\(\int \frac {x^4}{\text {arccosh}(a x)^{7/2}} \, dx\) [109]

Optimal result

Integrand size = 12, antiderivative size = 300 \[ \int \frac {x^4}{\text {arccosh}(a x)^{7/2}} \, dx=-\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{5 a \text {arccosh}(a x)^{5/2}}+\frac {16 x^3}{15 a^2 \text {arccosh}(a x)^{3/2}}-\frac {4 x^5}{3 \text {arccosh}(a x)^{3/2}}+\frac {32 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a^3 \sqrt {\text {arccosh}(a x)}}-\frac {40 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \sqrt {\text {arccosh}(a x)}}+\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{30 a^5}+\frac {9 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{20 a^5}+\frac {5 \sqrt {5 \pi } \text {erf}\left (\sqrt {5} \sqrt {\text {arccosh}(a x)}\right )}{12 a^5}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{30 a^5}+\frac {9 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{20 a^5}+\frac {5 \sqrt {5 \pi } \text {erfi}\left (\sqrt {5} \sqrt {\text {arccosh}(a x)}\right )}{12 a^5} \] Output:

-2/5*x^4*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)^(5/2)+16/15*x^3/a^2/ar 
ccosh(a*x)^(3/2)-4/3*x^5/arccosh(a*x)^(3/2)+32/5*x^2*(a*x-1)^(1/2)*(a*x+1) 
^(1/2)/a^3/arccosh(a*x)^(1/2)-40/3*x^4*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arcco 
sh(a*x)^(1/2)+1/30*Pi^(1/2)*erf(arccosh(a*x)^(1/2))/a^5+9/20*3^(1/2)*Pi^(1 
/2)*erf(3^(1/2)*arccosh(a*x)^(1/2))/a^5+5/12*5^(1/2)*Pi^(1/2)*erf(5^(1/2)* 
arccosh(a*x)^(1/2))/a^5+1/30*Pi^(1/2)*erfi(arccosh(a*x)^(1/2))/a^5+9/20*3^ 
(1/2)*Pi^(1/2)*erfi(3^(1/2)*arccosh(a*x)^(1/2))/a^5+5/12*5^(1/2)*Pi^(1/2)* 
erfi(5^(1/2)*arccosh(a*x)^(1/2))/a^5
 

Mathematica [A] (warning: unable to verify)

Time = 1.30 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.25 \[ \int \frac {x^4}{\text {arccosh}(a x)^{7/2}} \, dx=\frac {-6 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)-2 e^{-\text {arccosh}(a x)} \text {arccosh}(a x)-2 e^{\text {arccosh}(a x)} \text {arccosh}(a x)+4 e^{-\text {arccosh}(a x)} \text {arccosh}(a x)^2-4 e^{\text {arccosh}(a x)} \text {arccosh}(a x)^2+4 (-\text {arccosh}(a x))^{5/2} \Gamma \left (\frac {1}{2},-\text {arccosh}(a x)\right )-4 \text {arccosh}(a x)^{5/2} \Gamma \left (\frac {1}{2},\text {arccosh}(a x)\right )-5 \text {arccosh}(a x) \left (e^{-5 \text {arccosh}(a x)} (1-10 \text {arccosh}(a x))+e^{5 \text {arccosh}(a x)} (1+10 \text {arccosh}(a x))+10 \sqrt {5} (-\text {arccosh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-5 \text {arccosh}(a x)\right )+10 \sqrt {5} \text {arccosh}(a x)^{3/2} \Gamma \left (\frac {1}{2},5 \text {arccosh}(a x)\right )\right )-9 e^{-3 \text {arccosh}(a x)} \left (\text {arccosh}(a x)+e^{6 \text {arccosh}(a x)} \text {arccosh}(a x)-6 \text {arccosh}(a x)^2+6 e^{6 \text {arccosh}(a x)} \text {arccosh}(a x)^2-6 \sqrt {3} e^{3 \text {arccosh}(a x)} (-\text {arccosh}(a x))^{5/2} \Gamma \left (\frac {1}{2},-3 \text {arccosh}(a x)\right )+6 \sqrt {3} e^{3 \text {arccosh}(a x)} \text {arccosh}(a x)^{5/2} \Gamma \left (\frac {1}{2},3 \text {arccosh}(a x)\right )+e^{3 \text {arccosh}(a x)} \sinh (3 \text {arccosh}(a x))\right )-3 \sinh (5 \text {arccosh}(a x))}{120 a^5 \text {arccosh}(a x)^{5/2}} \] Input:

Integrate[x^4/ArcCosh[a*x]^(7/2),x]
 

Output:

