\(\int \frac {x^4}{\text {arccosh}(a x)^{5/2}} \, dx\) [103]

Optimal result

Integrand size = 12, antiderivative size = 228 \[ \int \frac {x^4}{\text {arccosh}(a x)^{5/2}} \, dx=-\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {16 x^3}{3 a^2 \sqrt {\text {arccosh}(a x)}}-\frac {20 x^5}{3 \sqrt {\text {arccosh}(a x)}}-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{12 a^5}-\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{8 a^5}-\frac {5 \sqrt {5 \pi } \text {erf}\left (\sqrt {5} \sqrt {\text {arccosh}(a x)}\right )}{24 a^5}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{12 a^5}+\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{8 a^5}+\frac {5 \sqrt {5 \pi } \text {erfi}\left (\sqrt {5} \sqrt {\text {arccosh}(a x)}\right )}{24 a^5} \] Output:

-2/3*x^4*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)^(3/2)+16/3*x^3/a^2/arc 
cosh(a*x)^(1/2)-20/3*x^5/arccosh(a*x)^(1/2)-1/12*Pi^(1/2)*erf(arccosh(a*x) 
^(1/2))/a^5-3/8*3^(1/2)*Pi^(1/2)*erf(3^(1/2)*arccosh(a*x)^(1/2))/a^5-5/24* 
5^(1/2)*Pi^(1/2)*erf(5^(1/2)*arccosh(a*x)^(1/2))/a^5+1/12*Pi^(1/2)*erfi(ar 
ccosh(a*x)^(1/2))/a^5+3/8*3^(1/2)*Pi^(1/2)*erfi(3^(1/2)*arccosh(a*x)^(1/2) 
)/a^5+5/24*5^(1/2)*Pi^(1/2)*erfi(5^(1/2)*arccosh(a*x)^(1/2))/a^5
 

Mathematica [A] (warning: unable to verify)

Time = 0.99 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.25 \[ \int \frac {x^4}{\text {arccosh}(a x)^{5/2}} \, dx=\frac {-5 e^{-5 \text {arccosh}(a x)} \text {arccosh}(a x)-5 e^{5 \text {arccosh}(a x)} \text {arccosh}(a x)-5 \sqrt {5} (-\text {arccosh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-5 \text {arccosh}(a x)\right )-2 \left (\sqrt {\frac {-1+a x}{1+a x}} (1+a x)+e^{-\text {arccosh}(a x)} \text {arccosh}(a x)+e^{\text {arccosh}(a x)} \text {arccosh}(a x)+(-\text {arccosh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-\text {arccosh}(a x)\right )-\text {arccosh}(a x)^{3/2} \Gamma \left (\frac {1}{2},\text {arccosh}(a x)\right )\right )+5 \sqrt {5} \text {arccosh}(a x)^{3/2} \Gamma \left (\frac {1}{2},5 \text {arccosh}(a x)\right )-3 \left (3 e^{-3 \text {arccosh}(a x)} \text {arccosh}(a x)+3 e^{3 \text {arccosh}(a x)} \text {arccosh}(a x)+3 \sqrt {3} (-\text {arccosh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-3 \text {arccosh}(a x)\right )-3 \sqrt {3} \text {arccosh}(a x)^{3/2} \Gamma \left (\frac {1}{2},3 \text {arccosh}(a x)\right )+\sinh (3 \text {arccosh}(a x))\right )-\sinh (5 \text {arccosh}(a x))}{24 a^5 \text {arccosh}(a x)^{3/2}} \] Input:

Integrate[x^4/ArcCosh[a*x]^(5/2),x]
 

Output:

((-5*ArcCosh[a*x])/E^(5*ArcCosh[a*x]) - 5*E^(5*ArcCosh[a*x])*ArcCosh[a*x] 
- 5*Sqrt[5]*(-ArcCosh[a*x])^(3/2)*Gamma[1/2, -5*ArcCosh[a*x]] - 2*(Sqrt[(- 
1 + a*x)/(1 + a*x)]*(1 + a*x) + ArcCosh[a*x]/E^ArcCosh[a*x] + E^ArcCosh[a* 
x]*ArcCosh[a*x] + (-ArcCosh[a*x])^(3/2)*Gamma[1/2, -ArcCosh[a*x]] - ArcCos 
h[a*x]^(3/2)*Gamma[1/2, ArcCosh[a*x]]) + 5*Sqrt[5]*ArcCosh[a*x]^(3/2)*Gamm 
a[1/2, 5*ArcCosh[a*x]] - 3*((3*ArcCosh[a*x])/E^(3*ArcCosh[a*x]) + 3*E^(3*A 
rcCosh[a*x])*ArcCosh[a*x] + 3*Sqrt[3]*(-ArcCosh[a*x])^(3/2)*Gamma[1/2, -3* 
ArcCosh[a*x]] - 3*Sqrt[3]*ArcCosh[a*x]^(3/2)*Gamma[1/2, 3*ArcCosh[a*x]] + 
Sinh[3*ArcCosh[a*x]]) - Sinh[5*ArcCosh[a*x]])/(24*a^5*ArcCosh[a*x]^(3/2))
 

Rubi [A] (verified)

Time = 1.50 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.45, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6301, 6366, 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.

