3.218 \(\int \frac{(e+f x)^2 \text{csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx\)

Optimal. Leaf size=368 \[ \frac{3 f (e+f x) \text{PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}-\frac{3 f (e+f x) \text{PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac{4 i f^2 \text{PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac{i f^2 \text{PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac{3 f^2 \text{PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac{3 f^2 \text{PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac{4 i f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac{2 i f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}-\frac{f (e+f x) \text{csch}(c+d x)}{a d^2}-\frac{f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac{3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{i (e+f x)^2 \tanh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right )}{a d}+\frac{i (e+f x)^2 \coth (c+d x)}{a d}-\frac{(e+f x)^2 \coth (c+d x) \text{csch}(c+d x)}{2 a d}+\frac{2 i (e+f x)^2}{a d} \]

[Out]

((2*I)*(e + f*x)^2)/(a*d) + (3*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a*d) - (f^2*ArcTanh[Cosh[c + d*x]])/(a*d^3)
+ (I*(e + f*x)^2*Coth[c + d*x])/(a*d) - (f*(e + f*x)*Csch[c + d*x])/(a*d^2) - ((e + f*x)^2*Coth[c + d*x]*Csch[
c + d*x])/(2*a*d) - ((4*I)*f*(e + f*x)*Log[1 + I*E^(c + d*x)])/(a*d^2) - ((2*I)*f*(e + f*x)*Log[1 - E^(2*(c +
d*x))])/(a*d^2) + (3*f*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a*d^2) - ((4*I)*f^2*PolyLog[2, (-I)*E^(c + d*x)])/
(a*d^3) - (3*f*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a*d^2) - (I*f^2*PolyLog[2, E^(2*(c + d*x))])/(a*d^3) - (3*f
^2*PolyLog[3, -E^(c + d*x)])/(a*d^3) + (3*f^2*PolyLog[3, E^(c + d*x)])/(a*d^3) + (I*(e + f*x)^2*Tanh[c/2 + (I/
4)*Pi + (d*x)/2])/(a*d)

________________________________________________________________________________________

Rubi [A]  time = 0.855921, antiderivative size = 368, normalized size of antiderivative = 1., number of steps used = 30, number of rules used = 13, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.419, Rules used = {5575, 4186, 3770, 4182, 2531, 2282, 6589, 4184, 3716, 2190, 2279, 2391, 3318} \[ \frac{3 f (e+f x) \text{PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}-\frac{3 f (e+f x) \text{PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac{4 i f^2 \text{PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac{i f^2 \text{PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac{3 f^2 \text{PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac{3 f^2 \text{PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac{4 i f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac{2 i f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}-\frac{f (e+f x) \text{csch}(c+d x)}{a d^2}-\frac{f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac{3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{i (e+f x)^2 \tanh \left (\frac{c}{2}+\frac{d x}{2}+\frac{i \pi }{4}\right )}{a d}+\frac{i (e+f x)^2 \coth (c+d x)}{a d}-\frac{(e+f x)^2 \coth (c+d x) \text{csch}(c+d x)}{2 a d}+\frac{2 i (e+f x)^2}{a d} \]

Antiderivative was successfully verified.

[In]

Int[((e + f*x)^2*Csch[c + d*x]^3)/(a + I*a*Sinh[c + d*x]),x]

[Out]

((2*I)*(e + f*x)^2)/(a*d) + (3*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a*d) - (f^2*ArcTanh[Cosh[c + d*x]])/(a*d^3)
+ (I*(e + f*x)^2*Coth[c + d*x])/(a*d) - (f*(e + f*x)*Csch[c + d*x])/(a*d^2) - ((e + f*x)^2*Coth[c + d*x]*Csch[
c + d*x])/(2*a*d) - ((4*I)*f*(e + f*x)*Log[1 + I*E^(c + d*x)])/(a*d^2) - ((2*I)*f*(e + f*x)*Log[1 - E^(2*(c +
d*x))])/(a*d^2) + (3*f*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a*d^2) - ((4*I)*f^2*PolyLog[2, (-I)*E^(c + d*x)])/
(a*d^3) - (3*f*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a*d^2) - (I*f^2*PolyLog[2, E^(2*(c + d*x))])/(a*d^3) - (3*f
^2*PolyLog[3, -E^(c + d*x)])/(a*d^3) + (3*f^2*PolyLog[3, E^(c + d*x)])/(a*d^3) + (I*(e + f*x)^2*Tanh[c/2 + (I/
4)*Pi + (d*x)/2])/(a*d)