(-6*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x) - (2*ArcCosh[a*x])/E^ArcCosh[a*x] 
 - 2*E^ArcCosh[a*x]*ArcCosh[a*x] + (4*ArcCosh[a*x]^2)/E^ArcCosh[a*x] - 4*E 
^ArcCosh[a*x]*ArcCosh[a*x]^2 + 4*(-ArcCosh[a*x])^(5/2)*Gamma[1/2, -ArcCosh 
[a*x]] - 4*ArcCosh[a*x]^(5/2)*Gamma[1/2, ArcCosh[a*x]] - 5*ArcCosh[a*x]*(( 
1 - 10*ArcCosh[a*x])/E^(5*ArcCosh[a*x]) + E^(5*ArcCosh[a*x])*(1 + 10*ArcCo 
sh[a*x]) + 10*Sqrt[5]*(-ArcCosh[a*x])^(3/2)*Gamma[1/2, -5*ArcCosh[a*x]] + 
10*Sqrt[5]*ArcCosh[a*x]^(3/2)*Gamma[1/2, 5*ArcCosh[a*x]]) - (9*(ArcCosh[a* 
x] + E^(6*ArcCosh[a*x])*ArcCosh[a*x] - 6*ArcCosh[a*x]^2 + 6*E^(6*ArcCosh[a 
*x])*ArcCosh[a*x]^2 - 6*Sqrt[3]*E^(3*ArcCosh[a*x])*(-ArcCosh[a*x])^(5/2)*G 
amma[1/2, -3*ArcCosh[a*x]] + 6*Sqrt[3]*E^(3*ArcCosh[a*x])*ArcCosh[a*x]^(5/ 
2)*Gamma[1/2, 3*ArcCosh[a*x]] + E^(3*ArcCosh[a*x])*Sinh[3*ArcCosh[a*x]]))/ 
E^(3*ArcCosh[a*x]) - 3*Sinh[5*ArcCosh[a*x]])/(120*a^5*ArcCosh[a*x]^(5/2))
 

Rubi [A] (verified)

Time = 1.28 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.36, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6301, 6366, 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.

\(\displaystyle \int \frac {x^4}{\text {arccosh}(a x)^{7/2}} \, dx\)

\(\Big \downarrow \) 6301

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

\(\Big \downarrow \) 6366

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

\(\Big \downarrow \) 6300

\(\displaystyle 2 a \left (\frac {10 \left (-\frac {2 \int \left (-\frac {a x}{8 \sqrt {\text {arccosh}(a x)}}-\frac {9 \cosh (3 \text {arccosh}(a x))}{16 \sqrt {\text {arccosh}(a x)}}-\frac {5 \cosh (5 \text {arccosh}(a x))}{16 \sqrt {\text {arccosh}(a x)}}\right )d\text {arccosh}(a x)}{a^5}-\frac {2 x^4 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{3 a}-\frac {2 x^5}{3 a \text {arccosh}(a x)^{3/2}}\right )-\frac {8 \left (\frac {2 \left (-\frac {2 \int \left (-\frac {a x}{4 \sqrt {\text {arccosh}(a x)}}-\frac {3 \cosh (3 \text {arccosh}(a x))}{4 \sqrt {\text {arccosh}(a x)}}\right )d\text {arccosh}(a x)}{a^3}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {\text {arccosh}(a x)}}\right )}{a}-\frac {2 x^3}{3 a \text {arccosh}(a x)^{3/2}}\right )}{5 a}-\frac {2 x^4 \sqrt {a x-1} \sqrt {a x+1}}{5 a \text {arccosh}(a x)^{5/2}}\)

\(\Big \downarrow \) 2009

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

Input:

Int[x^4/ArcCosh[a*x]^(7/2),x]
 

Output:

(-2*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(5*a*ArcCosh[a*x]^(5/2)) - (8*((-2*x 
^3)/(3*a*ArcCosh[a*x]^(3/2)) + (2*((-2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/( 
a*Sqrt[ArcCosh[a*x]]) - (2*(-1/8*(Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]]) - (Sqr 
t[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/8 - (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a 
*x]]])/8 - (Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/8))/a^3))/a))/(5* 
a) + 2*a*((-2*x^5)/(3*a*ArcCosh[a*x]^(3/2)) + (10*((-2*x^4*Sqrt[-1 + a*x]* 
Sqrt[1 + a*x])/(a*Sqrt[ArcCosh[a*x]]) - (2*(-1/16*(Sqrt[Pi]*Erf[Sqrt[ArcCo 
sh[a*x]]]) - (3*Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/32 - (Sqrt[5*P 
i]*Erf[Sqrt[5]*Sqrt[ArcCosh[a*x]]])/32 - (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]] 
])/16 - (3*Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/32 - (Sqrt[5*Pi]*E 
rfi[Sqrt[5]*Sqrt[ArcCosh[a*x]]])/32))/a^5))/(3*a))
 