\(\displaystyle \int \frac {x^4}{\text {arccosh}(a x)^{5/2}} \, dx\)

\(\Big \downarrow \) 6301

\(\displaystyle \frac {10}{3} a \int \frac {x^5}{\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}dx-\frac {8 \int \frac {x^3}{\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{3/2}}dx}{3 a}-\frac {2 x^4 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 6366

\(\displaystyle \frac {10}{3} a \left (\frac {10 \int \frac {x^4}{\sqrt {\text {arccosh}(a x)}}dx}{a}-\frac {2 x^5}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {8 \left (\frac {6 \int \frac {x^2}{\sqrt {\text {arccosh}(a x)}}dx}{a}-\frac {2 x^3}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x^4 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 6302

\(\displaystyle \frac {10}{3} a \left (\frac {10 \int \frac {a^4 x^4 \sqrt {\frac {a x-1}{a x+1}} (a x+1)}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^6}-\frac {2 x^5}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {8 \left (\frac {6 \int \frac {a^2 x^2 \sqrt {\frac {a x-1}{a x+1}} (a x+1)}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^4}-\frac {2 x^3}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x^4 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {10}{3} a \left (\frac {10 \int \left (\frac {\sqrt {\frac {a x-1}{a x+1}} (a x+1)}{8 \sqrt {\text {arccosh}(a x)}}+\frac {3 \sinh (3 \text {arccosh}(a x))}{16 \sqrt {\text {arccosh}(a x)}}+\frac {\sinh (5 \text {arccosh}(a x))}{16 \sqrt {\text {arccosh}(a x)}}\right )d\text {arccosh}(a x)}{a^6}-\frac {2 x^5}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {8 \left (\frac {6 \int \left (\frac {\sqrt {\frac {a x-1}{a x+1}} (a x+1)}{4 \sqrt {\text {arccosh}(a x)}}+\frac {\sinh (3 \text {arccosh}(a x))}{4 \sqrt {\text {arccosh}(a x)}}\right )d\text {arccosh}(a x)}{a^4}-\frac {2 x^3}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x^4 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {10}{3} a \left (\frac {10 \left (-\frac {1}{16} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{16} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )+\frac {1}{32} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^6}-\frac {2 x^5}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {8 \left (\frac {6 \left (-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )\right )}{a^4}-\frac {2 x^3}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x^4 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\)

Input:

Int[x^4/ArcCosh[a*x]^(5/2),x]
 

Output:

(-2*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]^(3/2)) - (8*((-2*x 
^3)/(a*Sqrt[ArcCosh[a*x]]) + (6*(-1/8*(Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]]) - 
 (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/8 + (Sqrt[Pi]*Erfi[Sqrt[ArcC 
osh[a*x]]])/8 + (Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/8))/a^4))/(3 
*a) + (10*a*((-2*x^5)/(a*Sqrt[ArcCosh[a*x]]) + (10*(-1/16*(Sqrt[Pi]*Erf[Sq 
rt[ArcCosh[a*x]]]) - (Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/32 - (Sq 
rt[Pi/5]*Erf[Sqrt[5]*Sqrt[ArcCosh[a*x]]])/32 + (Sqrt[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^6))/3
 

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 6301

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[c*((m + 1)/(b*(n + 1)))   Int[x^(m + 1)*((a + b*ArcCosh[c*x 
])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] + Simp[m/(b*c*(n + 1)) 
 Int[x^(m - 1)*((a + b*ArcCosh[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]) 
), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
 

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]
 

rule 6366

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1 
_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(f*x)^m*((a 
 + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x 
]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], x] - Simp[f*(m/(b*c*(n + 1)))*Simp 
[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]   Int[ 
(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1, e 
1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && LtQ[n, -1]
 

Maple [F]

\[\int \frac {x^{4}}{\operatorname {arccosh}\left (a x \right )^{\frac {5}{2}}}d x\]

Input:

int(x^4/arccosh(a*x)^(5/2),x)
 

