\(\int \frac {x^2 (1-c^2 x^2)^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx\) [279]

Optimal result

Integrand size = 28, antiderivative size = 354 \[ \int \frac {x^2 \left (1-c^2 x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}-\frac {\sqrt {1-c x} \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {4 a}{b}\right )}{4 b^2 c^3 \sqrt {-1+c x}}+\frac {3 \sqrt {1-c x} \text {Chi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {6 a}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{4 b^2 c^3 \sqrt {-1+c x}}-\frac {3 \sqrt {1-c x} \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}} \] Output:

-x^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(3/2)/b/c/(a+b*arccosh(c*x)) 
-1/16*(-c*x+1)^(1/2)*Chi(2*(a+b*arccosh(c*x))/b)*sinh(2*a/b)/b^2/c^3/(c*x- 
1)^(1/2)-1/4*(-c*x+1)^(1/2)*Chi(4*(a+b*arccosh(c*x))/b)*sinh(4*a/b)/b^2/c^ 
3/(c*x-1)^(1/2)+3/16*(-c*x+1)^(1/2)*Chi(6*(a+b*arccosh(c*x))/b)*sinh(6*a/b 
)/b^2/c^3/(c*x-1)^(1/2)+1/16*(-c*x+1)^(1/2)*cosh(2*a/b)*Shi(2*(a+b*arccosh 
(c*x))/b)/b^2/c^3/(c*x-1)^(1/2)+1/4*(-c*x+1)^(1/2)*cosh(4*a/b)*Shi(4*(a+b* 
arccosh(c*x))/b)/b^2/c^3/(c*x-1)^(1/2)-3/16*(-c*x+1)^(1/2)*cosh(6*a/b)*Shi 
(6*(a+b*arccosh(c*x))/b)/b^2/c^3/(c*x-1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.81 (sec) , antiderivative size = 338, normalized size of antiderivative = 0.95 \[ \int \frac {x^2 \left (1-c^2 x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (16 b c^2 x^2-32 b c^4 x^4+16 b c^6 x^6-(a+b \text {arccosh}(c x)) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )-4 (a+b \text {arccosh}(c x)) \text {Chi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {4 a}{b}\right )+3 a \text {Chi}\left (6 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {6 a}{b}\right )+3 b \text {arccosh}(c x) \text {Chi}\left (6 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {6 a}{b}\right )+a \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+b \text {arccosh}(c x) \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+4 a \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+4 b \text {arccosh}(c x) \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-3 a \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-3 b \text {arccosh}(c x) \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{16 b^2 c^3 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \] Input:

Integrate[(x^2*(1 - c^2*x^2)^(3/2))/(a + b*ArcCosh[c*x])^2,x]
 

Output:

-1/16*(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(16*b*c^2*x^2 - 32*b*c^4*x^4 + 16*b*c^ 
6*x^6 - (a + b*ArcCosh[c*x])*CoshIntegral[2*(a/b + ArcCosh[c*x])]*Sinh[(2* 
a)/b] - 4*(a + b*ArcCosh[c*x])*CoshIntegral[4*(a/b + ArcCosh[c*x])]*Sinh[( 
4*a)/b] + 3*a*CoshIntegral[6*(a/b + ArcCosh[c*x])]*Sinh[(6*a)/b] + 3*b*Arc 
Cosh[c*x]*CoshIntegral[6*(a/b + ArcCosh[c*x])]*Sinh[(6*a)/b] + a*Cosh[(2*a 
)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] + b*ArcCosh[c*x]*Cosh[(2*a)/b]*S 
inhIntegral[2*(a/b + ArcCosh[c*x])] + 4*a*Cosh[(4*a)/b]*SinhIntegral[4*(a/ 
b + ArcCosh[c*x])] + 4*b*ArcCosh[c*x]*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + 
ArcCosh[c*x])] - 3*a*Cosh[(6*a)/b]*SinhIntegral[6*(a/b + ArcCosh[c*x])] - 
3*b*ArcCosh[c*x]*Cosh[(6*a)/b]*SinhIntegral[6*(a/b + ArcCosh[c*x])]))/(b^2 
*c^3*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x]))
 

Rubi [A] (verified)

Time = 1.33 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.88, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6357, 25, 6327, 6367, 25, 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.

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

\(\Big \downarrow \) 6357

\(\displaystyle -\frac {6 c \sqrt {1-c x} \int -\frac {x^3 (1-c x) (c x+1)}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}+\frac {2 \sqrt {1-c x} \int -\frac {x (1-c x) (c x+1)}{a+b \text {arccosh}(c x)}dx}{b c \sqrt {c x-1}}-\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {6 c \sqrt {1-c x} \int \frac {x^3 (1-c x) (c x+1)}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {2 \sqrt {1-c x} \int \frac {x (1-c x) (c x+1)}{a+b \text {arccosh}(c x)}dx}{b c \sqrt {c x-1}}-\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 6327

