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

Optimal. Leaf size=1428 \[ \frac {(e+f x)^3 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) b^3}{a^2 \left (a^2+b^2\right ) d}+\frac {(e+f x)^3 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) b^3}{a^2 \left (a^2+b^2\right ) d}-\frac {(e+f x)^3 \log \left (1+e^{2 (c+d x)}\right ) b^3}{a^2 \left (a^2+b^2\right ) d}+\frac {3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^2}+\frac {3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^2}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right ) b^3}{2 a^2 \left (a^2+b^2\right ) d^2}-\frac {6 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^3}-\frac {6 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^3}+\frac {3 f^2 (e+f x) \text {Li}_3\left (-e^{2 (c+d x)}\right ) b^3}{2 a^2 \left (a^2+b^2\right ) d^3}+\frac {6 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^4}+\frac {6 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^4}-\frac {3 f^3 \text {Li}_4\left (-e^{2 (c+d x)}\right ) b^3}{4 a^2 \left (a^2+b^2\right ) d^4}+\frac {2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d^2}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d^2}+\frac {6 i f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d^3}-\frac {6 i f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d^3}-\frac {6 i f^3 \text {Li}_4\left (-i e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d^4}+\frac {6 i f^3 \text {Li}_4\left (i e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d^4}+\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right ) b}{a^2 d}+\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{2 c+2 d x}\right ) b}{2 a^2 d^2}-\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right ) b}{2 a^2 d^2}-\frac {3 f^2 (e+f x) \text {Li}_3\left (-e^{2 c+2 d x}\right ) b}{2 a^2 d^3}+\frac {3 f^2 (e+f x) \text {Li}_3\left (e^{2 c+2 d x}\right ) b}{2 a^2 d^3}+\frac {3 f^3 \text {Li}_4\left (-e^{2 c+2 d x}\right ) b}{4 a^2 d^4}-\frac {3 f^3 \text {Li}_4\left (e^{2 c+2 d x}\right ) b}{4 a^2 d^4}-\frac {2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {6 f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac {(e+f x)^3 \text {csch}(c+d x)}{a d}-\frac {6 f^2 (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^3}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{a d^2}+\frac {6 f^2 (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^3}+\frac {6 f^3 \text {Li}_3\left (-e^{c+d x}\right )}{a d^4}-\frac {6 i f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{a d^3}+\frac {6 i f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{a d^3}-\frac {6 f^3 \text {Li}_3\left (e^{c+d x}\right )}{a d^4}+\frac {6 i f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{a d^4}-\frac {6 i f^3 \text {Li}_4\left (i e^{c+d x}\right )}{a d^4} \]

[Out]

-b^3*(f*x+e)^3*ln(1+exp(2*d*x+2*c))/a^2/(a^2+b^2)/d+b^3*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/(
a^2+b^2)/d+b^3*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)/d+6*b^3*f^3*polylog(4,-b*exp(d*x
+c)/(a-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)/d^4+6*b^3*f^3*polylog(4,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2
)/d^4+2*b^2*(f*x+e)^3*arctan(exp(d*x+c))/a/(a^2+b^2)/d+3/2*b*f*(f*x+e)^2*polylog(2,-exp(2*d*x+2*c))/a^2/d^2-3/
2*b*f^2*(f*x+e)*polylog(3,-exp(2*d*x+2*c))/a^2/d^3+6*I*f^3*polylog(4,-I*exp(d*x+c))/a/d^4-3/4*b^3*f^3*polylog(
4,-exp(2*d*x+2*c))/a^2/(a^2+b^2)/d^4-3*I*f*(f*x+e)^2*polylog(2,I*exp(d*x+c))/a/d^2-6*I*f^2*(f*x+e)*polylog(3,-
I*exp(d*x+c))/a/d^3-3/2*b*f*(f*x+e)^2*polylog(2,exp(2*d*x+2*c))/a^2/d^2+3/2*b*f^2*(f*x+e)*polylog(3,exp(2*d*x+
2*c))/a^2/d^3+3*I*f*(f*x+e)^2*polylog(2,-I*exp(d*x+c))/a/d^2-3/2*b^3*f*(f*x+e)^2*polylog(2,-exp(2*d*x+2*c))/a^
2/(a^2+b^2)/d^2+6*I*f^2*(f*x+e)*polylog(3,I*exp(d*x+c))/a/d^3+3/2*b^3*f^2*(f*x+e)*polylog(3,-exp(2*d*x+2*c))/a
^2/(a^2+b^2)/d^3-6*I*b^2*f^3*polylog(4,-I*exp(d*x+c))/a/(a^2+b^2)/d^4+2*b*(f*x+e)^3*arctanh(exp(2*d*x+2*c))/a^
2/d+3/4*b*f^3*polylog(4,-exp(2*d*x+2*c))/a^2/d^4-6*I*f^3*polylog(4,I*exp(d*x+c))/a/d^4-6*f*(f*x+e)^2*arctanh(e
xp(d*x+c))/a/d^2-6*f^2*(f*x+e)*polylog(2,-exp(d*x+c))/a/d^3+6*f^2*(f*x+e)*polylog(2,exp(d*x+c))/a/d^3-3/4*b*f^
3*polylog(4,exp(2*d*x+2*c))/a^2/d^4+6*I*b^2*f^3*polylog(4,I*exp(d*x+c))/a/(a^2+b^2)/d^4-3*I*b^2*f*(f*x+e)^2*po
lylog(2,-I*exp(d*x+c))/a/(a^2+b^2)/d^2-6*I*b^2*f^2*(f*x+e)*polylog(3,I*exp(d*x+c))/a/(a^2+b^2)/d^3-(f*x+e)^3*c
sch(d*x+c)/a/d+6*f^3*polylog(3,-exp(d*x+c))/a/d^4-6*f^3*polylog(3,exp(d*x+c))/a/d^4-2*(f*x+e)^3*arctan(exp(d*x
+c))/a/d+3*I*b^2*f*(f*x+e)^2*polylog(2,I*exp(d*x+c))/a/(a^2+b^2)/d^2+6*I*b^2*f^2*(f*x+e)*polylog(3,-I*exp(d*x+
c))/a/(a^2+b^2)/d^3+3*b^3*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)/d^2+3*b^3*f*(
f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)/d^2-6*b^3*f^2*(f*x+e)*polylog(3,-b*exp(d*x
+c)/(a-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)/d^3-6*b^3*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/
(a^2+b^2)/d^3