Rule 5575

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

Rule 4186

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

Rule 3770

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

Rule 4182

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 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 4184

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 3716

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c
+ d*x)^(m + 1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

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

Rule 2279

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 2391

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 3318

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(2*a)^n, Int[(c
+ d*x)^m*Sin[(1*(e + (Pi*a)/(2*b)))/2 + (f*x)/2]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2
- b^2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])

Rubi steps

\begin{align*} \int \frac{(e+f x)^2 \text{csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\left (i \int \frac{(e+f x)^2 \text{csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx\right )+\frac{\int (e+f x)^2 \text{csch}^3(c+d x) \, dx}{a}\\ &=-\frac{f (e+f x) \text{csch}(c+d x)}{a d^2}-\frac{(e+f x)^2 \coth (c+d x) \text{csch}(c+d x)}{2 a d}-\frac{i \int (e+f x)^2 \text{csch}^2(c+d x) \, dx}{a}-\frac{\int (e+f x)^2 \text{csch}(c+d x) \, dx}{2 a}+\frac{f^2 \int \text{csch}(c+d x) \, dx}{a d^2}-\int \frac{(e+f x)^2 \text{csch}(c+d x)}{a+i a \sinh (c+d x)} \, dx\\ &=\frac{(e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac{i (e+f x)^2 \coth (c+d x)}{a d}-\frac{f (e+f x) \text{csch}(c+d x)}{a d^2}-\frac{(e+f x)^2 \coth (c+d x) \text{csch}(c+d x)}{2 a d}+i \int \frac{(e+f x)^2}{a+i a \sinh (c+d x)} \, dx-\frac{\int (e+f x)^2 \text{csch}(c+d x) \, dx}{a}-\frac{(2 i f) \int (e+f x) \coth (c+d x) \, dx}{a d}+\frac{f \int (e+f x) \log \left (1-e^{c+d x}\right ) \, dx}{a d}-\frac{f \int (e+f x) \log \left (1+e^{c+d x}\right ) \, dx}{a d}\\ &=\frac{i (e+f x)^2}{a d}+\frac{3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac{i (e+f x)^2 \coth (c+d x)}{a d}-\frac{f (e+f x) \text{csch}(c+d x)}{a d^2}-\frac{(e+f x)^2 \coth (c+d x) \text{csch}(c+d x)}{2 a d}+\frac{f (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac{f (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac{i \int (e+f x)^2 \csc ^2\left (\frac{1}{2} \left (i c+\frac{\pi }{2}\right )+\frac{i d x}{2}\right ) \, dx}{2 a}+\frac{(4 i f) \int \frac{e^{2 (c+d x)} (e+f x)}{1-e^{2 (c+d x)}} \, dx}{a d}+\frac{(2 f) \int (e+f x) \log \left (1-e^{c+d x}\right ) \, dx}{a d}-\frac{(2 f) \int (e+f x) \log \left (1+e^{c+d x}\right ) \, dx}{a d}-\frac{f^2 \int \text{Li}_2\left (-e^{c+d x}\right ) \, dx}{a d^2}+\frac{f^2 \int \text{Li}_2\left (e^{c+d x}\right ) \, dx}{a d^2}\\ &=\frac{i (e+f x)^2}{a d}+\frac{3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac{i (e+f x)^2 \coth (c+d x)}{a d}-\frac{f (e+f x) \text{csch}(c+d x)}{a d^2}-\frac{(e+f x)^2 \coth (c+d x) \text{csch}(c+d x)}{2 a d}-\frac{2 i f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac{3 f (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac{3 f (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac{i (e+f x)^2 \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{a d}-\frac{(2 i f) \int (e+f x) \coth \left (\frac{c}{2}-\frac{i \pi }{4}+\frac{d x}{2}\right ) \, dx}{a d}-\frac{f^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}+\frac{f^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}+\frac{\left (2 i f^2\right ) \int \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a d^2}-\frac{\left (2 f^2\right ) \int \text{Li}_2\left (-e^{c+d x}\right ) \, dx}{a d^2}+\frac{\left (2 f^2\right ) \int \text{Li}_2\left (e^{c+d x}\right ) \, dx}{a d^2}\\ &=\frac{2 i (e+f x)^2}{a d}+\frac{3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac{i (e+f x)^2 \coth (c+d x)}{a d}-\frac{f (e+f x) \text{csch}(c+d x)}{a d^2}-\frac{(e+f x)^2 \coth (c+d x) \text{csch}(c+d x)}{2 a d}-\frac{2 i f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac{3 f (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac{3 f (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac{f^2 \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac{f^2 \text{Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac{i (e+f x)^2 \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{a d}+\frac{(4 f) \int \frac{e^{2 \left (\frac{c}{2}+\frac{d x}{2}\right )} (e+f x)}{1+i e^{2 \left (\frac{c}{2}+\frac{d x}{2}\right )}} \, dx}{a d}+\frac{\left (i f^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{a d^3}-\frac{\left (2 f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}+\frac{\left (2 f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}\\ &=\frac{2 i (e+f x)^2}{a d}+\frac{3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac{i (e+f x)^2 \coth (c+d x)}{a d}-\frac{f (e+f x) \text{csch}(c+d x)}{a d^2}-\frac{(e+f x)^2 \coth (c+d x) \text{csch}(c+d x)}{2 a d}-\frac{4 i f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac{2 i f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac{3 f (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac{3 f (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac{i f^2 \text{Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}-\frac{3 f^2 \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac{3 f^2 \text{Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac{i (e+f x)^2 \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{a d}+\frac{\left (4 i f^2\right ) \int \log \left (1+i e^{2 \left (\frac{c}{2}+\frac{d x}{2}\right )}\right ) \, dx}{a d^2}\\ &=\frac{2 i (e+f x)^2}{a d}+\frac{3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac{i (e+f x)^2 \coth (c+d x)}{a d}-\frac{f (e+f x) \text{csch}(c+d x)}{a d^2}-\frac{(e+f x)^2 \coth (c+d x) \text{csch}(c+d x)}{2 a d}-\frac{4 i f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac{2 i f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac{3 f (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac{3 f (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac{i f^2 \text{Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}-\frac{3 f^2 \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac{3 f^2 \text{Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac{i (e+f x)^2 \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{a d}+\frac{\left (4 i f^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{2 \left (\frac{c}{2}+\frac{d x}{2}\right )}\right )}{a d^3}\\ &=\frac{2 i (e+f x)^2}{a d}+\frac{3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac{i (e+f x)^2 \coth (c+d x)}{a d}-\frac{f (e+f x) \text{csch}(c+d x)}{a d^2}-\frac{(e+f x)^2 \coth (c+d x) \text{csch}(c+d x)}{2 a d}-\frac{4 i f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac{2 i f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac{3 f (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac{4 i f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a d^3}-\frac{3 f (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac{i f^2 \text{Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}-\frac{3 f^2 \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac{3 f^2 \text{Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac{i (e+f x)^2 \tanh \left (\frac{c}{2}+\frac{i \pi }{4}+\frac{d x}{2}\right )}{a d}\\ \end{align*}

