\(\int \frac {x^4}{\sqrt {\text {arccosh}(a x)}} \, dx\) [90]

Optimal result

Integrand size = 12, antiderivative size = 163 \[ \int \frac {x^4}{\sqrt {\text {arccosh}(a x)}} \, dx=-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{16 a^5}-\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{32 a^5}-\frac {\sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arccosh}(a x)}\right )}{32 a^5}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{16 a^5}+\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{32 a^5}+\frac {\sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arccosh}(a x)}\right )}{32 a^5} \] Output:

-1/16*Pi^(1/2)*erf(arccosh(a*x)^(1/2))/a^5-1/32*3^(1/2)*Pi^(1/2)*erf(3^(1/ 
2)*arccosh(a*x)^(1/2))/a^5-1/160*5^(1/2)*Pi^(1/2)*erf(5^(1/2)*arccosh(a*x) 
^(1/2))/a^5+1/16*Pi^(1/2)*erfi(arccosh(a*x)^(1/2))/a^5+1/32*3^(1/2)*Pi^(1/ 
2)*erfi(3^(1/2)*arccosh(a*x)^(1/2))/a^5+1/160*5^(1/2)*Pi^(1/2)*erfi(5^(1/2 
)*arccosh(a*x)^(1/2))/a^5
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.92 \[ \int \frac {x^4}{\sqrt {\text {arccosh}(a x)}} \, dx=\frac {\frac {\sqrt {5} \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-5 \text {arccosh}(a x)\right )}{\sqrt {\text {arccosh}(a x)}}+\frac {5 \sqrt {3} \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-3 \text {arccosh}(a x)\right )}{\sqrt {\text {arccosh}(a x)}}+\frac {10 \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-\text {arccosh}(a x)\right )}{\sqrt {\text {arccosh}(a x)}}+10 \Gamma \left (\frac {1}{2},\text {arccosh}(a x)\right )+5 \sqrt {3} \Gamma \left (\frac {1}{2},3 \text {arccosh}(a x)\right )+\sqrt {5} \Gamma \left (\frac {1}{2},5 \text {arccosh}(a x)\right )}{160 a^5} \] Input:

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

Output:

((Sqrt[5]*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -5*ArcCosh[a*x]])/Sqrt[ArcCosh[a* 
x]] + (5*Sqrt[3]*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -3*ArcCosh[a*x]])/Sqrt[Arc 
Cosh[a*x]] + (10*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -ArcCosh[a*x]])/Sqrt[ArcCo 
sh[a*x]] + 10*Gamma[1/2, ArcCosh[a*x]] + 5*Sqrt[3]*Gamma[1/2, 3*ArcCosh[a* 
x]] + Sqrt[5]*Gamma[1/2, 5*ArcCosh[a*x]])/(160*a^5)
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.91, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6302, 5971, 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.

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

\(\Big \downarrow \) 6302

\(\displaystyle \frac {\int \frac {a^4 x^4 \sqrt {\frac {a x-1}{a x+1}} (a x+1)}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^5}\)

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {1}{16} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{16} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )+\frac {1}{32} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arccosh}(a x)}\right )}{a^5}\)

Input:

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

Output:

(-1/16*(Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]]) - (Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[A 
rcCosh[a*x]]])/32 - (Sqrt[Pi/5]*Erf[Sqrt[5]*Sqrt[ArcCosh[a*x]]])/32 + (Sqr 
t[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/16 + (Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcCosh 
[a*x]]])/32 + (Sqrt[Pi/5]*Erfi[Sqrt[5]*Sqrt[ArcCosh[a*x]]])/32)/a^5
 

Defintions of rubi rules used

rule 2009

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

rule 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 0]
 

rule 6302

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

Maple [F]

\[\int \frac {x^{4}}{\sqrt {\operatorname {arccosh}\left (a x \right )}}d x\]

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^4}{\sqrt {\text {arccosh}(a x)}} \, dx=\frac {2 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, a^{4} x^{4}-\frac {8 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}}{3}-\frac {4 \left (\int \frac {\sqrt {\mathit {acosh} \left (a x \right )}}{\mathit {acosh} \left (a x \right ) a^{2} x^{2}-\mathit {acosh} \left (a x \right )}d x \right ) a}{3}+\frac {4 \left (\int \frac {\sqrt {\mathit {acosh} \left (a x \right )}\, x^{2}}{\mathit {acosh} \left (a x \right ) a^{2} x^{2}-\mathit {acosh} \left (a x \right )}d x \right ) a^{3}}{3}-10 \left (\int \frac {\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x^{5}}{a^{2} x^{2}-1}d x \right ) a^{6}+8 \left (\int \frac {\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x^{3}}{a^{2} x^{2}-1}d x \right ) a^{4}+\frac {8 \left (\int \frac {\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x}{a^{2} x^{2}-1}d x \right ) a^{2}}{3}}{a^{5}} \] Input:

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

Output:

(2*(3*sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt(acosh(a*x))*a**4*x**4 - 4*sqrt(a*x 
+ 1)*sqrt(a*x - 1)*sqrt(acosh(a*x)) - 2*int(sqrt(acosh(a*x))/(acosh(a*x)*a 
**2*x**2 - acosh(a*x)),x)*a + 2*int((sqrt(acosh(a*x))*x**2)/(acosh(a*x)*a* 
*2*x**2 - acosh(a*x)),x)*a**3 - 15*int((sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt(a 
cosh(a*x))*x**5)/(a**2*x**2 - 1),x)*a**6 + 12*int((sqrt(a*x + 1)*sqrt(a*x 
- 1)*sqrt(acosh(a*x))*x**3)/(a**2*x**2 - 1),x)*a**4 + 4*int((sqrt(a*x + 1) 
*sqrt(a*x - 1)*sqrt(acosh(a*x))*x)/(a**2*x**2 - 1),x)*a**2))/(3*a**5)