\(\int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{5/2}} \, dx\) [117]

Optimal result

Integrand size = 24, antiderivative size = 201 \[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{5/2}} \, dx=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {c-a^2 c x^2}}{3 a \text {arccosh}(a x)^{3/2}}-\frac {8 x \sqrt {c-a^2 c x^2}}{3 \sqrt {\text {arccosh}(a x)}}+\frac {2 \sqrt {2 \pi } \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {2 \sqrt {2 \pi } \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{3 a \sqrt {-1+a x} \sqrt {1+a x}} \] Output:

-2/3*(a*x-1)^(1/2)*(a*x+1)^(1/2)*(-a^2*c*x^2+c)^(1/2)/a/arccosh(a*x)^(3/2) 
-8/3*x*(-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(1/2)+2/3*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)+2/3*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 = 0.26 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{5/2}} \, dx=-\frac {2 \sqrt {c-a^2 c x^2} \left ((1+a x) \left (-1+a x+4 a x \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)\right )+\sqrt {2} (-\text {arccosh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-2 \text {arccosh}(a x)\right )+\sqrt {2} \text {arccosh}(a x)^{3/2} \Gamma \left (\frac {1}{2},2 \text {arccosh}(a x)\right )\right )}{3 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {arccosh}(a x)^{3/2}} \] Input:

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

Output:

(-2*Sqrt[c - a^2*c*x^2]*((1 + a*x)*(-1 + a*x + 4*a*x*Sqrt[(-1 + a*x)/(1 + 
a*x)]*ArcCosh[a*x]) + Sqrt[2]*(-ArcCosh[a*x])^(3/2)*Gamma[1/2, -2*ArcCosh[ 
a*x]] + Sqrt[2]*ArcCosh[a*x]^(3/2)*Gamma[1/2, 2*ArcCosh[a*x]]))/(3*a*Sqrt[ 
(-1 + a*x)/(1 + a*x)]*(1 + a*x)*ArcCosh[a*x]^(3/2))
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.91, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6319, 6300, 25, 3042, 3788, 26, 2611, 2633, 2634}

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.

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

\(\Big \downarrow \) 6319

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

\(\Big \downarrow \) 6300

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3788

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 2611

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 2634

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

Input:

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

Output:

(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[c - a^2*c*x^2])/(3*a*ArcCosh[a*x]^(3 
/2)) + (4*a*Sqrt[c - a^2*c*x^2]*((-2*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*Sq 
rt[ArcCosh[a*x]]) - (2*(-1/2*(Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]]) 
- (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/2))/a^2))/(3*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 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 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 3788

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

rule 6300

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[1/(b^2*c^(m + 1)*(n + 1))   Subst[Int[ExpandTrigReduce[x^(n + 
 1), Cosh[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cosh[-a/b + x/b]^2), x], x], x, 
a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] 
&& LtQ[n, -1]
 

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]
 

Maple [F]

\[\int \frac {\sqrt {-a^{2} c \,x^{2}+c}}{\operatorname {arccosh}\left (a x \right )^{\frac {5}{2}}}d x\]

Input:

int((-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(5/2),x)
 

Output:

int((-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(5/2),x)
 

Fricas [F(-2)]

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

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

\[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{5/2}} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}{\operatorname {acosh}^{\frac {5}{2}}{\left (a x \right )}}\, dx \] Input:

integrate((-a**2*c*x**2+c)**(1/2)/acosh(a*x)**(5/2),x)
 

Output:

Integral(sqrt(-c*(a*x - 1)*(a*x + 1))/acosh(a*x)**(5/2), x)
 

Maxima [F]

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

integrate((-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(5/2),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

integrate(sqrt(-a^2*c*x^2 + c)/arccosh(a*x)^(5/2), x)
 

Giac [F]

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

integrate((-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(5/2),x, algorithm="giac")
 

Output:

integrate(sqrt(-a^2*c*x^2 + c)/arccosh(a*x)^(5/2), x)
 

Mupad [F(-1)]

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

int((c - a^2*c*x^2)^(1/2)/acosh(a*x)^(5/2),x)
 

Output:

int((c - a^2*c*x^2)^(1/2)/acosh(a*x)^(5/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {c-a^2 c x^2}}{\text {arccosh}(a x)^{5/2}} \, dx=\frac {2 \sqrt {c}\, \left (2 \mathit {acosh} \left (a x \right )^{2} \left (\int \frac {\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x}{\mathit {acosh} \left (a x \right )^{2} a^{2} x^{2}-\mathit {acosh} \left (a x \right )^{2}}d x \right ) a^{2}-\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acosh} \left (a x \right )}\right )}{3 \mathit {acosh} \left (a x \right )^{2} a} \] Input:

int((-a^2*c*x^2+c)^(1/2)/acosh(a*x)^(5/2),x)
 

Output:

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