3.2.5 \(\int \frac {x^2}{\text {arccosh}(a x)^{5/2}} \, dx\) [105]

3.2.5.1 Optimal result

Integrand size = 12, antiderivative size = 166 \[ \int \frac {x^2}{\text {arccosh}(a x)^{5/2}} \, dx=-\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^{3/2}}+\frac {8 x}{3 a^2 \sqrt {\text {arccosh}(a x)}}-\frac {4 x^3}{\sqrt {\text {arccosh}(a x)}}-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )}{6 a^3}-\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{2 a^3}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )}{6 a^3}+\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arccosh}(a x)}\right )}{2 a^3} \]

output

-1/6*erf(arccosh(a*x)^(1/2))*Pi^(1/2)/a^3+1/6*erfi(arccosh(a*x)^(1/2))*Pi^ 
(1/2)/a^3-1/2*erf(3^(1/2)*arccosh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^3+1/2*erf 
i(3^(1/2)*arccosh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^3-2/3*x^2*(a*x-1)^(1/2)*( 
a*x+1)^(1/2)/a/arccosh(a*x)^(3/2)+8/3*x/a^2/arccosh(a*x)^(1/2)-4*x^3/arcco 
sh(a*x)^(1/2)
 

3.2.5.2 Mathematica [A] (warning: unable to verify)

Time = 0.45 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.17 \[ \int \frac {x^2}{\text {arccosh}(a x)^{5/2}} \, dx=\frac {-\sqrt {\frac {-1+a x}{1+a x}} (1+a x)-3 e^{-3 \text {arccosh}(a x)} \text {arccosh}(a x)-e^{-\text {arccosh}(a x)} \text {arccosh}(a x)-e^{\text {arccosh}(a x)} \text {arccosh}(a x)-3 e^{3 \text {arccosh}(a x)} \text {arccosh}(a x)-3 \sqrt {3} (-\text {arccosh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-3 \text {arccosh}(a x)\right )-(-\text {arccosh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-\text {arccosh}(a x)\right )+\text {arccosh}(a x)^{3/2} \Gamma \left (\frac {1}{2},\text {arccosh}(a x)\right )+3 \sqrt {3} \text {arccosh}(a x)^{3/2} \Gamma \left (\frac {1}{2},3 \text {arccosh}(a x)\right )-\sinh (3 \text {arccosh}(a x))}{6 a^3 \text {arccosh}(a x)^{3/2}} \]

input

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

output

(-(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)) - (3*ArcCosh[a*x])/E^(3*ArcCosh[a 
*x]) - ArcCosh[a*x]/E^ArcCosh[a*x] - E^ArcCosh[a*x]*ArcCosh[a*x] - 3*E^(3* 
ArcCosh[a*x])*ArcCosh[a*x] - 3*Sqrt[3]*(-ArcCosh[a*x])^(3/2)*Gamma[1/2, -3 
*ArcCosh[a*x]] - (-ArcCosh[a*x])^(3/2)*Gamma[1/2, -ArcCosh[a*x]] + ArcCosh 
[a*x]^(3/2)*Gamma[1/2, ArcCosh[a*x]] + 3*Sqrt[3]*ArcCosh[a*x]^(3/2)*Gamma[ 
1/2, 3*ArcCosh[a*x]] - Sinh[3*ArcCosh[a*x]])/(6*a^3*ArcCosh[a*x]^(3/2))
 

3.2.5.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.39 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.36, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6301, 6366, 6296, 3042, 26, 3789, 2611, 2633, 2634, 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.

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

\(\Big \downarrow \) 6301

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

\(\Big \downarrow \) 6366

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

\(\Big \downarrow \) 6296

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3789

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

\(\Big \downarrow \) 2611

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

\(\Big \downarrow \) 2633

\(\displaystyle -\frac {4 \left (-\frac {2 x}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 i \left (\frac {1}{2} i \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )-i \int e^{-\text {arccosh}(a x)}d\sqrt {\text {arccosh}(a x)}\right )}{a^2}\right )}{3 a}+2 a \left (\frac {6 \int \frac {x^2}{\sqrt {\text {arccosh}(a x)}}dx}{a}-\frac {2 x^3}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 2634

\(\displaystyle 2 a \left (\frac {6 \int \frac {x^2}{\sqrt {\text {arccosh}(a x)}}dx}{a}-\frac {2 x^3}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {4 \left (-\frac {2 x}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 i \left (\frac {1}{2} i \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{2} i \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )\right )}{a^2}\right )}{3 a}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 6302

\(\displaystyle 2 a \left (\frac {6 \int \frac {a^2 x^2 \sqrt {\frac {a x-1}{a x+1}} (a x+1)}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a^4}-\frac {2 x^3}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {4 \left (-\frac {2 x}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 i \left (\frac {1}{2} i \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{2} i \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )\right )}{a^2}\right )}{3 a}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 5971

\(\displaystyle 2 a \left (\frac {6 \int \left (\frac {\sqrt {\frac {a x-1}{a x+1}} (a x+1)}{4 \sqrt {\text {arccosh}(a x)}}+\frac {\sinh (3 \text {arccosh}(a x))}{4 \sqrt {\text {arccosh}(a x)}}\right )d\text {arccosh}(a x)}{a^4}-\frac {2 x^3}{a \sqrt {\text {arccosh}(a x)}}\right )-\frac {4 \left (-\frac {2 x}{a \sqrt {\text {arccosh}(a x)}}-\frac {2 i \left (\frac {1}{2} i \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arccosh}(a x)}\right )-\frac {1}{2} i \sqrt {\pi } \text {erf}\left (\sqrt {\text {arccosh}(a x)}\right )\right )}{a^2}\right )}{3 a}-\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^{3/2}}\)

\(\Big \downarrow \) 2009

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

input

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

output

(-2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]^(3/2)) - (4*((-2*x 
)/(a*Sqrt[ArcCosh[a*x]]) - ((2*I)*((-1/2*I)*Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x] 
]] + (I/2)*Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]]))/a^2))/(3*a) + 2*a*((-2*x^3) 
/(a*Sqrt[ArcCosh[a*x]]) + (6*(-1/8*(Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]]) - (S 
qrt[Pi/3]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/8 + (Sqrt[Pi]*Erfi[Sqrt[ArcCosh 
[a*x]]])/8 + (Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/8))/a^4)
 

3.2.5.3.1 Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2009

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

rule 2611

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : 
> Simp[2/d   Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d 
*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]
 

rule 2633

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ 
F, a, b, c, d}, x] && PosQ[b]
 

rule 2634

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr 
eeQ[{F, a, b, c, d}, x] && NegQ[b]
 

rule 3042

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

rule 3789

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I 
/2   Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2   Int[(c + d*x)^m*E 
^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
 

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 6296

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c)   S 
ubst[Int[x^n*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, 
b, c, n}, x]
 

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]
 

3.2.5.4 Maple [F]

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

input

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

output

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

3.2.5.5 Fricas [F(-2)]

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

input

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

output

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

3.2.5.6 Sympy [F]

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

input

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

output

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

3.2.5.7 Maxima [F]

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

input

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

output

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

3.2.5.8 Giac [F]

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

input

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

output

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

3.2.5.9 Mupad [F(-1)]

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

input

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

output

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