Mathematica [B]  time = 17.0436, size = 1378, normalized size = 3.74 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Integrate[((e + f*x)^2*Csch[c + d*x]^3)/(a + I*a*Sinh[c + d*x]),x]

[Out]

(-2*d*(e + f*x)*(d*(e + f*x) + 2*(1 + I*E^c)*f*Log[1 - I*E^(-c - d*x)]) + 4*(1 + I*E^c)*f^2*PolyLog[2, I*E^(-c
- d*x)])/(a*d^3*(-I + E^c)) + ((2*I)*d^2*(e + f*x)^2*(-1 + Coth[c]) + (3*d^2*e^2 + (4*I)*d*e*f - 2*f^2)*(d*x
- Log[1 - Cosh[c + d*x] - Sinh[c + d*x]]) + 2*d*(3*d*e - (2*I)*f)*f*x*Log[1 + Cosh[c + d*x] - Sinh[c + d*x]] +
3*d^2*f^2*x^2*Log[1 + Cosh[c + d*x] - Sinh[c + d*x]] - 2*d*(3*d*e + (2*I)*f)*f*x*Log[1 - Cosh[c + d*x] + Sinh
[c + d*x]] - 3*d^2*f^2*x^2*Log[1 - Cosh[c + d*x] + Sinh[c + d*x]] - (3*d^2*e^2 - (4*I)*d*e*f - 2*f^2)*(d*x - L
og[1 + Cosh[c + d*x] + Sinh[c + d*x]]) + 2*(3*d*e + (2*I)*f)*f*PolyLog[2, Cosh[c + d*x] - Sinh[c + d*x]] - 2*(
3*d*e - (2*I)*f)*f*PolyLog[2, -Cosh[c + d*x] + Sinh[c + d*x]] + 6*f^2*(d*x*PolyLog[2, Cosh[c + d*x] - Sinh[c +
d*x]] + PolyLog[3, Cosh[c + d*x] - Sinh[c + d*x]]) - 6*f^2*(d*x*PolyLog[2, -Cosh[c + d*x] + Sinh[c + d*x]] +
PolyLog[3, -Cosh[c + d*x] + Sinh[c + d*x]]))/(2*a*d^3) + (Csch[c]*Csch[c + d*x]^2*(2*e*f*Cosh[(d*x)/2] + 2*f^2
*x*Cosh[(d*x)/2] + 2*e*f*Cosh[(3*d*x)/2] + 2*f^2*x*Cosh[(3*d*x)/2] + (5*I)*d*e^2*Cosh[c - (d*x)/2] + (10*I)*d*
e*f*x*Cosh[c - (d*x)/2] + (5*I)*d*f^2*x^2*Cosh[c - (d*x)/2] - I*d*e^2*Cosh[c + (d*x)/2] - (2*I)*d*e*f*x*Cosh[c
+ (d*x)/2] - I*d*f^2*x^2*Cosh[c + (d*x)/2] - 2*e*f*Cosh[2*c + (d*x)/2] - 2*f^2*x*Cosh[2*c + (d*x)/2] + I*d*e^
2*Cosh[c + (3*d*x)/2] + (2*I)*d*e*f*x*Cosh[c + (3*d*x)/2] + I*d*f^2*x^2*Cosh[c + (3*d*x)/2] - 2*e*f*Cosh[2*c +
(3*d*x)/2] - 2*f^2*x*Cosh[2*c + (3*d*x)/2] - (3*I)*d*e^2*Cosh[3*c + (3*d*x)/2] - (6*I)*d*e*f*x*Cosh[3*c + (3*
d*x)/2] - (3*I)*d*f^2*x^2*Cosh[3*c + (3*d*x)/2] - (4*I)*d*e^2*Cosh[c + (5*d*x)/2] - (8*I)*d*e*f*x*Cosh[c + (5*
d*x)/2] - (4*I)*d*f^2*x^2*Cosh[c + (5*d*x)/2] + (2*I)*d*e^2*Cosh[3*c + (5*d*x)/2] + (4*I)*d*e*f*x*Cosh[3*c + (
5*d*x)/2] + (2*I)*d*f^2*x^2*Cosh[3*c + (5*d*x)/2] - d*e^2*Sinh[(d*x)/2] - 2*d*e*f*x*Sinh[(d*x)/2] - d*f^2*x^2*
Sinh[(d*x)/2] - d*e^2*Sinh[(3*d*x)/2] - 2*d*e*f*x*Sinh[(3*d*x)/2] - d*f^2*x^2*Sinh[(3*d*x)/2] + (2*I)*e*f*Sinh
[c - (d*x)/2] + (2*I)*f^2*x*Sinh[c - (d*x)/2] + (2*I)*e*f*Sinh[c + (d*x)/2] + (2*I)*f^2*x*Sinh[c + (d*x)/2] -
3*d*e^2*Sinh[2*c + (d*x)/2] - 6*d*e*f*x*Sinh[2*c + (d*x)/2] - 3*d*f^2*x^2*Sinh[2*c + (d*x)/2] + (2*I)*e*f*Sinh
[c + (3*d*x)/2] + (2*I)*f^2*x*Sinh[c + (3*d*x)/2] - d*e^2*Sinh[2*c + (3*d*x)/2] - 2*d*e*f*x*Sinh[2*c + (3*d*x)
/2] - d*f^2*x^2*Sinh[2*c + (3*d*x)/2] - (2*I)*e*f*Sinh[3*c + (3*d*x)/2] - (2*I)*f^2*x*Sinh[3*c + (3*d*x)/2] +
2*d*e^2*Sinh[2*c + (5*d*x)/2] + 4*d*e*f*x*Sinh[2*c + (5*d*x)/2] + 2*d*f^2*x^2*Sinh[2*c + (5*d*x)/2]))/(8*a*d^2
*(Cosh[c/2] + I*Sinh[c/2])*(Cosh[c/2 + (d*x)/2] + I*Sinh[c/2 + (d*x)/2]))

