\(\int \frac {x^4}{\text {arccosh}(a x)^{3/2}} \, dx\) [97]

Optimal result

Integrand size = 12, antiderivative size = 193 \[ \int \frac {x^4}{\text {arccosh}(a x)^{3/2}} \, dx=-\frac {2 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {\text {arccosh}(a x)}}+\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{8 a^5}+\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{16 a^5}+\frac {\sqrt {5 \pi } \text {erf}\left (\sqrt {5} \sqrt {\text {arccosh}(a x)}\right )}{16 a^5}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{8 a^5}+\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{16 a^5}+\frac {\sqrt {5 \pi } \text {erfi}\left (\sqrt {5} \sqrt {\text {arccosh}(a x)}\right )}{16 a^5} \] Output:

-2*x^4*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)^(1/2)+1/8*Pi^(1/2)*erf(a 
rccosh(a*x)^(1/2))/a^5+3/16*3^(1/2)*Pi^(1/2)*erf(3^(1/2)*arccosh(a*x)^(1/2 
))/a^5+1/16*5^(1/2)*Pi^(1/2)*erf(5^(1/2)*arccosh(a*x)^(1/2))/a^5+1/8*Pi^(1 
/2)*erfi(arccosh(a*x)^(1/2))/a^5+3/16*3^(1/2)*Pi^(1/2)*erfi(3^(1/2)*arccos 
h(a*x)^(1/2))/a^5+1/16*5^(1/2)*Pi^(1/2)*erfi(5^(1/2)*arccosh(a*x)^(1/2))/a 
^5
 

Mathematica [A] (warning: unable to verify)

Time = 0.22 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.04 \[ \int \frac {x^4}{\text {arccosh}(a x)^{3/2}} \, dx=-\frac {4 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)-\sqrt {5} \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-5 \text {arccosh}(a x)\right )-3 \sqrt {3} \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-3 \text {arccosh}(a x)\right )-2 \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-\text {arccosh}(a x)\right )+2 \sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},\text {arccosh}(a x)\right )+3 \sqrt {3} \sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},3 \text {arccosh}(a x)\right )+\sqrt {5} \sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},5 \text {arccosh}(a x)\right )+6 \sinh (3 \text {arccosh}(a x))+2 \sinh (5 \text {arccosh}(a x))}{16 a^5 \sqrt {\text {arccosh}(a x)}} \] Input:

Integrate[x^4/ArcCosh[a*x]^(3/2),x]
 

Output:

-1/16*(4*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x) - Sqrt[5]*Sqrt[-ArcCosh[a*x] 
]*Gamma[1/2, -5*ArcCosh[a*x]] - 3*Sqrt[3]*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, - 
3*ArcCosh[a*x]] - 2*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -ArcCosh[a*x]] + 2*Sqrt 
[ArcCosh[a*x]]*Gamma[1/2, ArcCosh[a*x]] + 3*Sqrt[3]*Sqrt[ArcCosh[a*x]]*Gam 
ma[1/2, 3*ArcCosh[a*x]] + Sqrt[5]*Sqrt[ArcCosh[a*x]]*Gamma[1/2, 5*ArcCosh[ 
a*x]] + 6*Sinh[3*ArcCosh[a*x]] + 2*Sinh[5*ArcCosh[a*x]])/(a^5*Sqrt[ArcCosh 
[a*x]])
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.94, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, 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.

\(\displaystyle \int \frac {x^4}{\text {arccosh}(a x)^{3/2}} \, dx\)

\(\Big \downarrow \) 6300

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

\(\Big \downarrow \) 2009

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

Input:

Int[x^4/ArcCosh[a*x]^(3/2),x]
 

Output:

(-2*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*Sqrt[ArcCosh[a*x]]) - (2*(-1/16*( 
Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]]) - (3*Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcCosh 
[a*x]]])/32 - (Sqrt[5*Pi]*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]*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 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 [F]

\[\int \frac {x^{4}}{\operatorname {arccosh}\left (a x \right )^{\frac {3}{2}}}d x\]

Input:

int(x^4/arccosh(a*x)^(3/2),x)
 

Output:

int(x^4/arccosh(a*x)^(3/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^4}{\text {arccosh}(a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^4/arccosh(a*x)^(3/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {x^4}{\text {arccosh}(a x)^{3/2}} \, dx=\int \frac {x^{4}}{\operatorname {acosh}^{\frac {3}{2}}{\left (a x \right )}}\, dx \] Input:

integrate(x**4/acosh(a*x)**(3/2),x)
 

Output:

Integral(x**4/acosh(a*x)**(3/2), x)
 

Maxima [F]

\[ \int \frac {x^4}{\text {arccosh}(a x)^{3/2}} \, dx=\int { \frac {x^{4}}{\operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(x^4/arccosh(a*x)^(3/2),x, algorithm="maxima")
 

Output:

integrate(x^4/arccosh(a*x)^(3/2), x)
 

Giac [F]

\[ \int \frac {x^4}{\text {arccosh}(a x)^{3/2}} \, dx=\int { \frac {x^{4}}{\operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(x^4/arccosh(a*x)^(3/2),x, algorithm="giac")
 

Output:

integrate(x^4/arccosh(a*x)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4}{\text {arccosh}(a x)^{3/2}} \, dx=\int \frac {x^4}{{\mathrm {acosh}\left (a\,x\right )}^{3/2}} \,d x \] Input:

int(x^4/acosh(a*x)^(3/2),x)
 

Output:

int(x^4/acosh(a*x)^(3/2), x)
 

Reduce [F]

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

int(x^4/acosh(a*x)^(3/2),x)
 

Output:

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