________________________________________________________________________________________

Rubi [A]  time = 2.28, antiderivative size = 1428, normalized size of antiderivative = 1.00, number of steps used = 64, number of rules used = 20, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.588, Rules used = {5589, 2621, 321, 207, 5462, 6741, 12, 6742, 5205, 4180, 2531, 6609, 2282, 6589, 4182, 5461, 5573, 5561, 2190, 3718} \[ \text {result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(e + f*x)^3*ArcTan[E^(c + d*x)])/(a*d) + (2*b^2*(e + f*x)^3*ArcTan[E^(c + d*x)])/(a*(a^2 + b^2)*d) - (6*f*
(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a*d^2) + (2*b*(e + f*x)^3*ArcTanh[E^(2*c + 2*d*x)])/(a^2*d) - ((e + f*x)^3*
Csch[c + d*x])/(a*d) + (b^3*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d) +
(b^3*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d) - (b^3*(e + f*x)^3*Log[1
+ E^(2*(c + d*x))])/(a^2*(a^2 + b^2)*d) - (6*f^2*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a*d^3) + ((3*I)*f*(e + f
*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^2) - ((3*I)*b^2*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/(a*(a^2 +
 b^2)*d^2) - ((3*I)*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)])/(a*d^2) + ((3*I)*b^2*f*(e + f*x)^2*PolyLog[2, I*E
^(c + d*x)])/(a*(a^2 + b^2)*d^2) + (6*f^2*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a*d^3) + (3*b^3*f*(e + f*x)^2*Po
lyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^2) + (3*b^3*f*(e + f*x)^2*PolyLog[2, -(
(b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^2) - (3*b^3*f*(e + f*x)^2*PolyLog[2, -E^(2*(c + d*
x))])/(2*a^2*(a^2 + b^2)*d^2) + (3*b*f*(e + f*x)^2*PolyLog[2, -E^(2*c + 2*d*x)])/(2*a^2*d^2) - (3*b*f*(e + f*x
)^2*PolyLog[2, E^(2*c + 2*d*x)])/(2*a^2*d^2) + (6*f^3*PolyLog[3, -E^(c + d*x)])/(a*d^4) - ((6*I)*f^2*(e + f*x)
*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^3) + ((6*I)*b^2*f^2*(e + f*x)*PolyLog[3, (-I)*E^(c + d*x)])/(a*(a^2 + b^2)
*d^3) + ((6*I)*f^2*(e + f*x)*PolyLog[3, I*E^(c + d*x)])/(a*d^3) - ((6*I)*b^2*f^2*(e + f*x)*PolyLog[3, I*E^(c +
 d*x)])/(a*(a^2 + b^2)*d^3) - (6*f^3*PolyLog[3, E^(c + d*x)])/(a*d^4) - (6*b^3*f^2*(e + f*x)*PolyLog[3, -((b*E
^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^3) - (6*b^3*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))
/(a + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^3) + (3*b^3*f^2*(e + f*x)*PolyLog[3, -E^(2*(c + d*x))])/(2*a^2*(a
^2 + b^2)*d^3) - (3*b*f^2*(e + f*x)*PolyLog[3, -E^(2*c + 2*d*x)])/(2*a^2*d^3) + (3*b*f^2*(e + f*x)*PolyLog[3,
E^(2*c + 2*d*x)])/(2*a^2*d^3) + ((6*I)*f^3*PolyLog[4, (-I)*E^(c + d*x)])/(a*d^4) - ((6*I)*b^2*f^3*PolyLog[4, (
-I)*E^(c + d*x)])/(a*(a^2 + b^2)*d^4) - ((6*I)*f^3*PolyLog[4, I*E^(c + d*x)])/(a*d^4) + ((6*I)*b^2*f^3*PolyLog
[4, I*E^(c + d*x)])/(a*(a^2 + b^2)*d^4) + (6*b^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^
2*(a^2 + b^2)*d^4) + (6*b^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^4) -
(3*b^3*f^3*PolyLog[4, -E^(2*(c + d*x))])/(4*a^2*(a^2 + b^2)*d^4) + (3*b*f^3*PolyLog[4, -E^(2*c + 2*d*x)])/(4*a
^2*d^4) - (3*b*f^3*PolyLog[4, E^(2*c + 2*d*x)])/(4*a^2*d^4)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
 FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
 c, n, m, p, x]

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 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 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 2621

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(f*a^n)^(-1), Subst
[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && Integer
Q[(n + 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 3718

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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))), x],
x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 4180

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

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 5205

Int[((a_.) + ArcTan[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(a + b*ArcTan[
u]))/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/(1 + u^2), x], x]
, x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)^(m +
1), u, x] && FalseQ[PowerVariableExpn[u, m + 1, x]]

Rule 5461

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dis
t[2^n, Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]

Rule 5462

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Wit
h[{u = IntHide[Csch[a + b*x]^n*Sech[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - Dist[d*m, Int[(c + d*x)^(m - 1)
*u, x], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n, p]

Rule 5561

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :
> -Simp[(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(c + d*x))/(a - Rt[a^2 + b^2, 2] + b*E^(c +
d*x)), x] + Int[((e + f*x)^m*E^(c + d*x))/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x)), x]) /; FreeQ[{a, b, c, d, e,
 f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]

Rule 5573

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[b^2/(a^2 + b^2), Int[((e + f*x)^m*Sech[c + d*x]^(n - 2))/(a + b*Sinh[c + d*x]), x], x] + Dist[1/(
a^2 + b^2), Int[(e + f*x)^m*Sech[c + d*x]^n*(a - b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && I
GtQ[m, 0] && NeQ[a^2 + b^2, 0] && IGtQ[n, 0]

Rule 5589

Int[(Csch[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(p_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[(e + f*x)^m*Sech[c + d*x]^p*Csch[c + d*x]^n, x], x] - Dis
t[b/a, Int[((e + f*x)^m*Sech[c + d*x]^p*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] && IGtQ[p, 0]

