\(\int \frac {(d+e x^2)^2 (a+b \text {csch}^{-1}(c x))}{x^4} \, dx\) [92]

Optimal result

Integrand size = 21, antiderivative size = 164 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^4} \, dx=-\frac {2 b c d \left (c^2 d-9 e\right ) \sqrt {-1-c^2 x^2}}{9 \sqrt {-c^2 x^2}}+\frac {b c d^2 \sqrt {-1-c^2 x^2}}{9 x^2 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b e^2 x \arctan \left (\frac {c x}{\sqrt {-1-c^2 x^2}}\right )}{\sqrt {-c^2 x^2}} \] Output:

-2/9*b*c*d*(c^2*d-9*e)*(-c^2*x^2-1)^(1/2)/(-c^2*x^2)^(1/2)+1/9*b*c*d^2*(-c 
^2*x^2-1)^(1/2)/x^2/(-c^2*x^2)^(1/2)-1/3*d^2*(a+b*arccsch(c*x))/x^3-2*d*e* 
(a+b*arccsch(c*x))/x+e^2*x*(a+b*arccsch(c*x))-b*e^2*x*arctan(c*x/(-c^2*x^2 
-1)^(1/2))/(-c^2*x^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.75 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^4} \, dx=\frac {b c d \sqrt {1+\frac {1}{c^2 x^2}} x \left (d-2 c^2 d x^2+18 e x^2\right )-3 a \left (d^2+6 d e x^2-3 e^2 x^4\right )}{9 x^3}-\frac {b \left (d^2+6 d e x^2-3 e^2 x^4\right ) \text {csch}^{-1}(c x)}{3 x^3}+\frac {b e^2 \log \left (\left (1+\sqrt {1+\frac {1}{c^2 x^2}}\right ) x\right )}{c} \] Input:

Integrate[((d + e*x^2)^2*(a + b*ArcCsch[c*x]))/x^4,x]
 

Output:

(b*c*d*Sqrt[1 + 1/(c^2*x^2)]*x*(d - 2*c^2*d*x^2 + 18*e*x^2) - 3*a*(d^2 + 6 
*d*e*x^2 - 3*e^2*x^4))/(9*x^3) - (b*(d^2 + 6*d*e*x^2 - 3*e^2*x^4)*ArcCsch[ 
c*x])/(3*x^3) + (b*e^2*Log[(1 + Sqrt[1 + 1/(c^2*x^2)])*x])/c
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.92, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6856, 27, 1588, 25, 358, 224, 216}

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.

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

\(\Big \downarrow \) 6856

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1588

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 358

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 216

\(\displaystyle -\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b c x \left (\frac {1}{3} \left (-\frac {9 e^2 \arctan \left (\frac {c x}{\sqrt {-c^2 x^2-1}}\right )}{c}-\frac {2 d \sqrt {-c^2 x^2-1} \left (c^2 d-9 e\right )}{x}\right )+\frac {d^2 \sqrt {-c^2 x^2-1}}{3 x^3}\right )}{3 \sqrt {-c^2 x^2}}\)

Input:

Int[((d + e*x^2)^2*(a + b*ArcCsch[c*x]))/x^4,x]
 

Output:

-1/3*(d^2*(a + b*ArcCsch[c*x]))/x^3 - (2*d*e*(a + b*ArcCsch[c*x]))/x + e^2 
*x*(a + b*ArcCsch[c*x]) + (b*c*x*((d^2*Sqrt[-1 - c^2*x^2])/(3*x^3) + ((-2* 
d*(c^2*d - 9*e)*Sqrt[-1 - c^2*x^2])/x - (9*e^2*ArcTan[(c*x)/Sqrt[-1 - c^2* 
x^2]])/c)/3))/(3*Sqrt[-(c^2*x^2)])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 358

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x_ 
Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + S 
imp[d/e^2   Int[(e*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e 
, m, p}, x] && NeQ[b*c - a*d, 0] && EqQ[Simplify[m + 2*p + 3], 0] && NeQ[m, 
 -1]
 

rule 1588

Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c 
_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + 
c*x^4)^p, f*x, x], R = PolynomialRemainder[(a + b*x^2 + c*x^4)^p, f*x, x]}, 
 Simp[R*(f*x)^(m + 1)*((d + e*x^2)^(q + 1)/(d*f*(m + 1))), x] + Simp[1/(d*f 
^2*(m + 1))   Int[(f*x)^(m + 2)*(d + e*x^2)^q*ExpandToSum[d*f*(m + 1)*(Qx/x 
) - e*R*(m + 2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && Ne 
Q[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[m, -1]
 

rule 6856

Int[((a_.) + ArcCsch[(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]}, Si 
mp[(a + b*ArcCsch[c*x])   u, x] - Simp[b*c*(x/Sqrt[(-c^2)*x^2])   Int[Simpl 
ifyIntegrand[u/(x*Sqrt[-1 - c^2*x^2]), x], x], x]] /; FreeQ[{a, b, c, d, e, 
 f, m, p}, x] && ((IGtQ[p, 0] &&  !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 
0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0])) || (I 
LtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))
 