________________________________________________________________________________________

Maple [B]  time = 0.192, size = 1107, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)^2*csch(d*x+c)^3/(a+I*a*sinh(d*x+c)),x)

[Out]

-3/a/d^2*ln(1-exp(d*x+c))*c*e*f+3/a/d*ln(exp(d*x+c)+1)*e*f*x-3/a/d*ln(1-exp(d*x+c))*e*f*x+3/a/d^2*e*f*c*ln(exp
(d*x+c)-1)-4*I/a/d^2*ln(exp(d*x+c)-I)*e*f-4*I/a/d^2*f^2*ln(1+I*exp(d*x+c))*x-4*I/a/d^3*f^2*ln(1+I*exp(d*x+c))*
c+4*I/a/d^3*f^2*c*ln(exp(d*x+c)-I)-4*I*f^2*polylog(2,-I*exp(d*x+c))/a/d^3+3*f^2*polylog(3,exp(d*x+c))/a/d^3-3*
f^2*polylog(3,-exp(d*x+c))/a/d^3-1/a/d^3*f^2*ln(exp(d*x+c)+1)+1/a/d^3*f^2*ln(exp(d*x+c)-1)+4*I/a/d^3*c^2*f^2-2
*I/a/d^3*f^2*polylog(2,-exp(d*x+c))-2*I/a/d^3*f^2*polylog(2,exp(d*x+c))+4*I/a/d*f^2*x^2-3/2/a/d*e^2*ln(exp(d*x
+c)-1)+3/2/a/d*e^2*ln(exp(d*x+c)+1)-2*I/a/d^2*f*e*ln(exp(d*x+c)+1)+2*I/a/d^3*f^2*c*ln(exp(d*x+c)-1)-8*I/a/d^3*
f^2*c*ln(exp(d*x+c))-2*I/a/d^3*ln(1-exp(d*x+c))*c*f^2+8*I/a/d^2*c*f^2*x-2*I/a/d^2*ln(exp(d*x+c)+1)*f^2*x-2*I/a
/d^2*ln(1-exp(d*x+c))*f^2*x+8*I/a/d^2*e*f*ln(exp(d*x+c))-2*I/a/d^2*f*e*ln(exp(d*x+c)-1)+3/a/d^2*f^2*polylog(2,
-exp(d*x+c))*x+3/2/a/d^3*f^2*c^2*ln(1-exp(d*x+c))-3/2/a/d*f^2*ln(1-exp(d*x+c))*x^2-3/2/a/d^3*f^2*c^2*ln(exp(d*
x+c)-1)+3/a/d^2*e*f*polylog(2,-exp(d*x+c))-3/a/d^2*e*f*polylog(2,exp(d*x+c))-3/a/d^2*f^2*polylog(2,exp(d*x+c))
*x+3/2/a/d*f^2*ln(exp(d*x+c)+1)*x^2-(6*d*e*f*x*exp(4*d*x+4*c)+4*d*e^2-5*d*e^2*exp(2*d*x+2*c)+2*I*d*e*f*x*exp(d
*x+c)+2*I*exp(d*x+c)*e*f-10*d*e*f*x*exp(2*d*x+2*c)-3*I*d*e^2*exp(3*d*x+3*c)+I*d*f^2*x^2*exp(d*x+c)+8*d*e*f*x+4
*d*f^2*x^2+3*d*e^2*exp(4*d*x+4*c)+2*f^2*x*exp(4*d*x+4*c)+2*e*f*exp(4*d*x+4*c)-2*f^2*x*exp(2*d*x+2*c)-2*e*f*exp
(2*d*x+2*c)+3*d*f^2*x^2*exp(4*d*x+4*c)+I*d*e^2*exp(d*x+c)-3*I*d*f^2*x^2*exp(3*d*x+3*c)-2*I*e*f*exp(3*d*x+3*c)-
6*I*d*e*f*x*exp(3*d*x+3*c)+2*I*f^2*x*exp(d*x+c)-2*I*f^2*x*exp(3*d*x+3*c)-5*d*f^2*x^2*exp(2*d*x+2*c))/(exp(2*d*
x+2*c)-1)^2/d^2/(exp(d*x+c)-I)/a

