3.3.18 \(\int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx\) [218]

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

[Out]

2*I*(f*x+e)^2/a/d+3*(f*x+e)^2*arctanh(exp(d*x+c))/a/d-f^2*arctanh(cosh(d*x+c))/a/d^3+I*(f*x+e)^2*coth(d*x+c)/a
/d-f*(f*x+e)*csch(d*x+c)/a/d^2-1/2*(f*x+e)^2*coth(d*x+c)*csch(d*x+c)/a/d-4*I*f*(f*x+e)*ln(1+I*exp(d*x+c))/a/d^
2-2*I*f*(f*x+e)*ln(1-exp(2*d*x+2*c))/a/d^2+3*f*(f*x+e)*polylog(2,-exp(d*x+c))/a/d^2-4*I*f^2*polylog(2,-I*exp(d
*x+c))/a/d^3-3*f*(f*x+e)*polylog(2,exp(d*x+c))/a/d^2-I*f^2*polylog(2,exp(2*d*x+2*c))/a/d^3-3*f^2*polylog(3,-ex
p(d*x+c))/a/d^3+3*f^2*polylog(3,exp(d*x+c))/a/d^3+I*(f*x+e)^2*tanh(1/2*c+1/4*I*Pi+1/2*d*x)/a/d

________________________________________________________________________________________

Rubi [A]
time = 0.60, antiderivative size = 368, normalized size of antiderivative = 1.00, 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 = {5694, 4271, 3855, 4267, 2611, 2320, 6724, 4269, 3797, 2221, 2317, 2438, 3399} \begin {gather*} -\frac {4 i f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}-\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 {f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\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 {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 {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} \end {gather*}

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 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 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 2320

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 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 2611

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 3399

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(2*a)^n, Int[(c
+ d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) + 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])

Rule 3797

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))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*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 4271

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]) /; Free
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 5694

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 6724

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]

