\(\int \frac {(c-a^2 c x^2)^{5/2}}{\text {arccosh}(a x)^{3/2}} \, dx\) [110]

Optimal result

Integrand size = 24, antiderivative size = 433 \[ \int \frac {\left (c-a^2 c x^2\right )^{5/2}}{\text {arccosh}(a x)^{3/2}} \, dx=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \left (c-a^2 c x^2\right )^{5/2}}{a \sqrt {\text {arccosh}(a x)}}+\frac {3 c^2 \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{8 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 c^2 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {c^2 \sqrt {\frac {3 \pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {6} \sqrt {\text {arccosh}(a x)}\right )}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {3 c^2 \sqrt {\pi } \sqrt {c-a^2 c x^2} \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )}{8 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {15 c^2 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )}{16 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {c^2 \sqrt {\frac {3 \pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {6} \sqrt {\text {arccosh}(a x)}\right )}{16 a \sqrt {-1+a x} \sqrt {1+a x}} \] Output:

-2*(a*x-1)^(1/2)*(a*x+1)^(1/2)*(-a^2*c*x^2+c)^(5/2)/a/arccosh(a*x)^(1/2)+3 
/8*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)-15/32*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)-1/32*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)-3/8*c^2*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)+15/32*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)+ 
1/32*c^2*6^(1/2)*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)*erfi(6^(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 = 0.91 (sec) , antiderivative size = 411, normalized size of antiderivative = 0.95 \[ \int \frac {\left (c-a^2 c x^2\right )^{5/2}}{\text {arccosh}(a x)^{3/2}} \, dx=\frac {c^2 e^{-6 \text {arccosh}(a x)} \sqrt {c-a^2 c x^2} \left (-1+6 e^{2 \text {arccosh}(a x)}+e^{4 \text {arccosh}(a x)}+52 e^{6 \text {arccosh}(a x)}+e^{8 \text {arccosh}(a x)}+6 e^{10 \text {arccosh}(a x)}-e^{12 \text {arccosh}(a x)}-64 a^2 e^{6 \text {arccosh}(a x)} x^2-16 e^{6 \text {arccosh}(a x)} \sqrt {2 \pi } \sqrt {\text {arccosh}(a x)} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+16 e^{6 \text {arccosh}(a x)} \sqrt {2 \pi } \sqrt {\text {arccosh}(a x)} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\sqrt {6} e^{6 \text {arccosh}(a x)} \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-6 \text {arccosh}(a x)\right )-12 e^{6 \text {arccosh}(a x)} \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-4 \text {arccosh}(a x)\right )-\sqrt {2} e^{6 \text {arccosh}(a x)} \sqrt {-\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},-2 \text {arccosh}(a x)\right )-\sqrt {2} e^{6 \text {arccosh}(a x)} \sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},2 \text {arccosh}(a x)\right )-12 e^{6 \text {arccosh}(a x)} \sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},4 \text {arccosh}(a x)\right )+\sqrt {6} e^{6 \text {arccosh}(a x)} \sqrt {\text {arccosh}(a x)} \Gamma \left (\frac {1}{2},6 \text {arccosh}(a x)\right )\right )}{32 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \sqrt {\text {arccosh}(a x)}} \] Input:

Integrate[(c - a^2*c*x^2)^(5/2)/ArcCosh[a*x]^(3/2),x]
 

Output:

(c^2*Sqrt[c - a^2*c*x^2]*(-1 + 6*E^(2*ArcCosh[a*x]) + E^(4*ArcCosh[a*x]) + 
 52*E^(6*ArcCosh[a*x]) + E^(8*ArcCosh[a*x]) + 6*E^(10*ArcCosh[a*x]) - E^(1 
2*ArcCosh[a*x]) - 64*a^2*E^(6*ArcCosh[a*x])*x^2 - 16*E^(6*ArcCosh[a*x])*Sq 
rt[2*Pi]*Sqrt[ArcCosh[a*x]]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + 16*E^(6*ArcC 
osh[a*x])*Sqrt[2*Pi]*Sqrt[ArcCosh[a*x]]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + 
 Sqrt[6]*E^(6*ArcCosh[a*x])*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -6*ArcCosh[a*x] 
] - 12*E^(6*ArcCosh[a*x])*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -4*ArcCosh[a*x]] 
- Sqrt[2]*E^(6*ArcCosh[a*x])*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -2*ArcCosh[a*x 
]] - Sqrt[2]*E^(6*ArcCosh[a*x])*Sqrt[ArcCosh[a*x]]*Gamma[1/2, 2*ArcCosh[a* 
x]] - 12*E^(6*ArcCosh[a*x])*Sqrt[ArcCosh[a*x]]*Gamma[1/2, 4*ArcCosh[a*x]] 
+ Sqrt[6]*E^(6*ArcCosh[a*x])*Sqrt[ArcCosh[a*x]]*Gamma[1/2, 6*ArcCosh[a*x]] 
))/(32*a*E^(6*ArcCosh[a*x])*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*Sqrt[ArcC 
osh[a*x]])
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.56, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6319, 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.

