\(\int \frac {(d-c^2 d x^2)^3 (a+b \text {arccosh}(c x))}{x^4} \, dx\) [27]

Optimal result

Integrand size = 25, antiderivative size = 198 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^4} \, dx=-\frac {25}{9} b c^3 d^3 \sqrt {-1+c x} \sqrt {1+c x}+\frac {b c d^3 \sqrt {-1+c x} \sqrt {1+c x}}{6 x^2}+\frac {1}{9} b c^5 d^3 x^2 \sqrt {-1+c x} \sqrt {1+c x}-\frac {d^3 (a+b \text {arccosh}(c x))}{3 x^3}+\frac {3 c^2 d^3 (a+b \text {arccosh}(c x))}{x}+3 c^4 d^3 x (a+b \text {arccosh}(c x))-\frac {1}{3} c^6 d^3 x^3 (a+b \text {arccosh}(c x))-\frac {17}{6} b c^3 d^3 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \] Output:

-25/9*b*c^3*d^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)+1/6*b*c*d^3*(c*x-1)^(1/2)*(c*x 
+1)^(1/2)/x^2+1/9*b*c^5*d^3*x^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)-1/3*d^3*(a+b*a 
rccosh(c*x))/x^3+3*c^2*d^3*(a+b*arccosh(c*x))/x+3*c^4*d^3*x*(a+b*arccosh(c 
*x))-1/3*c^6*d^3*x^3*(a+b*arccosh(c*x))-17/6*b*c^3*d^3*arctan((c*x-1)^(1/2 
)*(c*x+1)^(1/2))
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.72 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^4} \, dx=\frac {d^3 \left (-6 a+54 a c^2 x^2+54 a c^4 x^4-6 a c^6 x^6+b c x \sqrt {-1+c x} \sqrt {1+c x} \left (3-50 c^2 x^2+2 c^4 x^4\right )-6 b \left (1-9 c^2 x^2-9 c^4 x^4+c^6 x^6\right ) \text {arccosh}(c x)+51 b c^3 x^3 \arctan \left (\frac {1}{\sqrt {-1+c x} \sqrt {1+c x}}\right )\right )}{18 x^3} \] Input:

Integrate[((d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]))/x^4,x]
 

Output:

(d^3*(-6*a + 54*a*c^2*x^2 + 54*a*c^4*x^4 - 6*a*c^6*x^6 + b*c*x*Sqrt[-1 + c 
*x]*Sqrt[1 + c*x]*(3 - 50*c^2*x^2 + 2*c^4*x^4) - 6*b*(1 - 9*c^2*x^2 - 9*c^ 
4*x^4 + c^6*x^6)*ArcCosh[c*x] + 51*b*c^3*x^3*ArcTan[1/(Sqrt[-1 + c*x]*Sqrt 
[1 + c*x])]))/(18*x^3)
 

Rubi [A] (warning: unable to verify)

Time = 0.81 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.94, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {6336, 27, 2113, 2331, 2124, 27, 1192, 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.

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

\(\Big \downarrow \) 6336

\(\displaystyle -b c \int -\frac {d^3 \left (c^6 x^6-9 c^4 x^4-9 c^2 x^2+1\right )}{3 x^3 \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{3} c^6 d^3 x^3 (a+b \text {arccosh}(c x))+3 c^4 d^3 x (a+b \text {arccosh}(c x))+\frac {3 c^2 d^3 (a+b \text {arccosh}(c x))}{x}-\frac {d^3 (a+b \text {arccosh}(c x))}{3 x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} b c d^3 \int \frac {c^6 x^6-9 c^4 x^4-9 c^2 x^2+1}{x^3 \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{3} c^6 d^3 x^3 (a+b \text {arccosh}(c x))+3 c^4 d^3 x (a+b \text {arccosh}(c x))+\frac {3 c^2 d^3 (a+b \text {arccosh}(c x))}{x}-\frac {d^3 (a+b \text {arccosh}(c x))}{3 x^3}\)

