\(\int \frac {x^3 (a+b \text {arccosh}(c x))}{(d-c^2 d x^2)^{3/2}} \, dx\) [111]

Optimal result

Integrand size = 27, antiderivative size = 148 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {b x \sqrt {-1+c x} \sqrt {1+c x}}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^4 d^2}+\frac {b \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{c^4 d \sqrt {d-c^2 d x^2}} \] Output:

b*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3/d/(-c^2*d*x^2+d)^(1/2)+(a+b*arccosh(c* 
x))/c^4/d/(-c^2*d*x^2+d)^(1/2)+(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/c^4 
/d^2+b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arctanh(c*x)/c^4/d/(-c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.66 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {2 a-a c^2 x^2+b c x \sqrt {-1+c x} \sqrt {1+c x}+b \left (2-c^2 x^2\right ) \text {arccosh}(c x)+b \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{c^4 d \sqrt {d-c^2 d x^2}} \] Input:

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

Output:

(2*a - a*c^2*x^2 + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + b*(2 - c^2*x^2)*Ar 
cCosh[c*x] + b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTanh[c*x])/(c^4*d*Sqrt[d - 
c^2*d*x^2])
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.76, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6337, 27, 299, 219}

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.

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

\(\Big \downarrow \) 6337

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(a + b*ArcCosh[c*x])/(c^4*d*Sqrt[d - c^2*d*x^2]) + (Sqrt[d - c^2*d*x^2]*(a 
 + b*ArcCosh[c*x]))/(c^4*d^2) - (b*Sqrt[d - c^2*d*x^2]*(x + ArcTanh[c*x]/c 
))/(c^3*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 0]
 

rule 6337

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_ 
), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcCo 
sh[c*x])   u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c 
*x])]   Int[SimplifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b 
, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] 
 && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
 

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.72

method	result	size
default	\(a \left (-\frac {x^{2}}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2}{d \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}\right )+\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}-c^{3} x^{3}+\ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) x^{2} c^{2}-\ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+c x -\ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )+\ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{\left (c^{2} x^{2}-1\right )^{2} d^{2} c^{4}}\)	\(255\)
parts	\(a \left (-\frac {x^{2}}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2}{d \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}\right )+\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}-c^{3} x^{3}+\ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) x^{2} c^{2}-\ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+c x -\ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )+\ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{\left (c^{2} x^{2}-1\right )^{2} d^{2} c^{4}}\)	\(255\)

Input:

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

Output:

a*(-x^2/c^2/d/(-c^2*d*x^2+d)^(1/2)+2/d/c^4/(-c^2*d*x^2+d)^(1/2))+b*(c*x-1) 
^(1/2)*(c*x+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*(arccosh(c*x)*(c*x+1)^(1/2)*(c 
*x-1)^(1/2)*c^2*x^2-c^3*x^3+ln((c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x-1)*x^2*c^2- 
ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*x^2*c^2-2*arccosh(c*x)*(c*x-1)^(1/2) 
*(c*x+1)^(1/2)+c*x-ln((c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x-1)+ln(1+c*x+(c*x-1)^ 
(1/2)*(c*x+1)^(1/2)))/(c^2*x^2-1)^2/d^2/c^4
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 429, normalized size of antiderivative = 2.90 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\left [-\frac {4 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} b c x - 4 \, {\left (b c^{2} x^{2} - 2 \, b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (b c^{2} x^{2} - b\right )} \sqrt {-d} \log \left (-\frac {c^{6} d x^{6} + 5 \, c^{4} d x^{4} - 5 \, c^{2} d x^{2} - 4 \, {\left (c^{3} x^{3} + c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} \sqrt {-d} - d}{c^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 1}\right ) - 4 \, {\left (a c^{2} x^{2} - 2 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{4 \, {\left (c^{6} d^{2} x^{2} - c^{4} d^{2}\right )}}, -\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} b c x + {\left (b c^{2} x^{2} - b\right )} \sqrt {d} \arctan \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} c \sqrt {d} x}{c^{4} d x^{4} - d}\right ) - 2 \, {\left (b c^{2} x^{2} - 2 \, b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 2 \, {\left (a c^{2} x^{2} - 2 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{2 \, {\left (c^{6} d^{2} x^{2} - c^{4} d^{2}\right )}}\right ] \] Input:

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

Output:

[-1/4*(4*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*b*c*x - 4*(b*c^2*x^2 - 2*b 
)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) + (b*c^2*x^2 - b)*sqrt 
(-d)*log(-(c^6*d*x^6 + 5*c^4*d*x^4 - 5*c^2*d*x^2 - 4*(c^3*x^3 + c*x)*sqrt( 
-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*sqrt(-d) - d)/(c^6*x^6 - 3*c^4*x^4 + 3*c 
^2*x^2 - 1)) - 4*(a*c^2*x^2 - 2*a)*sqrt(-c^2*d*x^2 + d))/(c^6*d^2*x^2 - c^ 
4*d^2), -1/2*(2*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*b*c*x + (b*c^2*x^2 
- b)*sqrt(d)*arctan(2*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*c*sqrt(d)*x/( 
c^4*d*x^4 - d)) - 2*(b*c^2*x^2 - 2*b)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt( 
c^2*x^2 - 1)) - 2*(a*c^2*x^2 - 2*a)*sqrt(-c^2*d*x^2 + d))/(c^6*d^2*x^2 - c 
^4*d^2)]
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.06 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {1}{2} \, b c {\left (\frac {2 \, \sqrt {-d} x}{c^{4} d^{2}} + \frac {\sqrt {-d} \log \left (c x + 1\right )}{c^{5} d^{2}} - \frac {\sqrt {-d} \log \left (c x - 1\right )}{c^{5} d^{2}}\right )} - b {\left (\frac {x^{2}}{\sqrt {-c^{2} d x^{2} + d} c^{2} d} - \frac {2}{\sqrt {-c^{2} d x^{2} + d} c^{4} d}\right )} \operatorname {arcosh}\left (c x\right ) - a {\left (\frac {x^{2}}{\sqrt {-c^{2} d x^{2} + d} c^{2} d} - \frac {2}{\sqrt {-c^{2} d x^{2} + d} c^{4} d}\right )} \] Input:

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

Output:

-1/2*b*c*(2*sqrt(-d)*x/(c^4*d^2) + sqrt(-d)*log(c*x + 1)/(c^5*d^2) - sqrt( 
-d)*log(c*x - 1)/(c^5*d^2)) - b*(x^2/(sqrt(-c^2*d*x^2 + d)*c^2*d) - 2/(sqr 
t(-c^2*d*x^2 + d)*c^4*d))*arccosh(c*x) - a*(x^2/(sqrt(-c^2*d*x^2 + d)*c^2* 
d) - 2/(sqrt(-c^2*d*x^2 + d)*c^4*d))
 

Giac [F(-2)]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - sqrt( - c**2*x**2 + 1)*int((acosh(c*x)*x**3)/(sqrt( - c**2*x**2 + 1)*c 
**2*x**2 - sqrt( - c**2*x**2 + 1)),x)*b*c**4 - a*c**2*x**2 + 2*a)/(sqrt(d) 
*sqrt( - c**2*x**2 + 1)*c**4*d)