\(\displaystyle -\frac {2 \sqrt {1-c x} \int \frac {x \left (1-c^2 x^2\right )}{a+b \text {arccosh}(c x)}dx}{b c \sqrt {c x-1}}+\frac {6 c \sqrt {1-c x} \int \frac {x^3 \left (1-c^2 x^2\right )}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 6367

\(\displaystyle -\frac {6 \sqrt {1-c x} \int -\frac {\cosh ^3\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^3 \sqrt {c x-1}}+\frac {2 \sqrt {1-c x} \int -\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^3 \sqrt {c x-1}}-\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {6 \sqrt {1-c x} \int \frac {\cosh ^3\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^3 \sqrt {c x-1}}-\frac {2 \sqrt {1-c x} \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^3 \sqrt {c x-1}}-\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 5971

\(\displaystyle -\frac {2 \sqrt {1-c x} \int \left (\frac {\sinh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{8 (a+b \text {arccosh}(c x))}-\frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{4 (a+b \text {arccosh}(c x))}\right )d(a+b \text {arccosh}(c x))}{b^2 c^3 \sqrt {c x-1}}+\frac {6 \sqrt {1-c x} \int \left (\frac {\sinh \left (\frac {6 a}{b}-\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{32 (a+b \text {arccosh}(c x))}-\frac {3 \sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{32 (a+b \text {arccosh}(c x))}\right )d(a+b \text {arccosh}(c x))}{b^2 c^3 \sqrt {c x-1}}-\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \sqrt {1-c x} \left (\frac {1}{4} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{4} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^3 \sqrt {c x-1}}-\frac {6 \sqrt {1-c x} \left (\frac {3}{32} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{32} \sinh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )-\frac {3}{32} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{32} \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^3 \sqrt {c x-1}}-\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\)

Input:

Int[(x^2*(1 - c^2*x^2)^(3/2))/(a + b*ArcCosh[c*x])^2,x]
 

Output:

-((x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(3/2))/(b*c*(a + b*ArcCo 
sh[c*x]))) + (2*Sqrt[1 - c*x]*((CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b]*S 
inh[(2*a)/b])/4 - (CoshIntegral[(4*(a + b*ArcCosh[c*x]))/b]*Sinh[(4*a)/b]) 
/8 - (Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b])/4 + (Cosh[(4 
*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c*x]))/b])/8))/(b^2*c^3*Sqrt[-1 + c* 
x]) - (6*Sqrt[1 - c*x]*((3*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b]*Sinh[( 
2*a)/b])/32 - (CoshIntegral[(6*(a + b*ArcCosh[c*x]))/b]*Sinh[(6*a)/b])/32 
- (3*Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b])/32 + (Cosh[(6 
*a)/b]*SinhIntegral[(6*(a + b*ArcCosh[c*x]))/b])/32))/(b^2*c^3*Sqrt[-1 + c 
*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 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 6357

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^m*Simp[Sqrt[1 + c*x]*Sqrt[-1 + c* 
x]*(d + e*x^2)^p]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (Simp[ 
f*(m/(b*c*(n + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]   Int[(f 
*x)^(m - 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^( 
n + 1), x], x] - Simp[c*((m + 2*p + 1)/(b*f*(n + 1)))*Simp[(d + e*x^2)^p/(( 
1 + c*x)^p*(-1 + c*x)^p)]   Int[(f*x)^(m + 1)*(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, 
f, m, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 
2*p + 1, 0] && IGtQ[m, -3]
 

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 [A] (verified)

Time = 0.30 (sec) , antiderivative size = 599, normalized size of antiderivative = 1.69

method	result	size
default	\(\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-32 \sqrt {c x -1}\, \sqrt {c x +1}\, b \,c^{6} x^{6}-32 b \,c^{7} x^{7}+64 \sqrt {c x -1}\, \sqrt {c x +1}\, b \,c^{4} x^{4}+64 b \,c^{5} x^{5}-32 \sqrt {c x -1}\, \sqrt {c x +1}\, b \,c^{2} x^{2}-32 b \,c^{3} x^{3}+4 \,\operatorname {arccosh}\left (c x \right ) b \,\operatorname {expIntegral}_{1}\left (-4 \,\operatorname {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}+\operatorname {arccosh}\left (c x \right ) b \,\operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}-3 \,\operatorname {arccosh}\left (c x \right ) b \,\operatorname {expIntegral}_{1}\left (-6 \,\operatorname {arccosh}\left (c x \right )-\frac {6 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+6 a}{b}}+3 \,\operatorname {expIntegral}_{1}\left (6 \,\operatorname {arccosh}\left (c x \right )+\frac {6 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+6 a}{b}} b \,\operatorname {arccosh}\left (c x \right )-4 \,\operatorname {expIntegral}_{1}\left (4 \,\operatorname {arccosh}\left (c x \right )+\frac {4 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}} b \,\operatorname {arccosh}\left (c x \right )-\operatorname {expIntegral}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}} b \,\operatorname {arccosh}\left (c x \right )+4 a \,\operatorname {expIntegral}_{1}\left (-4 \,\operatorname {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}+a \,\operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}-3 a \,\operatorname {expIntegral}_{1}\left (-6 \,\operatorname {arccosh}\left (c x \right )-\frac {6 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+6 a}{b}}+3 \,\operatorname {expIntegral}_{1}\left (6 \,\operatorname {arccosh}\left (c x \right )+\frac {6 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+6 a}{b}} a -4 \,\operatorname {expIntegral}_{1}\left (4 \,\operatorname {arccosh}\left (c x \right )+\frac {4 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}} a -\operatorname {expIntegral}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}} a \right )}{32 \left (c x +1\right ) c^{3} \left (c x -1\right ) b^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}\)	\(599\)

