\(\int x^3 (d+e x^2)^2 (a+b \text {csch}^{-1}(c x)) \, dx\) [95]

Optimal result

Integrand size = 21, antiderivative size = 250 \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=-\frac {b \left (6 c^4 d^2-8 c^2 d e+3 e^2\right ) x \sqrt {-1-c^2 x^2}}{24 c^7 \sqrt {-c^2 x^2}}-\frac {b \left (6 c^4 d^2-16 c^2 d e+9 e^2\right ) x \left (-1-c^2 x^2\right )^{3/2}}{72 c^7 \sqrt {-c^2 x^2}}+\frac {b \left (8 c^2 d-9 e\right ) e x \left (-1-c^2 x^2\right )^{5/2}}{120 c^7 \sqrt {-c^2 x^2}}-\frac {b e^2 x \left (-1-c^2 x^2\right )^{7/2}}{56 c^7 \sqrt {-c^2 x^2}}+\frac {1}{4} d^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \text {csch}^{-1}(c x)\right ) \] Output:

-1/24*b*(6*c^4*d^2-8*c^2*d*e+3*e^2)*x*(-c^2*x^2-1)^(1/2)/c^7/(-c^2*x^2)^(1 
/2)-1/72*b*(6*c^4*d^2-16*c^2*d*e+9*e^2)*x*(-c^2*x^2-1)^(3/2)/c^7/(-c^2*x^2 
)^(1/2)+1/120*b*(8*c^2*d-9*e)*e*x*(-c^2*x^2-1)^(5/2)/c^7/(-c^2*x^2)^(1/2)- 
1/56*b*e^2*x*(-c^2*x^2-1)^(7/2)/c^7/(-c^2*x^2)^(1/2)+1/4*d^2*x^4*(a+b*arcc 
sch(c*x))+1/3*d*e*x^6*(a+b*arccsch(c*x))+1/8*e^2*x^8*(a+b*arccsch(c*x))
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.64 \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {x \left (105 a x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )+\frac {b \sqrt {1+\frac {1}{c^2 x^2}} \left (-144 e^2+8 c^2 e \left (56 d+9 e x^2\right )-2 c^4 \left (210 d^2+112 d e x^2+27 e^2 x^4\right )+3 c^6 \left (70 d^2 x^2+56 d e x^4+15 e^2 x^6\right )\right )}{c^7}+105 b x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right ) \text {csch}^{-1}(c x)\right )}{2520} \] Input:

Integrate[x^3*(d + e*x^2)^2*(a + b*ArcCsch[c*x]),x]
 

Output:

(x*(105*a*x^3*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4) + (b*Sqrt[1 + 1/(c^2*x^2)]*( 
-144*e^2 + 8*c^2*e*(56*d + 9*e*x^2) - 2*c^4*(210*d^2 + 112*d*e*x^2 + 27*e^ 
2*x^4) + 3*c^6*(70*d^2*x^2 + 56*d*e*x^4 + 15*e^2*x^6)))/c^7 + 105*b*x^3*(6 
*d^2 + 8*d*e*x^2 + 3*e^2*x^4)*ArcCsch[c*x]))/2520
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.85, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6856, 27, 1578, 1195, 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.

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

\(\Big \downarrow \) 6856

\(\displaystyle -\frac {b c x \int \frac {x^3 \left (3 e^2 x^4+8 d e x^2+6 d^2\right )}{24 \sqrt {-c^2 x^2-1}}dx}{\sqrt {-c^2 x^2}}+\frac {1}{4} d^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \text {csch}^{-1}(c x)\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b c x \int \frac {x^3 \left (3 e^2 x^4+8 d e x^2+6 d^2\right )}{\sqrt {-c^2 x^2-1}}dx}{24 \sqrt {-c^2 x^2}}+\frac {1}{4} d^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \text {csch}^{-1}(c x)\right )\)

\(\Big \downarrow \) 1578

\(\displaystyle -\frac {b c x \int \frac {x^2 \left (3 e^2 x^4+8 d e x^2+6 d^2\right )}{\sqrt {-c^2 x^2-1}}dx^2}{48 \sqrt {-c^2 x^2}}+\frac {1}{4} d^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \text {csch}^{-1}(c x)\right )\)

