\(\int (e+f x)^3 (a+b \text {csch}^{-1}(c+d x))^2 \, dx\) [7]

Optimal result

Integrand size = 20, antiderivative size = 501 \[ \int (e+f x)^3 \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\frac {b^2 f^2 (d e-c f) x}{d^3}+\frac {b^2 f^3 (c+d x)^2}{12 d^4}-\frac {b f^3 (c+d x) \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{3 d^4}+\frac {3 b f (d e-c f)^2 (c+d x) \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{d^4}+\frac {b f^2 (d e-c f) (c+d x)^2 \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{d^4}+\frac {b f^3 (c+d x)^3 \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{6 d^4}-\frac {(d e-c f)^4 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{4 d^4 f}+\frac {(e+f x)^4 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{4 f}-\frac {2 b f^2 (d e-c f) \left (a+b \text {csch}^{-1}(c+d x)\right ) \text {arctanh}\left (e^{\text {csch}^{-1}(c+d x)}\right )}{d^4}+\frac {4 b (d e-c f)^3 \left (a+b \text {csch}^{-1}(c+d x)\right ) \text {arctanh}\left (e^{\text {csch}^{-1}(c+d x)}\right )}{d^4}-\frac {b^2 f^3 \log (c+d x)}{3 d^4}+\frac {3 b^2 f (d e-c f)^2 \log (c+d x)}{d^4}-\frac {b^2 f^2 (d e-c f) \operatorname {PolyLog}\left (2,-e^{\text {csch}^{-1}(c+d x)}\right )}{d^4}+\frac {2 b^2 (d e-c f)^3 \operatorname {PolyLog}\left (2,-e^{\text {csch}^{-1}(c+d x)}\right )}{d^4}+\frac {b^2 f^2 (d e-c f) \operatorname {PolyLog}\left (2,e^{\text {csch}^{-1}(c+d x)}\right )}{d^4}-\frac {2 b^2 (d e-c f)^3 \operatorname {PolyLog}\left (2,e^{\text {csch}^{-1}(c+d x)}\right )}{d^4} \] Output:

b^2*f^2*(-c*f+d*e)*x/d^3+1/12*b^2*f^3*(d*x+c)^2/d^4-1/3*b*f^3*(d*x+c)*(1+1 
/(d*x+c)^2)^(1/2)*(a+b*arccsch(d*x+c))/d^4+3*b*f*(-c*f+d*e)^2*(d*x+c)*(1+1 
/(d*x+c)^2)^(1/2)*(a+b*arccsch(d*x+c))/d^4+b*f^2*(-c*f+d*e)*(d*x+c)^2*(1+1 
/(d*x+c)^2)^(1/2)*(a+b*arccsch(d*x+c))/d^4+1/6*b*f^3*(d*x+c)^3*(1+1/(d*x+c 
)^2)^(1/2)*(a+b*arccsch(d*x+c))/d^4-1/4*(-c*f+d*e)^4*(a+b*arccsch(d*x+c))^ 
2/d^4/f+1/4*(f*x+e)^4*(a+b*arccsch(d*x+c))^2/f-2*b*f^2*(-c*f+d*e)*(a+b*arc 
csch(d*x+c))*arctanh(1/(d*x+c)+(1+1/(d*x+c)^2)^(1/2))/d^4+4*b*(-c*f+d*e)^3 
*(a+b*arccsch(d*x+c))*arctanh(1/(d*x+c)+(1+1/(d*x+c)^2)^(1/2))/d^4-1/3*b^2 
*f^3*ln(d*x+c)/d^4+3*b^2*f*(-c*f+d*e)^2*ln(d*x+c)/d^4-b^2*f^2*(-c*f+d*e)*p 
olylog(2,-1/(d*x+c)-(1+1/(d*x+c)^2)^(1/2))/d^4+2*b^2*(-c*f+d*e)^3*polylog( 
2,-1/(d*x+c)-(1+1/(d*x+c)^2)^(1/2))/d^4+b^2*f^2*(-c*f+d*e)*polylog(2,1/(d* 
x+c)+(1+1/(d*x+c)^2)^(1/2))/d^4-2*b^2*(-c*f+d*e)^3*polylog(2,1/(d*x+c)+(1+ 
1/(d*x+c)^2)^(1/2))/d^4
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 9.98 (sec) , antiderivative size = 1429, normalized size of antiderivative = 2.85 \[ \int (e+f x)^3 \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