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 \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}+\frac {f^2 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1370\) vs. \(2(368)=736\).
time = 21.34, size = 1370, normalized size = 3.72 \begin {gather*} \frac {2 f \left (d \left (d e^c x (2 e+f x)-2 \left (-i+e^c\right ) (e+f x) \log \left (1+i e^{c+d x}\right )\right )-2 \left (-i+e^c\right ) f \text {PolyLog}\left (2,-i e^{c+d x}\right )\right )}{a d^3 \left (-1-i e^c\right )}+\frac {8 i d e e^{2 c} f x-8 i d e \left (-1+e^{2 c}\right ) f x+4 i d e^{2 c} f^2 x^2-4 i d \left (-1+e^{2 c}\right ) f^2 x^2+6 d e^2 \left (-1+e^{2 c}\right ) \tanh ^{-1}\left (e^{c+d x}\right )-\frac {4 \left (-1+e^{2 c}\right ) f^2 \tanh ^{-1}\left (e^{c+d x}\right )}{d}+4 i e \left (-1+e^{2 c}\right ) f \left (2 d x-\log \left (1-e^{2 (c+d x)}\right )\right )+6 e \left (-1+e^{2 c}\right ) f \left (d x \left (-\log \left (1-e^{c+d x}\right )+\log \left (1+e^{c+d x}\right )\right )+\text {PolyLog}\left (2,-e^{c+d x}\right )-\text {PolyLog}\left (2,e^{c+d x}\right )\right )+\frac {2 i \left (-1+e^{2 c}\right ) f^2 \left (2 d x \left (d x-\log \left (1-e^{2 (c+d x)}\right )\right )-\text {PolyLog}\left (2,e^{2 (c+d x)}\right )\right )}{d}+\frac {3 \left (-1+e^{2 c}\right ) f^2 \left (-d^2 x^2 \log \left (1-e^{c+d x}\right )+d^2 x^2 \log \left (1+e^{c+d x}\right )+2 d x \text {PolyLog}\left (2,-e^{c+d x}\right )-2 d x \text {PolyLog}\left (2,e^{c+d x}\right )-2 \text {PolyLog}\left (3,-e^{c+d x}\right )+2 \text {PolyLog}\left (3,e^{c+d x}\right )\right )}{d}}{2 a d^2 \left (-1+e^{2 c}\right )}+\frac {\text {csch}(c) \text {csch}^2(c+d x) \left (2 e f \cosh \left (\frac {d x}{2}\right )+2 f^2 x \cosh \left (\frac {d x}{2}\right )+2 e f \cosh \left (\frac {3 d x}{2}\right )+2 f^2 x \cosh \left (\frac {3 d x}{2}\right )+5 i d e^2 \cosh \left (c-\frac {d x}{2}\right )+10 i d e f x \cosh \left (c-\frac {d x}{2}\right )+5 i d f^2 x^2 \cosh \left (c-\frac {d x}{2}\right )-i d e^2 \cosh \left (c+\frac {d x}{2}\right )-2 i d e f x \cosh \left (c+\frac {d x}{2}\right )-i d f^2 x^2 \cosh \left (c+\frac {d x}{2}\right )-2 e f \cosh \left (2 c+\frac {d x}{2}\right )-2 f^2 x \cosh \left (2 c+\frac {d x}{2}\right )+i d e^2 \cosh \left (c+\frac {3 d x}{2}\right )+2 i d e f x \cosh \left (c+\frac {3 d x}{2}\right )+i d f^2 x^2 \cosh \left (c+\frac {3 d x}{2}\right )-2 e f \cosh \left (2 c+\frac {3 d x}{2}\right )-2 f^2 x \cosh \left (2 c+\frac {3 d x}{2}\right )-3 i d e^2 \cosh \left (3 c+\frac {3 d x}{2}\right )-6 i d e f x \cosh \left (3 c+\frac {3 d x}{2}\right )-3 i d f^2 x^2 \cosh \left (3 c+\frac {3 