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 6609

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

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {(e+f x)^3 \text {csch}^2(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^3 \text {csch}^2(c+d x) \text {sech}(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=-\frac {(e+f x)^3 \tan ^{-1}(\sinh (c+d x))}{a d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{a d}-\frac {b \int (e+f x)^3 \text {csch}(c+d x) \text {sech}(c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}-\frac {(3 f) \int (e+f x)^2 \left (-\frac {\tan ^{-1}(\sinh (c+d x))}{d}-\frac {\text {csch}(c+d x)}{d}\right ) \, dx}{a}\\ &=-\frac {(e+f x)^3 \tan ^{-1}(\sinh (c+d x))}{a d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{a d}-\frac {(2 b) \int (e+f x)^3 \text {csch}(2 c+2 d x) \, dx}{a^2}+\frac {b^2 \int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x)) \, dx}{a^2 \left (a^2+b^2\right )}+\frac {b^4 \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2 \left (a^2+b^2\right )}-\frac {(3 f) \int \frac {(e+f x)^2 \left (-\tan ^{-1}(\sinh (c+d x))-\text {csch}(c+d x)\right )}{d} \, dx}{a}\\ &=-\frac {b^3 (e+f x)^4}{4 a^2 \left (a^2+b^2\right ) f}-\frac {(e+f x)^3 \tan ^{-1}(\sinh (c+d x))}{a d}+\frac {2 b (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{a d}+\frac {b^2 \int \left (a (e+f x)^3 \text {sech}(c+d x)-b (e+f x)^3 \tanh (c+d x)\right ) \, dx}{a^2 \left (a^2+b^2\right )}+\frac {b^4 \int \frac {e^{c+d x} (e+f x)^3}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^2 \left (a^2+b^2\right )}+\frac {b^4 \int \frac {e^{c+d x} (e+f x)^3}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^2 \left (a^2+b^2\right )}-\frac {(3 f) \int (e+f x)^2 \left (-\tan ^{-1}(\sinh (c+d x))-\text {csch}(c+d x)\right ) \, dx}{a d}+\frac {(3 b f) \int (e+f x)^2 \log \left (1-e^{2 c+2 d x}\right ) \, dx}{a^2 d}-\frac {(3 b f) \int (e+f x)^2 \log \left (1+e^{2 c+2 d x}\right ) \, dx}{a^2 d}\\ &=-\frac {b^3 (e+f x)^4}{4 a^2 \left (a^2+b^2\right ) f}-\frac {(e+f x)^3 \tan ^{-1}(\sinh (c+d x))}{a d}+\frac {2 b (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{a d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {3 b f (e+f x)^2 \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac {3 b f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a^2 d^2}+\frac {b^2 \int (e+f x)^3 \text {sech}(c+d x) \, dx}{a \left (a^2+b^2\right )}-\frac {b^3 \int (e+f x)^3 \tanh (c+d x) \, dx}{a^2 \left (a^2+b^2\right )}-\frac {(3 f) \int \left (-(e+f x)^2 \tan ^{-1}(\sinh (c+d x))-(e+f x)^2 \text {csch}(c+d x)\right ) \, dx}{a d}-\frac {\left (3 b^3 f\right ) \int (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d}-\frac {\left (3 b^3 f\right ) \int (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d}-\frac {\left (3 b f^2\right ) \int (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right ) \, dx}{a^2 d^2}+\frac {\left (3 b f^2\right ) \int (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right ) \, dx}{a^2 d^2}\\ &=\frac {2 b^2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}-\frac {(e+f x)^3 \tan ^{-1}(\sinh (c+d x))}{a d}+\frac {2 b (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{a d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {3 b f (e+f x)^2 \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac {3 b f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac {3 b f^2 (e+f x) \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a^2 d^3}+\frac {3 b f^2 (e+f x) \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a^2 d^3}-\frac {\left (2 b^3\right ) \int \frac {e^{2 (c+d x)} (e+f x)^3}{1+e^{2 (c+d x)}} \, dx}{a^2 \left (a^2+b^2\right )}+\frac {(3 f) \int (e+f x)^2 \tan ^{-1}(\sinh (c+d x)) \, dx}{a d}+\frac {(3 f) \int (e+f x)^2 \text {csch}(c+d x) \, dx}{a d}-\frac {\left (3 i b^2 f\right ) \int (e+f x)^2 \log \left (1-i e^{c+d x}\right ) \, dx}{a \left (a^2+b^2\right ) d}+\frac {\left (3 i b^2 f\right ) \int (e+f x)^2 \log \left (1+i e^{c+d x}\right ) \, dx}{a \left (a^2+b^2\right ) d}-\frac {\left (6 b^3 f^2\right ) \int (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d^2}-\frac {\left (6 b^3 f^2\right ) \int (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d^2}+\frac {\left (3 b f^3\right ) \int \text {Li}_3\left (-e^{2 c+2 d x}\right ) \, dx}{2 a^2 d^3}-\frac {\left (3 b f^3\right ) \int \text {Li}_3\left (e^{2 c+2 d x}\right ) \, dx}{2 a^2 d^3}\\ &=\frac {2 b^2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}-\frac {6 f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac {2 b (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{a d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {b^3 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {3 i b^2 f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {3 i b^2 f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {3 b f (e+f x)^2 \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac {3 b f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac {6 b^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {6 b^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {3 b f^2 (e+f x) \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a^2 d^3}+\frac {3 b f^2 (e+f x) \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a^2 d^3}-\frac {\int d (e+f x)^3 \text {sech}(c+d x) \, dx}{a d}+\frac {\left (3 b^3 f\right ) \int (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d}-\frac {\left (6 f^2\right ) \int (e+f x) \log \left (1-e^{c+d x}\right ) \, dx}{a d^2}+\frac {\left (6 f^2\right ) \int (e+f x) \log \left (1+e^{c+d x}\right ) \, dx}{a d^2}+\frac {\left (6 i b^2 f^2\right ) \int (e+f x) \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{a \left (a^2+b^2\right ) d^2}-\frac {\left (6 i b^2 f^2\right ) \int (e+f x) \text {Li}_2\left (i e^{c+d x}\right ) \, dx}{a \left (a^2+b^2\right ) d^2}+\frac {\left (3 b f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{4 a^2 d^4}-\frac {\left (3 b f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{4 a^2 d^4}+\frac {\left (6 b^3 f^3\right ) \int \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d^3}+\frac {\left (6 b^3 f^3\right ) \int \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d^3}\\ &=\frac {2 b^2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}-\frac {6 f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac {2 b (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{a d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {b^3 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {6 f^2 (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^3}-\frac {3 i b^2 f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {3 i b^2 f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {6 f^2 (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^3}+\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 a^2 \left (a^2+b^2\right ) d^2}+\frac {3 b f (e+f x)^2 \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac {3 b f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a^2 d^2}+\frac {6 i b^2 f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {6 i b^2 f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {6 b^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {6 b^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {3 b f^2 (e+f x) \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a^2 d^3}+\frac {3 b f^2 (e+f x) \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a^2 d^3}+\frac {3 b f^3 \text {Li}_4\left (-e^{2 c+2 d x}\right )}{4 a^2 d^4}-\frac {3 b f^3 \text {Li}_4\left (e^{2 c+2 d x}\right )}{4 a^2 d^4}-\frac {\int (e+f x)^3 \text {sech}(c+d x) \, dx}{a}+\frac {\left (3 b^3 f^2\right ) \int (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d^2}+\frac {\left (6 b^3 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^4}+\frac {\left (6 b^3 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^4}+\frac {\left (6 f^3\right ) \int \text {Li}_2\left (-e^{c+d x}\right ) \, dx}{a d^3}-\frac {\left (6 f^3\right ) \int \text {Li}_2\left (e^{c+d x}\right ) \, dx}{a d^3}-\frac {\left (6 i b^2 f^3\right ) \int \text {Li}_3\left (-i e^{c+d x}\right ) \, dx}{a \left (a^2+b^2\right ) d^3}+\frac {\left (6 i b^2 f^3\right ) \int \text {Li}_3\left (i e^{c+d x}\right ) \, dx}{a \left (a^2+b^2\right ) d^3}\\ &=-\frac {2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {2 b^2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}-\frac {6 f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac {2 b (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{a d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {b^3 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {6 f^2 (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^3}-\frac {3 i b^2 f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {3 i b^2 f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {6 f^2 (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^3}+\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 a^2 \left (a^2+b^2\right ) d^2}+\frac {3 b f (e+f x)^2 \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac {3 b f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a^2 d^2}+\frac {6 i b^2 f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {6 i b^2 f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {6 b^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {6 b^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {3 b^3 f^2 (e+f x) \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 a^2 \left (a^2+b^2\right ) d^3}-\frac {3 b f^2 (e+f x) \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a^2 d^3}+\frac {3 b f^2 (e+f x) \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a^2 d^3}+\frac {6 b^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^4}+\frac {6 b^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^4}+\frac {3 b f^3 \text {Li}_4\left (-e^{2 c+2 d x}\right )}{4 a^2 d^4}-\frac {3 b f^3 \text {Li}_4\left (e^{2 c+2 d x}\right )}{4 a^2 d^4}+\frac {(3 i f) \int (e+f x)^2 \log \left (1-i e^{c+d x}\right ) \, dx}{a d}-\frac {(3 i f) \int (e+f x)^2 \log \left (1+i e^{c+d x}\right ) \, dx}{a d}+\frac {\left (6 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}-\frac {\left (6 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}-\frac {\left (6 i b^2 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}+\frac {\left (6 i b^2 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}-\frac {\left (3 b^3 f^3\right ) \int \text {Li}_3\left (-e^{2 (c+d x)}\right ) \, dx}{2 a^2 \left (a^2+b^2\right ) d^3}\\ &=-\frac {2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {2 b^2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}-\frac {6 f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac {2 b (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{a d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {b^3 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {6 f^2 (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^3}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a d^2}-\frac {3 i b^2 f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{a d^2}+\frac {3 i b^2 f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {6 f^2 (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^3}+\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 a^2 \left (a^2+b^2\right ) d^2}+\frac {3 b f (e+f x)^2 \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac {3 b f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a^2 d^2}+\frac {6 f^3 \text {Li}_3\left (-e^{c+d x}\right )}{a d^4}+\frac {6 i b^2 f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {6 i b^2 f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {6 f^3 \text {Li}_3\left (e^{c+d x}\right )}{a d^4}-\frac {6 b^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {6 b^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {3 b^3 f^2 (e+f x) \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 a^2 \left (a^2+b^2\right ) d^3}-\frac {3 b f^2 (e+f x) \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a^2 d^3}+\frac {3 b f^2 (e+f x) \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a^2 d^3}-\frac {6 i b^2 f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}+\frac {6 i b^2 f^3 \text {Li}_4\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}+\frac {6 b^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^4}+\frac {6 b^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^4}+\frac {3 b f^3 \text {Li}_4\left (-e^{2 c+2 d x}\right )}{4 a^2 d^4}-\frac {3 b f^3 \text {Li}_4\left (e^{2 c+2 d x}\right )}{4 a^2 d^4}-\frac {\left (6 i f^2\right ) \int (e+f x) \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{a d^2}+\frac {\left (6 i f^2\right ) \int (e+f x) \text {Li}_2\left (i e^{c+d x}\right ) \, dx}{a d^2}-\frac {\left (3 b^3 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{4 a^2 \left (a^2+b^2\right ) d^4}\\ &=-\frac {2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {2 b^2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}-\frac {6 f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac {2 b (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{a d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {b^3 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {6 f^2 (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^3}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a d^2}-\frac {3 i b^2 f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{a d^2}+\frac {3 i b^2 f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {6 f^2 (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^3}+\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 a^2 \left (a^2+b^2\right ) d^2}+\frac {3 b f (e+f x)^2 \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac {3 b f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a^2 d^2}+\frac {6 f^3 \text {Li}_3\left (-e^{c+d x}\right )}{a d^4}-\frac {6 i f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{a d^3}+\frac {6 i b^2 f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {6 i f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{a d^3}-\frac {6 i b^2 f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {6 f^3 \text {Li}_3\left (e^{c+d x}\right )}{a d^4}-\frac {6 b^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {6 b^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {3 b^3 f^2 (e+f x) \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 a^2 \left (a^2+b^2\right ) d^3}-\frac {3 b f^2 (e+f x) \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a^2 d^3}+\frac {3 b f^2 (e+f x) \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a^2 d^3}-\frac {6 i b^2 f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}+\frac {6 i b^2 f^3 \text {Li}_4\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}+\frac {6 b^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^4}+\frac {6 b^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^4}-\frac {3 b^3 f^3 \text {Li}_4\left (-e^{2 (c+d x)}\right )}{4 a^2 \left (a^2+b^2\right ) d^4}+\frac {3 b f^3 \text {Li}_4\left (-e^{2 c+2 d x}\right )}{4 a^2 d^4}-\frac {3 b f^3 \text {Li}_4\left (e^{2 c+2 d x}\right )}{4 a^2 d^4}+\frac {\left (6 i f^3\right ) \int \text {Li}_3\left (-i e^{c+d x}\right ) \, dx}{a d^3}-\frac {\left (6 i f^3\right ) \int \text {Li}_3\left (i e^{c+d x}\right ) \, dx}{a d^3}\\ &=-\frac {2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {2 b^2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}-\frac {6 f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac {2 b (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{a d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {b^3 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {6 f^2 (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^3}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a d^2}-\frac {3 i b^2 f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{a d^2}+\frac {3 i b^2 f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {6 f^2 (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^3}+\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 a^2 \left (a^2+b^2\right ) d^2}+\frac {3 b f (e+f x)^2 \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac {3 b f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a^2 d^2}+\frac {6 f^3 \text {Li}_3\left (-e^{c+d x}\right )}{a d^4}-\frac {6 i f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{a d^3}+\frac {6 i b^2 f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {6 i f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{a d^3}-\frac {6 i b^2 f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {6 f^3 \text {Li}_3\left (e^{c+d x}\right )}{a d^4}-\frac {6 b^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {6 b^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {3 b^3 f^2 (e+f x) \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 a^2 \left (a^2+b^2\right ) d^3}-\frac {3 b f^2 (e+f x) \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a^2 d^3}+\frac {3 b f^2 (e+f x) \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a^2 d^3}-\frac {6 i b^2 f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}+\frac {6 i b^2 f^3 \text {Li}_4\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}+\frac {6 b^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^4}+\frac {6 b^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^4}-\frac {3 b^3 f^3 \text {Li}_4\left (-e^{2 (c+d x)}\right )}{4 a^2 \left (a^2+b^2\right ) d^4}+\frac {3 b f^3 \text {Li}_4\left (-e^{2 c+2 d x}\right )}{4 a^2 d^4}-\frac {3 b f^3 \text {Li}_4\left (e^{2 c+2 d x}\right )}{4 a^2 d^4}+\frac {\left (6 i f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}-\frac {\left (6 i f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}\\ &=-\frac {2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {2 b^2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}-\frac {6 f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac {2 b (e+f x)^3 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {(e+f x)^3 \text {csch}(c+d x)}{a d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {b^3 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {6 f^2 (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^3}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a d^2}-\frac {3 i b^2 f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{a d^2}+\frac {3 i b^2 f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {6 f^2 (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^3}+\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 a^2 \left (a^2+b^2\right ) d^2}+\frac {3 b f (e+f x)^2 \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac {3 b f (e+f x)^2 \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a^2 d^2}+\frac {6 f^3 \text {Li}_3\left (-e^{c+d x}\right )}{a d^4}-\frac {6 i f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{a d^3}+\frac {6 i b^2 f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {6 i f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{a d^3}-\frac {6 i b^2 f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {6 f^3 \text {Li}_3\left (e^{c+d x}\right )}{a d^4}-\frac {6 b^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {6 b^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {3 b^3 f^2 (e+f x) \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 a^2 \left (a^2+b^2\right ) d^3}-\frac {3 b f^2 (e+f x) \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a^2 d^3}+\frac {3 b f^2 (e+f x) \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a^2 d^3}+\frac {6 i f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{a d^4}-\frac {6 i b^2 f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}-\frac {6 i f^3 \text {Li}_4\left (i e^{c+d x}\right )}{a d^4}+\frac {6 i b^2 f^3 \text {Li}_4\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}+\frac {6 b^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^4}+\frac {6 b^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^4}-\frac {3 b^3 f^3 \text {Li}_4\left (-e^{2 (c+d x)}\right )}{4 a^2 \left (a^2+b^2\right ) d^4}+\frac {3 b f^3 \text {Li}_4\left (-e^{2 c+2 d x}\right )}{4 a^2 d^4}-\frac {3 b f^3 \text {Li}_4\left (e^{2 c+2 d x}\right )}{4 a^2 d^4}\\ \end {align*}