\(\Big \downarrow \) 1195

\(\displaystyle -\frac {b c x \int \left (-\frac {3 e^2 \left (-c^2 x^2-1\right )^{5/2}}{c^6}+\frac {\left (8 c^2 d-9 e\right ) e \left (-c^2 x^2-1\right )^{3/2}}{c^6}+\frac {\left (-6 d^2 c^4+16 d e c^2-9 e^2\right ) \sqrt {-c^2 x^2-1}}{c^6}+\frac {-6 d^2 c^4+8 d e c^2-3 e^2}{c^6 \sqrt {-c^2 x^2-1}}\right )dx^2}{48 \sqrt {-c^2 x^2}}+\frac {1}{4} d^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \text {csch}^{-1}(c x)\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{4} d^2 x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b c x \left (-\frac {2 e \left (-c^2 x^2-1\right )^{5/2} \left (8 c^2 d-9 e\right )}{5 c^8}+\frac {6 e^2 \left (-c^2 x^2-1\right )^{7/2}}{7 c^8}+\frac {2 \left (-c^2 x^2-1\right )^{3/2} \left (6 c^4 d^2-16 c^2 d e+9 e^2\right )}{3 c^8}+\frac {2 \sqrt {-c^2 x^2-1} \left (6 c^4 d^2-8 c^2 d e+3 e^2\right )}{c^8}\right )}{48 \sqrt {-c^2 x^2}}\)

Input:

Int[x^3*(d + e*x^2)^2*(a + b*ArcCsch[c*x]),x]
 

Output:

-1/48*(b*c*x*((2*(6*c^4*d^2 - 8*c^2*d*e + 3*e^2)*Sqrt[-1 - c^2*x^2])/c^8 + 
 (2*(6*c^4*d^2 - 16*c^2*d*e + 9*e^2)*(-1 - c^2*x^2)^(3/2))/(3*c^8) - (2*(8 
*c^2*d - 9*e)*e*(-1 - c^2*x^2)^(5/2))/(5*c^8) + (6*e^2*(-1 - c^2*x^2)^(7/2 
))/(7*c^8)))/Sqrt[-(c^2*x^2)] + (d^2*x^4*(a + b*ArcCsch[c*x]))/4 + (d*e*x^ 
6*(a + b*ArcCsch[c*x]))/3 + (e^2*x^8*(a + b*ArcCsch[c*x]))/8
 

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 1195

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

rule 1578

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ 
)^4)^(p_.), x_Symbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a 
+ b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int 
egerQ[(m - 1)/2]
 

rule 2009

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

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 = 0.62 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.79

