\(\int (e+f x)^2 (a+b \text {csch}^{-1}(c+d x))^2 \, dx\) [8]

Optimal result

Integrand size = 20, antiderivative size = 351 \[ \int (e+f x)^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\frac {b^2 f^2 x}{3 d^2}+\frac {2 b f (d e-c f) (c+d x) \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{d^3}+\frac {b f^2 (c+d x)^2 \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{3 d^3}-\frac {(d e-c f)^3 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac {(e+f x)^3 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{3 f}-\frac {2 b f^2 \left (a+b \text {csch}^{-1}(c+d x)\right ) \text {arctanh}\left (e^{\text {csch}^{-1}(c+d x)}\right )}{3 d^3}+\frac {4 b (d e-c f)^2 \left (a+b \text {csch}^{-1}(c+d x)\right ) \text {arctanh}\left (e^{\text {csch}^{-1}(c+d x)}\right )}{d^3}+\frac {2 b^2 f (d e-c f) \log (c+d x)}{d^3}-\frac {b^2 f^2 \operatorname {PolyLog}\left (2,-e^{\text {csch}^{-1}(c+d x)}\right )}{3 d^3}+\frac {2 b^2 (d e-c f)^2 \operatorname {PolyLog}\left (2,-e^{\text {csch}^{-1}(c+d x)}\right )}{d^3}+\frac {b^2 f^2 \operatorname {PolyLog}\left (2,e^{\text {csch}^{-1}(c+d x)}\right )}{3 d^3}-\frac {2 b^2 (d e-c f)^2 \operatorname {PolyLog}\left (2,e^{\text {csch}^{-1}(c+d x)}\right )}{d^3} \] Output:

1/3*b^2*f^2*x/d^2+2*b*f*(-c*f+d*e)*(d*x+c)*(1+1/(d*x+c)^2)^(1/2)*(a+b*arcc 
sch(d*x+c))/d^3+1/3*b*f^2*(d*x+c)^2*(1+1/(d*x+c)^2)^(1/2)*(a+b*arccsch(d*x 
+c))/d^3-1/3*(-c*f+d*e)^3*(a+b*arccsch(d*x+c))^2/d^3/f+1/3*(f*x+e)^3*(a+b* 
arccsch(d*x+c))^2/f-2/3*b*f^2*(a+b*arccsch(d*x+c))*arctanh(1/(d*x+c)+(1+1/ 
(d*x+c)^2)^(1/2))/d^3+4*b*(-c*f+d*e)^2*(a+b*arccsch(d*x+c))*arctanh(1/(d*x 
+c)+(1+1/(d*x+c)^2)^(1/2))/d^3+2*b^2*f*(-c*f+d*e)*ln(d*x+c)/d^3-1/3*b^2*f^ 
2*polylog(2,-1/(d*x+c)-(1+1/(d*x+c)^2)^(1/2))/d^3+2*b^2*(-c*f+d*e)^2*polyl 
og(2,-1/(d*x+c)-(1+1/(d*x+c)^2)^(1/2))/d^3+1/3*b^2*f^2*polylog(2,1/(d*x+c) 
+(1+1/(d*x+c)^2)^(1/2))/d^3-2*b^2*(-c*f+d*e)^2*polylog(2,1/(d*x+c)+(1+1/(d 
*x+c)^2)^(1/2))/d^3
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 7.87 (sec) , antiderivative size = 864, normalized size of antiderivative = 2.46 \[ \int (e+f x)^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

a^2*e^2*x + a^2*e*f*x^2 + (a^2*f^2*x^3)/3 + (a*b*(2*x*(3*e^2 + 3*e*f*x + f 
^2*x^2)*ArcCsch[c + d*x] + (-(f*(c + d*x)*Sqrt[(1 + c^2 + 2*c*d*x + d^2*x^ 
2)/(c + d*x)^2]*(5*c*f - d*(6*e + f*x))) + 2*c*(3*d^2*e^2 - 3*c*d*e*f + c^ 
2*f^2)*ArcSinh[(c + d*x)^(-1)] + (6*d^2*e^2 - 12*c*d*e*f + (-1 + 6*c^2)*f^ 
2)*Log[(c + d*x)*(1 + Sqrt[(1 + c^2 + 2*c*d*x + d^2*x^2)/(c + d*x)^2])])/d 
^3))/3 - (b^2*e^2*(-(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*PolyLo 
g[2, -E^(-ArcCsch[c + d*x])] - 2*PolyLog[2, E^(-ArcCsch[c + d*x])]))/d - ( 
2*b^2*d*e*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[Arc 
Csch[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*f^2*(2*(-2 + 12*c*ArcCsch[c + d*x] + Arc 
Csch[c + d*x]^2 - 6*c^2*ArcCsch[c + d*x]^2)*Coth[ArcCsch[c + d*x]/2] + 2*A 
rcCsch[c + d*x]*(-1 + 3*c*ArcCsch[c + d*x])*Csch[ArcCsch[c + d*x]/2]^2 - ( 
ArcCsch[c + d*x]^2*Csch[ArcCsch[c + d*x]/2]^4)/(2*(c + d*x)) - 48*c*Log[(c 
 + d*x)^(-1)] + 8*(-1 + 6*c^2)*(ArcCsch[c + d*x]*(Log[1 - E^(-ArcCsch[c + 
d*x])] - Log[1 + E^(-ArcCsch[c + d*x])]) + PolyLog[2, -E^(-ArcCsch[c + ...
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 336, normalized size of antiderivative = 0.96, 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.

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

\(\Big \downarrow \) 6876

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

\(\Big \downarrow \) 5992

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4678

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((-1/3*((d*e - c*f + f*(c + d*x))^3*(a + b*ArcCsch[c + d*x])^2)/f + (2*b* 
(-1/2*(b*f^3*(c + d*x)) - 3*f^2*(d*e - c*f)*(c + d*x)*Sqrt[1 + (c + d*x)^( 
-2)]*(a + b*ArcCsch[c + d*x]) - (f^3*(c + d*x)^2*Sqrt[1 + (c + d*x)^(-2)]* 
(a + b*ArcCsch[c + d*x]))/2 + ((d*e - c*f)^3*(a + b*ArcCsch[c + d*x])^2)/( 
2*b) + f^3*(a + b*ArcCsch[c + d*x])*ArcTanh[E^ArcCsch[c + d*x]] - 6*f*(d*e 
 - c*f)^2*(a + b*ArcCsch[c + d*x])*ArcTanh[E^ArcCsch[c + d*x]] + 3*b*f^2*( 
d*e - c*f)*Log[(c + d*x)^(-1)] + (b*f^3*PolyLog[2, -E^ArcCsch[c + d*x]])/2 
 - 3*b*f*(d*e - c*f)^2*PolyLog[2, -E^ArcCsch[c + d*x]] - (b*f^3*PolyLog[2, 
 E^ArcCsch[c + d*x]])/2 + 3*b*f*(d*e - c*f)^2*PolyLog[2, E^ArcCsch[c + d*x 
]]))/(3*f))/d^3)
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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