\(\int (d+e x^2)^2 (a+b \text {csch}^{-1}(c x)) \, dx\) [90]

Optimal result

Integrand size = 18, antiderivative size = 197 \[ \int \left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {b \left (40 c^2 d-9 e\right ) e x^2 \sqrt {-1-c^2 x^2}}{120 c^3 \sqrt {-c^2 x^2}}+\frac {b e^2 x^4 \sqrt {-1-c^2 x^2}}{20 c \sqrt {-c^2 x^2}}+d^2 x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b \left (120 c^4 d^2-40 c^2 d e+9 e^2\right ) x \arctan \left (\frac {c x}{\sqrt {-1-c^2 x^2}}\right )}{120 c^4 \sqrt {-c^2 x^2}} \] Output:

1/120*b*(40*c^2*d-9*e)*e*x^2*(-c^2*x^2-1)^(1/2)/c^3/(-c^2*x^2)^(1/2)+1/20* 
b*e^2*x^4*(-c^2*x^2-1)^(1/2)/c/(-c^2*x^2)^(1/2)+d^2*x*(a+b*arccsch(c*x))+2 
/3*d*e*x^3*(a+b*arccsch(c*x))+1/5*e^2*x^5*(a+b*arccsch(c*x))-1/120*b*(120* 
c^4*d^2-40*c^2*d*e+9*e^2)*x*arctan(c*x/(-c^2*x^2-1)^(1/2))/c^4/(-c^2*x^2)^ 
(1/2)
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.76 \[ \int \left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {c^2 x \left (8 a c^3 \left (15 d^2+10 d e x^2+3 e^2 x^4\right )+b e \sqrt {1+\frac {1}{c^2 x^2}} x \left (-9 e+c^2 \left (40 d+6 e x^2\right )\right )\right )+8 b c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right ) \text {csch}^{-1}(c x)+b \left (120 c^4 d^2-40 c^2 d e+9 e^2\right ) \log \left (\left (1+\sqrt {1+\frac {1}{c^2 x^2}}\right ) x\right )}{120 c^5} \] Input:

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

Output:

(c^2*x*(8*a*c^3*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) + b*e*Sqrt[1 + 1/(c^2*x^ 
2)]*x*(-9*e + c^2*(40*d + 6*e*x^2))) + 8*b*c^5*x*(15*d^2 + 10*d*e*x^2 + 3* 
e^2*x^4)*ArcCsch[c*x] + b*(120*c^4*d^2 - 40*c^2*d*e + 9*e^2)*Log[(1 + Sqrt 
[1 + 1/(c^2*x^2)])*x])/(120*c^5)
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.93, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {6846, 27, 1473, 25, 299, 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.

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

\(\Big \downarrow \) 6846

\(\displaystyle -\frac {b c x \int \frac {3 e^2 x^4+10 d e x^2+15 d^2}{15 \sqrt {-c^2 x^2-1}}dx}{\sqrt {-c^2 x^2}}+d^2 x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {csch}^{-1}(c x)\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b c x \int \frac {3 e^2 x^4+10 d e x^2+15 d^2}{\sqrt {-c^2 x^2-1}}dx}{15 \sqrt {-c^2 x^2}}+d^2 x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {csch}^{-1}(c x)\right )\)

\(\Big \downarrow \) 1473

\(\displaystyle -\frac {b c x \left (-\frac {\int -\frac {60 c^2 d^2+\left (40 c^2 d-9 e\right ) e x^2}{\sqrt {-c^2 x^2-1}}dx}{4 c^2}-\frac {3 e^2 x^3 \sqrt {-c^2 x^2-1}}{4 c^2}\right )}{15 \sqrt {-c^2 x^2}}+d^2 x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {csch}^{-1}(c x)\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {b c x \left (\frac {\int \frac {60 c^2 d^2+\left (40 c^2 d-9 e\right ) e x^2}{\sqrt {-c^2 x^2-1}}dx}{4 c^2}-\frac {3 e^2 x^3 \sqrt {-c^2 x^2-1}}{4 c^2}\right )}{15 \sqrt {-c^2 x^2}}+d^2 x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {csch}^{-1}(c x)\right )\)