\(\Big \downarrow \) 2113

\(\displaystyle \frac {b c d^3 \sqrt {c^2 x^2-1} \int \frac {c^6 x^6-9 c^4 x^4-9 c^2 x^2+1}{x^3 \sqrt {c^2 x^2-1}}dx}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3} c^6 d^3 x^3 (a+b \text {arccosh}(c x))+3 c^4 d^3 x (a+b \text {arccosh}(c x))+\frac {3 c^2 d^3 (a+b \text {arccosh}(c x))}{x}-\frac {d^3 (a+b \text {arccosh}(c x))}{3 x^3}\)

\(\Big \downarrow \) 2331

\(\displaystyle \frac {b c d^3 \sqrt {c^2 x^2-1} \int \frac {c^6 x^6-9 c^4 x^4-9 c^2 x^2+1}{x^4 \sqrt {c^2 x^2-1}}dx^2}{6 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3} c^6 d^3 x^3 (a+b \text {arccosh}(c x))+3 c^4 d^3 x (a+b \text {arccosh}(c x))+\frac {3 c^2 d^3 (a+b \text {arccosh}(c x))}{x}-\frac {d^3 (a+b \text {arccosh}(c x))}{3 x^3}\)

\(\Big \downarrow \) 2124

\(\displaystyle \frac {b c d^3 \sqrt {c^2 x^2-1} \left (\int -\frac {-2 x^4 c^6+18 x^2 c^4+17 c^2}{2 x^2 \sqrt {c^2 x^2-1}}dx^2+\frac {\sqrt {c^2 x^2-1}}{x^2}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3} c^6 d^3 x^3 (a+b \text {arccosh}(c x))+3 c^4 d^3 x (a+b \text {arccosh}(c x))+\frac {3 c^2 d^3 (a+b \text {arccosh}(c x))}{x}-\frac {d^3 (a+b \text {arccosh}(c x))}{3 x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b c d^3 \sqrt {c^2 x^2-1} \left (\frac {\sqrt {c^2 x^2-1}}{x^2}-\frac {1}{2} \int \frac {-2 x^4 c^6+18 x^2 c^4+17 c^2}{x^2 \sqrt {c^2 x^2-1}}dx^2\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3} c^6 d^3 x^3 (a+b \text {arccosh}(c x))+3 c^4 d^3 x (a+b \text {arccosh}(c x))+\frac {3 c^2 d^3 (a+b \text {arccosh}(c x))}{x}-\frac {d^3 (a+b \text {arccosh}(c x))}{3 x^3}\)

\(\Big \downarrow \) 1192

\(\displaystyle \frac {b c d^3 \sqrt {c^2 x^2-1} \left (\frac {\sqrt {c^2 x^2-1}}{x^2}-\frac {\int \frac {-2 c^6 x^8+14 c^6 x^4+33 c^6}{x^4+1}d\sqrt {c^2 x^2-1}}{c^4}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3} c^6 d^3 x^3 (a+b \text {arccosh}(c x))+3 c^4 d^3 x (a+b \text {arccosh}(c x))+\frac {3 c^2 d^3 (a+b \text {arccosh}(c x))}{x}-\frac {d^3 (a+b \text {arccosh}(c x))}{3 x^3}\)