d x}{2}\right )-4 i d e^2 \cosh \left (c+\frac {5 d x}{2}\right )-8 i d e f x \cosh \left (c+\frac {5 d x}{2}\right )-4 i d f^2 x^2 \cosh \left (c+\frac {5 d x}{2}\right )+2 i d e^2 \cosh \left (3 c+\frac {5 d x}{2}\right )+4 i d e f x \cosh \left (3 c+\frac {5 d x}{2}\right )+2 i d f^2 x^2 \cosh \left (3 c+\frac {5 d x}{2}\right )-d e^2 \sinh \left (\frac {d x}{2}\right )-2 d e f x \sinh \left (\frac {d x}{2}\right )-d f^2 x^2 \sinh \left (\frac {d x}{2}\right )-d e^2 \sinh \left (\frac {3 d x}{2}\right )-2 d e f x \sinh \left (\frac {3 d x}{2}\right )-d f^2 x^2 \sinh \left (\frac {3 d x}{2}\right )+2 i e f \sinh \left (c-\frac {d x}{2}\right )+2 i f^2 x \sinh \left (c-\frac {d x}{2}\right )+2 i e f \sinh \left (c+\frac {d x}{2}\right )+2 i f^2 x \sinh \left (c+\frac {d x}{2}\right )-3 d e^2 \sinh \left (2 c+\frac {d x}{2}\right )-6 d e f x \sinh \left (2 c+\frac {d x}{2}\right )-3 d f^2 x^2 \sinh \left (2 c+\frac {d x}{2}\right )+2 i e f \sinh \left (c+\frac {3 d x}{2}\right )+2 i f^2 x \sinh \left (c+\frac {3 d x}{2}\right )-d e^2 \sinh \left (2 c+\frac {3 d x}{2}\right )-2 d e f x \sinh \left (2 c+\frac {3 d x}{2}\right )-d f^2 x^2 \sinh \left (2 c+\frac {3 d x}{2}\right )-2 i e f \sinh \left (3 c+\frac {3 d x}{2}\right )-2 i f^2 x \sinh \left (3 c+\frac {3 d x}{2}\right )+2 d e^2 \sinh \left (2 c+\frac {5 d x}{2}\right )+4 d e f x \sinh \left (2 c+\frac {5 d x}{2}\right )+2 d f^2 x^2 \sinh \left (2 c+\frac {5 d x}{2}\right )\right )}{8 a d^2 \left (\cosh \left (\frac {c}{2}\right )+i \sinh \left (\frac {c}{2}\right )\right ) \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )+i \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*f*(d*(d*E^c*x*(2*e + f*x) - 2*(-I + E^c)*(e + f*x)*Log[1 + I*E^(c + d*x)]) - 2*(-I + E^c)*f*PolyLog[2, (-I)
*E^(c + d*x)]))/(a*d^3*(-1 - I*E^c)) + ((8*I)*d*e*E^(2*c)*f*x - (8*I)*d*e*(-1 + E^(2*c))*f*x + (4*I)*d*E^(2*c)
*f^2*x^2 - (4*I)*d*(-1 + E^(2*c))*f^2*x^2 + 6*d*e^2*(-1 + E^(2*c))*ArcTanh[E^(c + d*x)] - (4*(-1 + E^(2*c))*f^
2*ArcTanh[E^(c + d*x)])/d + (4*I)*e*(-1 + E^(2*c))*f*(2*d*x - Log[1 - E^(2*(c + d*x))]) + 6*e*(-1 + E^(2*c))*f
*(d*x*(-Log[1 - E^(c + d*x)] + Log[1 + E^(c + d*x)]) + PolyLog[2, -E^(c + d*x)] - PolyLog[2, E^(c + d*x)]) + (
(2*I)*(-1 + E^(2*c))*f^2*(2*d*x*(d*x - Log[1 - E^(2*(c + d*x))]) - PolyLog[2, E^(2*(c + d*x))]))/d + (3*(-1 +
E^(2*c))*f^2*(-(d^2*x^2*Log[1 - E^(c + d*x)]) + d^2*x^2*Log[1 + E^(c + d*x)] + 2*d*x*PolyLog[2, -E^(c + d*x)]
- 2*d*x*PolyLog[2, E^(c + d*x)] - 2*PolyLog[3, -E^(c + d*x)] + 2*PolyLog[3, E^(c + d*x)]))/d)/(2*a*d^2*(-1 + E
^(2*c))) + (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]))

