 
-8*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^4*x^4+8*a^5*x^5+8*a^2*x^2*(a*x-1)^(1/2)*( 
a*x+1)^(1/2)-12*a^3*x^3-(a*x-1)^(1/2)*(a*x+1)^(1/2)+4*a*x)*(1+4*arccosh(a* 
x))/a^5/(a^2*x^2-1)
 

Fricas [F]

\[ \int \frac {x^4 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{4} \operatorname {arcosh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:

integrate(x^4*arccosh(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
 

Output:

integral(-sqrt(-a^2*x^2 + 1)*x^4*arccosh(a*x)/(a^2*x^2 - 1), x)
 

Sympy [F]

\[ \int \frac {x^4 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^{4} \operatorname {acosh}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \] Input:

integrate(x**4*acosh(a*x)/(-a**2*x**2+1)**(1/2),x)
 

Output:

Integral(x**4*acosh(a*x)/sqrt(-(a*x - 1)*(a*x + 1)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^4 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^4*arccosh(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [F]

\[ \int \frac {x^4 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{4} \operatorname {arcosh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:

integrate(x^4*arccosh(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^4\,\mathrm {acosh}\left (a\,x\right )}{\sqrt {1-a^2\,x^2}} \,d x \] Input:

int((x^4*acosh(a*x))/(1 - a^2*x^2)^(1/2),x)
 

Output:

int((x^4*acosh(a*x))/(1 - a^2*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^4 \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathit {acosh} \left (a x \right ) x^{4}}{\sqrt {-a^{2} x^{2}+1}}d x \] Input:

int(x^4*acosh(a*x)/(-a^2*x^2+1)^(1/2),x)
 

Output:

int((acosh(a*x)*x**4)/sqrt( - a**2*x**2 + 1),x)