\(\int \frac {x^4}{\text {arccosh}(a x)^2} \, dx\) [51]

Optimal result

Integrand size = 10, antiderivative size = 73 \[ \int \frac {x^4}{\text {arccosh}(a x)^2} \, dx=-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{a \text {arccosh}(a x)}+\frac {\text {Chi}(\text {arccosh}(a x))}{8 a^5}+\frac {9 \text {Chi}(3 \text {arccosh}(a x))}{16 a^5}+\frac {5 \text {Chi}(5 \text {arccosh}(a x))}{16 a^5} \] Output:

-x^4*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)+1/8*Chi(arccosh(a*x))/a^5+ 
9/16*Chi(3*arccosh(a*x))/a^5+5/16*Chi(5*arccosh(a*x))/a^5
 

Mathematica [A] (warning: unable to verify)

Time = 0.16 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.38 \[ \int \frac {x^4}{\text {arccosh}(a x)^2} \, dx=\frac {-16 a^4 x^4 \sqrt {\frac {-1+a x}{1+a x}}-16 a^5 x^5 \sqrt {\frac {-1+a x}{1+a x}}+2 \text {arccosh}(a x) \text {Chi}(\text {arccosh}(a x))+9 \text {arccosh}(a x) \text {Chi}(3 \text {arccosh}(a x))+5 \text {arccosh}(a x) \text {Chi}(5 \text {arccosh}(a x))}{16 a^5 \text {arccosh}(a x)} \] Input:

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

Output:

(-16*a^4*x^4*Sqrt[(-1 + a*x)/(1 + a*x)] - 16*a^5*x^5*Sqrt[(-1 + a*x)/(1 + 
a*x)] + 2*ArcCosh[a*x]*CoshIntegral[ArcCosh[a*x]] + 9*ArcCosh[a*x]*CoshInt 
egral[3*ArcCosh[a*x]] + 5*ArcCosh[a*x]*CoshIntegral[5*ArcCosh[a*x]])/(16*a 
^5*ArcCosh[a*x])
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6300, 2009}

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.

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

\(\Big \downarrow \) 6300

\(\displaystyle -\frac {\int \left (-\frac {a x}{8 \text {arccosh}(a x)}-\frac {9 \cosh (3 \text {arccosh}(a x))}{16 \text {arccosh}(a x)}-\frac {5 \cosh (5 \text {arccosh}(a x))}{16 \text {arccosh}(a x)}\right )d\text {arccosh}(a x)}{a^5}-\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {-\frac {1}{8} \text {Chi}(\text {arccosh}(a x))-\frac {9}{16} \text {Chi}(3 \text {arccosh}(a x))-\frac {5}{16} \text {Chi}(5 \text {arccosh}(a x))}{a^5}-\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}\)

Input:

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

Output:

-((x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x])) - (-1/8*CoshIntegra 
l[ArcCosh[a*x]] - (9*CoshIntegral[3*ArcCosh[a*x]])/16 - (5*CoshIntegral[5* 
ArcCosh[a*x]])/16)/a^5
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6300

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1) 
)), x] + Simp[1/(b^2*c^(m + 1)*(n + 1))   Subst[Int[ExpandTrigReduce[x^(n + 
 1), Cosh[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cosh[-a/b + x/b]^2), x], x], x, 
a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] 
&& LtQ[n, -1]
 

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.14

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{8 \,\operatorname {arccosh}\left (a x \right )}+\frac {\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right )}{8}-\frac {3 \sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{16 \,\operatorname {arccosh}\left (a x \right )}+\frac {9 \,\operatorname {Chi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{16}-\frac {\sinh \left (5 \,\operatorname {arccosh}\left (a x \right )\right )}{16 \,\operatorname {arccosh}\left (a x \right )}+\frac {5 \,\operatorname {Chi}\left (5 \,\operatorname {arccosh}\left (a x \right )\right )}{16}}{a^{5}}\)	\(83\)
default	\(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{8 \,\operatorname {arccosh}\left (a x \right )}+\frac {\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right )}{8}-\frac {3 \sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{16 \,\operatorname {arccosh}\left (a x \right )}+\frac {9 \,\operatorname {Chi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{16}-\frac {\sinh \left (5 \,\operatorname {arccosh}\left (a x \right )\right )}{16 \,\operatorname {arccosh}\left (a x \right )}+\frac {5 \,\operatorname {Chi}\left (5 \,\operatorname {arccosh}\left (a x \right )\right )}{16}}{a^{5}}\)	\(83\)

Input:

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

Output:

1/a^5*(-1/8/arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)+1/8*Chi(arccosh(a*x)) 
-3/16/arccosh(a*x)*sinh(3*arccosh(a*x))+9/16*Chi(3*arccosh(a*x))-1/16/arcc 
osh(a*x)*sinh(5*arccosh(a*x))+5/16*Chi(5*arccosh(a*x)))
 

Fricas [F]

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

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

Output:

integral(x^4/arccosh(a*x)^2, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-(a^3*x^7 - a*x^5 + (a^2*x^6 - x^4)*sqrt(a*x + 1)*sqrt(a*x - 1))/((a^3*x^2 
 + sqrt(a*x + 1)*sqrt(a*x - 1)*a^2*x - a)*log(a*x + sqrt(a*x + 1)*sqrt(a*x 
 - 1))) + integrate((5*a^5*x^8 - 10*a^3*x^6 + 5*a*x^4 + (5*a^3*x^6 - 3*a*x 
^4)*(a*x + 1)*(a*x - 1) + (10*a^4*x^7 - 13*a^2*x^5 + 4*x^3)*sqrt(a*x + 1)* 
sqrt(a*x - 1))/((a^5*x^4 + (a*x + 1)*(a*x - 1)*a^3*x^2 - 2*a^3*x^2 + 2*(a^ 
4*x^3 - a^2*x)*sqrt(a*x + 1)*sqrt(a*x - 1) + a)*log(a*x + sqrt(a*x + 1)*sq 
rt(a*x - 1))), x)
 

Giac [F]

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

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

Output:

integrate(x^4/arccosh(a*x)^2, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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