Output:

int(x^4/arccosh(a*x)^(5/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^4}{\text {arccosh}(a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^4/arccosh(a*x)^(5/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {x^4}{\text {arccosh}(a x)^{5/2}} \, dx=\int \frac {x^{4}}{\operatorname {acosh}^{\frac {5}{2}}{\left (a x \right )}}\, dx \] Input:

integrate(x**4/acosh(a*x)**(5/2),x)
 

Output:

Integral(x**4/acosh(a*x)**(5/2), x)
 

Maxima [F]

\[ \int \frac {x^4}{\text {arccosh}(a x)^{5/2}} \, dx=\int { \frac {x^{4}}{\operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^4/arccosh(a*x)^(5/2),x, algorithm="maxima")
 

Output:

integrate(x^4/arccosh(a*x)^(5/2), x)
 

Giac [F]

\[ \int \frac {x^4}{\text {arccosh}(a x)^{5/2}} \, dx=\int { \frac {x^{4}}{\operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^4/arccosh(a*x)^(5/2),x, algorithm="giac")
 

Output:

integrate(x^4/arccosh(a*x)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4}{\text {arccosh}(a x)^{5/2}} \, dx=\int \frac {x^4}{{\mathrm {acosh}\left (a\,x\right )}^{5/2}} \,d x \] Input:

int(x^4/acosh(a*x)^(5/2),x)
 

Output:

int(x^4/acosh(a*x)^(5/2), x)
 

Reduce [F]

\[ \int \frac {x^4}{\text {arccosh}(a x)^{5/2}} \, dx=\frac {-\frac {4 \mathit {acosh} \left (a x \right )^{2} \left (\int \frac {\sqrt {\mathit {acosh} \left (a x \right )}}{\mathit {acosh} \left (a x \right )^{3} a^{2} x^{2}-\mathit {acosh} \left (a x \right )^{3}}d x \right ) a}{3}+\frac {4 \mathit {acosh} \left (a x \right )^{2} \left (\int \frac {\sqrt {\mathit {acosh} \left (a x \right )}\, x^{2}}{\mathit {acosh} \left (a x \right )^{3} a^{2} x^{2}-\mathit {acosh} \left (a x \right )^{3}}d x \right ) a^{3}}{3}+\frac {10 \mathit {acosh} \left (a x \right )^{2} \left (\int \frac {\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x^{5}}{\mathit {acosh} \left (a x \right )^{2} a^{2} x^{2}-\mathit {acosh} \left (a x \right )^{2}}d x \right ) a^{6}}{3}-\frac {8 \mathit {acosh} \left (a x \right )^{2} \left (\int \frac {\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x^{3}}{\mathit {acosh} \left (a x \right )^{2} a^{2} x^{2}-\mathit {acosh} \left (a x \right )^{2}}d x \right ) a^{4}}{3}-\frac {8 \mathit {acosh} \left (a x \right )^{2} \left (\int \frac {\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x}{\mathit {acosh} \left (a x \right )^{2} a^{2} x^{2}-\mathit {acosh} \left (a x \right )^{2}}d x \right ) a^{2}}{9}-\frac {2 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, a^{4} x^{4}}{3}+\frac {8 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}}{9}}{\mathit {acosh} \left (a x \right )^{2} a^{5}} \] Input:

int(x^4/acosh(a*x)^(5/2),x)
 

Output:

(2*( - 6*acosh(a*x)**2*int(sqrt(acosh(a*x))/(acosh(a*x)**3*a**2*x**2 - aco 
sh(a*x)**3),x)*a + 6*acosh(a*x)**2*int((sqrt(acosh(a*x))*x**2)/(acosh(a*x) 
**3*a**2*x**2 - acosh(a*x)**3),x)*a**3 + 15*acosh(a*x)**2*int((sqrt(a*x + 
1)*sqrt(a*x - 1)*sqrt(acosh(a*x))*x**5)/(acosh(a*x)**2*a**2*x**2 - acosh(a 
*x)**2),x)*a**6 - 12*acosh(a*x)**2*int((sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt(a 
cosh(a*x))*x**3)/(acosh(a*x)**2*a**2*x**2 - acosh(a*x)**2),x)*a**4 - 4*aco 
sh(a*x)**2*int((sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt(acosh(a*x))*x)/(acosh(a*x 
)**2*a**2*x**2 - acosh(a*x)**2),x)*a**2 - 3*sqrt(a*x + 1)*sqrt(a*x - 1)*sq 
rt(acosh(a*x))*a**4*x**4 + 4*sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt(acosh(a*x))) 
)/(9*acosh(a*x)**2*a**5)