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

\(\Big \downarrow \) 6319

\(\displaystyle \frac {12 a c^2 \sqrt {c-a^2 c x^2} \int \frac {x (1-a x)^2 (a x+1)^2}{\sqrt {\text {arccosh}(a x)}}dx}{\sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \left (c-a^2 c x^2\right )^{5/2}}{a \sqrt {\text {arccosh}(a x)}}\)

\(\Big \downarrow \) 6327

\(\displaystyle \frac {12 a c^2 \sqrt {c-a^2 c x^2} \int \frac {x \left (1-a^2 x^2\right )^2}{\sqrt {\text {arccosh}(a x)}}dx}{\sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \left (c-a^2 c x^2\right )^{5/2}}{a \sqrt {\text {arccosh}(a x)}}\)

\(\Big \downarrow \) 6367

\(\displaystyle \frac {12 c^2 \sqrt {c-a^2 c x^2} \int \frac {a x \left (\frac {a x-1}{a x+1}\right )^{5/2} (a x+1)^5}{\sqrt {\text {arccosh}(a x)}}d\text {arccosh}(a x)}{a \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \left (c-a^2 c x^2\right )^{5/2}}{a \sqrt {\text {arccosh}(a x)}}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {12 c^2 \sqrt {c-a^2 c x^2} \int \left (\frac {5 \sinh (2 \text {arccosh}(a x))}{32 \sqrt {\text {arccosh}(a x)}}-\frac {\sinh (4 \text {arccosh}(a x))}{8 \sqrt {\text {arccosh}(a x)}}+\frac {\sinh (6 \text {arccosh}(a x))}{32 \sqrt {\text {arccosh}(a x)}}\right )d\text {arccosh}(a x)}{a \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \left (c-a^2 c x^2\right )^{5/2}}{a \sqrt {\text {arccosh}(a x)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {12 c^2 \sqrt {c-a^2 c x^2} \left (\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}(a x)}\right )-\frac {5}{64} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{64} \sqrt {\frac {\pi }{6}} \text {erf}\left (\sqrt {6} \sqrt {\text {arccosh}(a x)}\right )-\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}(a x)}\right )+\frac {5}{64} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}(a x)}\right )+\frac {1}{64} \sqrt {\frac {\pi }{6}} \text {erfi}\left (\sqrt {6} \sqrt {\text {arccosh}(a x)}\right )\right )}{a \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \left (c-a^2 c x^2\right )^{5/2}}{a \sqrt {\text {arccosh}(a x)}}\)

Input:

Int[(c - a^2*c*x^2)^(5/2)/ArcCosh[a*x]^(3/2),x]
 

Output:

(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(c - a^2*c*x^2)^(5/2))/(a*Sqrt[ArcCosh[a* 
x]]) + (12*c^2*Sqrt[c - a^2*c*x^2]*((Sqrt[Pi]*Erf[2*Sqrt[ArcCosh[a*x]]])/3 
2 - (5*Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/64 - (Sqrt[Pi/6]*Erf[Sq 
rt[6]*Sqrt[ArcCosh[a*x]]])/64 - (Sqrt[Pi]*Erfi[2*Sqrt[ArcCosh[a*x]]])/32 + 
 (5*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/64 + (Sqrt[Pi/6]*Erfi[Sqr 
t[6]*Sqrt[ArcCosh[a*x]]])/64))/(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 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]

\[\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{\operatorname {arccosh}\left (a x \right )^{\frac {3}{2}}}d x\]

Input:

int((-a^2*c*x^2+c)^(5/2)/arccosh(a*x)^(3/2),x)
 

Output:

int((-a^2*c*x^2+c)^(5/2)/arccosh(a*x)^(3/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {\left (c-a^2 c x^2\right )^{5/2}}{\text {arccosh}(a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Timed out. \[ \int \frac {\left (c-a^2 c x^2\right )^{5/2}}{\text {arccosh}(a x)^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((-a**2*c*x**2+c)**(5/2)/acosh(a*x)**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((-a^2*c*x^2+c)^(5/2)/arccosh(a*x)^(3/2),x, algorithm="maxima")
 

Output:

integrate((-a^2*c*x^2 + c)^(5/2)/arccosh(a*x)^(3/2), x)
 

Giac [F]

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

integrate((-a^2*c*x^2+c)^(5/2)/arccosh(a*x)^(3/2),x, algorithm="giac")
 

Output:

integrate((-a^2*c*x^2 + c)^(5/2)/arccosh(a*x)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c-a^2 c x^2\right )^{5/2}}{\text {arccosh}(a x)^{3/2}} \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^{5/2}}{{\mathrm {acosh}\left (a\,x\right )}^{3/2}} \,d x \] Input:

int((c - a^2*c*x^2)^(5/2)/acosh(a*x)^(3/2),x)
 

Output:

int((c - a^2*c*x^2)^(5/2)/acosh(a*x)^(3/2), x)
 

Reduce [F]

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

int((-a^2*c*x^2+c)^(5/2)/acosh(a*x)^(3/2),x)
 

Output:

(2*sqrt(c)*c**2*(6*acosh(a*x)*int((sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt( - a** 
2*x**2 + 1)*sqrt(acosh(a*x))*x**5)/(acosh(a*x)*a**2*x**2 - acosh(a*x)),x)* 
a**6 - 12*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 + 6* 
acosh(a*x)*int((sqrt(a*x + 1)*sqrt(a*x - 1)*sqrt( - a**2*x**2 + 1)*sqrt(ac 
osh(a*x))*x)/(acosh(a*x)*a**2*x**2 - acosh(a*x)),x)*a**2 - sqrt(a*x + 1)*s 
qrt(a*x - 1)*sqrt( - a**2*x**2 + 1)*sqrt(acosh(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 - s 
qrt(a*x + 1)*sqrt(a*x - 1)*sqrt( - a**2*x**2 + 1)*sqrt(acosh(a*x))))/(acos 
h(a*x)*a)