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Mathematica [B]  time = 17.19, size = 4010, normalized size = 2.81 \[ \text {Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-8*b*d^4*e^3*E^(2*c)*x - 12*b*d^4*e^2*E^(2*c)*f*x^2 - 8*b*d^4*e*E^(2*c)*f^2*x^3 - 2*b*d^4*E^(2*c)*f^3*x^4 - 8
*a*d^3*e^3*ArcTan[E^(c + d*x)] - 8*a*d^3*e^3*E^(2*c)*ArcTan[E^(c + d*x)] - (12*I)*a*d^3*e^2*f*x*Log[1 - I*E^(c
 + d*x)] - (12*I)*a*d^3*e^2*E^(2*c)*f*x*Log[1 - I*E^(c + d*x)] - (12*I)*a*d^3*e*f^2*x^2*Log[1 - I*E^(c + d*x)]
 - (12*I)*a*d^3*e*E^(2*c)*f^2*x^2*Log[1 - I*E^(c + d*x)] - (4*I)*a*d^3*f^3*x^3*Log[1 - I*E^(c + d*x)] - (4*I)*
a*d^3*E^(2*c)*f^3*x^3*Log[1 - I*E^(c + d*x)] + (12*I)*a*d^3*e^2*f*x*Log[1 + I*E^(c + d*x)] + (12*I)*a*d^3*e^2*
E^(2*c)*f*x*Log[1 + I*E^(c + d*x)] + (12*I)*a*d^3*e*f^2*x^2*Log[1 + I*E^(c + d*x)] + (12*I)*a*d^3*e*E^(2*c)*f^
2*x^2*Log[1 + I*E^(c + d*x)] + (4*I)*a*d^3*f^3*x^3*Log[1 + I*E^(c + d*x)] + (4*I)*a*d^3*E^(2*c)*f^3*x^3*Log[1
+ I*E^(c + d*x)] + 4*b*d^3*e^3*Log[1 + E^(2*(c + d*x))] + 4*b*d^3*e^3*E^(2*c)*Log[1 + E^(2*(c + d*x))] + 12*b*
d^3*e^2*f*x*Log[1 + E^(2*(c + d*x))] + 12*b*d^3*e^2*E^(2*c)*f*x*Log[1 + E^(2*(c + d*x))] + 12*b*d^3*e*f^2*x^2*
Log[1 + E^(2*(c + d*x))] + 12*b*d^3*e*E^(2*c)*f^2*x^2*Log[1 + E^(2*(c + d*x))] + 4*b*d^3*f^3*x^3*Log[1 + E^(2*
(c + d*x))] + 4*b*d^3*E^(2*c)*f^3*x^3*Log[1 + E^(2*(c + d*x))] + (12*I)*a*d^2*(1 + E^(2*c))*f*(e + f*x)^2*Poly
Log[2, (-I)*E^(c + d*x)] - (12*I)*a*d^2*(1 + E^(2*c))*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)] + 6*b*d^2*e^2*f*
PolyLog[2, -E^(2*(c + d*x))] + 6*b*d^2*e^2*E^(2*c)*f*PolyLog[2, -E^(2*(c + d*x))] + 12*b*d^2*e*f^2*x*PolyLog[2
, -E^(2*(c + d*x))] + 12*b*d^2*e*E^(2*c)*f^2*x*PolyLog[2, -E^(2*(c + d*x))] + 6*b*d^2*f^3*x^2*PolyLog[2, -E^(2
*(c + d*x))] + 6*b*d^2*E^(2*c)*f^3*x^2*PolyLog[2, -E^(2*(c + d*x))] - (24*I)*a*d*e*f^2*PolyLog[3, (-I)*E^(c +
d*x)] - (24*I)*a*d*e*E^(2*c)*f^2*PolyLog[3, (-I)*E^(c + d*x)] - (24*I)*a*d*f^3*x*PolyLog[3, (-I)*E^(c + d*x)]
- (24*I)*a*d*E^(2*c)*f^3*x*PolyLog[3, (-I)*E^(c + d*x)] + (24*I)*a*d*e*f^2*PolyLog[3, I*E^(c + d*x)] + (24*I)*
a*d*e*E^(2*c)*f^2*PolyLog[3, I*E^(c + d*x)] + (24*I)*a*d*f^3*x*PolyLog[3, I*E^(c + d*x)] + (24*I)*a*d*E^(2*c)*
f^3*x*PolyLog[3, I*E^(c + d*x)] - 6*b*d*e*f^2*PolyLog[3, -E^(2*(c + d*x))] - 6*b*d*e*E^(2*c)*f^2*PolyLog[3, -E
^(2*(c + d*x))] - 6*b*d*f^3*x*PolyLog[3, -E^(2*(c + d*x))] - 6*b*d*E^(2*c)*f^3*x*PolyLog[3, -E^(2*(c + d*x))]
+ (24*I)*a*f^3*PolyLog[4, (-I)*E^(c + d*x)] + (24*I)*a*E^(2*c)*f^3*PolyLog[4, (-I)*E^(c + d*x)] - (24*I)*a*f^3
*PolyLog[4, I*E^(c + d*x)] - (24*I)*a*E^(2*c)*f^3*PolyLog[4, I*E^(c + d*x)] + 3*b*f^3*PolyLog[4, -E^(2*(c + d*
x))] + 3*b*E^(2*c)*f^3*PolyLog[4, -E^(2*(c + d*x))])/(4*(a^2 + b^2)*d^4*(1 + E^(2*c))) - (b^3*(4*e^3*E^(2*c)*x
 + 6*e^2*E^(2*c)*f*x^2 + 4*e*E^(2*c)*f^2*x^3 + E^(2*c)*f^3*x^4 + (4*a*Sqrt[-(a^2 + b^2)^2]*e^3*E^(2*c)*ArcTan[
(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/((a^2 + b^2)^(3/2)*d) + (4*a*Sqrt[-(a^2 + b^2)^2]*e^3*E^(2*c)*ArcTanh[(
a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/((-a^2 - b^2)^(3/2)*d) + (2*e^3*Log[b - 2*a*E^(c + d*x) - b*E^(2*(c + d*x
))])/d - (2*e^3*E^(2*c)*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))])/d + (6*e^2*f*x*Log[1 + (b*E^(2*c + d*
x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e^2*E^(2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2
+ b^2)*E^(2*c)])])/d + (6*e*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e*E
^(2*c)*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (2*f^3*x^3*Log[1 + (b*E^(2*
c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (2*E^(2*c)*f^3*x^3*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt
[(a^2 + b^2)*E^(2*c)])])/d + (6*e^2*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6
*e^2*E^(2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (6*e*f^2*x^2*Log[1 + (b*E
^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e*E^(2*c)*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c
+ Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (2*f^3*x^3*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/
d - (2*E^(2*c)*f^3*x^3*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*(-1 + E^(2*c))*f
*(e + f*x)^2*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^2 - (6*(-1 + E^(2*c))*f*(
e + f*x)^2*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^2 - (12*e*f^2*PolyLog[3, -(
(b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (12*e*E^(2*c)*f^2*PolyLog[3, -((b*E^(2*c + d*x)
)/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 - (12*f^3*x*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b
^2)*E^(2*c)]))])/d^3 + (12*E^(2*c)*f^3*x*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])
/d^3 - (12*e*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (12*e*E^(2*c)*f^2
*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 - (12*f^3*x*PolyLog[3, -((b*E^(2*c
+ d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (12*E^(2*c)*f^3*x*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c +
 Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (12*f^3*PolyLog[4, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]
))])/d^4 - (12*E^(2*c)*f^3*PolyLog[4, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^4 + (12*f^3
*PolyLog[4, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^4 - (12*E^(2*c)*f^3*PolyLog[4, -((b*E
^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^4))/(2*a^2*(a^2 + b^2)*(-1 + E^(2*c))) + ((b*(e + f*x)^
4*(-1 + Coth[c]))/(2*f) + (2*e^2*(b*d*e - 3*a*f)*(d*x - Log[1 - Cosh[c + d*x] - Sinh[c + d*x]]))/d^2 - (6*e*f*
(b*d*e + 2*a*f)*x*Log[1 + Cosh[c + d*x] - Sinh[c + d*x]])/d^2 - (6*f^2*(b*d*e + a*f)*x^2*Log[1 + Cosh[c + d*x]
 - Sinh[c + d*x]])/d^2 - (2*b*f^3*x^3*Log[1 + Cosh[c + d*x] - Sinh[c + d*x]])/d - (6*e*f*(b*d*e - 2*a*f)*x*Log
[1 - Cosh[c + d*x] + Sinh[c + d*x]])/d^2 + (6*f^2*(-(b*d*e) + a*f)*x^2*Log[1 - Cosh[c + d*x] + Sinh[c + d*x]])
/d^2 - (2*b*f^3*x^3*Log[1 - Cosh[c + d*x] + Sinh[c + d*x]])/d + (2*e^2*(b*d*e + 3*a*f)*(d*x - Log[1 + Cosh[c +
 d*x] + Sinh[c + d*x]]))/d^2 + (6*e*f*(b*d*e - 2*a*f)*PolyLog[2, Cosh[c + d*x] - Sinh[c + d*x]])/d^3 + (6*e*f*
(b*d*e + 2*a*f)*PolyLog[2, -Cosh[c + d*x] + Sinh[c + d*x]])/d^3 + (12*f^2*(b*d*e - a*f)*(d*x*PolyLog[2, Cosh[c
 + d*x] - Sinh[c + d*x]] + PolyLog[3, Cosh[c + d*x] - Sinh[c + d*x]]))/d^4 + (12*f^2*(b*d*e + a*f)*(d*x*PolyLo
g[2, -Cosh[c + d*x] + Sinh[c + d*x]] + PolyLog[3, -Cosh[c + d*x] + Sinh[c + d*x]]))/d^4 + (6*b*f^3*(d^2*x^2*Po
lyLog[2, Cosh[c + d*x] - Sinh[c + d*x]] + 2*(d*x*PolyLog[3, Cosh[c + d*x] - Sinh[c + d*x]] + PolyLog[4, Cosh[c
 + d*x] - Sinh[c + d*x]])))/d^4 + (6*b*f^3*(d^2*x^2*PolyLog[2, -Cosh[c + d*x] + Sinh[c + d*x]] + 2*(d*x*PolyLo
g[3, -Cosh[c + d*x] + Sinh[c + d*x]] + PolyLog[4, -Cosh[c + d*x] + Sinh[c + d*x]])))/d^4)/(2*a^2) + ((-4*a*b*d
*e^3*x - 6*a*b*d*e^2*f*x^2 - 4*a*b*d*e*f^2*x^3 - a*b*d*f^3*x^4 - 4*a^2*e^3*Cosh[c] - 4*b^2*e^3*Cosh[c] - 12*a^
2*e^2*f*x*Cosh[c] - 12*b^2*e^2*f*x*Cosh[c] - 12*a^2*e*f^2*x^2*Cosh[c] - 12*b^2*e*f^2*x^2*Cosh[c] - 4*a^2*f^3*x
^3*Cosh[c] - 4*b^2*f^3*x^3*Cosh[c])*Csch[c/2]*Sech[c/2]*Sech[c])/(8*a*(a^2 + b^2)*d) + (Csch[c/2]*Csch[c/2 + (
d*x)/2]*(e^3*Sinh[(d*x)/2] + 3*e^2*f*x*Sinh[(d*x)/2] + 3*e*f^2*x^2*Sinh[(d*x)/2] + f^3*x^3*Sinh[(d*x)/2]))/(2*
a*d) + (Sech[c/2]*Sech[c/2 + (d*x)/2]*(e^3*Sinh[(d*x)/2] + 3*e^2*f*x*Sinh[(d*x)/2] + 3*e*f^2*x^2*Sinh[(d*x)/2]
 + f^3*x^3*Sinh[(d*x)/2]))/(2*a*d)

