\(\int \frac {(c-a^2 c x^2)^{5/2}}{\sqrt {\text {arccosh}(a x)}} \, dx\) [104]

Optimal result

Integrand size = 24, antiderivative size = 438 \[ \int \frac {\left (c-a^2 c x^2\right )^{5/2}}{\sqrt {\text {arccosh}(a x)}} \, dx=-\frac {5 c^2 \sqrt {c-a^2 c x^2} \sqrt {\text {arccosh}(a x)}}{8 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {3 c^2 \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {15 c^2 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {c^2 \sqrt {\frac {\pi }{6}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {6} \sqrt {\text {arccosh}(a x)}\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {3 c^2 \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {15 c^2 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {c^2 \sqrt {\frac {\pi }{6}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {6} \sqrt {\text {arccosh}(a x)}\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}} \] Output:

-5/8*c^2*(-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^( 
1/2)-3/64*c^2*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)+15/128*c^2*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)+1/384*c^2*6 
^(1/2)*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)*erf(6^(1/2)*arccosh(a*x)^(1/2))/a/(a* 
x-1)^(1/2)/(a*x+1)^(1/2)-3/64*c^2*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)*erfi(2*arc 
cosh(a*x)^(1/2))/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)+15/128*c^2*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)+1/384*c^2*6^(1/2)*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)*erfi(6^(1/2)*ar 
ccosh(a*x)^(1/2))/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.48 \[ \int \frac {\left (c-a^2 c x^2\right )^{5/2}}{\sqrt {\text {arccosh}(a x)}} \, dx=-\frac {c^2 \sqrt {c-a^2 c x^2} \left (240 \text {arccosh}(a x)-\sqrt {6} \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-6 \text {arccosh}(a x)\right )+18 \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-4 \text {arccosh}(a x)\right )-45 \sqrt {2} \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-2 \text {arccosh}(a x)\right )+45 \sqrt {2} \sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},2 \text {arccosh}(a x)\right )-18 \sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},4 \text {arccosh}(a x)\right )+\sqrt {6} \sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},6 \text {arccosh}(a x)\right )\right )}{384 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \sqrt {\text {arccosh}(a x)}} \] Input:

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

Output:

-1/384*(c^2*Sqrt[c - a^2*c*x^2]*(240*ArcCosh[a*x] - Sqrt[6]*Sqrt[-ArcCosh[ 
a*x]]*Gamma[1/2, -6*ArcCosh[a*x]] + 18*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -4*A 
rcCosh[a*x]] - 45*Sqrt[2]*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -2*ArcCosh[a*x]] 
+ 45*Sqrt[2]*Sqrt[ArcCosh[a*x]]*Gamma[1/2, 2*ArcCosh[a*x]] - 18*Sqrt[ArcCo 
sh[a*x]]*Gamma[1/2, 4*ArcCosh[a*x]] + Sqrt[6]*Sqrt[ArcCosh[a*x]]*Gamma[1/2 
, 6*ArcCosh[a*x]]))/(a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*Sqrt[ArcCosh[a 
*x]])
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.47, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6321, 3042, 25, 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.

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

\(\Big \downarrow \) 6321

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {c^2 \sqrt {c-a^2 c x^2} \int -\frac {\sin (i \text {arccosh}(a x))^6}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a \sqrt {a x-1} \sqrt {a x+1}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {c^2 \sqrt {c-a^2 c x^2} \int \frac {\sin (i \text {arccosh}(a x))^6}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a \sqrt {a x-1} \sqrt {a x+1}}\)

\(\Big \downarrow \) 3793

\(\displaystyle -\frac {c^2 \sqrt {c-a^2 c x^2} \int \left (-\frac {15 \cosh (2 \text {arccosh}(a x))}{32 \sqrt {\text {arccosh}(a x)}}+\frac {3 \cosh (4 \text {arccosh}(a x))}{16 \sqrt {\text {arccosh}(a x)}}-\frac {\cosh (6 \text {arccosh}(a x))}{32 \sqrt {\text {arccosh}(a x)}}+\frac {5}{16 \sqrt {\text {arccosh}(a x)}}\right )d\text {arccosh}(a x)}{a \sqrt {a x-1} \sqrt {a x+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {c^2 \sqrt {c-a^2 c x^2} \left (-\frac {3}{64} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )+\frac {15}{64} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{64} \sqrt {\frac {\pi }{6}} \text {erf}\left (\sqrt {6} \sqrt {\text {arccosh}(a x)}\right )-\frac {3}{64} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )+\frac {15}{64} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{64} \sqrt {\frac {\pi }{6}} \text {erfi}\left (\sqrt {6} \sqrt {\text {arccosh}(a x)}\right )-\frac {5}{8} \sqrt {\text {arccosh}(a x)}\right )}{a \sqrt {a x-1} \sqrt {a x+1}}\)

Input:

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

Output:

(c^2*Sqrt[c - a^2*c*x^2]*((-5*Sqrt[ArcCosh[a*x]])/8 - (3*Sqrt[Pi]*Erf[2*Sq 
rt[ArcCosh[a*x]]])/64 + (15*Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/64 
 + (Sqrt[Pi/6]*Erf[Sqrt[6]*Sqrt[ArcCosh[a*x]]])/64 - (3*Sqrt[Pi]*Erfi[2*Sq 
rt[ArcCosh[a*x]]])/64 + (15*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/6 
4 + (Sqrt[Pi/6]*Erfi[Sqrt[6]*Sqrt[ArcCosh[a*x]]])/64))/(a*Sqrt[-1 + a*x]*S 
qrt[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 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]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

integrate((-a^2*c*x^2+c)^(5/2)/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(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (c-a^2 c x^2\right )^{5/2}}{\sqrt {\text {arccosh}(a x)}} \, dx=\frac {2 \sqrt {c}\, c^{2} \left (\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, a^{4} x^{4}-2 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, a^{2} x^{2}+\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acosh} \left (a x \right )}-6 \left (\int \frac {\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x^{5}}{a^{2} x^{2}-1}d x \right ) a^{6}+12 \left (\int \frac {\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x^{3}}{a^{2} x^{2}-1}d x \right ) a^{4}-6 \left (\int \frac {\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acosh} \left (a x \right )}\, x}{a^{2} x^{2}-1}d x \right ) a^{2}\right )}{a} \] Input:

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

Output:

(2*sqrt(c)*c**2*(sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt( - a**2*x**2 + 1)*sqrt(a 
cosh(a*x))*a**4*x**4 - 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)) - 6*int((sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt( - a**2 
*x**2 + 1)*sqrt(acosh(a*x))*x**5)/(a**2*x**2 - 1),x)*a**6 + 12*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 - 6*int((sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt( - a**2*x**2 + 
 1)*sqrt(acosh(a*x))*x)/(a**2*x**2 - 1),x)*a**2))/a