\(\int x \text {arccosh}(a x)^{5/2} \, dx\) [87]

Optimal result

Integrand size = 10, antiderivative size = 157 \[ \int x \text {arccosh}(a x)^{5/2} \, dx=-\frac {15 \sqrt {\text {arccosh}(a x)}}{64 a^2}+\frac {15}{32} x^2 \sqrt {\text {arccosh}(a x)}-\frac {5 x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^{3/2}}{8 a}-\frac {\text {arccosh}(a x)^{5/2}}{4 a^2}+\frac {1}{2} x^2 \text {arccosh}(a x)^{5/2}-\frac {15 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{256 a^2}-\frac {15 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{256 a^2} \] Output:

-15/64*arccosh(a*x)^(1/2)/a^2+15/32*x^2*arccosh(a*x)^(1/2)-5/8*x*(a*x-1)^( 
1/2)*(a*x+1)^(1/2)*arccosh(a*x)^(3/2)/a-1/4*arccosh(a*x)^(5/2)/a^2+1/2*x^2 
*arccosh(a*x)^(5/2)-15/512*2^(1/2)*Pi^(1/2)*erf(2^(1/2)*arccosh(a*x)^(1/2) 
)/a^2-15/512*2^(1/2)*Pi^(1/2)*erfi(2^(1/2)*arccosh(a*x)^(1/2))/a^2
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.59 \[ \int x \text {arccosh}(a x)^{5/2} \, dx=\frac {8 \sqrt {\text {arccosh}(a x)} \left (15+16 \text {arccosh}(a x)^2\right ) \cosh (2 \text {arccosh}(a x))-15 \sqrt {2 \pi } \left (\text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )-160 \text {arccosh}(a x)^{3/2} \sinh (2 \text {arccosh}(a x))}{512 a^2} \] Input:

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

Output:

(8*Sqrt[ArcCosh[a*x]]*(15 + 16*ArcCosh[a*x]^2)*Cosh[2*ArcCosh[a*x]] - 15*S 
qrt[2*Pi]*(Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x 
]]]) - 160*ArcCosh[a*x]^(3/2)*Sinh[2*ArcCosh[a*x]])/(512*a^2)
 

Rubi [A] (verified)

Time = 2.09 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6299, 6354, 6299, 6308, 6368, 3042, 3793, 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.

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

\(\Big \downarrow \) 6299

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

\(\Big \downarrow \) 6354

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

\(\Big \downarrow \) 6299

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

\(\Big \downarrow \) 6308

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

\(\Big \downarrow \) 6368

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 6299

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
x^(m + 1)*((a + b*ArcCosh[c*x])^n/(m + 1)), x] - Simp[b*c*(n/(m + 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] && GtQ[n, 0]
 

rule 6308

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq 
rt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + 
 c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*ArcCosh[ 
c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1 
] && EqQ[e2, (-c)*d2] && NeQ[n, -1]
 

rule 6354

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 
1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 
1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1*e2*( 
m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1)))   Int[(f*x)^(m 
 - 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*f 
*(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/( 
-1 + c*x)^p]   Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*( 
a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, 
p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IGtQ[m, 1] && N 
eQ[m + 2*p + 1, 0]
 

rule 6368

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x 
_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))* 
Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p]   Subst[In 
t[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c 
*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[ 
e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]
 

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.89

method	result	size
default	\(-\frac {\sqrt {2}\, \left (-128 \operatorname {arccosh}\left (a x \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }\, a^{2} x^{2}+160 \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, \sqrt {a x +1}\, \sqrt {a x -1}\, a x -120 \sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }\, a^{2} x^{2}+64 \operatorname {arccosh}\left (a x \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }+60 \sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\, \sqrt {\pi }+15 \pi \,\operatorname {erf}\left (\sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\right )+15 \pi \,\operatorname {erfi}\left (\sqrt {2}\, \sqrt {\operatorname {arccosh}\left (a x \right )}\right )\right )}{512 \sqrt {\pi }\, a^{2}}\)	\(139\)

Input:

int(x*arccosh(a*x)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/512*2^(1/2)*(-128*arccosh(a*x)^(5/2)*2^(1/2)*Pi^(1/2)*a^2*x^2+160*arcco 
sh(a*x)^(3/2)*2^(1/2)*Pi^(1/2)*(a*x+1)^(1/2)*(a*x-1)^(1/2)*a*x-120*2^(1/2) 
*arccosh(a*x)^(1/2)*Pi^(1/2)*a^2*x^2+64*arccosh(a*x)^(5/2)*2^(1/2)*Pi^(1/2 
)+60*2^(1/2)*arccosh(a*x)^(1/2)*Pi^(1/2)+15*Pi*erf(2^(1/2)*arccosh(a*x)^(1 
/2))+15*Pi*erfi(2^(1/2)*arccosh(a*x)^(1/2)))/Pi^(1/2)/a^2
 

Fricas [F(-2)]

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

integrate(x*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(-1)]

Timed out. \[ \int x \text {arccosh}(a x)^{5/2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(acosh(a*x))*acosh(a*x)**2*x,x)