method	result	size
parts	\(a \left (\frac {1}{8} e^{2} x^{8}+\frac {1}{3} d e \,x^{6}+\frac {1}{4} d^{2} x^{4}\right )+\frac {b \left (\frac {c^{4} \operatorname {arccsch}\left (c x \right ) e^{2} x^{8}}{8}+\frac {c^{4} \operatorname {arccsch}\left (c x \right ) d e \,x^{6}}{3}+\frac {\operatorname {arccsch}\left (c x \right ) d^{2} c^{4} x^{4}}{4}+\frac {\left (c^{2} x^{2}+1\right ) \left (45 c^{6} e^{2} x^{6}+168 c^{6} d e \,x^{4}+210 c^{6} d^{2} x^{2}-54 c^{4} e^{2} x^{4}-224 c^{4} d e \,x^{2}-420 c^{4} d^{2}+72 c^{2} e^{2} x^{2}+448 c^{2} d e -144 e^{2}\right )}{2520 c^{5} \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x}\right )}{c^{4}}\)	\(198\)
derivativedivides	\(\frac {-\frac {a \left (\frac {c^{2} d \left (e \,c^{2} x^{2}+c^{2} d \right )^{3}}{3}-\frac {\left (e \,c^{2} x^{2}+c^{2} d \right )^{4}}{4}\right )}{2 c^{4} e^{2}}+\frac {b \left (-\frac {\operatorname {arccsch}\left (c x \right ) c^{8} d^{4}}{24 e^{2}}+\frac {\operatorname {arccsch}\left (c x \right ) c^{8} d^{2} x^{4}}{4}+\frac {e \,\operatorname {arccsch}\left (c x \right ) c^{8} d \,x^{6}}{3}+\frac {e^{2} \operatorname {arccsch}\left (c x \right ) c^{8} x^{8}}{8}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (105 c^{8} d^{4} \operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )+210 c^{6} d^{2} e^{2} x^{2} \sqrt {c^{2} x^{2}+1}+168 c^{6} d \,e^{3} x^{4} \sqrt {c^{2} x^{2}+1}+45 e^{4} c^{6} x^{6} \sqrt {c^{2} x^{2}+1}-420 c^{4} d^{2} e^{2} \sqrt {c^{2} x^{2}+1}-224 c^{4} d \,e^{3} x^{2} \sqrt {c^{2} x^{2}+1}-54 e^{4} c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+448 c^{2} d \,e^{3} \sqrt {c^{2} x^{2}+1}+72 e^{4} c^{2} x^{2} \sqrt {c^{2} x^{2}+1}-144 e^{4} \sqrt {c^{2} x^{2}+1}\right )}{2520 e^{2} \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{4}}}{c^{4}}\)	\(377\)
default	\(\frac {-\frac {a \left (\frac {c^{2} d \left (e \,c^{2} x^{2}+c^{2} d \right )^{3}}{3}-\frac {\left (e \,c^{2} x^{2}+c^{2} d \right )^{4}}{4}\right )}{2 c^{4} e^{2}}+\frac {b \left (-\frac {\operatorname {arccsch}\left (c x \right ) c^{8} d^{4}}{24 e^{2}}+\frac {\operatorname {arccsch}\left (c x \right ) c^{8} d^{2} x^{4}}{4}+\frac {e \,\operatorname {arccsch}\left (c x \right ) c^{8} d \,x^{6}}{3}+\frac {e^{2} \operatorname {arccsch}\left (c x \right ) c^{8} x^{8}}{8}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (105 c^{8} d^{4} \operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )+210 c^{6} d^{2} e^{2} x^{2} \sqrt {c^{2} x^{2}+1}+168 c^{6} d \,e^{3} x^{4} \sqrt {c^{2} x^{2}+1}+45 e^{4} c^{6} x^{6} \sqrt {c^{2} x^{2}+1}-420 c^{4} d^{2} e^{2} \sqrt {c^{2} x^{2}+1}-224 c^{4} d \,e^{3} x^{2} \sqrt {c^{2} x^{2}+1}-54 e^{4} c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+448 c^{2} d \,e^{3} \sqrt {c^{2} x^{2}+1}+72 e^{4} c^{2} x^{2} \sqrt {c^{2} x^{2}+1}-144 e^{4} \sqrt {c^{2} x^{2}+1}\right )}{2520 e^{2} \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{4}}}{c^{4}}\)	\(377\)

Input:

int(x^3*(e*x^2+d)^2*(a+b*arccsch(c*x)),x,method=_RETURNVERBOSE)
 

Output:

a*(1/8*e^2*x^8+1/3*d*e*x^6+1/4*d^2*x^4)+b/c^4*(1/8*c^4*arccsch(c*x)*e^2*x^ 
8+1/3*c^4*arccsch(c*x)*d*e*x^6+1/4*arccsch(c*x)*d^2*c^4*x^4+1/2520/c^5*(c^ 
2*x^2+1)*(45*c^6*e^2*x^6+168*c^6*d*e*x^4+210*c^6*d^2*x^2-54*c^4*e^2*x^4-22 
4*c^4*d*e*x^2-420*c^4*d^2+72*c^2*e^2*x^2+448*c^2*d*e-144*e^2)/((c^2*x^2+1) 
/c^2/x^2)^(1/2)/x)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.90 \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {315 \, a c^{7} e^{2} x^{8} + 840 \, a c^{7} d e x^{6} + 630 \, a c^{7} d^{2} x^{4} + 105 \, {\left (3 \, b c^{7} e^{2} x^{8} + 8 \, b c^{7} d e x^{6} + 6 \, b c^{7} d^{2} x^{4}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (45 \, b c^{6} e^{2} x^{7} + 6 \, {\left (28 \, b c^{6} d e - 9 \, b c^{4} e^{2}\right )} x^{5} + 2 \, {\left (105 \, b c^{6} d^{2} - 112 \, b c^{4} d e + 36 \, b c^{2} e^{2}\right )} x^{3} - 4 \, {\left (105 \, b c^{4} d^{2} - 112 \, b c^{2} d e + 36 \, b e^{2}\right )} x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{2520 \, c^{7}} \] Input:

integrate(x^3*(e*x^2+d)^2*(a+b*arccsch(c*x)),x, algorithm="fricas")
 

Output:

1/2520*(315*a*c^7*e^2*x^8 + 840*a*c^7*d*e*x^6 + 630*a*c^7*d^2*x^4 + 105*(3 
*b*c^7*e^2*x^8 + 8*b*c^7*d*e*x^6 + 6*b*c^7*d^2*x^4)*log((c*x*sqrt((c^2*x^2 
 + 1)/(c^2*x^2)) + 1)/(c*x)) + (45*b*c^6*e^2*x^7 + 6*(28*b*c^6*d*e - 9*b*c 
^4*e^2)*x^5 + 2*(105*b*c^6*d^2 - 112*b*c^4*d*e + 36*b*c^2*e^2)*x^3 - 4*(10 
5*b*c^4*d^2 - 112*b*c^2*d*e + 36*b*e^2)*x)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))/ 
c^7
                                                                                    
                                                                                    
 

Sympy [F]

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

integrate(x**3*(e*x**2+d)**2*(a+b*acsch(c*x)),x)
 

Output:

Integral(x**3*(a + b*acsch(c*x))*(d + e*x**2)**2, x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.98 \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {1}{8} \, a e^{2} x^{8} + \frac {1}{3} \, a d e x^{6} + \frac {1}{4} \, a d^{2} x^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \operatorname {arcsch}\left (c x\right ) + \frac {c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b d^{2} + \frac {1}{45} \, {\left (15 \, x^{6} \operatorname {arcsch}\left (c x\right ) + \frac {3 \, c^{4} x^{5} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 10 \, c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b d e + \frac {1}{280} \, {\left (35 \, x^{8} \operatorname {arcsch}\left (c x\right ) + \frac {5 \, c^{6} x^{7} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {7}{2}} - 21 \, c^{4} x^{5} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 35 \, c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 35 \, x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{7}}\right )} b e^{2} \] Input:

integrate(x^3*(e*x^2+d)^2*(a+b*arccsch(c*x)),x, algorithm="maxima")
 

Output:

1/8*a*e^2*x^8 + 1/3*a*d*e*x^6 + 1/4*a*d^2*x^4 + 1/12*(3*x^4*arccsch(c*x) + 
 (c^2*x^3*(1/(c^2*x^2) + 1)^(3/2) - 3*x*sqrt(1/(c^2*x^2) + 1))/c^3)*b*d^2 
+ 1/45*(15*x^6*arccsch(c*x) + (3*c^4*x^5*(1/(c^2*x^2) + 1)^(5/2) - 10*c^2* 
x^3*(1/(c^2*x^2) + 1)^(3/2) + 15*x*sqrt(1/(c^2*x^2) + 1))/c^5)*b*d*e + 1/2 
80*(35*x^8*arccsch(c*x) + (5*c^6*x^7*(1/(c^2*x^2) + 1)^(7/2) - 21*c^4*x^5* 
(1/(c^2*x^2) + 1)^(5/2) + 35*c^2*x^3*(1/(c^2*x^2) + 1)^(3/2) - 35*x*sqrt(1 
/(c^2*x^2) + 1))/c^7)*b*e^2
 

Giac [F]

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

integrate(x^3*(e*x^2+d)^2*(a+b*arccsch(c*x)),x, algorithm="giac")
 

Output:

integrate((e*x^2 + d)^2*(b*arccsch(c*x) + a)*x^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int x^3\,{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right ) \,d x \] Input:

int(x^3*(d + e*x^2)^2*(a + b*asinh(1/(c*x))),x)
 

Output:

int(x^3*(d + e*x^2)^2*(a + b*asinh(1/(c*x))), x)
 

Reduce [F]

\[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\left (\int \mathit {acsch} \left (c x \right ) x^{7}d x \right ) b \,e^{2}+2 \left (\int \mathit {acsch} \left (c x \right ) x^{5}d x \right ) b d e +\left (\int \mathit {acsch} \left (c x \right ) x^{3}d x \right ) b \,d^{2}+\frac {a \,d^{2} x^{4}}{4}+\frac {a d e \,x^{6}}{3}+\frac {a \,e^{2} x^{8}}{8} \] Input:

int(x^3*(e*x^2+d)^2*(a+b*acsch(c*x)),x)
 

Output:

(24*int(acsch(c*x)*x**7,x)*b*e**2 + 48*int(acsch(c*x)*x**5,x)*b*d*e + 24*i 
nt(acsch(c*x)*x**3,x)*b*d**2 + 6*a*d**2*x**4 + 8*a*d*e*x**6 + 3*a*e**2*x** 
8)/24