a^2*e^3*x + (3*a^2*e^2*f*x^2)/2 + a^2*e*f^2*x^3 + (a^2*f^3*x^4)/4 + (a*b*( 
3*x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3)*ArcCsch[c + d*x] + (f*(c + 
 d*x)*Sqrt[(1 + c^2 + 2*c*d*x + d^2*x^2)/(c + d*x)^2]*((-2 + 13*c^2)*f^2 - 
 2*c*d*f*(15*e + 2*f*x) + d^2*(18*e^2 + 6*e*f*x + f^2*x^2)) - 3*c*(-4*d^3* 
e^3 + 6*c*d^2*e^2*f - 4*c^2*d*e*f^2 + c^3*f^3)*ArcSinh[(c + d*x)^(-1)] + 6 
*(2*d^3*e^3 - 6*c*d^2*e^2*f + (-1 + 6*c^2)*d*e*f^2 + c*(1 - 2*c^2)*f^3)*Lo 
g[(c + d*x)*(1 + Sqrt[(1 + c^2 + 2*c*d*x + d^2*x^2)/(c + d*x)^2])])/d^4))/ 
6 - (b^2*e^3*(-(ArcCsch[c + d*x]*((c + d*x)*ArcCsch[c + d*x] - 2*Log[1 - E 
^(-ArcCsch[c + d*x])] + 2*Log[1 + E^(-ArcCsch[c + d*x])])) + 2*PolyLog[2, 
-E^(-ArcCsch[c + d*x])] - 2*PolyLog[2, E^(-ArcCsch[c + d*x])]))/d - (3*b^2 
*d*e^2*f*x*(((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]*ArcCsch[c + d*x])/d^2 + (( 
c + d*x)^2*ArcCsch[c + d*x]^2)/(2*d^2) - (c*ArcCsch[c + d*x]^2*Coth[ArcCsc 
h[c + d*x]/2])/(2*d^2) - Log[(c + d*x)^(-1)]/d^2 - ((2*I)*c*(I*ArcCsch[c + 
 d*x]*(Log[1 - E^(-ArcCsch[c + d*x])] - Log[1 + E^(-ArcCsch[c + d*x])]) + 
I*(PolyLog[2, -E^(-ArcCsch[c + d*x])] - PolyLog[2, E^(-ArcCsch[c + d*x])]) 
))/d^2 + (c*ArcCsch[c + d*x]^2*Tanh[ArcCsch[c + d*x]/2])/(2*d^2)))/((c + d 
*x)*(-1 + c/(c + d*x))) - (b^2*e*f^2*(2*(-2 + 12*c*ArcCsch[c + d*x] + ArcC 
sch[c + d*x]^2 - 6*c^2*ArcCsch[c + d*x]^2)*Coth[ArcCsch[c + d*x]/2] + 2*Ar 
cCsch[c + d*x]*(-1 + 3*c*ArcCsch[c + d*x])*Csch[ArcCsch[c + d*x]/2]^2 - (A 
rcCsch[c + d*x]^2*Csch[ArcCsch[c + d*x]/2]^4)/(2*(c + d*x)) - 48*c*Log[...
 

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 476, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6876, 5992, 3042, 4678, 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.

