\(\int \frac {(c-a^2 c x^2)^{3/2}}{\text {arccosh}(a x)^{3/2}} \, dx\) [111]

Optimal result

Integrand size = 24, antiderivative size = 286 \[ \int \frac {\left (c-a^2 c x^2\right )^{3/2}}{\text {arccosh}(a x)^{3/2}} \, dx=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \left (c-a^2 c x^2\right )^{3/2}}{a \sqrt {\text {arccosh}(a x)}}+\frac {c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{4 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {c \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{4 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{a \sqrt {-1+a x} \sqrt {1+a x}} \] Output:

-2*(a*x-1)^(1/2)*(a*x+1)^(1/2)*(-a^2*c*x^2+c)^(3/2)/a/arccosh(a*x)^(1/2)+1 
/4*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)-1/2*c*2^(1/2)*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)*erf(2^(1/2)*a 
rccosh(a*x)^(1/2))/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)-1/4*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)+1/2*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] (warning: unable to verify)

Time = 0.41 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.84 \[ \int \frac {\left (c-a^2 c x^2\right )^{3/2}}{\text {arccosh}(a x)^{3/2}} \, dx=-\frac {c e^{-4 \text {arccosh}(a x)} \sqrt {c-a^2 c x^2} \left (-1-14 e^{4 \text {arccosh}(a x)}-e^{8 \text {arccosh}(a x)}+16 a^2 e^{4 \text {arccosh}(a x)} x^2+4 e^{4 \text {arccosh}(a x)} \sqrt {2 \pi } \sqrt {\text {arccosh}(a x)} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-4 e^{4 \text {arccosh}(a x)} \sqrt {2 \pi } \sqrt {\text {arccosh}(a x)} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+2 e^{4 \text {arccosh}(a x)} \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-4 \text {arccosh}(a x)\right )+2 e^{4 \text {arccosh}(a x)} \sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},4 \text {arccosh}(a x)\right )\right )}{8 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \sqrt {\text {arccosh}(a x)}} \] Input:

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

Output:

-1/8*(c*Sqrt[c - a^2*c*x^2]*(-1 - 14*E^(4*ArcCosh[a*x]) - E^(8*ArcCosh[a*x 
]) + 16*a^2*E^(4*ArcCosh[a*x])*x^2 + 4*E^(4*ArcCosh[a*x])*Sqrt[2*Pi]*Sqrt[ 
ArcCosh[a*x]]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] - 4*E^(4*ArcCosh[a*x])*Sqrt[ 
2*Pi]*Sqrt[ArcCosh[a*x]]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + 2*E^(4*ArcCosh 
[a*x])*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -4*ArcCosh[a*x]] + 2*E^(4*ArcCosh[a* 
x])*Sqrt[ArcCosh[a*x]]*Gamma[1/2, 4*ArcCosh[a*x]]))/(a*E^(4*ArcCosh[a*x])* 
Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*Sqrt[ArcCosh[a*x]])
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.64, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6319, 25, 6327, 6367, 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.

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

\(\Big \downarrow \) 6319

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 6327

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

\(\Big \downarrow \) 6367

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {8 c \sqrt {c-a^2 c x^2} \left (-\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )\right )}{a \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \left (c-a^2 c x^2\right )^{3/2}}{a \sqrt {\text {arccosh}(a x)}}\)

Input:

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

Output:

(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(c - a^2*c*x^2)^(3/2))/(a*Sqrt[ArcCosh[a* 
x]]) - (8*c*Sqrt[c - a^2*c*x^2]*(-1/32*(Sqrt[Pi]*Erf[2*Sqrt[ArcCosh[a*x]]] 
) + (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/8 + (Sqrt[Pi]*Erfi[2*Sqrt 
[ArcCosh[a*x]]])/32 - (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/8))/(a 
*Sqrt[-1 + a*x]*Sqrt[1 + a*x])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

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

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 6319

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[Simp[Sqrt[1 + c*x]*Sqrt[-1 + c*x]*(d + e*x^2)^p]*((a + b*A 
rcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp[c*((2*p + 1)/(b*(n + 1)))*Si 
mp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]   Int[x*(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, 
d, e, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IntegerQ[2*p]
 

rule 6327

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d1_) + ( 
e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(f*x)^m*(d1 
*d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2 
, e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]
 

rule 6367

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) 
^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/((1 + c*x 
)^p*(-1 + c*x)^p)]   Subst[Int[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, d, e, n}, x] && Eq 
Q[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

integrate((-a^2*c*x^2+c)^(3/2)/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]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*sqrt(c)*c*( - 4*acosh(a*x)*int((sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt( - a** 
2*x**2 + 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( - a**2*x**2 + 1 
)*sqrt(acosh(a*x))*x)/(acosh(a*x)*a**2*x**2 - acosh(a*x)),x)*a**2 + sqrt(a 
*x + 1)*sqrt(a*x - 1)*sqrt( - a**2*x**2 + 1)*sqrt(acosh(a*x))*a**2*x**2 - 
sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt( - a**2*x**2 + 1)*sqrt(acosh(a*x))))/(aco 
sh(a*x)*a)