\(\Big \downarrow \) 1467

\(\displaystyle \frac {b c d^3 \sqrt {c^2 x^2-1} \left (\frac {\sqrt {c^2 x^2-1}}{x^2}-\frac {\int \left (-2 x^4 c^6+\frac {17 c^6}{x^4+1}+16 c^6\right )d\sqrt {c^2 x^2-1}}{c^4}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3} c^6 d^3 x^3 (a+b \text {arccosh}(c x))+3 c^4 d^3 x (a+b \text {arccosh}(c x))+\frac {3 c^2 d^3 (a+b \text {arccosh}(c x))}{x}-\frac {d^3 (a+b \text {arccosh}(c x))}{3 x^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {1}{3} c^6 d^3 x^3 (a+b \text {arccosh}(c x))+3 c^4 d^3 x (a+b \text {arccosh}(c x))+\frac {3 c^2 d^3 (a+b \text {arccosh}(c x))}{x}-\frac {d^3 (a+b \text {arccosh}(c x))}{3 x^3}+\frac {b c d^3 \sqrt {c^2 x^2-1} \left (\frac {\sqrt {c^2 x^2-1}}{x^2}-\frac {17 c^6 \arctan \left (\sqrt {c^2 x^2-1}\right )-\frac {2}{3} c^6 x^6+16 c^6 \sqrt {c^2 x^2-1}}{c^4}\right )}{6 \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

Int[((d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]))/x^4,x]
 

Output:

-1/3*(d^3*(a + b*ArcCosh[c*x]))/x^3 + (3*c^2*d^3*(a + b*ArcCosh[c*x]))/x + 
 3*c^4*d^3*x*(a + b*ArcCosh[c*x]) - (c^6*d^3*x^3*(a + b*ArcCosh[c*x]))/3 + 
 (b*c*d^3*Sqrt[-1 + c^2*x^2]*(Sqrt[-1 + c^2*x^2]/x^2 - ((-2*c^6*x^6)/3 + 1 
6*c^6*Sqrt[-1 + c^2*x^2] + 17*c^6*ArcTan[Sqrt[-1 + c^2*x^2]])/c^4))/(6*Sqr 
t[-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 1192

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2/e^(n + 2*p + 1)   Subst[Int[x^( 
2*m + 1)*(e*f - d*g + g*x^2)^n*(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + 
 c*x^4)^p, x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && 
IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m + 1/2]
 

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 2113

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_. 
)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*x)^FracPart[m]*((c + d*x)^FracPart[ 
m]/(a*c + b*d*x^2)^FracPart[m])   Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a 
*d, 0] && EqQ[m, n] &&  !IntegerQ[m]
 

rule 2124

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px 
, a + b*x, x]}, Simp[R*(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((m + 1)*(b*c - 
 a*d))), x] + Simp[1/((m + 1)*(b*c - a*d))   Int[(a + b*x)^(m + 1)*(c + d*x 
)^n*ExpandToSum[(m + 1)*(b*c - a*d)*Qx - d*R*(m + n + 2), x], x], x]] /; Fr 
eeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && LtQ[m, -1] && (IntegerQ[m] ||  ! 
ILtQ[n, -1])
 

rule 2331

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2   S 
ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; 
 FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
 

rule 6336

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

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.98

method	result	size
parts	\(-d^{3} a \left (\frac {c^{6} x^{3}}{3}-3 c^{4} x -\frac {3 c^{2}}{x}+\frac {1}{3 x^{3}}\right )-d^{3} b \,c^{3} \left (\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-3 c x \,\operatorname {arccosh}\left (c x \right )-\frac {3 \,\operatorname {arccosh}\left (c x \right )}{c x}+\frac {\operatorname {arccosh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (2 c^{4} x^{4} \sqrt {c^{2} x^{2}-1}+51 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{2} x^{2}-50 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+3 \sqrt {c^{2} x^{2}-1}\right )}{18 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}}\right )\)	\(195\)
derivativedivides	\(c^{3} \left (-d^{3} a \left (\frac {c^{3} x^{3}}{3}-3 c x -\frac {3}{c x}+\frac {1}{3 c^{3} x^{3}}\right )-d^{3} b \left (\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-3 c x \,\operatorname {arccosh}\left (c x \right )-\frac {3 \,\operatorname {arccosh}\left (c x \right )}{c x}+\frac {\operatorname {arccosh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (2 c^{4} x^{4} \sqrt {c^{2} x^{2}-1}+51 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{2} x^{2}-50 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+3 \sqrt {c^{2} x^{2}-1}\right )}{18 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}}\right )\right )\)	\(197\)
default	\(c^{3} \left (-d^{3} a \left (\frac {c^{3} x^{3}}{3}-3 c x -\frac {3}{c x}+\frac {1}{3 c^{3} x^{3}}\right )-d^{3} b \left (\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-3 c x \,\operatorname {arccosh}\left (c x \right )-\frac {3 \,\operatorname {arccosh}\left (c x \right )}{c x}+\frac {\operatorname {arccosh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (2 c^{4} x^{4} \sqrt {c^{2} x^{2}-1}+51 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) c^{2} x^{2}-50 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+3 \sqrt {c^{2} x^{2}-1}\right )}{18 c^{2} x^{2} \sqrt {c^{2} x^{2}-1}}\right )\right )\)	\(197\)