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

\(\Big \downarrow \) 6876

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

\(\Big \downarrow \) 5992

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4678

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\frac {b \left (4 f^3 (d e-c f) \text {arctanh}\left (e^{\text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )-8 f (d e-c f)^3 \text {arctanh}\left (e^{\text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )-2 f^3 (c+d x)^2 \sqrt {\frac {1}{(c+d x)^2}+1} (d e-c f) \left (a+b \text {csch}^{-1}(c+d x)\right )-6 f^2 (c+d x) \sqrt {\frac {1}{(c+d x)^2}+1} (d e-c f)^2 \left (a+b \text {csch}^{-1}(c+d x)\right )+\frac {(d e-c f)^4 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 b}-\frac {1}{3} f^4 (c+d x)^3 \sqrt {\frac {1}{(c+d x)^2}+1} \left (a+b \text {csch}^{-1}(c+d x)\right )+\frac {2}{3} f^4 (c+d x) \sqrt {\frac {1}{(c+d x)^2}+1} \left (a+b \text {csch}^{-1}(c+d x)\right )+2 b f^3 (d e-c f) \operatorname {PolyLog}\left (2,-e^{\text {csch}^{-1}(c+d x)}\right )-2 b f^3 (d e-c f) \operatorname {PolyLog}\left (2,e^{\text {csch}^{-1}(c+d x)}\right )-2 b f^3 (c+d x) (d e-c f)+6 b f^2 (d e-c f)^2 \log \left (\frac {1}{c+d x}\right )-4 b f (d e-c f)^3 \operatorname {PolyLog}\left (2,-e^{\text {csch}^{-1}(c+d x)}\right )+4 b f (d e-c f)^3 \operatorname {PolyLog}\left (2,e^{\text {csch}^{-1}(c+d x)}\right )-\frac {1}{6} b f^4 (c+d x)^2-\frac {2}{3} b f^4 \log \left (\frac {1}{c+d x}\right )\right )}{2 f}-\frac {(f (c+d x)-c f+d e)^4 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{4 f}}{d^4}\)

Input:

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

Output:

-((-1/4*((d*e - c*f + f*(c + d*x))^4*(a + b*ArcCsch[c + d*x])^2)/f + (b*(- 
2*b*f^3*(d*e - c*f)*(c + d*x) - (b*f^4*(c + d*x)^2)/6 + (2*f^4*(c + d*x)*S 
qrt[1 + (c + d*x)^(-2)]*(a + b*ArcCsch[c + d*x]))/3 - 6*f^2*(d*e - c*f)^2* 
(c + d*x)*Sqrt[1 + (c + d*x)^(-2)]*(a + b*ArcCsch[c + d*x]) - 2*f^3*(d*e - 
 c*f)*(c + d*x)^2*Sqrt[1 + (c + d*x)^(-2)]*(a + b*ArcCsch[c + d*x]) - (f^4 
*(c + d*x)^3*Sqrt[1 + (c + d*x)^(-2)]*(a + b*ArcCsch[c + d*x]))/3 + ((d*e 
- c*f)^4*(a + b*ArcCsch[c + d*x])^2)/(2*b) + 4*f^3*(d*e - c*f)*(a + b*ArcC 
sch[c + d*x])*ArcTanh[E^ArcCsch[c + d*x]] - 8*f*(d*e - c*f)^3*(a + b*ArcCs 
ch[c + d*x])*ArcTanh[E^ArcCsch[c + d*x]] - (2*b*f^4*Log[(c + d*x)^(-1)])/3 
 + 6*b*f^2*(d*e - c*f)^2*Log[(c + d*x)^(-1)] + 2*b*f^3*(d*e - c*f)*PolyLog 
[2, -E^ArcCsch[c + d*x]] - 4*b*f*(d*e - c*f)^3*PolyLog[2, -E^ArcCsch[c + d 
*x]] - 2*b*f^3*(d*e - c*f)*PolyLog[2, E^ArcCsch[c + d*x]] + 4*b*f*(d*e - c 
*f)^3*PolyLog[2, E^ArcCsch[c + d*x]]))/(2*f))/d^4)
 

Defintions of rubi rules used

rule 2009

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4678

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

rule 5992

Int[Coth[(c_.) + (d_.)*(x_)]*Csch[(c_.) + (d_.)*(x_)]*(Csch[(c_.) + (d_.)*( 
x_)]*(b_.) + (a_))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(e 
 + f*x)^m)*((a + b*Csch[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[f*(m/(b 
*d*(n + 1)))   Int[(e + f*x)^(m - 1)*(a + b*Csch[c + d*x])^(n + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
 

rule 6876

Int[((a_.) + ArcCsch[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( 
m_.), x_Symbol] :> Simp[-(d^(m + 1))^(-1)   Subst[Int[(a + b*x)^p*Csch[x]*C 
oth[x]*(d*e - c*f + f*Csch[x])^m, x], x, ArcCsch[c + d*x]], x] /; FreeQ[{a, 
 b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]
 