________________________________________________________________________________________

Maxima [B]  time = 2.44375, size = 1165, normalized size = 3.17 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^2*csch(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-1/2*e^2*(16*(-I*e^(-d*x - c) - 5*e^(-2*d*x - 2*c) + 3*I*e^(-3*d*x - 3*c) + 3*e^(-4*d*x - 4*c) + 4)/((8*a*e^(-
d*x - c) - 16*I*a*e^(-2*d*x - 2*c) - 16*a*e^(-3*d*x - 3*c) + 8*I*a*e^(-4*d*x - 4*c) + 8*a*e^(-5*d*x - 5*c) + 8
*I*a)*d) - 3*log(e^(-d*x - c) + 1)/(a*d) + 3*log(e^(-d*x - c) - 1)/(a*d)) + 2*I*f^2*x^2/(a*d) + 4*I*e*f*x/(a*d
) - (4*d*f^2*x^2 + 8*d*e*f*x + (3*d*f^2*x^2*e^(4*c) + 2*e*f*e^(4*c) + 2*(3*d*e*f + f^2)*x*e^(4*c))*e^(4*d*x) +
(-3*I*d*f^2*x^2*e^(3*c) - 2*I*e*f*e^(3*c) + (-6*I*d*e*f - 2*I*f^2)*x*e^(3*c))*e^(3*d*x) - (5*d*f^2*x^2*e^(2*c
) + 2*e*f*e^(2*c) + 2*(5*d*e*f + f^2)*x*e^(2*c))*e^(2*d*x) + (I*d*f^2*x^2*e^c + 2*I*e*f*e^c + (2*I*d*e*f + 2*I
*f^2)*x*e^c)*e^(d*x))/(a*d^2*e^(5*d*x + 5*c) - I*a*d^2*e^(4*d*x + 4*c) - 2*a*d^2*e^(3*d*x + 3*c) + 2*I*a*d^2*e
^(2*d*x + 2*c) + a*d^2*e^(d*x + c) - I*a*d^2) - 4*I*e*f*log(I*e^(d*x + c) + 1)/(a*d^2) + 3/2*(d^2*x^2*log(e^(d
*x + c) + 1) + 2*d*x*dilog(-e^(d*x + c)) - 2*polylog(3, -e^(d*x + c)))*f^2/(a*d^3) - 3/2*(d^2*x^2*log(-e^(d*x
+ c) + 1) + 2*d*x*dilog(e^(d*x + c)) - 2*polylog(3, e^(d*x + c)))*f^2/(a*d^3) - 4*I*(d*x*log(I*e^(d*x + c) + 1
) + dilog(-I*e^(d*x + c)))*f^2/(a*d^3) + (2*I*d*e*f + f^2)*x/(a*d^2) + (2*I*d*e*f - f^2)*x/(a*d^2) + (3*d*e*f
- 2*I*f^2)*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))/(a*d^3) - (3*d*e*f + 2*I*f^2)*(d*x*log(-e^(d*x + c
) + 1) + dilog(e^(d*x + c)))/(a*d^3) - (2*I*d*e*f + f^2)*log(e^(d*x + c) + 1)/(a*d^3) - (2*I*d*e*f - f^2)*log(
e^(d*x + c) - 1)/(a*d^3) + 1/2*(d^3*f^2*x^3 + (3*d*e*f + 2*I*f^2)*d^2*x^2)/(a*d^3) - 1/2*(d^3*f^2*x^3 + (3*d*e
*f - 2*I*f^2)*d^2*x^2)/(a*d^3)

