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

Optimal result

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

[Out]

1/3*b^2*f^2*x/d^2-1/3*(-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))*arc
tanh(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*polylog(2,-1/(d*x+c)-(1+1/(d*x+c)^2)^(1/2))/d^3+1/3*b^2*f^2*polylo
g(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+2*b
*f*(-c*f+d*e)*(d*x+c)*(a+b*arccsch(d*x+c))*(1+1/(d*x+c)^2)^(1/2)/d^3+1/3*b*f^2*(d*x+c)^2*(a+b*arccsch(d*x+c))*
(1+1/(d*x+c)^2)^(1/2)/d^3

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6457, 5577, 4275, 4267, 2317, 2438, 4269, 3556, 4270} \[ \int (e+f x)^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\frac {4 b (d e-c f)^2 \text {arctanh}\left (e^{\text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )}{d^3}-\frac {2 b f^2 \text {arctanh}\left (e^{\text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )}{3 d^3}-\frac {(d e-c f)^3 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac {2 b f (c+d x) \sqrt {\frac {1}{(c+d x)^2}+1} (d e-c f) \left (a+b \text {csch}^{-1}(c+d x)\right )}{d^3}+\frac {b f^2 (c+d x)^2 \sqrt {\frac {1}{(c+d x)^2}+1} \left (a+b \text {csch}^{-1}(c+d x)\right )}{3 d^3}+\frac {(e+f x)^3 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{3 f}+\frac {2 b^2 (d e-c f)^2 \operatorname {PolyLog}\left (2,-e^{\text {csch}^{-1}(c+d x)}\right )}{d^3}-\frac {2 b^2 (d e-c f)^2 \operatorname {PolyLog}\left (2,e^{\text {csch}^{-1}(c+d x)}\right )}{d^3}+\frac {2 b^2 f (d e-c f) \log (c+d x)}{d^3}-\frac {b^2 f^2 \operatorname {PolyLog}\left (2,-e^{\text {csch}^{-1}(c+d x)}\right )}{3 d^3}+\frac {b^2 f^2 \operatorname {PolyLog}\left (2,e^{\text {csch}^{-1}(c+d x)}\right )}{3 d^3}+\frac {b^2 f^2 x}{3 d^2} \]

[In]

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

[Out]

(b^2*f^2*x)/(3*d^2) + (2*b*f*(d*e - c*f)*(c + d*x)*Sqrt[1 + (c + d*x)^(-2)]*(a + b*ArcCsch[c + d*x]))/d^3 + (b
*f^2*(c + d*x)^2*Sqrt[1 + (c + d*x)^(-2)]*(a + b*ArcCsch[c + d*x]))/(3*d^3) - ((d*e - c*f)^3*(a + b*ArcCsch[c
+ d*x])^2)/(3*d^3*f) + ((e + f*x)^3*(a + b*ArcCsch[c + d*x])^2)/(3*f) - (2*b*f^2*(a + b*ArcCsch[c + d*x])*ArcT
anh[E^ArcCsch[c + d*x]])/(3*d^3) + (4*b*(d*e - c*f)^2*(a + b*ArcCsch[c + d*x])*ArcTanh[E^ArcCsch[c + d*x]])/d^
3 + (2*b^2*f*(d*e - c*f)*Log[c + d*x])/d^3 - (b^2*f^2*PolyLog[2, -E^ArcCsch[c + d*x]])/(3*d^3) + (2*b^2*(d*e -
c*f)^2*PolyLog[2, -E^ArcCsch[c + d*x]])/d^3 + (b^2*f^2*PolyLog[2, E^ArcCsch[c + d*x]])/(3*d^3) - (2*b^2*(d*e
- c*f)^2*PolyLog[2, E^ArcCsch[c + d*x]])/d^3

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 4269

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Cot[e + f*x]/f), x
] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 4270

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(-b^2)*(c + d*x)*Cot[e + f*x]
*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (Dist[b^2*((n - 2)/(n - 1)), Int[(c + d*x)*(b*Csc[e + f*x])^(n -
2), x], x] - Simp[b^2*d*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x]) /; FreeQ[{b, c, d, e, f}, x] &&
GtQ[n, 1] && NeQ[n, 2]