Input:

int((-c^2*d*x^2+d)^3*(a+b*arccosh(c*x))/x^4,x,method=_RETURNVERBOSE)
 

Output:

-d^3*a*(1/3*c^6*x^3-3*c^4*x-3*c^2/x+1/3/x^3)-d^3*b*c^3*(1/3*c^3*x^3*arccos 
h(c*x)-3*c*x*arccosh(c*x)-3*arccosh(c*x)/c/x+1/3*arccosh(c*x)/c^3/x^3-1/18 
*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(2*c^4*x^4*(c^2*x^2-1)^(1/2)+51*arctan(1/(c^2 
*x^2-1)^(1/2))*c^2*x^2-50*c^2*x^2*(c^2*x^2-1)^(1/2)+3*(c^2*x^2-1)^(1/2))/c 
^2/x^2/(c^2*x^2-1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.28 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^4} \, dx=-\frac {6 \, a c^{6} d^{3} x^{6} - 54 \, a c^{4} d^{3} x^{4} + 102 \, b c^{3} d^{3} x^{3} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 54 \, a c^{2} d^{3} x^{2} - 6 \, {\left (b c^{6} - 9 \, b c^{4} - 9 \, b c^{2} + b\right )} d^{3} x^{3} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 6 \, a d^{3} + 6 \, {\left (b c^{6} d^{3} x^{6} - 9 \, b c^{4} d^{3} x^{4} - 9 \, b c^{2} d^{3} x^{2} - {\left (b c^{6} - 9 \, b c^{4} - 9 \, b c^{2} + b\right )} d^{3} x^{3} + b d^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (2 \, b c^{5} d^{3} x^{5} - 50 \, b c^{3} d^{3} x^{3} + 3 \, b c d^{3} x\right )} \sqrt {c^{2} x^{2} - 1}}{18 \, x^{3}} \] Input:

integrate((-c^2*d*x^2+d)^3*(a+b*arccosh(c*x))/x^4,x, algorithm="fricas")
 

Output:

-1/18*(6*a*c^6*d^3*x^6 - 54*a*c^4*d^3*x^4 + 102*b*c^3*d^3*x^3*arctan(-c*x 
+ sqrt(c^2*x^2 - 1)) - 54*a*c^2*d^3*x^2 - 6*(b*c^6 - 9*b*c^4 - 9*b*c^2 + b 
)*d^3*x^3*log(-c*x + sqrt(c^2*x^2 - 1)) + 6*a*d^3 + 6*(b*c^6*d^3*x^6 - 9*b 
*c^4*d^3*x^4 - 9*b*c^2*d^3*x^2 - (b*c^6 - 9*b*c^4 - 9*b*c^2 + b)*d^3*x^3 + 
 b*d^3)*log(c*x + sqrt(c^2*x^2 - 1)) - (2*b*c^5*d^3*x^5 - 50*b*c^3*d^3*x^3 
 + 3*b*c*d^3*x)*sqrt(c^2*x^2 - 1))/x^3
 

Sympy [F]

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

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

Output:

-d**3*(Integral(-3*a*c**4, x) + Integral(-a/x**4, x) + Integral(3*a*c**2/x 
**2, x) + Integral(a*c**6*x**2, x) + Integral(-3*b*c**4*acosh(c*x), x) + I 
ntegral(-b*acosh(c*x)/x**4, x) + Integral(3*b*c**2*acosh(c*x)/x**2, x) + I 
ntegral(b*c**6*x**2*acosh(c*x), x))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.05 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^4} \, dx=-\frac {1}{3} \, a c^{6} d^{3} x^{3} - \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b c^{6} d^{3} + 3 \, a c^{4} d^{3} x + 3 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b c^{3} d^{3} + 3 \, {\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b c^{2} d^{3} - \frac {1}{6} \, {\left ({\left (c^{2} \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) - \frac {\sqrt {c^{2} x^{2} - 1}}{x^{2}}\right )} c + \frac {2 \, \operatorname {arcosh}\left (c x\right )}{x^{3}}\right )} b d^{3} + \frac {3 \, a c^{2} d^{3}}{x} - \frac {a d^{3}}{3 \, x^{3}} \] Input:

integrate((-c^2*d*x^2+d)^3*(a+b*arccosh(c*x))/x^4,x, algorithm="maxima")
 

Output:

-1/3*a*c^6*d^3*x^3 - 1/9*(3*x^3*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x^2/c^ 
2 + 2*sqrt(c^2*x^2 - 1)/c^4))*b*c^6*d^3 + 3*a*c^4*d^3*x + 3*(c*x*arccosh(c 
*x) - sqrt(c^2*x^2 - 1))*b*c^3*d^3 + 3*(c*arcsin(1/(c*abs(x))) + arccosh(c 
*x)/x)*b*c^2*d^3 - 1/6*((c^2*arcsin(1/(c*abs(x))) - sqrt(c^2*x^2 - 1)/x^2) 
*c + 2*arccosh(c*x)/x^3)*b*d^3 + 3*a*c^2*d^3/x - 1/3*a*d^3/x^3
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^4} \, dx=\text {Exception raised: TypeError} \] Input:

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.92 \[ \int \frac {\left (d-c^2 d x^2\right )^3 (a+b \text {arccosh}(c x))}{x^4} \, dx=\frac {d^{3} \left (-6 \mathit {acosh} \left (c x \right ) b \,c^{6} x^{6}+54 \mathit {acosh} \left (c x \right ) b \,c^{4} x^{4}+54 \mathit {acosh} \left (c x \right ) b \,c^{2} x^{2}-6 \mathit {acosh} \left (c x \right ) b +102 \mathit {atan} \left (\sqrt {c^{2} x^{2}-1}+c x \right ) b \,c^{3} x^{3}+2 \sqrt {c^{2} x^{2}-1}\, b \,c^{5} x^{5}+4 \sqrt {c^{2} x^{2}-1}\, b \,c^{3} x^{3}-3 \sqrt {c^{2} x^{2}-1}\, b c x -54 \sqrt {c x +1}\, \sqrt {c x -1}\, b \,c^{3} x^{3}-6 a \,c^{6} x^{6}+54 a \,c^{4} x^{4}+54 a \,c^{2} x^{2}-6 a \right )}{18 x^{3}} \] Input:

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

Output:

(d**3*( - 6*acosh(c*x)*b*c**6*x**6 + 54*acosh(c*x)*b*c**4*x**4 + 54*acosh( 
c*x)*b*c**2*x**2 - 6*acosh(c*x)*b + 102*atan(sqrt(c**2*x**2 - 1) + c*x)*b* 
c**3*x**3 + 2*sqrt(c**2*x**2 - 1)*b*c**5*x**5 + 4*sqrt(c**2*x**2 - 1)*b*c* 
*3*x**3 - 3*sqrt(c**2*x**2 - 1)*b*c*x - 54*sqrt(c*x + 1)*sqrt(c*x - 1)*b*c 
**3*x**3 - 6*a*c**6*x**6 + 54*a*c**4*x**4 + 54*a*c**2*x**2 - 6*a))/(18*x** 
3)