3.3.68 \(\int \frac {x^3 \sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx\) [268]

3.3.68.1 Optimal result

Integrand size = 28, antiderivative size = 297 \[ \int \frac {x^3 \sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=-\frac {\sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b c^4 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c^4 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c^4 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b c^4 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c^4 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c^4 \sqrt {-1+c x}} \]

output

-1/8*Chi((a+b*arccosh(c*x))/b)*cosh(a/b)*(-c*x+1)^(1/2)/b/c^4/(c*x-1)^(1/2 
)+1/16*Chi(3*(a+b*arccosh(c*x))/b)*cosh(3*a/b)*(-c*x+1)^(1/2)/b/c^4/(c*x-1 
)^(1/2)+1/16*Chi(5*(a+b*arccosh(c*x))/b)*cosh(5*a/b)*(-c*x+1)^(1/2)/b/c^4/ 
(c*x-1)^(1/2)+1/8*Shi((a+b*arccosh(c*x))/b)*sinh(a/b)*(-c*x+1)^(1/2)/b/c^4 
/(c*x-1)^(1/2)-1/16*Shi(3*(a+b*arccosh(c*x))/b)*sinh(3*a/b)*(-c*x+1)^(1/2) 
/b/c^4/(c*x-1)^(1/2)-1/16*Shi(5*(a+b*arccosh(c*x))/b)*sinh(5*a/b)*(-c*x+1) 
^(1/2)/b/c^4/(c*x-1)^(1/2)
 

3.3.68.2 Mathematica [A] (warning: unable to verify)

Time = 0.34 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.58 \[ \int \frac {x^3 \sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\frac {\sqrt {1-c^2 x^2} \left (-2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+\cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+\cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )-\sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-\sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{16 c^4 \sqrt {\frac {-1+c x}{1+c x}} (b+b c x)} \]

input

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

output

(Sqrt[1 - c^2*x^2]*(-2*Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c*x]] + Cosh[( 
3*a)/b]*CoshIntegral[3*(a/b + ArcCosh[c*x])] + Cosh[(5*a)/b]*CoshIntegral[ 
5*(a/b + ArcCosh[c*x])] + 2*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] - S 
inh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] - Sinh[(5*a)/b]*SinhInte 
gral[5*(a/b + ArcCosh[c*x])]))/(16*c^4*Sqrt[(-1 + c*x)/(1 + c*x)]*(b + b*c 
*x))
 

3.3.68.3 Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.58, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {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.

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

\(\Big \downarrow \) 6367

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

\(\Big \downarrow \) 5971

\(\displaystyle \frac {\sqrt {1-c x} \int \left (\frac {\cosh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 (a+b \text {arccosh}(c x))}+\frac {\cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 (a+b \text {arccosh}(c x))}-\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 (a+b \text {arccosh}(c x))}\right )d(a+b \text {arccosh}(c x))}{b c^4 \sqrt {c x-1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {1-c x} \left (-\frac {1}{8} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {1}{16} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{16} \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{8} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {1}{16} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{16} \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b c^4 \sqrt {c x-1}}\)

input

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

output

(Sqrt[1 - c*x]*(-1/8*(Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b]) + (C 
osh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b])/16 + (Cosh[(5*a)/b] 
*CoshIntegral[(5*(a + b*ArcCosh[c*x]))/b])/16 + (Sinh[a/b]*SinhIntegral[(a 
 + b*ArcCosh[c*x])/b])/8 - (Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c 
*x]))/b])/16 - (Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c*x]))/b])/16 
))/(b*c^4*Sqrt[-1 + c*x])
 

3.3.68.3.1 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 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]
 

3.3.68.4 Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.84

method	result	size
default	\(\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {Ei}_{1}\left (5 \,\operatorname {arccosh}\left (c x \right )+\frac {5 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+5 a}{b}}+\operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}}+\operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}}+\operatorname {Ei}_{1}\left (-5 \,\operatorname {arccosh}\left (c x \right )-\frac {5 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+5 a}{b}}-2 \,\operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) {\mathrm e}^{\frac {a +b \,\operatorname {arccosh}\left (c x \right )}{b}}-2 \,\operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+a}{b}}\right )}{32 \left (c x +1\right ) c^{4} \left (c x -1\right ) b}\)	\(250\)

input

int(x^3*(-c^2*x^2+1)^(1/2)/(a+b*arccosh(c*x)),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)*(Ei(1 
,5*arccosh(c*x)+5*a/b)*exp((b*arccosh(c*x)+5*a)/b)+Ei(1,3*arccosh(c*x)+3*a 
/b)*exp((b*arccosh(c*x)+3*a)/b)+Ei(1,-3*arccosh(c*x)-3*a/b)*exp(-(-b*arcco 
sh(c*x)+3*a)/b)+Ei(1,-5*arccosh(c*x)-5*a/b)*exp(-(-b*arccosh(c*x)+5*a)/b)- 
2*Ei(1,arccosh(c*x)+a/b)*exp((a+b*arccosh(c*x))/b)-2*Ei(1,-arccosh(c*x)-a/ 
b)*exp(-(-b*arccosh(c*x)+a)/b))/(c*x+1)/c^4/(c*x-1)/b
 

3.3.68.5 Fricas [F]

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

input

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

output

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

3.3.68.6 Sympy [F]

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

input

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

output

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

3.3.68.7 Maxima [F]

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

input

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

output

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

3.3.68.8 Giac [F(-2)]

Exception generated. \[ \int \frac {x^3 \sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\text {Exception raised: TypeError} \]

input

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

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

3.3.68.9 Mupad [F(-1)]

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

input

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

output

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