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fricas [C]  time = 1.02, size = 9779, normalized size = 6.85 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*((a^3 + a*b^2)*d^3*f^3*x^3 + 3*(a^3 + a*b^2)*d^3*e*f^2*x^2 + 3*(a^3 + a*b^2)*d^3*e^2*f*x + (a^3 + a*b^2)*d
^3*e^3)*cosh(d*x + c) + 3*(b^3*d^2*f^3*x^2 + 2*b^3*d^2*e*f^2*x + b^3*d^2*e^2*f - (b^3*d^2*f^3*x^2 + 2*b^3*d^2*
e*f^2*x + b^3*d^2*e^2*f)*cosh(d*x + c)^2 - 2*(b^3*d^2*f^3*x^2 + 2*b^3*d^2*e*f^2*x + b^3*d^2*e^2*f)*cosh(d*x +
c)*sinh(d*x + c) - (b^3*d^2*f^3*x^2 + 2*b^3*d^2*e*f^2*x + b^3*d^2*e^2*f)*sinh(d*x + c)^2)*dilog((a*cosh(d*x +
c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 3*(b^3*d^2*f^3*
x^2 + 2*b^3*d^2*e*f^2*x + b^3*d^2*e^2*f - (b^3*d^2*f^3*x^2 + 2*b^3*d^2*e*f^2*x + b^3*d^2*e^2*f)*cosh(d*x + c)^
2 - 2*(b^3*d^2*f^3*x^2 + 2*b^3*d^2*e*f^2*x + b^3*d^2*e^2*f)*cosh(d*x + c)*sinh(d*x + c) - (b^3*d^2*f^3*x^2 + 2
*b^3*d^2*e*f^2*x + b^3*d^2*e^2*f)*sinh(d*x + c)^2)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c)
 + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - 3*((a^2*b + b^3)*d^2*f^3*x^2 + (a^2*b + b^3)*d^2*e^2*f
 - 2*(a^3 + a*b^2)*d*e*f^2 - ((a^2*b + b^3)*d^2*f^3*x^2 + (a^2*b + b^3)*d^2*e^2*f - 2*(a^3 + a*b^2)*d*e*f^2 +
2*((a^2*b + b^3)*d^2*e*f^2 - (a^3 + a*b^2)*d*f^3)*x)*cosh(d*x + c)^2 - 2*((a^2*b + b^3)*d^2*f^3*x^2 + (a^2*b +
 b^3)*d^2*e^2*f - 2*(a^3 + a*b^2)*d*e*f^2 + 2*((a^2*b + b^3)*d^2*e*f^2 - (a^3 + a*b^2)*d*f^3)*x)*cosh(d*x + c)
*sinh(d*x + c) - ((a^2*b + b^3)*d^2*f^3*x^2 + (a^2*b + b^3)*d^2*e^2*f - 2*(a^3 + a*b^2)*d*e*f^2 + 2*((a^2*b +
b^3)*d^2*e*f^2 - (a^3 + a*b^2)*d*f^3)*x)*sinh(d*x + c)^2 + 2*((a^2*b + b^3)*d^2*e*f^2 - (a^3 + a*b^2)*d*f^3)*x
)*dilog(cosh(d*x + c) + sinh(d*x + c)) - (3*I*a^3*d^2*f^3*x^2 - 3*a^2*b*d^2*f^3*x^2 + 6*I*a^3*d^2*e*f^2*x - 6*
a^2*b*d^2*e*f^2*x + 3*I*a^3*d^2*e^2*f - 3*a^2*b*d^2*e^2*f + (-3*I*a^3*d^2*f^3*x^2 + 3*a^2*b*d^2*f^3*x^2 - 6*I*
a^3*d^2*e*f^2*x + 6*a^2*b*d^2*e*f^2*x - 3*I*a^3*d^2*e^2*f + 3*a^2*b*d^2*e^2*f)*cosh(d*x + c)^2 + (-6*I*a^3*d^2
*f^3*x^2 + 6*a^2*b*d^2*f^3*x^2 - 12*I*a^3*d^2*e*f^2*x + 12*a^2*b*d^2*e*f^2*x - 6*I*a^3*d^2*e^2*f + 6*a^2*b*d^2
*e^2*f)*cosh(d*x + c)*sinh(d*x + c) + (-3*I*a^3*d^2*f^3*x^2 + 3*a^2*b*d^2*f^3*x^2 - 6*I*a^3*d^2*e*f^2*x + 6*a^
2*b*d^2*e*f^2*x - 3*I*a^3*d^2*e^2*f + 3*a^2*b*d^2*e^2*f)*sinh(d*x + c)^2)*dilog(I*cosh(d*x + c) + I*sinh(d*x +
 c)) - (-3*I*a^3*d^2*f^3*x^2 - 3*a^2*b*d^2*f^3*x^2 - 6*I*a^3*d^2*e*f^2*x - 6*a^2*b*d^2*e*f^2*x - 3*I*a^3*d^2*e
^2*f - 3*a^2*b*d^2*e^2*f + (3*I*a^3*d^2*f^3*x^2 + 3*a^2*b*d^2*f^3*x^2 + 6*I*a^3*d^2*e*f^2*x + 6*a^2*b*d^2*e*f^
2*x + 3*I*a^3*d^2*e^2*f + 3*a^2*b*d^2*e^2*f)*cosh(d*x + c)^2 + (6*I*a^3*d^2*f^3*x^2 + 6*a^2*b*d^2*f^3*x^2 + 12
*I*a^3*d^2*e*f^2*x + 12*a^2*b*d^2*e*f^2*x + 6*I*a^3*d^2*e^2*f + 6*a^2*b*d^2*e^2*f)*cosh(d*x + c)*sinh(d*x + c)
 + (3*I*a^3*d^2*f^3*x^2 + 3*a^2*b*d^2*f^3*x^2 + 6*I*a^3*d^2*e*f^2*x + 6*a^2*b*d^2*e*f^2*x + 3*I*a^3*d^2*e^2*f
+ 3*a^2*b*d^2*e^2*f)*sinh(d*x + c)^2)*dilog(-I*cosh(d*x + c) - I*sinh(d*x + c)) - 3*((a^2*b + b^3)*d^2*f^3*x^2
 + (a^2*b + b^3)*d^2*e^2*f + 2*(a^3 + a*b^2)*d*e*f^2 - ((a^2*b + b^3)*d^2*f^3*x^2 + (a^2*b + b^3)*d^2*e^2*f +
2*(a^3 + a*b^2)*d*e*f^2 + 2*((a^2*b + b^3)*d^2*e*f^2 + (a^3 + a*b^2)*d*f^3)*x)*cosh(d*x + c)^2 - 2*((a^2*b + b
^3)*d^2*f^3*x^2 + (a^2*b + b^3)*d^2*e^2*f + 2*(a^3 + a*b^2)*d*e*f^2 + 2*((a^2*b + b^3)*d^2*e*f^2 + (a^3 + a*b^
2)*d*f^3)*x)*cosh(d*x + c)*sinh(d*x + c) - ((a^2*b + b^3)*d^2*f^3*x^2 + (a^2*b + b^3)*d^2*e^2*f + 2*(a^3 + a*b
^2)*d*e*f^2 + 2*((a^2*b + b^3)*d^2*e*f^2 + (a^3 + a*b^2)*d*f^3)*x)*sinh(d*x + c)^2 + 2*((a^2*b + b^3)*d^2*e*f^
2 + (a^3 + a*b^2)*d*f^3)*x)*dilog(-cosh(d*x + c) - sinh(d*x + c)) + (b^3*d^3*e^3 - 3*b^3*c*d^2*e^2*f + 3*b^3*c