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.13

method	result	size
parts	\(a \left (e^{2} x -\frac {d^{2}}{3 x^{3}}-\frac {2 d e}{x}\right )+b \,c^{3} \left (\frac {\operatorname {arccsch}\left (c x \right ) e^{2} x}{c^{3}}-\frac {\operatorname {arccsch}\left (c x \right ) d^{2}}{3 c^{3} x^{3}}-\frac {2 \,\operatorname {arccsch}\left (c x \right ) d e}{c^{3} x}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (-2 \sqrt {c^{2} x^{2}+1}\, c^{6} d^{2} x^{2}+c^{4} d^{2} \sqrt {c^{2} x^{2}+1}+18 c^{4} d e \sqrt {c^{2} x^{2}+1}\, x^{2}+9 e^{2} \operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}\right )}{9 c^{8} \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x^{4}}\right )\)	\(186\)
derivativedivides	\(c^{3} \left (\frac {a \left (e^{2} c x -\frac {2 c d e}{x}-\frac {c \,d^{2}}{3 x^{3}}\right )}{c^{4}}+\frac {b \left (\operatorname {arccsch}\left (c x \right ) e^{2} c x -\frac {2 \,\operatorname {arccsch}\left (c x \right ) c d e}{x}-\frac {\operatorname {arccsch}\left (c x \right ) c \,d^{2}}{3 x^{3}}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (-2 \sqrt {c^{2} x^{2}+1}\, c^{6} d^{2} x^{2}+c^{4} d^{2} \sqrt {c^{2} x^{2}+1}+18 c^{4} d e \sqrt {c^{2} x^{2}+1}\, x^{2}+9 e^{2} \operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}\right )}{9 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )}{c^{4}}\right )\)	\(190\)
default	\(c^{3} \left (\frac {a \left (e^{2} c x -\frac {2 c d e}{x}-\frac {c \,d^{2}}{3 x^{3}}\right )}{c^{4}}+\frac {b \left (\operatorname {arccsch}\left (c x \right ) e^{2} c x -\frac {2 \,\operatorname {arccsch}\left (c x \right ) c d e}{x}-\frac {\operatorname {arccsch}\left (c x \right ) c \,d^{2}}{3 x^{3}}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (-2 \sqrt {c^{2} x^{2}+1}\, c^{6} d^{2} x^{2}+c^{4} d^{2} \sqrt {c^{2} x^{2}+1}+18 c^{4} d e \sqrt {c^{2} x^{2}+1}\, x^{2}+9 e^{2} \operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}\right )}{9 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )}{c^{4}}\right )\)	\(190\)

Input:

int((e*x^2+d)^2*(a+b*arccsch(c*x))/x^4,x,method=_RETURNVERBOSE)
 

Output:

a*(e^2*x-1/3*d^2/x^3-2*d*e/x)+b*c^3*(1/c^3*arccsch(c*x)*e^2*x-1/3*arccsch( 
c*x)*d^2/c^3/x^3-2/c^3*arccsch(c*x)*d*e/x+1/9/c^8*(c^2*x^2+1)^(1/2)*(-2*(c 
^2*x^2+1)^(1/2)*c^6*d^2*x^2+c^4*d^2*(c^2*x^2+1)^(1/2)+18*c^4*d*e*(c^2*x^2+ 
1)^(1/2)*x^2+9*e^2*arcsinh(c*x)*c^3*x^3)/((c^2*x^2+1)/c^2/x^2)^(1/2)/x^4)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 334 vs. \(2 (146) = 292\).

Time = 0.12 (sec) , antiderivative size = 334, normalized size of antiderivative = 2.04 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^4} \, dx=\frac {9 \, a c e^{2} x^{4} - 9 \, b e^{2} x^{3} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) - 18 \, a c d e x^{2} - 3 \, {\left (b c d^{2} + 6 \, b c d e - 3 \, b c e^{2}\right )} x^{3} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) + 3 \, {\left (b c d^{2} + 6 \, b c d e - 3 \, b c e^{2}\right )} x^{3} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) - 3 \, a c d^{2} - 2 \, {\left (b c^{4} d^{2} - 9 \, b c^{2} d e\right )} x^{3} + 3 \, {\left (3 \, b c e^{2} x^{4} - 6 \, b c d e x^{2} - b c d^{2} + {\left (b c d^{2} + 6 \, b c d e - 3 \, b c e^{2}\right )} x^{3}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (b c^{2} d^{2} x - 2 \, {\left (b c^{4} d^{2} - 9 \, b c^{2} d e\right )} x^{3}\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{9 \, c x^{3}} \] Input:

integrate((e*x^2+d)^2*(a+b*arccsch(c*x))/x^4,x, algorithm="fricas")
 

