\(\int \frac {x^5 (a+b \text {arccosh}(c x))}{(d-c^2 d x^2)^{3/2}} \, dx\) [109]

Optimal result

Integrand size = 27, antiderivative size = 232 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {5 b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x}}{9 c^3 d \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{c^6 d \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 c^6 d^3}+\frac {b \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{c^6 d \sqrt {d-c^2 d x^2}} \] Output:

5/3*b*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^5/d/(-c^2*d*x^2+d)^(1/2)+1/9*b*x^3*( 
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 
^6/d/(-c^2*d*x^2+d)^(1/2)+2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/c^6/d^ 
2-1/3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/c^6/d^3+b*(c*x-1)^(1/2)*(c*x 
+1)^(1/2)*arctanh(c*x)/c^6/d/(-c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.62 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {24 a-12 a c^2 x^2-3 a c^4 x^4+15 b c x \sqrt {-1+c x} \sqrt {1+c x}+b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x}-3 b \left (-8+4 c^2 x^2+c^4 x^4\right ) \text {arccosh}(c x)+9 b \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{9 c^6 d \sqrt {d-c^2 d x^2}} \] Input:

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

Output:

(24*a - 12*a*c^2*x^2 - 3*a*c^4*x^4 + 15*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] 
 + b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 3*b*(-8 + 4*c^2*x^2 + c^4*x^4) 
*ArcCosh[c*x] + 9*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTanh[c*x])/(9*c^6*d*Sq 
rt[d - c^2*d*x^2])
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.70, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6337, 27, 1467, 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.

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

\(\Big \downarrow \) 6337

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

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b \sqrt {d-c^2 d x^2} \int \frac {-c^4 x^4-4 c^2 x^2+8}{1-c^2 x^2}dx}{3 c^5 d^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 c^6 d^3}+\frac {2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^6 d^2}+\frac {a+b \text {arccosh}(c x)}{c^6 d \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 1467

\(\displaystyle -\frac {b \sqrt {d-c^2 d x^2} \int \left (c^2 x^2+\frac {3}{1-c^2 x^2}+5\right )dx}{3 c^5 d^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 c^6 d^3}+\frac {2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^6 d^2}+\frac {a+b \text {arccosh}(c x)}{c^6 d \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{3 c^6 d^3}+\frac {2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^6 d^2}+\frac {a+b \text {arccosh}(c x)}{c^6 d \sqrt {d-c^2 d x^2}}-\frac {b \left (\frac {3 \text {arctanh}(c x)}{c}+\frac {c^2 x^3}{3}+5 x\right ) \sqrt {d-c^2 d x^2}}{3 c^5 d^2 \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

(a + b*ArcCosh[c*x])/(c^6*d*Sqrt[d - c^2*d*x^2]) + (2*Sqrt[d - c^2*d*x^2]* 
(a + b*ArcCosh[c*x]))/(c^6*d^2) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c* 
x]))/(3*c^6*d^3) - (b*Sqrt[d - c^2*d*x^2]*(5*x + (c^2*x^3)/3 + (3*ArcTanh[ 
c*x])/c))/(3*c^5*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 1467

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), 
 x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], 
x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e 
 + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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 = 0.45 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.40

method	result	size
default	\(a \left (-\frac {x^{4}}{3 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {-\frac {4 x^{2}}{3 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {8}{3 d \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}}{c^{2}}\right )+\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) x^{4} c^{4}-c^{5} x^{5}+12 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}-14 c^{3} x^{3}-9 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+9 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) x^{2} c^{2}-24 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+15 c x +9 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-9 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )\right )}{9 \left (c^{2} x^{2}-1\right )^{2} d^{2} c^{6}}\)	\(325\)
parts	\(a \left (-\frac {x^{4}}{3 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {-\frac {4 x^{2}}{3 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {8}{3 d \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}}{c^{2}}\right )+\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) x^{4} c^{4}-c^{5} x^{5}+12 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}-14 c^{3} x^{3}-9 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}+9 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) x^{2} c^{2}-24 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+15 c x +9 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-9 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )\right )}{9 \left (c^{2} x^{2}-1\right )^{2} d^{2} c^{6}}\)	\(325\)

Input:

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

Output:

a*(-1/3*x^4/c^2/d/(-c^2*d*x^2+d)^(1/2)+4/3/c^2*(-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)))+1/9*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(-d 
*(c^2*x^2-1))^(1/2)*(3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)*x^4*c^4-c^ 
5*x^5+12*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^2*x^2-14*c^3*x^3-9*ln( 
1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*x^2*c^2+9*ln((c*x-1)^(1/2)*(c*x+1)^(1/2 
)+c*x-1)*x^2*c^2-24*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)+15*c*x+9*ln(1 
+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-9*ln((c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x-1)) 
/(c^2*x^2-1)^2/d^2/c^6
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 489, normalized size of antiderivative = 2.11 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\left [\frac {12 \, {\left (b c^{4} x^{4} + 4 \, b c^{2} x^{2} - 8 \, b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 9 \, {\left (b c^{2} x^{2} - b\right )} \sqrt {-d} \log \left (-\frac {c^{6} d x^{6} + 5 \, c^{4} d x^{4} - 5 \, c^{2} d x^{2} - 4 \, {\left (c^{3} x^{3} + c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} \sqrt {-d} - d}{c^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 1}\right ) - 4 \, {\left (b c^{3} x^{3} + 15 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 12 \, {\left (a c^{4} x^{4} + 4 \, a c^{2} x^{2} - 8 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{36 \, {\left (c^{8} d^{2} x^{2} - c^{6} d^{2}\right )}}, -\frac {9 \, {\left (b c^{2} x^{2} - b\right )} \sqrt {d} \arctan \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} c \sqrt {d} x}{c^{4} d x^{4} - d}\right ) - 6 \, {\left (b c^{4} x^{4} + 4 \, b c^{2} x^{2} - 8 \, b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, {\left (b c^{3} x^{3} + 15 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 6 \, {\left (a c^{4} x^{4} + 4 \, a c^{2} x^{2} - 8 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{18 \, {\left (c^{8} d^{2} x^{2} - c^{6} d^{2}\right )}}\right ] \] Input:

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

Output:

[1/36*(12*(b*c^4*x^4 + 4*b*c^2*x^2 - 8*b)*sqrt(-c^2*d*x^2 + d)*log(c*x + s 
qrt(c^2*x^2 - 1)) - 9*(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*(b*c^3*x^3 + 15 
*b*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 12*(a*c^4*x^4 + 4*a*c^2*x 
^2 - 8*a)*sqrt(-c^2*d*x^2 + d))/(c^8*d^2*x^2 - c^6*d^2), -1/18*(9*(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)) - 6*(b*c^4*x^4 + 4*b*c^2*x^2 - 8*b)*sqrt(-c^2*d*x^2 + d 
)*log(c*x + sqrt(c^2*x^2 - 1)) + 2*(b*c^3*x^3 + 15*b*c*x)*sqrt(-c^2*d*x^2 
+ d)*sqrt(c^2*x^2 - 1) - 6*(a*c^4*x^4 + 4*a*c^2*x^2 - 8*a)*sqrt(-c^2*d*x^2 
 + d))/(c^8*d^2*x^2 - c^6*d^2)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/3*a*(x^4/(sqrt(-c^2*d*x^2 + d)*c^2*d) + 4*x^2/(sqrt(-c^2*d*x^2 + d)*c^4 
*d) - 8/(sqrt(-c^2*d*x^2 + d)*c^6*d)) + 1/9*b*(((c^4*sqrt(d)*x^4 + 16*c^2* 
sqrt(d)*x^2 - 8*sqrt(d))*sqrt(c*x + 1)*sqrt(c*x - 1)/sqrt(-c*x + 1) - 3*(c 
^5*sqrt(d)*x^5 + 4*c^3*sqrt(d)*x^3 - 8*c*sqrt(d)*x + (c^4*sqrt(d)*x^4 + 4* 
c^2*sqrt(d)*x^2 - 8*sqrt(d))*sqrt(c*x + 1)*sqrt(c*x - 1))*log(c*x + sqrt(c 
*x + 1)*sqrt(c*x - 1))/sqrt(-c*x + 1))/(sqrt(c*x + 1)*c^7*d^2*x + (c*x + 1 
)*sqrt(c*x - 1)*c^6*d^2) + 9*integrate(1/9*(3*c^7*sqrt(d)*x^7 + 9*c^5*sqrt 
(d)*x^5 - 36*c^3*sqrt(d)*x^3 + 24*c*sqrt(d)*x + (3*c^6*sqrt(d)*x^6 + 8*c^4 
*sqrt(d)*x^4 - 52*c^2*sqrt(d)*x^2 + 32*sqrt(d))*e^(1/2*log(c*x + 1) + 1/2* 
log(c*x - 1)))/(sqrt(-c*x + 1)*((c^7*d^2*x^2 - c^5*d^2)*e^(3/2*log(c*x + 1 
) + log(c*x - 1)) + 2*(c^8*d^2*x^3 - c^6*d^2*x)*e^(log(c*x + 1) + 1/2*log( 
c*x - 1)) + (c^9*d^2*x^4 - c^7*d^2*x^2)*sqrt(c*x + 1))), x))
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {-3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right ) x^{5}}{\sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{6}-a \,c^{4} x^{4}-4 a \,c^{2} x^{2}+8 a}{3 \sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, c^{6} d} \] Input:

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

Output:

( - 3*sqrt( - c**2*x**2 + 1)*int((acosh(c*x)*x**5)/(sqrt( - c**2*x**2 + 1) 
*c**2*x**2 - sqrt( - c**2*x**2 + 1)),x)*b*c**6 - a*c**4*x**4 - 4*a*c**2*x* 
*2 + 8*a)/(3*sqrt(d)*sqrt( - c**2*x**2 + 1)*c**6*d)