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1106 vs. \(2 (343 ) = 686\).
time = 3.25, size = 1107, normalized size = 3.01

method	result	size
risch	\(-\frac {2 i \ln \left (1-{\mathrm e}^{d x +c}\right ) c \,f^{2}}{a \,d^{3}}-\frac {3 f^{2} \polylog \left (3, -{\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {2 i f^{2} c \ln \left ({\mathrm e}^{d x +c}-1\right )}{a \,d^{3}}-\frac {4 i f^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right ) x}{a \,d^{2}}-\frac {8 i f^{2} c \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{3}}-\frac {4 i f^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right ) c}{a \,d^{3}}+\frac {4 i f^{2} x^{2}}{a d}+\frac {4 i c^{2} f^{2}}{a \,d^{3}}-\frac {6 d e f x \,{\mathrm e}^{4 d x +4 c}-2 i e f \,{\mathrm e}^{3 d x +3 c}+i d \,f^{2} x^{2} {\mathrm e}^{d x +c}+3 d \,f^{2} x^{2} {\mathrm e}^{4 d x +4 c}-6 i d e f x \,{\mathrm e}^{3 d x +3 c}+4 d \,e^{2}+2 i e f \,{\mathrm e}^{d x +c}+2 i d e f x \,{\mathrm e}^{d x +c}-3 i d \,f^{2} x^{2} {\mathrm e}^{3 d x +3 c}-2 i f^{2} x \,{\mathrm e}^{3 d x +3 c}-3 i d \,e^{2} {\mathrm e}^{3 d x +3 c}-5 d \,e^{2} {\mathrm e}^{2 d x +2 c}+8 d e f x +3 d \,e^{2} {\mathrm e}^{4 d x +4 c}+2 f^{2} x \,{\mathrm e}^{4 d x +4 c}+2 e f \,{\mathrm e}^{4 d x +4 c}-2 f^{2} x \,{\mathrm e}^{2 d x +2 c}-2 e f \,{\mathrm e}^{2 d x +2 c}+4 d \,f^{2} x^{2}-10 d e f x \,{\mathrm e}^{2 d x +2 c}-5 d \,f^{2} x^{2} {\mathrm e}^{2 d x +2 c}+2 i f^{2} x \,{\mathrm e}^{d x +c}+i d \,e^{2} {\mathrm e}^{d x +c}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2} d^{2} \left ({\mathrm e}^{d x +c}-i\right ) a}-\frac {2 i f^{2} \polylog \left (2, {\mathrm e}^{d x +c}\right )}{a \,d^{3}}-\frac {2 i f^{2} \polylog \left (2, -{\mathrm e}^{d x +c}\right )}{a \,d^{3}}-\frac {3 e f \polylog \left (2, {\mathrm e}^{d x +c}\right )}{a \,d^{2}}-\frac {3 e^{2} \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 a d}+\frac {3 e^{2} \ln \left ({\mathrm e}^{d x +c}+1\right )}{2 a d}+\frac {f^{2} \ln \left ({\mathrm e}^{d x +c}-1\right )}{a \,d^{3}}-\frac {f^{2} \ln \left ({\mathrm e}^{d x +c}+1\right )}{a \,d^{3}}+\frac {3 f^{2} \polylog \left (3, {\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {4 i f^{2} c \ln \left ({\mathrm e}^{d x +c}-i\right )}{a \,d^{3}}+\frac {8 i f^{2} c x}{a \,d^{2}}-\frac {2 i e f \ln \left ({\mathrm e}^{d x +c}+1\right )}{a \,d^{2}}-\frac {4 i e f \ln \left ({\mathrm e}^{d x +c}-i\right )}{a \,d^{2}}-\frac {2 i e f \ln \left ({\mathrm e}^{d x +c}-1\right )}{a \,d^{2}}-\frac {2 i \ln \left ({\mathrm e}^{d x +c}+1\right ) f^{2} x}{a \,d^{2}}-\frac {2 i \ln \left (1-{\mathrm e}^{d x +c}\right ) f^{2} x}{a \,d^{2}}-\frac {4 i f^{2} \polylog \left (2, -i {\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {8 i e f \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{2}}-\frac {3 f^{2} \polylog \left (2, {\mathrm e}^{d x +c}\right ) x}{a \,d^{2}}+\frac {3 \ln \left ({\mathrm e}^{d x +c}+1\right ) f^{2} x^{2}}{2 a d}-\frac {3 \ln \left (1-{\mathrm e}^{d x +c}\right ) f^{2} x^{2}}{2 a d}+\frac {3 \ln \left ({\mathrm e}^{d x +c}+1\right ) e f x}{a d}-\frac {3 \ln \left (1-{\mathrm e}^{d x +c}\right ) e f x}{a d}-\frac {3 \ln \left (1-{\mathrm e}^{d x +c}\right ) c e f}{a \,d^{2}}-\frac {3 f^{2} c^{2} \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 a \,d^{3}}+\frac {3 \ln \left (1-{\mathrm e}^{d x +c}\right ) c^{2} f^{2}}{2 a \,d^{3}}+\frac {3 e f c \ln \left ({\mathrm e}^{d x +c}-1\right )}{a \,d^{2}}+\frac {3 \polylog \left (2, -{\mathrm e}^{d x +c}\right ) f^{2} x}{a \,d^{2}}+\frac {3 e f \polylog \left (2, -{\mathrm e}^{d x +c}\right )}{a \,d^{2}}\)	\(1107\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 897 vs. \(2 (339) = 678\).