Rule 4275

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 5577

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] + Dist[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 6457

Int[((a_.) + ArcCsch[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[-(d^(m + 1)
)^(-1), Subst[Int[(a + b*x)^p*Csch[x]*Coth[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int (a+b x)^2 \coth (x) \text {csch}(x) (d e-c f+f \text {csch}(x))^2 \, dx,x,\text {csch}^{-1}(c+d x)\right )}{d^3} \\ & = \frac {(e+f x)^3 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{3 f}-\frac {(2 b) \text {Subst}\left (\int (a+b x) (d e-c f+f \text {csch}(x))^3 \, dx,x,\text {csch}^{-1}(c+d x)\right )}{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) \text {Subst}\left (\int \left (d^3 e^3 \left (1-\frac {c f \left (3 d^2 e^2-3 c d e f+c^2 f^2\right )}{d^3 e^3}\right ) (a+b x)+3 d^2 e^2 f \left (1+\frac {c f (-2 d e+c f)}{d^2 e^2}\right ) (a+b x) \text {csch}(x)+3 d e f^2 \left (1-\frac {c f}{d e}\right ) (a+b x) \text {csch}^2(x)+f^3 (a+b x) \text {csch}^3(x)\right ) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{3 d^3 f} \\ & = -\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 {\left (2 b f^2\right ) \text {Subst}\left (\int (a+b x) \text {csch}^3(x) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{3 d^3}-\frac {(2 b f (d e-c f)) \text {Subst}\left (\int (a+b x) \text {csch}^2(x) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{d^3}-\frac {\left (2 b (d e-c f)^2\right ) \text {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{d^3} \\ & = \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 {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 {\left (b f^2\right ) \text {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{3 d^3}-\frac {\left (2 b^2 f (d e-c f)\right ) \text {Subst}\left (\int \coth (x) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{d^3}+\frac {\left (2 b^2 (d e-c f)^2\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{d^3}-\frac {\left (2 b^2 (d e-c f)^2\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{d^3} \\ & = \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 {\left (b^2 f^2\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{3 d^3}+\frac {\left (b^2 f^2\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{3 d^3}+\frac {\left (2 b^2 (d e-c f)^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {csch}^{-1}(c+d x)}\right )}{d^3}-\frac {\left (2 b^2 (d e-c f)^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {csch}^{-1}(c+d x)}\right )}{d^3} \\ & = \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 {2 b^2 (d e-c f)^2 \operatorname {PolyLog}\left (2,-e^{\text {csch}^{-1}(c+d x)}\right )}{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 {\left (b^2 f^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {csch}^{-1}(c+d x)}\right )}{3 d^3}+\frac {\left (b^2 f^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {csch}^{-1}(c+d x)}\right )}{3 d^3} \\ & = \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} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 7.97 (sec) , antiderivative size = 893, normalized size of antiderivative = 2.54 \[ \int (e+f x)^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=a^2 e^2 x+a^2 e f x^2+\frac {1}{3} a^2 f^2 x^3+\frac {1}{3} a b \left (2 x \left (3 e^2+3 e f x+f^2 x^2\right ) \text {csch}^{-1}(c+d x)+\frac {-f (c+d x) \sqrt {\frac {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 \left (3 d^2 e^2-3 c d e f+c^2 f^2\right ) \text {arcsinh}\left (\frac {1}{c+d x}\right )+\left (6 d^2 e^2-12 c d e f+\left (-1+6 c^2\right ) f^2\right ) \log \left ((c+d x) \left (1+\sqrt {\frac {1+c^2+2 c d x+d^2 x^2}{(c+d x)^2}}\right )\right )}{d^3}\right )-\frac {b^2 e^2 \left (-\text {csch}^{-1}(c+d x) \left ((c+d x) \text {csch}^{-1}(c+d x)-2 \log \left (1-e^{-\text {csch}^{-1}(c+d x)}\right )+2 \log \left (1+e^{-\text {csch}^{-1}(c+d x)}\right )\right )+2 \operatorname {PolyLog}\left (2,-e^{-\text {csch}^{-1}(c+d x)}\right )-2 \operatorname {PolyLog}\left (2,e^{-\text {csch}^{-1}(c+d x)}\right )\right )}{d}-\frac {2 b^2 d e f x \left (\frac {(c+d x) \sqrt {1+\frac {1}{(c+d x)^2}} \text {csch}^{-1}(c+d x)}{d^2}+\frac {(c+d x)^2 \text {csch}^{-1}(c+d x)^2}{2 d^2}-\frac {c \text {csch}^{-1}(c+d x)^2 \coth \left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )}{2 d^2}-\frac {\log \left (\frac {1}{c+d x}\right )}{d^2}-\frac {2 i c \left (i \text {csch}^{-1}(c+d x) \left (\log \left (1-e^{-\text {csch}^{-1}(c+d x)}\right )-\log \left (1+e^{-\text {csch}^{-1}(c+d x)}\right )\right )+i \left (\operatorname {PolyLog}\left (2,-e^{-\text {csch}^{-1}(c+d x)}\right )-\operatorname {PolyLog}\left (2,e^{-\text {csch}^{-1}(c+d x)}\right )\right )\right )}{d^2}+\frac {c \text {csch}^{-1}(c+d x)^2 \tanh \left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )}{2 d^2}\right )}{(c+d x) \left (-1+\frac {c}{c+d x}\right )}-\frac {b^2 f^2 \left (2 \left (-2+12 c \text {csch}^{-1}(c+d x)+\text {csch}^{-1}(c+d x)^2-6 c^2 \text {csch}^{-1}(c+d x)^2\right ) \coth \left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )+2 \text {csch}^{-1}(c+d x) \left (-1+3 c \text {csch}^{-1}(c+d x)\right ) \text {csch}^2\left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )-\frac {\text {csch}^{-1}(c+d x)^2 \text {csch}^4\left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )}{2 (c+d x)}-48 c \left (\log \left (\frac {1}{(c+d x) \sqrt {1+\frac {1}{(c+d x)^2}}}\right )+\log \left (\sqrt {1+\frac {1}{(c+d x)^2}}\right )\right )+8 \left (-1+6 c^2\right ) \left (\text {csch}^{-1}(c+d x) \left (\log \left (1-e^{-\text {csch}^{-1}(c+d x)}\right )-\log \left (1+e^{-\text {csch}^{-1}(c+d x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-\text {csch}^{-1}(c+d x)}\right )-\operatorname {PolyLog}\left (2,e^{-\text {csch}^{-1}(c+d x)}\right )\right )-2 \text {csch}^{-1}(c+d x) \left (1+3 c \text {csch}^{-1}(c+d x)\right ) \text {sech}^2\left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )-8 (c+d x)^3 \text {csch}^{-1}(c+d x)^2 \sinh ^4\left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )+2 \left (2+12 c \text {csch}^{-1}(c+d x)-\text {csch}^{-1}(c+d x)^2+6 c^2 \text {csch}^{-1}(c+d x)^2\right ) \tanh \left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )\right )}{24 d^3} \]