\(\Big \downarrow \) 299

\(\displaystyle -\frac {b c x \left (\frac {\frac {\left (120 c^4 d^2-40 c^2 d e+9 e^2\right ) \int \frac {1}{\sqrt {-c^2 x^2-1}}dx}{2 c^2}-\frac {e x \sqrt {-c^2 x^2-1} \left (40 c^2 d-9 e\right )}{2 c^2}}{4 c^2}-\frac {3 e^2 x^3 \sqrt {-c^2 x^2-1}}{4 c^2}\right )}{15 \sqrt {-c^2 x^2}}+d^2 x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {csch}^{-1}(c x)\right )\)

\(\Big \downarrow \) 224

\(\displaystyle -\frac {b c x \left (\frac {\frac {\left (120 c^4 d^2-40 c^2 d e+9 e^2\right ) \int \frac {1}{\frac {c^2 x^2}{-c^2 x^2-1}+1}d\frac {x}{\sqrt {-c^2 x^2-1}}}{2 c^2}-\frac {e x \sqrt {-c^2 x^2-1} \left (40 c^2 d-9 e\right )}{2 c^2}}{4 c^2}-\frac {3 e^2 x^3 \sqrt {-c^2 x^2-1}}{4 c^2}\right )}{15 \sqrt {-c^2 x^2}}+d^2 x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {csch}^{-1}(c x)\right )\)

\(\Big \downarrow \) 216

\(\displaystyle d^2 x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {2}{3} d e x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b c x \left (\frac {\frac {\arctan \left (\frac {c x}{\sqrt {-c^2 x^2-1}}\right ) \left (120 c^4 d^2-40 c^2 d e+9 e^2\right )}{2 c^3}-\frac {e x \sqrt {-c^2 x^2-1} \left (40 c^2 d-9 e\right )}{2 c^2}}{4 c^2}-\frac {3 e^2 x^3 \sqrt {-c^2 x^2-1}}{4 c^2}\right )}{15 \sqrt {-c^2 x^2}}\)

Input:

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

Output:

d^2*x*(a + b*ArcCsch[c*x]) + (2*d*e*x^3*(a + b*ArcCsch[c*x]))/3 + (e^2*x^5 
*(a + b*ArcCsch[c*x]))/5 - (b*c*x*((-3*e^2*x^3*Sqrt[-1 - c^2*x^2])/(4*c^2) 
 + (-1/2*((40*c^2*d - 9*e)*e*x*Sqrt[-1 - c^2*x^2])/c^2 + ((120*c^4*d^2 - 4 
0*c^2*d*e + 9*e^2)*ArcTan[(c*x)/Sqrt[-1 - c^2*x^2]])/(2*c^3))/(4*c^2)))/(1 
5*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 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 0]
 

rule 1473

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), 
x_Symbol] :> Simp[c^p*x^(4*p - 1)*((d + e*x^2)^(q + 1)/(e*(4*p + 2*q + 1))) 
, x] + Simp[1/(e*(4*p + 2*q + 1))   Int[(d + e*x^2)^q*ExpandToSum[e*(4*p + 
2*q + 1)*(a + b*x^2 + c*x^4)^p - d*c^p*(4*p - 1)*x^(4*p - 2) - e*c^p*(4*p + 
 2*q + 1)*x^(4*p), x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] &&  !LtQ[q, -1]
 

rule 6846

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(p_.), x_Sym 
bol] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcCsch[c*x])   u 
, x] - Simp[b*c*(x/Sqrt[(-c^2)*x^2])   Int[SimplifyIntegrand[u/(x*Sqrt[-1 - 
 c^2*x^2]), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && (IGtQ[p, 0] || ILtQ 
[p + 1/2, 0])
 