Output:

1/9*(9*a*c*e^2*x^4 - 9*b*e^2*x^3*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c 
*x) - 18*a*c*d*e*x^2 - 3*(b*c*d^2 + 6*b*c*d*e - 3*b*c*e^2)*x^3*log(c*x*sqr 
t((c^2*x^2 + 1)/(c^2*x^2)) - c*x + 1) + 3*(b*c*d^2 + 6*b*c*d*e - 3*b*c*e^2 
)*x^3*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x - 1) - 3*a*c*d^2 - 2*(b* 
c^4*d^2 - 9*b*c^2*d*e)*x^3 + 3*(3*b*c*e^2*x^4 - 6*b*c*d*e*x^2 - b*c*d^2 + 
(b*c*d^2 + 6*b*c*d*e - 3*b*c*e^2)*x^3)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^ 
2)) + 1)/(c*x)) + (b*c^2*d^2*x - 2*(b*c^4*d^2 - 9*b*c^2*d*e)*x^3)*sqrt((c^ 
2*x^2 + 1)/(c^2*x^2)))/(c*x^3)
 

Sympy [F]

\[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^4} \, dx=\int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x^{4}}\, dx \] Input:

integrate((e*x**2+d)**2*(a+b*acsch(c*x))/x**4,x)
 

Output:

Integral((a + b*acsch(c*x))*(d + e*x**2)**2/x**4, x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.93 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^4} \, dx=2 \, {\left (c \sqrt {\frac {1}{c^{2} x^{2}} + 1} - \frac {\operatorname {arcsch}\left (c x\right )}{x}\right )} b d e + a e^{2} x + \frac {1}{9} \, b d^{2} {\left (\frac {c^{4} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, c^{4} \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {3 \, \operatorname {arcsch}\left (c x\right )}{x^{3}}\right )} + \frac {{\left (2 \, c x \operatorname {arcsch}\left (c x\right ) + \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )\right )} b e^{2}}{2 \, c} - \frac {2 \, a d e}{x} - \frac {a d^{2}}{3 \, x^{3}} \] Input:

integrate((e*x^2+d)^2*(a+b*arccsch(c*x))/x^4,x, algorithm="maxima")
 

Output:

2*(c*sqrt(1/(c^2*x^2) + 1) - arccsch(c*x)/x)*b*d*e + a*e^2*x + 1/9*b*d^2*( 
(c^4*(1/(c^2*x^2) + 1)^(3/2) - 3*c^4*sqrt(1/(c^2*x^2) + 1))/c - 3*arccsch( 
c*x)/x^3) + 1/2*(2*c*x*arccsch(c*x) + log(sqrt(1/(c^2*x^2) + 1) + 1) - log 
(sqrt(1/(c^2*x^2) + 1) - 1))*b*e^2/c - 2*a*d*e/x - 1/3*a*d^2/x^3
 

Giac [F]

\[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^4} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{x^{4}} \,d x } \] Input:

integrate((e*x^2+d)^2*(a+b*arccsch(c*x))/x^4,x, algorithm="giac")
 

Output:

integrate((e*x^2 + d)^2*(b*arccsch(c*x) + a)/x^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^4} \, dx=\int \frac {{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{x^4} \,d x \] Input:

int(((d + e*x^2)^2*(a + b*asinh(1/(c*x))))/x^4,x)
 

Output:

int(((d + e*x^2)^2*(a + b*asinh(1/(c*x))))/x^4, x)
 

Reduce [F]

\[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^4} \, dx=\frac {3 \left (\int \mathit {acsch} \left (c x \right )d x \right ) b \,e^{2} x^{3}+3 \left (\int \frac {\mathit {acsch} \left (c x \right )}{x^{4}}d x \right ) b \,d^{2} x^{3}+6 \left (\int \frac {\mathit {acsch} \left (c x \right )}{x^{2}}d x \right ) b d e \,x^{3}-a \,d^{2}-6 a d e \,x^{2}+3 a \,e^{2} x^{4}}{3 x^{3}} \] Input:

int((e*x^2+d)^2*(a+b*acsch(c*x))/x^4,x)
 

Output:

(3*int(acsch(c*x),x)*b*e**2*x**3 + 3*int(acsch(c*x)/x**4,x)*b*d**2*x**3 + 
6*int(acsch(c*x)/x**2,x)*b*d*e*x**3 - a*d**2 - 6*a*d*e*x**2 + 3*a*e**2*x** 
4)/(3*x**3)