________________________________________________________________________________________

Fricas [C]  time = 2.95854, size = 5378, normalized size = 14.61 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^2*csch(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-(8*d^2*e^2 - 16*c*d*e*f + 8*c^2*f^2 - (-8*I*f^2*e^(5*d*x + 5*c) - 8*f^2*e^(4*d*x + 4*c) + 16*I*f^2*e^(3*d*x +
3*c) + 16*f^2*e^(2*d*x + 2*c) - 8*I*f^2*e^(d*x + c) - 8*f^2)*dilog(-I*e^(d*x + c)) - (-6*I*d*f^2*x - 6*I*d*e*
f - 4*f^2 + 2*(3*d*f^2*x + 3*d*e*f - 2*I*f^2)*e^(5*d*x + 5*c) + (-6*I*d*f^2*x - 6*I*d*e*f - 4*f^2)*e^(4*d*x +
4*c) - 4*(3*d*f^2*x + 3*d*e*f - 2*I*f^2)*e^(3*d*x + 3*c) + (12*I*d*f^2*x + 12*I*d*e*f + 8*f^2)*e^(2*d*x + 2*c)
+ 2*(3*d*f^2*x + 3*d*e*f - 2*I*f^2)*e^(d*x + c))*dilog(-e^(d*x + c)) - (6*I*d*f^2*x + 6*I*d*e*f - 4*f^2 - 2*(
3*d*f^2*x + 3*d*e*f + 2*I*f^2)*e^(5*d*x + 5*c) + (6*I*d*f^2*x + 6*I*d*e*f - 4*f^2)*e^(4*d*x + 4*c) + 4*(3*d*f^
2*x + 3*d*e*f + 2*I*f^2)*e^(3*d*x + 3*c) + (-12*I*d*f^2*x - 12*I*d*e*f + 8*f^2)*e^(2*d*x + 2*c) - 2*(3*d*f^2*x
+ 3*d*e*f + 2*I*f^2)*e^(d*x + c))*dilog(e^(d*x + c)) - (8*I*d^2*f^2*x^2 + 16*I*d^2*e*f*x + 16*I*c*d*e*f - 8*I
*c^2*f^2)*e^(5*d*x + 5*c) - 2*(d^2*f^2*x^2 - 3*d^2*e^2 + 2*(4*c - 1)*d*e*f - 4*c^2*f^2 + 2*(d^2*e*f - d*f^2)*x
)*e^(4*d*x + 4*c) - (-10*I*d^2*f^2*x^2 + 6*I*d^2*e^2 + (-32*I*c + 4*I)*d*e*f + 16*I*c^2*f^2 + (-20*I*d^2*e*f +
4*I*d*f^2)*x)*e^(3*d*x + 3*c) + 2*(3*d^2*f^2*x^2 - 5*d^2*e^2 + 2*(8*c - 1)*d*e*f - 8*c^2*f^2 + 2*(3*d^2*e*f -
d*f^2)*x)*e^(2*d*x + 2*c) - (6*I*d^2*f^2*x^2 - 2*I*d^2*e^2 + (16*I*c - 4*I)*d*e*f - 8*I*c^2*f^2 + (12*I*d^2*e
*f - 4*I*d*f^2)*x)*e^(d*x + c) - (-3*I*d^2*f^2*x^2 - 3*I*d^2*e^2 - 4*d*e*f + 2*I*f^2 - 2*(3*I*d^2*e*f + 2*d*f^
2)*x + (3*d^2*f^2*x^2 + 3*d^2*e^2 - 4*I*d*e*f - 2*f^2 + (6*d^2*e*f - 4*I*d*f^2)*x)*e^(5*d*x + 5*c) + (-3*I*d^2
*f^2*x^2 - 3*I*d^2*e^2 - 4*d*e*f + 2*I*f^2 - 2*(3*I*d^2*e*f + 2*d*f^2)*x)*e^(4*d*x + 4*c) - (6*d^2*f^2*x^2 + 6
*d^2*e^2 - 8*I*d*e*f - 4*f^2 + (12*d^2*e*f - 8*I*d*f^2)*x)*e^(3*d*x + 3*c) + (6*I*d^2*f^2*x^2 + 6*I*d^2*e^2 +
8*d*e*f - 4*I*f^2 - 4*(-3*I*d^2*e*f - 2*d*f^2)*x)*e^(2*d*x + 2*c) + (3*d^2*f^2*x^2 + 3*d^2*e^2 - 4*I*d*e*f - 2
*f^2 + (6*d^2*e*f - 4*I*d*f^2)*x)*e^(d*x + c))*log(e^(d*x + c) + 1) + (8*d*e*f - 8*c*f^2 - (-8*I*d*e*f + 8*I*c
*f^2)*e^(5*d*x + 5*c) + 8*(d*e*f - c*f^2)*e^(4*d*x + 4*c) - (16*I*d*e*f - 16*I*c*f^2)*e^(3*d*x + 3*c) - 16*(d*