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.99

method	result	size
parts	\(a \left (\frac {1}{5} e^{2} x^{5}+\frac {2}{3} d e \,x^{3}+d^{2} x \right )+\frac {b \left (\frac {c \,\operatorname {arccsch}\left (c x \right ) e^{2} x^{5}}{5}+\frac {2 c \,\operatorname {arccsch}\left (c x \right ) d e \,x^{3}}{3}+\operatorname {arccsch}\left (c x \right ) c x \,d^{2}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (120 d^{2} c^{4} \operatorname {arcsinh}\left (c x \right )+40 d \,c^{3} e x \sqrt {c^{2} x^{2}+1}+6 e^{2} c^{3} x^{3} \sqrt {c^{2} x^{2}+1}-40 d \,c^{2} e \,\operatorname {arcsinh}\left (c x \right )-9 e^{2} c x \sqrt {c^{2} x^{2}+1}+9 e^{2} \operatorname {arcsinh}\left (c x \right )\right )}{120 c^{5} x \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}}\right )}{c}\)	\(195\)
derivativedivides	\(\frac {\frac {a \left (d^{2} c^{5} x +\frac {2}{3} d \,c^{5} e \,x^{3}+\frac {1}{5} e^{2} c^{5} x^{5}\right )}{c^{4}}+\frac {b \left (\operatorname {arccsch}\left (c x \right ) d^{2} c^{5} x +\frac {2 \,\operatorname {arccsch}\left (c x \right ) d \,c^{5} e \,x^{3}}{3}+\frac {\operatorname {arccsch}\left (c x \right ) e^{2} c^{5} x^{5}}{5}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (120 d^{2} c^{4} \operatorname {arcsinh}\left (c x \right )+40 d \,c^{3} e x \sqrt {c^{2} x^{2}+1}+6 e^{2} c^{3} x^{3} \sqrt {c^{2} x^{2}+1}-40 d \,c^{2} e \,\operatorname {arcsinh}\left (c x \right )-9 e^{2} c x \sqrt {c^{2} x^{2}+1}+9 e^{2} \operatorname {arcsinh}\left (c x \right )\right )}{120 c x \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}}\right )}{c^{4}}}{c}\)	\(217\)
default	\(\frac {\frac {a \left (d^{2} c^{5} x +\frac {2}{3} d \,c^{5} e \,x^{3}+\frac {1}{5} e^{2} c^{5} x^{5}\right )}{c^{4}}+\frac {b \left (\operatorname {arccsch}\left (c x \right ) d^{2} c^{5} x +\frac {2 \,\operatorname {arccsch}\left (c x \right ) d \,c^{5} e \,x^{3}}{3}+\frac {\operatorname {arccsch}\left (c x \right ) e^{2} c^{5} x^{5}}{5}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (120 d^{2} c^{4} \operatorname {arcsinh}\left (c x \right )+40 d \,c^{3} e x \sqrt {c^{2} x^{2}+1}+6 e^{2} c^{3} x^{3} \sqrt {c^{2} x^{2}+1}-40 d \,c^{2} e \,\operatorname {arcsinh}\left (c x \right )-9 e^{2} c x \sqrt {c^{2} x^{2}+1}+9 e^{2} \operatorname {arcsinh}\left (c x \right )\right )}{120 c x \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}}\right )}{c^{4}}}{c}\)	\(217\)

Input:

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

Output:

a*(1/5*e^2*x^5+2/3*d*e*x^3+d^2*x)+b/c*(1/5*c*arccsch(c*x)*e^2*x^5+2/3*c*ar 
ccsch(c*x)*d*e*x^3+arccsch(c*x)*c*x*d^2+1/120/c^5*(c^2*x^2+1)^(1/2)*(120*d 
^2*c^4*arcsinh(c*x)+40*d*c^3*e*x*(c^2*x^2+1)^(1/2)+6*e^2*c^3*x^3*(c^2*x^2+ 
1)^(1/2)-40*d*c^2*e*arcsinh(c*x)-9*e^2*c*x*(c^2*x^2+1)^(1/2)+9*e^2*arcsinh 
(c*x))/x/((c^2*x^2+1)/c^2/x^2)^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 353 vs. \(2 (175) = 350\).