time = 0.52, size = 897, normalized size = 2.44 \begin {gather*} \frac {2 i \, f^{2} x^{2}}{a d} - \frac {1}{2} \, {\left (\frac {2 \, {\left (-i \, e^{\left (-d x - c\right )} - 5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 i \, e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4\right )}}{{\left (a e^{\left (-d x - c\right )} - 2 i \, a e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, a e^{\left (-3 \, d x - 3 \, c\right )} + i \, a e^{\left (-4 \, d x - 4 \, c\right )} + a e^{\left (-5 \, d x - 5 \, c\right )} + i \, a\right )} d} - \frac {3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} + \frac {3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{a d}\right )} e^{2} + \frac {4 i \, f x e}{a d} - \frac {4 \, d f^{2} x^{2} + 8 \, d f x e + {\left (3 \, d f^{2} x^{2} e^{\left (4 \, c\right )} + 2 \, {\left (f^{2} e^{\left (4 \, c\right )} + 3 \, d f e^{\left (4 \, c + 1\right )}\right )} x + 2 \, f e^{\left (4 \, c + 1\right )}\right )} e^{\left (4 \, d x\right )} + {\left (-3 i \, d f^{2} x^{2} e^{\left (3 \, c\right )} - 2 \, {\left (i \, f^{2} e^{\left (3 \, c\right )} + 3 i \, d f e^{\left (3 \, c + 1\right )}\right )} x - 2 i \, f e^{\left (3 \, c + 1\right )}\right )} e^{\left (3 \, d x\right )} - {\left (5 \, d f^{2} x^{2} e^{\left (2 \, c\right )} + 2 \, {\left (f^{2} e^{\left (2 \, c\right )} + 5 \, d f e^{\left (2 \, c + 1\right )}\right )} x + 2 \, f e^{\left (2 \, c + 1\right )}\right )} e^{\left (2 \, d x\right )} + {\left (i \, d f^{2} x^{2} e^{c} - 2 \, {\left (-i \, d f e^{\left (c + 1\right )} - i \, f^{2} e^{c}\right )} x + 2 i \, f e^{\left (c + 1\right )}\right )} e^{\left (d x\right )}}{a d^{2} e^{\left (5 \, d x + 5 \, c\right )} - i \, a d^{2} e^{\left (4 \, d x + 4 \, c\right )} - 2 \, a d^{2} e^{\left (3 \, d x + 3 \, c\right )} + 2 i \, a d^{2} e^{\left (2 \, d x + 2 \, c\right )} + a d^{2} e^{\left (d x + c\right )} - i \, a d^{2}} - \frac {4 i \, f e \log \left (i \, e^{\left (d x + c\right )} + 1\right )}{a d^{2}} + \frac {3 \, {\left (d^{2} x^{2} \log \left (e^{\left (d x + c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (-e^{\left (d x + c\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (d x + c\right )})\right )} f^{2}}{2 \, a d^{3}} - \frac {3 \, {\left (d^{2} x^{2} \log \left (-e^{\left (d x + c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (e^{\left (d x + c\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (d x + c\right )})\right )} f^{2}}{2 \, a d^{3}} - \frac {4 i \, {\left (d x \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right )\right )} f^{2}}{a d^{3}} + \frac {{\left (2 i \, d f e + f^{2}\right )} x}{a d^{2}} + \frac {{\left (2 i \, d f e - f^{2}\right )} x}{a d^{2}} + \frac {{\left (3 \, d f e - 2 i \, f^{2}\right )} {\left (d x \log \left (e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (d x + c\right )}\right )\right )}}{a d^{3}} - \frac {{\left (3 \, d f e + 2 i \, f^{2}\right )} {\left (d x \log \left (-e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (d x + c\right )}\right )\right )}}{a d^{3}} - \frac {{\left (2 i \, d f e + f^{2}\right )} \log \left (e^{\left (d x + c\right )} + 1\right )}{a d^{3}} - \frac {{\left (2 i \, d f e - f^{2}\right )} \log \left (e^{\left (d x + c\right )} - 1\right )}{a d^{3}} + \frac {d^{3} f^{2} x^{3} + {\left (3 \, d f e + 2 i \, f^{2}\right )} d^{2} x^{2}}{2 \, a d^{3}} - \frac {d^{3} f^{2} x^{3} + {\left (3 \, d f e - 2 i \, f^{2}\right )} d^{2} x^{2}}{2 \, a d^{3}} \end {gather*}

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]

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

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2238 vs. \(2 (339) = 678\).
time = 0.38, size = 2238, normalized size = 6.08 \begin {gather*} \text {Too large to display} \end {gather*}

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]

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

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (e+f\,x\right )}^2}{{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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