\(\int \frac {x^3 (a+b \text {arccosh}(c x))}{(d-c^2 d x^2)^{5/2}} \, dx\) [121]

Optimal result

Integrand size = 27, antiderivative size = 158 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b x}{6 c^3 d^2 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{3 c^4 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 b \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{6 c^4 d^2 \sqrt {d-c^2 d x^2}} \] Output:

-1/6*b*x/c^3/d^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)+1/3*(a+b 
*arccosh(c*x))/c^4/d/(-c^2*d*x^2+d)^(3/2)-(a+b*arccosh(c*x))/c^4/d^2/(-c^2 
*d*x^2+d)^(1/2)-5/6*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arctanh(c*x)/c^4/d^2/(-c 
^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.77 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {4 a-6 a c^2 x^2-b c x \sqrt {-1+c x} \sqrt {1+c x}+b \left (4-6 c^2 x^2\right ) \text {arccosh}(c x)-5 b \sqrt {-1+c x} \sqrt {1+c x} \left (-1+c^2 x^2\right ) \text {arctanh}(c x)}{6 c^4 d^2 \left (-1+c^2 x^2\right ) \sqrt {d-c^2 d x^2}} \] Input:

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

Output:

(4*a - 6*a*c^2*x^2 - b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + b*(4 - 6*c^2*x^2 
)*ArcCosh[c*x] - 5*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-1 + c^2*x^2)*ArcTanh[c 
*x])/(6*c^4*d^2*(-1 + c^2*x^2)*Sqrt[d - c^2*d*x^2])
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.87, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6337, 27, 298, 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.

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

\(\Big \downarrow \) 6337

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 298

\(\displaystyle \frac {b \sqrt {d-c^2 d x^2} \left (\frac {5}{2} \int \frac {1}{1-c^2 x^2}dx-\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c^3 d^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {a+b \text {arccosh}(c x)}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{3 c^4 d \left (d-c^2 d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {a+b \text {arccosh}(c x)}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{3 c^4 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b \left (\frac {5 \text {arctanh}(c x)}{2 c}-\frac {x}{2 \left (1-c^2 x^2\right )}\right ) \sqrt {d-c^2 d x^2}}{3 c^3 d^3 \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

(a + b*ArcCosh[c*x])/(3*c^4*d*(d - c^2*d*x^2)^(3/2)) - (a + b*ArcCosh[c*x] 
)/(c^4*d^2*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[d - c^2*d*x^2]*(-1/2*x/(1 - c^2* 
x^2) + (5*ArcTanh[c*x])/(2*c)))/(3*c^3*d^3*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 298

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( 
b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 
2*p + 3))/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, 
 c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 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 = 0.42 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.63

method	result	size
default	\(a \left (\frac {x^{2}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {2}{3 d \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}\right )+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}\, c x -4 \,\operatorname {arccosh}\left (c x \right )\right )}{6 \left (c^{2} x^{2}-1\right )^{2} d^{3} c^{4}}+\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{6 d^{3} c^{4} \left (c^{2} x^{2}-1\right )}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )}{6 d^{3} c^{4} \left (c^{2} x^{2}-1\right )}\right )\)	\(257\)
parts	\(a \left (\frac {x^{2}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {2}{3 d \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}\right )+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}\, c x -4 \,\operatorname {arccosh}\left (c x \right )\right )}{6 \left (c^{2} x^{2}-1\right )^{2} d^{3} c^{4}}+\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{6 d^{3} c^{4} \left (c^{2} x^{2}-1\right )}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )}{6 d^{3} c^{4} \left (c^{2} x^{2}-1\right )}\right )\)	\(257\)

Input:

int(x^3*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

a*(x^2/c^2/d/(-c^2*d*x^2+d)^(3/2)-2/3/d/c^4/(-c^2*d*x^2+d)^(3/2))+b*(1/6*( 
-d*(c^2*x^2-1))^(1/2)*(6*c^2*x^2*arccosh(c*x)+(c*x-1)^(1/2)*(c*x+1)^(1/2)* 
c*x-4*arccosh(c*x))/(c^2*x^2-1)^2/d^3/c^4+5/6*(-d*(c^2*x^2-1))^(1/2)*(c*x- 
1)^(1/2)*(c*x+1)^(1/2)/d^3/c^4/(c^2*x^2-1)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^ 
(1/2))-5/6*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/c^4/(c^2 
*x^2-1)*ln((c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x-1))
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 469, normalized size of antiderivative = 2.97 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\left [\frac {4 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} b c x + 8 \, {\left (3 \, b c^{2} x^{2} - 2 \, b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 5 \, {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt {-d} \log \left (-\frac {c^{6} d x^{6} + 5 \, c^{4} d x^{4} - 5 \, c^{2} d x^{2} + 4 \, {\left (c^{3} x^{3} + c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} \sqrt {-d} - d}{c^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 1}\right ) + 8 \, {\left (3 \, a c^{2} x^{2} - 2 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{24 \, {\left (c^{8} d^{3} x^{4} - 2 \, c^{6} d^{3} x^{2} + c^{4} d^{3}\right )}}, \frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} b c x + 5 \, {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt {d} \arctan \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} c \sqrt {d} x}{c^{4} d x^{4} - d}\right ) + 4 \, {\left (3 \, b c^{2} x^{2} - 2 \, b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 4 \, {\left (3 \, a c^{2} x^{2} - 2 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{12 \, {\left (c^{8} d^{3} x^{4} - 2 \, c^{6} d^{3} x^{2} + c^{4} d^{3}\right )}}\right ] \] Input:

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

Output:

[1/24*(4*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*b*c*x + 8*(3*b*c^2*x^2 - 2 
*b)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) - 5*(b*c^4*x^4 - 2*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)) + 8*(3*a*c^2*x^2 - 2*a)*sqrt(-c^2*d*x^2 + d 
))/(c^8*d^3*x^4 - 2*c^6*d^3*x^2 + c^4*d^3), 1/12*(2*sqrt(-c^2*d*x^2 + d)*s 
qrt(c^2*x^2 - 1)*b*c*x + 5*(b*c^4*x^4 - 2*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)) + 4*(3 
*b*c^2*x^2 - 2*b)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) + 4*(3 
*a*c^2*x^2 - 2*a)*sqrt(-c^2*d*x^2 + d))/(c^8*d^3*x^4 - 2*c^6*d^3*x^2 + c^4 
*d^3)]
 

Sympy [F]

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

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

Output:

Integral(x**3*(a + b*acosh(c*x))/(-d*(c*x - 1)*(c*x + 1))**(5/2), x)
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.11 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {1}{12} \, b c {\left (\frac {2 \, \sqrt {-d} x}{c^{6} d^{3} x^{2} - c^{4} d^{3}} + \frac {5 \, \sqrt {-d} \log \left (c x + 1\right )}{c^{5} d^{3}} - \frac {5 \, \sqrt {-d} \log \left (c x - 1\right )}{c^{5} d^{3}}\right )} + \frac {1}{3} \, b {\left (\frac {3 \, x^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d} - \frac {2}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{4} d}\right )} \operatorname {arcosh}\left (c x\right ) + \frac {1}{3} \, a {\left (\frac {3 \, x^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d} - \frac {2}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{4} d}\right )} \] Input:

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

Output:

1/12*b*c*(2*sqrt(-d)*x/(c^6*d^3*x^2 - c^4*d^3) + 5*sqrt(-d)*log(c*x + 1)/( 
c^5*d^3) - 5*sqrt(-d)*log(c*x - 1)/(c^5*d^3)) + 1/3*b*(3*x^2/((-c^2*d*x^2 
+ d)^(3/2)*c^2*d) - 2/((-c^2*d*x^2 + d)^(3/2)*c^4*d))*arccosh(c*x) + 1/3*a 
*(3*x^2/((-c^2*d*x^2 + d)^(3/2)*c^2*d) - 2/((-c^2*d*x^2 + d)^(3/2)*c^4*d))
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

(3*sqrt( - c**2*x**2 + 1)*int((acosh(c*x)*x**3)/(sqrt( - c**2*x**2 + 1)*c* 
*4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2*x**2 + 1)),x)* 
b*c**6*x**2 - 3*sqrt( - c**2*x**2 + 1)*int((acosh(c*x)*x**3)/(sqrt( - c**2 
*x**2 + 1)*c**4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2*x 
**2 + 1)),x)*b*c**4 - 3*a*c**2*x**2 + 2*a)/(3*sqrt(d)*sqrt( - c**2*x**2 + 
1)*c**4*d**2*(c**2*x**2 - 1))