[In]

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

[Out]

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*PolyLog[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[ArcCsch[c + d*x]/2])/(2*d^2) - Lo
g[(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] + ArcCsch[c + d*x]^2 - 6*c^2*ArcCsch[c + d*x]^2)*Coth[ArcCsch[c + d*x]/2] + 2*ArcCsch[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[1/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)])] + Log[Sqrt[1 + (c + d*x)^(-2)]]) + 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 + d*x])
] - PolyLog[2, E^(-ArcCsch[c + d*x])]) - 2*ArcCsch[c + d*x]*(1 + 3*c*ArcCsch[c + d*x])*Sech[ArcCsch[c + d*x]/2
]^2 - 8*(c + d*x)^3*ArcCsch[c + d*x]^2*Sinh[ArcCsch[c + d*x]/2]^4 + 2*(2 + 12*c*ArcCsch[c + d*x] - ArcCsch[c +
d*x]^2 + 6*c^2*ArcCsch[c + d*x]^2)*Tanh[ArcCsch[c + d*x]/2]))/(24*d^3)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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)*arccsch(d*x + c), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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 + c^2 + 1)*((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 - 2*(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)))/(d^2*x^2 + 2*c*d*x + c^2 + (d^2*x^2
+ 2*c*d*x + c^2 + 1)^(3/2) + 1), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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