e*f - c*f^2)*e^(2*d*x + 2*c) - (-8*I*d*e*f + 8*I*c*f^2)*e^(d*x + c))*log(e^(d*x + c) - I) - (3*I*d^2*e^2 - 2*(
3*I*c + 2)*d*e*f + (3*I*c^2 + 4*c - 2*I)*f^2 - (3*d^2*e^2 - (6*c - 4*I)*d*e*f + (3*c^2 - 4*I*c - 2)*f^2)*e^(5*
d*x + 5*c) + (3*I*d^2*e^2 - 2*(3*I*c + 2)*d*e*f + (3*I*c^2 + 4*c - 2*I)*f^2)*e^(4*d*x + 4*c) + (6*d^2*e^2 - (1
2*c - 8*I)*d*e*f + 2*(3*c^2 - 4*I*c - 2)*f^2)*e^(3*d*x + 3*c) + (-6*I*d^2*e^2 - 4*(-3*I*c - 2)*d*e*f + (-6*I*c
^2 - 8*c + 4*I)*f^2)*e^(2*d*x + 2*c) - (3*d^2*e^2 - (6*c - 4*I)*d*e*f + (3*c^2 - 4*I*c - 2)*f^2)*e^(d*x + c))*
log(e^(d*x + c) - 1) + (8*d*f^2*x + 8*c*f^2 - (-8*I*d*f^2*x - 8*I*c*f^2)*e^(5*d*x + 5*c) + 8*(d*f^2*x + c*f^2)
*e^(4*d*x + 4*c) - (16*I*d*f^2*x + 16*I*c*f^2)*e^(3*d*x + 3*c) - 16*(d*f^2*x + c*f^2)*e^(2*d*x + 2*c) - (-8*I*
d*f^2*x - 8*I*c*f^2)*e^(d*x + c))*log(I*e^(d*x + c) + 1) - (3*I*d^2*f^2*x^2 + 6*I*c*d*e*f + (-3*I*c^2 - 4*c)*f
^2 - 2*(-3*I*d^2*e*f + 2*d*f^2)*x - (3*d^2*f^2*x^2 + 6*c*d*e*f - (3*c^2 - 4*I*c)*f^2 + (6*d^2*e*f + 4*I*d*f^2)
*x)*e^(5*d*x + 5*c) + (3*I*d^2*f^2*x^2 + 6*I*c*d*e*f + (-3*I*c^2 - 4*c)*f^2 - 2*(-3*I*d^2*e*f + 2*d*f^2)*x)*e^
(4*d*x + 4*c) + (6*d^2*f^2*x^2 + 12*c*d*e*f - 2*(3*c^2 - 4*I*c)*f^2 + (12*d^2*e*f + 8*I*d*f^2)*x)*e^(3*d*x + 3
*c) + (-6*I*d^2*f^2*x^2 - 12*I*c*d*e*f + (6*I*c^2 + 8*c)*f^2 - 4*(3*I*d^2*e*f - 2*d*f^2)*x)*e^(2*d*x + 2*c) -
(3*d^2*f^2*x^2 + 6*c*d*e*f - (3*c^2 - 4*I*c)*f^2 + (6*d^2*e*f + 4*I*d*f^2)*x)*e^(d*x + c))*log(-e^(d*x + c) +
1) + (6*f^2*e^(5*d*x + 5*c) - 6*I*f^2*e^(4*d*x + 4*c) - 12*f^2*e^(3*d*x + 3*c) + 12*I*f^2*e^(2*d*x + 2*c) + 6*
f^2*e^(d*x + c) - 6*I*f^2)*polylog(3, -e^(d*x + c)) - (6*f^2*e^(5*d*x + 5*c) - 6*I*f^2*e^(4*d*x + 4*c) - 12*f^
2*e^(3*d*x + 3*c) + 12*I*f^2*e^(2*d*x + 2*c) + 6*f^2*e^(d*x + c) - 6*I*f^2)*polylog(3, e^(d*x + c)))/(2*a*d^3*
e^(5*d*x + 5*c) - 2*I*a*d^3*e^(4*d*x + 4*c) - 4*a*d^3*e^(3*d*x + 3*c) + 4*I*a*d^3*e^(2*d*x + 2*c) + 2*a*d^3*e^
(d*x + c) - 2*I*a*d^3)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)**2*csch(d*x+c)**3/(a+I*a*sinh(d*x+c)),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )}^{2} \operatorname{csch}\left (d x + c\right )^{3}}{i \, a \sinh \left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^2*csch(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="giac")

[Out]

integrate((f*x + e)^2*csch(d*x + c)^3/(I*a*sinh(d*x + c) + a), x)