Input:

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

Output:

1/32*(-c^2*x^2+1)^(1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(-32* 
(c*x-1)^(1/2)*(c*x+1)^(1/2)*b*c^6*x^6-32*b*c^7*x^7+64*(c*x-1)^(1/2)*(c*x+1 
)^(1/2)*b*c^4*x^4+64*b*c^5*x^5-32*(c*x-1)^(1/2)*(c*x+1)^(1/2)*b*c^2*x^2-32 
*b*c^3*x^3+4*arccosh(c*x)*b*Ei(1,-4*arccosh(c*x)-4*a/b)*exp(-(-b*arccosh(c 
*x)+4*a)/b)+arccosh(c*x)*b*Ei(1,-2*arccosh(c*x)-2*a/b)*exp(-(-b*arccosh(c* 
x)+2*a)/b)-3*arccosh(c*x)*b*Ei(1,-6*arccosh(c*x)-6*a/b)*exp(-(-b*arccosh(c 
*x)+6*a)/b)+3*Ei(1,6*arccosh(c*x)+6*a/b)*exp((b*arccosh(c*x)+6*a)/b)*b*arc 
cosh(c*x)-4*Ei(1,4*arccosh(c*x)+4*a/b)*exp((b*arccosh(c*x)+4*a)/b)*b*arcco 
sh(c*x)-Ei(1,2*arccosh(c*x)+2*a/b)*exp((b*arccosh(c*x)+2*a)/b)*b*arccosh(c 
*x)+4*a*Ei(1,-4*arccosh(c*x)-4*a/b)*exp(-(-b*arccosh(c*x)+4*a)/b)+a*Ei(1,- 
2*arccosh(c*x)-2*a/b)*exp(-(-b*arccosh(c*x)+2*a)/b)-3*a*Ei(1,-6*arccosh(c* 
x)-6*a/b)*exp(-(-b*arccosh(c*x)+6*a)/b)+3*Ei(1,6*arccosh(c*x)+6*a/b)*exp(( 
b*arccosh(c*x)+6*a)/b)*a-4*Ei(1,4*arccosh(c*x)+4*a/b)*exp((b*arccosh(c*x)+ 
4*a)/b)*a-Ei(1,2*arccosh(c*x)+2*a/b)*exp((b*arccosh(c*x)+2*a)/b)*a)/(c*x+1 
)/c^3/(c*x-1)/b^2/(a+b*arccosh(c*x))
 

Fricas [F]

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

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

Output:

integral(-(c^2*x^4 - x^2)*sqrt(-c^2*x^2 + 1)/(b^2*arccosh(c*x)^2 + 2*a*b*a 
rccosh(c*x) + a^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

((c^4*x^6 - 2*c^2*x^4 + x^2)*(c*x + 1)*sqrt(c*x - 1) + (c^5*x^7 - 2*c^3*x^ 
5 + c*x^3)*sqrt(c*x + 1))*sqrt(-c*x + 1)/(a*b*c^3*x^2 + sqrt(c*x + 1)*sqrt 
(c*x - 1)*a*b*c^2*x - a*b*c + (b^2*c^3*x^2 + sqrt(c*x + 1)*sqrt(c*x - 1)*b 
^2*c^2*x - b^2*c)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))) - integrate(((6* 
c^5*x^6 - 7*c^3*x^4 + c*x^2)*(c*x + 1)^(3/2)*(c*x - 1) + 2*(6*c^6*x^7 - 11 
*c^4*x^5 + 6*c^2*x^3 - x)*(c*x + 1)*sqrt(c*x - 1) + 3*(2*c^7*x^8 - 5*c^5*x 
^6 + 4*c^3*x^4 - c*x^2)*sqrt(c*x + 1))*sqrt(-c*x + 1)/(a*b*c^5*x^4 + (c*x 
+ 1)*(c*x - 1)*a*b*c^3*x^2 - 2*a*b*c^3*x^2 + a*b*c + 2*(a*b*c^4*x^3 - a*b* 
c^2*x)*sqrt(c*x + 1)*sqrt(c*x - 1) + (b^2*c^5*x^4 + (c*x + 1)*(c*x - 1)*b^ 
2*c^3*x^2 - 2*b^2*c^3*x^2 + b^2*c + 2*(b^2*c^4*x^3 - b^2*c^2*x)*sqrt(c*x + 
 1)*sqrt(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))), x)
                                                                                    
                                                                                    
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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