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]
 

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 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]

\[\int \frac {x^{4}}{\operatorname {arccosh}\left (a x \right )^{\frac {7}{2}}}d x\]

Input:

int(x^4/arccosh(a*x)^(7/2),x)
 

Output:

int(x^4/arccosh(a*x)^(7/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^4}{\text {arccosh}(a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^4/arccosh(a*x)^(7/2),x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {x^4}{\text {arccosh}(a x)^{7/2}} \, dx=\text {Timed out} \] Input:

integrate(x**4/acosh(a*x)**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^4}{\text {arccosh}(a x)^{7/2}} \, dx=\int { \frac {x^{4}}{\operatorname {arcosh}\left (a x\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate(x^4/arccosh(a*x)^(7/2),x, algorithm="maxima")
 

Output:

integrate(x^4/arccosh(a*x)^(7/2), x)
 

Giac [F]

\[ \int \frac {x^4}{\text {arccosh}(a x)^{7/2}} \, dx=\int { \frac {x^{4}}{\operatorname {arcosh}\left (a x\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate(x^4/arccosh(a*x)^(7/2),x, algorithm="giac")
 

Output:

integrate(x^4/arccosh(a*x)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4}{\text {arccosh}(a x)^{7/2}} \, dx=\int \frac {x^4}{{\mathrm {acosh}\left (a\,x\right )}^{7/2}} \,d x \] Input:

int(x^4/acosh(a*x)^(7/2),x)
 

Output:

int(x^4/acosh(a*x)^(7/2), x)
 

Reduce [F]

\[ \int \frac {x^4}{\text {arccosh}(a x)^{7/2}} \, dx=\frac {-\frac {4 \mathit {acosh} \left (a x \right )^{3} \left (\int \frac {\sqrt {\mathit {acosh} \left (a x \right )}}{\mathit {acosh} \left (a x \right )^{4} a^{2} x^{2}-\mathit {acosh} \left (a x \right )^{4}}d x \right ) a}{3}+\frac {4 \mathit {acosh} \left (a x \right )^{3} \left (\int \frac {\sqrt {\mathit {acosh} \left (a x \right )}\, x^{2}}{\mathit {acosh} \left (a x \right )^{4} a^{2} x^{2}-\mathit {acosh} \left (a x \right )^{4}}d x \right ) a^{3}}{3}+2 \mathit {acosh} \left (a x \right )^{3} \left (\int \frac {\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x^{5}}{\mathit {acosh} \left (a x \right )^{3} a^{2} x^{2}-\mathit {acosh} \left (a x \right )^{3}}d x \right ) a^{6}-\frac {8 \mathit {acosh} \left (a x \right )^{3} \left (\int \frac {\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x^{3}}{\mathit {acosh} \left (a x \right )^{3} a^{2} x^{2}-\mathit {acosh} \left (a x \right )^{3}}d x \right ) a^{4}}{5}-\frac {8 \mathit {acosh} \left (a x \right )^{3} \left (\int \frac {\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x}{\mathit {acosh} \left (a x \right )^{3} a^{2} x^{2}-\mathit {acosh} \left (a x \right )^{3}}d x \right ) a^{2}}{15}-\frac {2 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, a^{4} x^{4}}{5}+\frac {8 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {\mathit {acosh} \left (a x \right )}}{15}}{\mathit {acosh} \left (a x \right )^{3} a^{5}} \] Input:

int(x^4/acosh(a*x)^(7/2),x)
 

Output:

(2*( - 10*acosh(a*x)**3*int(sqrt(acosh(a*x))/(acosh(a*x)**4*a**2*x**2 - ac 
osh(a*x)**4),x)*a + 10*acosh(a*x)**3*int((sqrt(acosh(a*x))*x**2)/(acosh(a* 
x)**4*a**2*x**2 - acosh(a*x)**4),x)*a**3 + 15*acosh(a*x)**3*int((sqrt(a*x 
+ 1)*sqrt(a*x - 1)*sqrt(acosh(a*x))*x**5)/(acosh(a*x)**3*a**2*x**2 - acosh 
(a*x)**3),x)*a**6 - 12*acosh(a*x)**3*int((sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt 
(acosh(a*x))*x**3)/(acosh(a*x)**3*a**2*x**2 - acosh(a*x)**3),x)*a**4 - 4*a 
cosh(a*x)**3*int((sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt(acosh(a*x))*x)/(acosh(a 
*x)**3*a**2*x**2 - acosh(a*x)**3),x)*a**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) 
)))/(15*acosh(a*x)**3*a**5)