Maple [F]

\[\int \left (f x +e \right )^{3} \left (a +b \,\operatorname {arccsch}\left (d x +c \right )\right )^{2}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/4*a^2*f^3*x^4 + a^2*e*f^2*x^3 + 3/2*a^2*e^2*f*x^2 + a^2*e^3*x + (2*(d*x 
+ c)*arccsch(d*x + c) + log(sqrt(1/(d*x + c)^2 + 1) + 1) - log(sqrt(1/(d*x 
 + c)^2 + 1) - 1))*a*b*e^3/d + 1/4*(b^2*f^3*x^4 + 4*b^2*e*f^2*x^3 + 6*b^2* 
e^2*f*x^2 + 4*b^2*e^3*x)*log(sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1) + 1)^2 - in 
tegrate(-1/2*(2*(b^2*d^2*f^3*x^5 + b^2*c^2*e^3 + b^2*e^3 + (3*b^2*d^2*e*f^ 
2 + 2*b^2*c*d*f^3)*x^4 + (6*b^2*c*d*e*f^2 + b^2*c^2*f^3 + (3*d^2*e^2*f + f 
^3)*b^2)*x^3 + (6*b^2*c*d*e^2*f + 3*b^2*c^2*e*f^2 + (d^2*e^3 + 3*e*f^2)*b^ 
2)*x^2 + (2*b^2*c*d*e^3 + 3*b^2*c^2*e^2*f + 3*b^2*e^2*f)*x)*log(d*x + c)^2 
 - 4*(a*b*d^2*f^3*x^5 + (3*a*b*d^2*e*f^2 + 2*a*b*c*d*f^3)*x^4 + (6*a*b*c*d 
*e*f^2 + a*b*c^2*f^3 + (3*d^2*e^2*f + f^3)*a*b)*x^3 + 3*(2*a*b*c*d*e^2*f + 
 a*b*c^2*e*f^2 + a*b*e*f^2)*x^2 + 3*(a*b*c^2*e^2*f + a*b*e^2*f)*x)*log(d*x 
 + c) + (4*a*b*d^2*f^3*x^5 + 4*(3*a*b*d^2*e*f^2 + 2*a*b*c*d*f^3)*x^4 + 4*( 
6*a*b*c*d*e*f^2 + a*b*c^2*f^3 + (3*d^2*e^2*f + f^3)*a*b)*x^3 + 12*(2*a*b*c 
*d*e^2*f + a*b*c^2*e*f^2 + a*b*e*f^2)*x^2 + 12*(a*b*c^2*e^2*f + a*b*e^2*f) 
*x - 4*(b^2*d^2*f^3*x^5 + b^2*c^2*e^3 + b^2*e^3 + (3*b^2*d^2*e*f^2 + 2*b^2 
*c*d*f^3)*x^4 + (6*b^2*c*d*e*f^2 + b^2*c^2*f^3 + (3*d^2*e^2*f + f^3)*b^2)* 
x^3 + (6*b^2*c*d*e^2*f + 3*b^2*c^2*e*f^2 + (d^2*e^3 + 3*e*f^2)*b^2)*x^2 + 
(2*b^2*c*d*e^3 + 3*b^2*c^2*e^2*f + 3*b^2*e^2*f)*x)*log(d*x + c) + ((4*a*b* 
d^2*f^3 - b^2*d^2*f^3)*x^5 + (12*a*b*d^2*e*f^2 - 4*b^2*d^2*e*f^2 + (8*a*b* 
d*f^3 - b^2*d*f^3)*c)*x^4 - 2*(3*b^2*d^2*e^2*f - 2*a*b*c^2*f^3 - 2*(3*d...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(8*int(acsch(c + d*x),x)*a*b*e**3 + 4*int(acsch(c + d*x)**2,x)*b**2*e**3 + 
 8*int(acsch(c + d*x)*x**3,x)*a*b*f**3 + 24*int(acsch(c + d*x)*x**2,x)*a*b 
*e*f**2 + 24*int(acsch(c + d*x)*x,x)*a*b*e**2*f + 4*int(acsch(c + d*x)**2* 
x**3,x)*b**2*f**3 + 12*int(acsch(c + d*x)**2*x**2,x)*b**2*e*f**2 + 12*int( 
acsch(c + d*x)**2*x,x)*b**2*e**2*f + 4*a**2*e**3*x + 6*a**2*e**2*f*x**2 + 
4*a**2*e*f**2*x**3 + a**2*f**3*x**4)/4