\(\int \frac {(d+e x^2)^2 (a+b \text {csch}^{-1}(c x))}{x^2} \, dx\) [91]

Optimal result

Integrand size = 21, antiderivative size = 170 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^2} \, dx=\frac {b c d^2 \sqrt {-1-c^2 x^2}}{\sqrt {-c^2 x^2}}+\frac {b e^2 x^2 \sqrt {-1-c^2 x^2}}{6 c \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x}+2 d e x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b \left (12 c^2 d-e\right ) e x \arctan \left (\frac {c x}{\sqrt {-1-c^2 x^2}}\right )}{6 c^2 \sqrt {-c^2 x^2}} \] Output:

b*c*d^2*(-c^2*x^2-1)^(1/2)/(-c^2*x^2)^(1/2)+1/6*b*e^2*x^2*(-c^2*x^2-1)^(1/ 
2)/c/(-c^2*x^2)^(1/2)-d^2*(a+b*arccsch(c*x))/x+2*d*e*x*(a+b*arccsch(c*x))+ 
1/3*e^2*x^3*(a+b*arccsch(c*x))-1/6*b*(12*c^2*d-e)*e*x*arctan(c*x/(-c^2*x^2 
-1)^(1/2))/c^2/(-c^2*x^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.79 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^2} \, dx=\frac {c^2 \left (b \sqrt {1+\frac {1}{c^2 x^2}} x \left (6 c^2 d^2+e^2 x^2\right )+2 a c \left (-3 d^2+6 d e x^2+e^2 x^4\right )\right )+2 b c^3 \left (-3 d^2+6 d e x^2+e^2 x^4\right ) \text {csch}^{-1}(c x)+b \left (12 c^2 d-e\right ) e x \log \left (\left (1+\sqrt {1+\frac {1}{c^2 x^2}}\right ) x\right )}{6 c^3 x} \] Input:

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

Output:

(c^2*(b*Sqrt[1 + 1/(c^2*x^2)]*x*(6*c^2*d^2 + e^2*x^2) + 2*a*c*(-3*d^2 + 6* 
d*e*x^2 + e^2*x^4)) + 2*b*c^3*(-3*d^2 + 6*d*e*x^2 + e^2*x^4)*ArcCsch[c*x] 
+ b*(12*c^2*d - e)*e*x*Log[(1 + Sqrt[1 + 1/(c^2*x^2)])*x])/(6*c^3*x)
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.89, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {6856, 27, 1588, 25, 27, 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.

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

\(\Big \downarrow \) 6856

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1588

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 299

\(\displaystyle \frac {b c x \left (\frac {3 d^2 \sqrt {-c^2 x^2-1}}{x}-e \left (\frac {1}{2} \left (12 d-\frac {e}{c^2}\right ) \int \frac {1}{\sqrt {-c^2 x^2-1}}dx-\frac {e x \sqrt {-c^2 x^2-1}}{2 c^2}\right )\right )}{3 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x}+2 d e x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \text {csch}^{-1}(c x)\right )\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {b c x \left (\frac {3 d^2 \sqrt {-c^2 x^2-1}}{x}-e \left (\frac {1}{2} \left (12 d-\frac {e}{c^2}\right ) \int \frac {1}{\frac {c^2 x^2}{-c^2 x^2-1}+1}d\frac {x}{\sqrt {-c^2 x^2-1}}-\frac {e x \sqrt {-c^2 x^2-1}}{2 c^2}\right )\right )}{3 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x}+2 d e x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \text {csch}^{-1}(c x)\right )\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x}+2 d e x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b c x \left (\frac {3 d^2 \sqrt {-c^2 x^2-1}}{x}-e \left (\frac {\arctan \left (\frac {c x}{\sqrt {-c^2 x^2-1}}\right ) \left (12 d-\frac {e}{c^2}\right )}{2 c}-\frac {e x \sqrt {-c^2 x^2-1}}{2 c^2}\right )\right )}{3 \sqrt {-c^2 x^2}}\)

Input:

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

Output:

-((d^2*(a + b*ArcCsch[c*x]))/x) + 2*d*e*x*(a + b*ArcCsch[c*x]) + (e^2*x^3* 
(a + b*ArcCsch[c*x]))/3 + (b*c*x*((3*d^2*Sqrt[-1 - c^2*x^2])/x - e*(-1/2*( 
e*x*Sqrt[-1 - c^2*x^2])/c^2 + ((12*d - e/c^2)*ArcTan[(c*x)/Sqrt[-1 - c^2*x 
^2]])/(2*c))))/(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 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 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 = 0.35 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.02