Time = 0.18 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.79 \[ \int \left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {24 \, a c^{5} e^{2} x^{5} + 80 \, a c^{5} d e x^{3} + 120 \, a c^{5} d^{2} x + 8 \, {\left (15 \, b c^{5} d^{2} + 10 \, b c^{5} d e + 3 \, b c^{5} e^{2}\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - {\left (120 \, b c^{4} d^{2} - 40 \, b c^{2} d e + 9 \, b e^{2}\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) - 8 \, {\left (15 \, b c^{5} d^{2} + 10 \, b c^{5} d e + 3 \, b c^{5} e^{2}\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + 8 \, {\left (3 \, b c^{5} e^{2} x^{5} + 10 \, b c^{5} d e x^{3} + 15 \, b c^{5} d^{2} x - 15 \, b c^{5} d^{2} - 10 \, b c^{5} d e - 3 \, b c^{5} e^{2}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (6 \, b c^{4} e^{2} x^{4} + {\left (40 \, b c^{4} d e - 9 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{120 \, c^{5}} \] Input:

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

Output:

1/120*(24*a*c^5*e^2*x^5 + 80*a*c^5*d*e*x^3 + 120*a*c^5*d^2*x + 8*(15*b*c^5 
*d^2 + 10*b*c^5*d*e + 3*b*c^5*e^2)*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - 
 c*x + 1) - (120*b*c^4*d^2 - 40*b*c^2*d*e + 9*b*e^2)*log(c*x*sqrt((c^2*x^2 
 + 1)/(c^2*x^2)) - c*x) - 8*(15*b*c^5*d^2 + 10*b*c^5*d*e + 3*b*c^5*e^2)*lo 
g(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x - 1) + 8*(3*b*c^5*e^2*x^5 + 10*b 
*c^5*d*e*x^3 + 15*b*c^5*d^2*x - 15*b*c^5*d^2 - 10*b*c^5*d*e - 3*b*c^5*e^2) 
*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + (6*b*c^4*e^2*x^4 + ( 
40*b*c^4*d*e - 9*b*c^2*e^2)*x^2)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))/c^5
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.46 \[ \int \left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {1}{5} \, a e^{2} x^{5} + \frac {2}{3} \, a d e x^{3} + \frac {1}{6} \, {\left (4 \, x^{3} \operatorname {arcsch}\left (c x\right ) + \frac {\frac {2 \, \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} + 1\right )} - c^{2}} - \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} + \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b d e + \frac {1}{80} \, {\left (16 \, x^{5} \operatorname {arcsch}\left (c x\right ) - \frac {\frac {2 \, {\left (3 \, {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{4} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{2} - 2 \, c^{4} {\left (\frac {1}{c^{2} x^{2}} + 1\right )} + c^{4}} - \frac {3 \, \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} + \frac {3 \, \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{4}}}{c}\right )} b e^{2} + a d^{2} x + \frac {{\left (2 \, c x \operatorname {arcsch}\left (c x\right ) + \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )\right )} b d^{2}}{2 \, c} \] Input:

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

Output:

1/5*a*e^2*x^5 + 2/3*a*d*e*x^3 + 1/6*(4*x^3*arccsch(c*x) + (2*sqrt(1/(c^2*x 
^2) + 1)/(c^2*(1/(c^2*x^2) + 1) - c^2) - log(sqrt(1/(c^2*x^2) + 1) + 1)/c^ 
2 + log(sqrt(1/(c^2*x^2) + 1) - 1)/c^2)/c)*b*d*e + 1/80*(16*x^5*arccsch(c* 
x) - (2*(3*(1/(c^2*x^2) + 1)^(3/2) - 5*sqrt(1/(c^2*x^2) + 1))/(c^4*(1/(c^2 
*x^2) + 1)^2 - 2*c^4*(1/(c^2*x^2) + 1) + c^4) - 3*log(sqrt(1/(c^2*x^2) + 1 
) + 1)/c^4 + 3*log(sqrt(1/(c^2*x^2) + 1) - 1)/c^4)/c)*b*e^2 + a*d^2*x + 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*d^2/c
 

Giac [F]

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

integrate((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)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(15*int(acsch(c*x),x)*b*d**2 + 15*int(acsch(c*x)*x**4,x)*b*e**2 + 30*int(a 
csch(c*x)*x**2,x)*b*d*e + 15*a*d**2*x + 10*a*d*e*x**3 + 3*a*e**2*x**5)/15