\(\int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2} \, dx\) [87]

Optimal result

Integrand size = 24, antiderivative size = 302 \[ \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2} \, dx=\frac {3 \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2}-\frac {\sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{5/2}}{5 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}} \] Output:

3/16*(-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2) 
-3/8*a*x^2*(-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^(1/2)/(a*x-1)^(1/2)/(a*x+1)^( 
1/2)+1/2*x*(-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^(3/2)-1/5*(-a^2*c*x^2+c)^(1/2 
)*arccosh(a*x)^(5/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)+3/128*2^(1/2)*Pi^(1/2)* 
(-a^2*c*x^2+c)^(1/2)*erf(2^(1/2)*arccosh(a*x)^(1/2))/a/(a*x-1)^(1/2)/(a*x+ 
1)^(1/2)+3/128*2^(1/2)*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)*erfi(2^(1/2)*arccosh( 
a*x)^(1/2))/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.45 \[ \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2} \, dx=\frac {\sqrt {c-a^2 c x^2} \left (15 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+15 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-8 \sqrt {\text {arccosh}(a x)} \left (16 \text {arccosh}(a x)^2+15 \cosh (2 \text {arccosh}(a x))-20 \text {arccosh}(a x) \sinh (2 \text {arccosh}(a x))\right )\right )}{640 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x)} \] Input:

Integrate[Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2),x]
 

Output:

(Sqrt[c - a^2*c*x^2]*(15*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + 15*S 
qrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]] - 8*Sqrt[ArcCosh[a*x]]*(16*ArcC 
osh[a*x]^2 + 15*Cosh[2*ArcCosh[a*x]] - 20*ArcCosh[a*x]*Sinh[2*ArcCosh[a*x] 
])))/(640*a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))
 

Rubi [A] (verified)

Time = 1.51 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.67, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6310, 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.

\(\displaystyle \int \text {arccosh}(a x)^{3/2} \sqrt {c-a^2 c x^2} \, dx\)

\(\Big \downarrow \) 6310

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

\(\Big \downarrow \) 6299

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

\(\Big \downarrow \) 6308

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

\(\Big \downarrow \) 6368

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

Input:

Int[Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2),x]
 

Output:

(x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2))/2 - (Sqrt[c - a^2*c*x^2]*ArcCos 
h[a*x]^(5/2))/(5*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (3*a*Sqrt[c - a^2*c*x^2 
]*((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*Sqrt[-1 + a*x]*Sqrt[1 + a*x])
 

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 6310

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcCosh[c*x])^n/2), x] + (-Simp[( 
1/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])]   Int[(a + b*ArcC 
osh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Simp[b*c*(n/2)*Simp[Sq 
rt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])]   Int[x*(a + b*ArcCosh[c*x])^ 
(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n 
, 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 [F]

\[\int \sqrt {-a^{2} c \,x^{2}+c}\, \operatorname {arccosh}\left (a x \right )^{\frac {3}{2}}d x\]

Input:

int((-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^(3/2),x)
 

Output:

int((-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^(3/2),x)
 

Fricas [F(-2)]

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

integrate((-a^2*c*x^2+c)^(1/2)*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]

\[ \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^{3/2} \, dx=\int \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {acosh}^{\frac {3}{2}}{\left (a x \right )}\, dx \] Input:

integrate((-a**2*c*x**2+c)**(1/2)*acosh(a*x)**(3/2),x)
 

Output:

Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*acosh(a*x)**(3/2), x)
 

Maxima [F]

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

integrate((-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^(3/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(-a^2*c*x^2 + c)*arccosh(a*x)^(3/2), x)
 

Giac [F(-2)]

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

integrate((-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^(3/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

int(acosh(a*x)^(3/2)*(c - a^2*c*x^2)^(1/2),x)
 

Output:

int(acosh(a*x)^(3/2)*(c - a^2*c*x^2)^(1/2), x)
 

Reduce [F]

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

int((-a^2*c*x^2+c)^(1/2)*acosh(a*x)^(3/2),x)
 

Output:

sqrt(c)*int(sqrt( - a**2*x**2 + 1)*sqrt(acosh(a*x))*acosh(a*x),x)