^2*d*e*f^2 - b^3*c^3*f^3 - (b^3*d^3*e^3 - 3*b^3*c*d^2*e^2*f + 3*b^3*c^2*d*e*f^2 - b^3*c^3*f^3)*cosh(d*x + c)^2
 - 2*(b^3*d^3*e^3 - 3*b^3*c*d^2*e^2*f + 3*b^3*c^2*d*e*f^2 - b^3*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c) - (b^3*d^
3*e^3 - 3*b^3*c*d^2*e^2*f + 3*b^3*c^2*d*e*f^2 - b^3*c^3*f^3)*sinh(d*x + c)^2)*log(2*b*cosh(d*x + c) + 2*b*sinh
(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + (b^3*d^3*e^3 - 3*b^3*c*d^2*e^2*f + 3*b^3*c^2*d*e*f^2 - b^3*c^3*
f^3 - (b^3*d^3*e^3 - 3*b^3*c*d^2*e^2*f + 3*b^3*c^2*d*e*f^2 - b^3*c^3*f^3)*cosh(d*x + c)^2 - 2*(b^3*d^3*e^3 - 3
*b^3*c*d^2*e^2*f + 3*b^3*c^2*d*e*f^2 - b^3*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c) - (b^3*d^3*e^3 - 3*b^3*c*d^2*e
^2*f + 3*b^3*c^2*d*e*f^2 - b^3*c^3*f^3)*sinh(d*x + c)^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt(
(a^2 + b^2)/b^2) + 2*a) + (b^3*d^3*f^3*x^3 + 3*b^3*d^3*e*f^2*x^2 + 3*b^3*d^3*e^2*f*x + 3*b^3*c*d^2*e^2*f - 3*b
^3*c^2*d*e*f^2 + b^3*c^3*f^3 - (b^3*d^3*f^3*x^3 + 3*b^3*d^3*e*f^2*x^2 + 3*b^3*d^3*e^2*f*x + 3*b^3*c*d^2*e^2*f
- 3*b^3*c^2*d*e*f^2 + b^3*c^3*f^3)*cosh(d*x + c)^2 - 2*(b^3*d^3*f^3*x^3 + 3*b^3*d^3*e*f^2*x^2 + 3*b^3*d^3*e^2*
f*x + 3*b^3*c*d^2*e^2*f - 3*b^3*c^2*d*e*f^2 + b^3*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c) - (b^3*d^3*f^3*x^3 + 3*
b^3*d^3*e*f^2*x^2 + 3*b^3*d^3*e^2*f*x + 3*b^3*c*d^2*e^2*f - 3*b^3*c^2*d*e*f^2 + b^3*c^3*f^3)*sinh(d*x + c)^2)*
log(-(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) +
(b^3*d^3*f^3*x^3 + 3*b^3*d^3*e*f^2*x^2 + 3*b^3*d^3*e^2*f*x + 3*b^3*c*d^2*e^2*f - 3*b^3*c^2*d*e*f^2 + b^3*c^3*f
^3 - (b^3*d^3*f^3*x^3 + 3*b^3*d^3*e*f^2*x^2 + 3*b^3*d^3*e^2*f*x + 3*b^3*c*d^2*e^2*f - 3*b^3*c^2*d*e*f^2 + b^3*
c^3*f^3)*cosh(d*x + c)^2 - 2*(b^3*d^3*f^3*x^3 + 3*b^3*d^3*e*f^2*x^2 + 3*b^3*d^3*e^2*f*x + 3*b^3*c*d^2*e^2*f -
3*b^3*c^2*d*e*f^2 + b^3*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c) - (b^3*d^3*f^3*x^3 + 3*b^3*d^3*e*f^2*x^2 + 3*b^3*
d^3*e^2*f*x + 3*b^3*c*d^2*e^2*f - 3*b^3*c^2*d*e*f^2 + b^3*c^3*f^3)*sinh(d*x + c)^2)*log(-(a*cosh(d*x + c) + a*
sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) - ((a^2*b + b^3)*d^3*f^3*x^3
 + (a^2*b + b^3)*d^3*e^3 + 3*(a^3 + a*b^2)*d^2*e^2*f + 3*((a^2*b + b^3)*d^3*e*f^2 + (a^3 + a*b^2)*d^2*f^3)*x^2
 - ((a^2*b + b^3)*d^3*f^3*x^3 + (a^2*b + b^3)*d^3*e^3 + 3*(a^3 + a*b^2)*d^2*e^2*f + 3*((a^2*b + b^3)*d^3*e*f^2
 + (a^3 + a*b^2)*d^2*f^3)*x^2 + 3*((a^2*b + b^3)*d^3*e^2*f + 2*(a^3 + a*b^2)*d^2*e*f^2)*x)*cosh(d*x + c)^2 - 2
*((a^2*b + b^3)*d^3*f^3*x^3 + (a^2*b + b^3)*d^3*e^3 + 3*(a^3 + a*b^2)*d^2*e^2*f + 3*((a^2*b + b^3)*d^3*e*f^2 +
 (a^3 + a*b^2)*d^2*f^3)*x^2 + 3*((a^2*b + b^3)*d^3*e^2*f + 2*(a^3 + a*b^2)*d^2*e*f^2)*x)*cosh(d*x + c)*sinh(d*
x + c) - ((a^2*b + b^3)*d^3*f^3*x^3 + (a^2*b + b^3)*d^3*e^3 + 3*(a^3 + a*b^2)*d^2*e^2*f + 3*((a^2*b + b^3)*d^3
*e*f^2 + (a^3 + a*b^2)*d^2*f^3)*x^2 + 3*((a^2*b + b^3)*d^3*e^2*f + 2*(a^3 + a*b^2)*d^2*e*f^2)*x)*sinh(d*x + c)
^2 + 3*((a^2*b + b^3)*d^3*e^2*f + 2*(a^3 + a*b^2)*d^2*e*f^2)*x)*log(cosh(d*x + c) + sinh(d*x + c) + 1) - (I*a^
3*d^3*e^3 - a^2*b*d^3*e^3 - 3*I*a^3*c*d^2*e^2*f + 3*a^2*b*c*d^2*e^2*f + 3*I*a^3*c^2*d*e*f^2 - 3*a^2*b*c^2*d*e*
f^2 - I*a^3*c^3*f^3 + a^2*b*c^3*f^3 + (-I*a^3*d^3*e^3 + a^2*b*d^3*e^3 + 3*I*a^3*c*d^2*e^2*f - 3*a^2*b*c*d^2*e^
2*f - 3*I*a^3*c^2*d*e*f^2 + 3*a^2*b*c^2*d*e*f^2 + I*a^3*c^3*f^3 - a^2*b*c^3*f^3)*cosh(d*x + c)^2 + (-2*I*a^3*d
^3*e^3 + 2*a^2*b*d^3*e^3 + 6*I*a^3*c*d^2*e^2*f - 6*a^2*b*c*d^2*e^2*f - 6*I*a^3*c^2*d*e*f^2 + 6*a^2*b*c^2*d*e*f
^2 + 2*I*a^3*c^3*f^3 - 2*a^2*b*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c) + (-I*a^3*d^3*e^3 + a^2*b*d^3*e^3 + 3*I*a^
3*c*d^2*e^2*f - 3*a^2*b*c*d^2*e^2*f - 3*I*a^3*c^2*d*e*f^2 + 3*a^2*b*c^2*d*e*f^2 + I*a^3*c^3*f^3 - a^2*b*c^3*f^
