\(\int \frac {(c-a^2 c x^2)^{3/2}}{\sqrt {\text {arccosh}(a x)}} \, dx\) [105]

Optimal result

Integrand size = 24, antiderivative size = 294 \[ \int \frac {\left (c-a^2 c x^2\right )^{3/2}}{\sqrt {\text {arccosh}(a x)}} \, dx=-\frac {3 c \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}}{4 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{4 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{4 a \sqrt {-1+a x} \sqrt {1+a x}} \] Output:

-3/4*c*(-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/ 
2)-1/32*c*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)*erf(2*arccosh(a*x)^(1/2))/a/(a*x-1 
)^(1/2)/(a*x+1)^(1/2)+1/8*c*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)-1/32*c*Pi^(1/2)*(-a^ 
2*c*x^2+c)^(1/2)*erfi(2*arccosh(a*x)^(1/2))/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)+ 
1/8*c*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 = 0.24 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.52 \[ \int \frac {\left (c-a^2 c x^2\right )^{3/2}}{\sqrt {\text {arccosh}(a x)}} \, dx=-\frac {c \sqrt {c-a^2 c x^2} \left (\sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-4 \text {arccosh}(a x)\right )-4 \sqrt {2} \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-2 \text {arccosh}(a x)\right )+\sqrt {\text {arccosh}(a x)} \left (24 \sqrt {\text {arccosh}(a x)}+4 \sqrt {2} \Gamma \left (\frac {1}{2},2 \text {arccosh}(a x)\right )-\Gamma \left (\frac {1}{2},4 \text {arccosh}(a x)\right )\right )\right )}{32 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \sqrt {\text {arccosh}(a x)}} \] Input:

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

Output:

-1/32*(c*Sqrt[c - a^2*c*x^2]*(Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -4*ArcCosh[a* 
x]] - 4*Sqrt[2]*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -2*ArcCosh[a*x]] + Sqrt[Arc 
Cosh[a*x]]*(24*Sqrt[ArcCosh[a*x]] + 4*Sqrt[2]*Gamma[1/2, 2*ArcCosh[a*x]] - 
 Gamma[1/2, 4*ArcCosh[a*x]])))/(a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*Sqr 
t[ArcCosh[a*x]])
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.50, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6321, 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.

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

\(\Big \downarrow \) 6321

\(\displaystyle -\frac {c \sqrt {c-a^2 c x^2} \int \frac {(a x-1)^2 (a x+1)^2}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a \sqrt {a x-1} \sqrt {a x+1}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {c \sqrt {c-a^2 c x^2} \int \frac {\sin (i \text {arccosh}(a x))^4}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a \sqrt {a x-1} \sqrt {a x+1}}\)

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((c*Sqrt[c - a^2*c*x^2]*((3*Sqrt[ArcCosh[a*x]])/4 + (Sqrt[Pi]*Erf[2*Sqrt[ 
ArcCosh[a*x]]])/32 - (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/4 + (Sqr 
t[Pi]*Erfi[2*Sqrt[ArcCosh[a*x]]])/32 - (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCo 
sh[a*x]]])/4))/(a*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 6321

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), 
x_Symbol] :> Simp[(1/(b*c))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] 
  Subst[Int[x^n*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] 
/; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*sqrt(c)*c*( - sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt( - a**2*x**2 + 1)*sqrt(a 
cosh(a*x))*a**2*x**2 + sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt( - a**2*x**2 + 1)* 
sqrt(acosh(a*x)) + 4*int((sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt( - a**2*x**2 + 
1)*sqrt(acosh(a*x))*x**3)/(a**2*x**2 - 1),x)*a**4 - 4*int((sqrt(a*x + 1)*s 
qrt(a*x - 1)*sqrt( - a**2*x**2 + 1)*sqrt(acosh(a*x))*x)/(a**2*x**2 - 1),x) 
*a**2))/a