method	result	size
parts	\(a \left (\frac {e^{2} x^{3}}{3}+2 d e x -\frac {d^{2}}{x}\right )+b c \left (\frac {\operatorname {arccsch}\left (c x \right ) e^{2} x^{3}}{3 c}+\frac {2 \,\operatorname {arccsch}\left (c x \right ) d e x}{c}-\frac {\operatorname {arccsch}\left (c x \right ) d^{2}}{c x}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (6 c^{4} d^{2} \sqrt {c^{2} x^{2}+1}+12 c^{3} d e \,\operatorname {arcsinh}\left (c x \right ) x +e^{2} c^{2} x^{2} \sqrt {c^{2} x^{2}+1}-\operatorname {arcsinh}\left (c x \right ) e^{2} c x \right )}{6 c^{6} x^{2} \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}}\right )\)	\(173\)
derivativedivides	\(c \left (\frac {a \left (2 c^{3} d e x +\frac {e^{2} c^{3} x^{3}}{3}-\frac {c^{3} d^{2}}{x}\right )}{c^{4}}+\frac {b \left (2 \,\operatorname {arccsch}\left (c x \right ) c^{3} d e x +\frac {e^{2} \operatorname {arccsch}\left (c x \right ) c^{3} x^{3}}{3}-\frac {\operatorname {arccsch}\left (c x \right ) c^{3} d^{2}}{x}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (6 c^{4} d^{2} \sqrt {c^{2} x^{2}+1}+12 c^{3} d e \,\operatorname {arcsinh}\left (c x \right ) x +e^{2} c^{2} x^{2} \sqrt {c^{2} x^{2}+1}-\operatorname {arcsinh}\left (c x \right ) e^{2} c x \right )}{6 c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}}\right )}{c^{4}}\right )\)	\(189\)
default	\(c \left (\frac {a \left (2 c^{3} d e x +\frac {e^{2} c^{3} x^{3}}{3}-\frac {c^{3} d^{2}}{x}\right )}{c^{4}}+\frac {b \left (2 \,\operatorname {arccsch}\left (c x \right ) c^{3} d e x +\frac {e^{2} \operatorname {arccsch}\left (c x \right ) c^{3} x^{3}}{3}-\frac {\operatorname {arccsch}\left (c x \right ) c^{3} d^{2}}{x}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (6 c^{4} d^{2} \sqrt {c^{2} x^{2}+1}+12 c^{3} d e \,\operatorname {arcsinh}\left (c x \right ) x +e^{2} c^{2} x^{2} \sqrt {c^{2} x^{2}+1}-\operatorname {arcsinh}\left (c x \right ) e^{2} c x \right )}{6 c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}}\right )}{c^{4}}\right )\)	\(189\)

Input:

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

Output:

a*(1/3*e^2*x^3+2*d*e*x-d^2/x)+b*c*(1/3/c*arccsch(c*x)*e^2*x^3+2/c*arccsch( 
c*x)*d*e*x-arccsch(c*x)*d^2/c/x+1/6/c^6*(c^2*x^2+1)^(1/2)*(6*c^4*d^2*(c^2* 
x^2+1)^(1/2)+12*c^3*d*e*arcsinh(c*x)*x+e^2*c^2*x^2*(c^2*x^2+1)^(1/2)-arcsi 
nh(c*x)*e^2*c*x)/x^2/((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. 347 vs. \(2 (152) = 304\).

Time = 0.17 (sec) , antiderivative size = 347, normalized size of antiderivative = 2.04 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^2} \, dx=\frac {2 \, a c^{3} e^{2} x^{4} + 6 \, b c^{4} d^{2} x + 12 \, a c^{3} d e x^{2} - 6 \, a c^{3} d^{2} - 2 \, {\left (3 \, b c^{3} d^{2} - 6 \, b c^{3} d e - b c^{3} e^{2}\right )} x \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - {\left (12 \, b c^{2} d e - b e^{2}\right )} x \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) + 2 \, {\left (3 \, b c^{3} d^{2} - 6 \, b c^{3} d e - b c^{3} e^{2}\right )} x \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + 2 \, {\left (b c^{3} e^{2} x^{4} + 6 \, b c^{3} d e x^{2} - 3 \, b c^{3} d^{2} + {\left (3 \, b c^{3} d^{2} - 6 \, b c^{3} d e - b c^{3} e^{2}\right )} x\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (6 \, b c^{4} d^{2} x + b c^{2} e^{2} x^{3}\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{6 \, c^{3} x} \] Input:

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

Output:

1/6*(2*a*c^3*e^2*x^4 + 6*b*c^4*d^2*x + 12*a*c^3*d*e*x^2 - 6*a*c^3*d^2 - 2* 
(3*b*c^3*d^2 - 6*b*c^3*d*e - b*c^3*e^2)*x*log(c*x*sqrt((c^2*x^2 + 1)/(c^2* 
x^2)) - c*x + 1) - (12*b*c^2*d*e - b*e^2)*x*log(c*x*sqrt((c^2*x^2 + 1)/(c^ 
2*x^2)) - c*x) + 2*(3*b*c^3*d^2 - 6*b*c^3*d*e - b*c^3*e^2)*x*log(c*x*sqrt( 
(c^2*x^2 + 1)/(c^2*x^2)) - c*x - 1) + 2*(b*c^3*e^2*x^4 + 6*b*c^3*d*e*x^2 - 
 3*b*c^3*d^2 + (3*b*c^3*d^2 - 6*b*c^3*d*e - b*c^3*e^2)*x)*log((c*x*sqrt((c 
^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + (6*b*c^4*d^2*x + b*c^2*e^2*x^3)*sqrt( 
(c^2*x^2 + 1)/(c^2*x^2)))/(c^3*x)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.12 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^2} \, dx=\frac {1}{3} \, a e^{2} x^{3} + {\left (c \sqrt {\frac {1}{c^{2} x^{2}} + 1} - \frac {\operatorname {arcsch}\left (c x\right )}{x}\right )} b d^{2} + \frac {1}{12} \, {\left (4 \, x^{3} \operatorname {arcsch}\left (c x\right ) + \frac {\frac {2 \, \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} + 1\right )} - c^{2}} - \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} + \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b e^{2} + 2 \, a d e x + \frac {{\left (2 \, c x \operatorname {arcsch}\left (c x\right ) + \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )\right )} b d e}{c} - \frac {a d^{2}}{x} \] Input:

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

Output:

1/3*a*e^2*x^3 + (c*sqrt(1/(c^2*x^2) + 1) - arccsch(c*x)/x)*b*d^2 + 1/12*(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*e^2 + 2*a*d*e*x + (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*e/c - a*d^2/x
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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