\(\int \frac {x (1-c^2 x^2)^{3/2}}{a+b \text {arccosh}(c x)} \, dx\) [240]

Optimal result

Integrand size = 26, antiderivative size = 297 \[ \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{a+b \text {arccosh}(c x)} \, dx=-\frac {\sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b c^2 \sqrt {-1+c x}}+\frac {3 \sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c^2 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c^2 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b c^2 \sqrt {-1+c x}}-\frac {3 \sqrt {1-c x} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c^2 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c^2 \sqrt {-1+c x}} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.57 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.58 \[ \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{a+b \text {arccosh}(c x)} \, dx=\frac {\sqrt {1-c^2 x^2} \left (-2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-\cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )-3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+\sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{16 c^2 \sqrt {\frac {-1+c x}{1+c x}} (b+b c x)} \] Input:

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

Output:

(Sqrt[1 - c^2*x^2]*(-2*Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c*x]] + 3*Cosh 
[(3*a)/b]*CoshIntegral[3*(a/b + ArcCosh[c*x])] - Cosh[(5*a)/b]*CoshIntegra 
l[5*(a/b + ArcCosh[c*x])] + 2*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] - 
 3*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + Sinh[(5*a)/b]*Sinh 
Integral[5*(a/b + ArcCosh[c*x])]))/(16*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(b + 
 b*c*x))
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.59, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, 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.

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

\(\Big \downarrow \) 6367

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

\(\Big \downarrow \) 5971

\(\displaystyle -\frac {\sqrt {1-c x} \int \left (\frac {\cosh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 (a+b \text {arccosh}(c x))}-\frac {3 \cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 (a+b \text {arccosh}(c x))}+\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 (a+b \text {arccosh}(c x))}\right )d(a+b \text {arccosh}(c x))}{b c^2 \sqrt {c x-1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt {1-c x} \left (\frac {1}{8} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {3}{16} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{16} \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {3}{16} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{16} \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b c^2 \sqrt {c x-1}}\)

Input:

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

Output:

-((Sqrt[1 - c*x]*((Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/8 - (3* 
Cosh[(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 + (3*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCos 
h[c*x]))/b])/16 - (Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c*x]))/b]) 
/16))/(b*c^2*Sqrt[-1 + c*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 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 = 0.54 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.85

method	result	size
default	\(-\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {expIntegral}_{1}\left (5 \,\operatorname {arccosh}\left (c x \right )+\frac {5 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+5 a}{b}}+\operatorname {expIntegral}_{1}\left (-5 \,\operatorname {arccosh}\left (c x \right )-\frac {5 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+5 a}{b}}-3 \,\operatorname {expIntegral}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}}+2 \,\operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) {\mathrm e}^{\frac {a +b \,\operatorname {arccosh}\left (c x \right )}{b}}+2 \,\operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+a}{b}}-3 \,\operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}}\right )}{32 \left (c x +1\right ) c^{2} \left (c x -1\right ) b}\)	\(252\)

Input:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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