3)*sinh(d*x + c)^2)*log(cosh(d*x + c) + sinh(d*x + c) + I) - (-I*a^3*d^3*e^3 - a^2*b*d^3*e^3 + 3*I*a^3*c*d^2*e
^2*f + 3*a^2*b*c*d^2*e^2*f - 3*I*a^3*c^2*d*e*f^2 - 3*a^2*b*c^2*d*e*f^2 + I*a^3*c^3*f^3 + a^2*b*c^3*f^3 + (I*a^
3*d^3*e^3 + a^2*b*d^3*e^3 - 3*I*a^3*c*d^2*e^2*f - 3*a^2*b*c*d^2*e^2*f + 3*I*a^3*c^2*d*e*f^2 + 3*a^2*b*c^2*d*e*
f^2 - I*a^3*c^3*f^3 - a^2*b*c^3*f^3)*cosh(d*x + c)^2 + (2*I*a^3*d^3*e^3 + 2*a^2*b*d^3*e^3 - 6*I*a^3*c*d^2*e^2*
f - 6*a^2*b*c*d^2*e^2*f + 6*I*a^3*c^2*d*e*f^2 + 6*a^2*b*c^2*d*e*f^2 - 2*I*a^3*c^3*f^3 - 2*a^2*b*c^3*f^3)*cosh(
d*x + c)*sinh(d*x + c) + (I*a^3*d^3*e^3 + a^2*b*d^3*e^3 - 3*I*a^3*c*d^2*e^2*f - 3*a^2*b*c*d^2*e^2*f + 3*I*a^3*
c^2*d*e*f^2 + 3*a^2*b*c^2*d*e*f^2 - I*a^3*c^3*f^3 - a^2*b*c^3*f^3)*sinh(d*x + c)^2)*log(cosh(d*x + c) + sinh(d
*x + c) - I) - ((a^2*b + b^3)*d^3*e^3 - 3*(a^3 + a*b^2 + (a^2*b + b^3)*c)*d^2*e^2*f + 3*((a^2*b + b^3)*c^2 + 2
*(a^3 + a*b^2)*c)*d*e*f^2 - ((a^2*b + b^3)*c^3 + 3*(a^3 + a*b^2)*c^2)*f^3 - ((a^2*b + b^3)*d^3*e^3 - 3*(a^3 +
a*b^2 + (a^2*b + b^3)*c)*d^2*e^2*f + 3*((a^2*b + b^3)*c^2 + 2*(a^3 + a*b^2)*c)*d*e*f^2 - ((a^2*b + b^3)*c^3 +
3*(a^3 + a*b^2)*c^2)*f^3)*cosh(d*x + c)^2 - 2*((a^2*b + b^3)*d^3*e^3 - 3*(a^3 + a*b^2 + (a^2*b + b^3)*c)*d^2*e
^2*f + 3*((a^2*b + b^3)*c^2 + 2*(a^3 + a*b^2)*c)*d*e*f^2 - ((a^2*b + b^3)*c^3 + 3*(a^3 + a*b^2)*c^2)*f^3)*cosh
(d*x + c)*sinh(d*x + c) - ((a^2*b + b^3)*d^3*e^3 - 3*(a^3 + a*b^2 + (a^2*b + b^3)*c)*d^2*e^2*f + 3*((a^2*b + b
^3)*c^2 + 2*(a^3 + a*b^2)*c)*d*e*f^2 - ((a^2*b + b^3)*c^3 + 3*(a^3 + a*b^2)*c^2)*f^3)*sinh(d*x + c)^2)*log(cos
h(d*x + c) + sinh(d*x + c) - 1) - (-I*a^3*d^3*f^3*x^3 - a^2*b*d^3*f^3*x^3 - 3*I*a^3*d^3*e*f^2*x^2 - 3*a^2*b*d^
3*e*f^2*x^2 - 3*I*a^3*d^3*e^2*f*x - 3*a^2*b*d^3*e^2*f*x - 3*I*a^3*c*d^2*e^2*f - 3*a^2*b*c*d^2*e^2*f + 3*I*a^3*
c^2*d*e*f^2 + 3*a^2*b*c^2*d*e*f^2 - I*a^3*c^3*f^3 - a^2*b*c^3*f^3 + (I*a^3*d^3*f^3*x^3 + a^2*b*d^3*f^3*x^3 + 3
*I*a^3*d^3*e*f^2*x^2 + 3*a^2*b*d^3*e*f^2*x^2 + 3*I*a^3*d^3*e^2*f*x + 3*a^2*b*d^3*e^2*f*x + 3*I*a^3*c*d^2*e^2*f
 + 3*a^2*b*c*d^2*e^2*f - 3*I*a^3*c^2*d*e*f^2 - 3*a^2*b*c^2*d*e*f^2 + I*a^3*c^3*f^3 + a^2*b*c^3*f^3)*cosh(d*x +
 c)^2 + (2*I*a^3*d^3*f^3*x^3 + 2*a^2*b*d^3*f^3*x^3 + 6*I*a^3*d^3*e*f^2*x^2 + 6*a^2*b*d^3*e*f^2*x^2 + 6*I*a^3*d
^3*e^2*f*x + 6*a^2*b*d^3*e^2*f*x + 6*I*a^3*c*d^2*e^2*f + 6*a^2*b*c*d^2*e^2*f - 6*I*a^3*c^2*d*e*f^2 - 6*a^2*b*c
^2*d*e*f^2 + 2*I*a^3*c^3*f^3 + 2*a^2*b*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c) + (I*a^3*d^3*f^3*x^3 + a^2*b*d^3*f
^3*x^3 + 3*I*a^3*d^3*e*f^2*x^2 + 3*a^2*b*d^3*e*f^2*x^2 + 3*I*a^3*d^3*e^2*f*x + 3*a^2*b*d^3*e^2*f*x + 3*I*a^3*c
*d^2*e^2*f + 3*a^2*b*c*d^2*e^2*f - 3*I*a^3*c^2*d*e*f^2 - 3*a^2*b*c^2*d*e*f^2 + I*a^3*c^3*f^3 + a^2*b*c^3*f^3)*
sinh(d*x + c)^2)*log(I*cosh(d*x + c) + I*sinh(d*x + c) + 1) - (I*a^3*d^3*f^3*x^3 - a^2*b*d^3*f^3*x^3 + 3*I*a^3
*d^3*e*f^2*x^2 - 3*a^2*b*d^3*e*f^2*x^2 + 3*I*a^3*d^3*e^2*f*x - 3*a^2*b*d^3*e^2*f*x + 3*I*a^3*c*d^2*e^2*f - 3*a
^2*b*c*d^2*e^2*f - 3*I*a^3*c^2*d*e*f^2 + 3*a^2*b*c^2*d*e*f^2 + I*a^3*c^3*f^3 - a^2*b*c^3*f^3 + (-I*a^3*d^3*f^3
*x^3 + a^2*b*d^3*f^3*x^3 - 3*I*a^3*d^3*e*f^2*x^2 + 3*a^2*b*d^3*e*f^2*x^2 - 3*I*a^3*d^3*e^2*f*x + 3*a^2*b*d^3*e
^2*f*x - 3*I*a^3*c*d^2*e^2*f + 3*a^2*b*c*d^2*e^2*f + 3*I*a^3*c^2*d*e*f^2 - 3*a^2*b*c^2*d*e*f^2 - I*a^3*c^3*f^3
 + a^2*b*c^3*f^3)*cosh(d*x + c)^2 + (-2*I*a^3*d^3*f^3*x^3 + 2*a^2*b*d^3*f^3*x^3 - 6*I*a^3*d^3*e*f^2*x^2 + 6*a^
2*b*d^3*e*f^2*x^2 - 6*I*a^3*d^3*e^2*f*x + 6*a^2*b*d^3*e^2*f*x - 6*I*a^3*c*d^2*e^2*f + 6*a^2*b*c*d^2*e^2*f + 6*
I*a^3*c^2*d*e*f^2 - 6*a^2*b*c^2*d*e*f^2 - 2*I*a^3*c^3*f^3 + 2*a^2*b*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c) + (-I
*a^3*d^3*f^3*x^3 + a^2*b*d^3*f^3*x^3 - 3*I*a^3*d^3*e*f^2*x^2 + 3*a^2*b*d^3*e*f^2*x^2 - 3*I*a^3*d^3*e^2*f*x + 3
*a^2*b*d^3*e^2*f*x - 3*I*a^3*c*d^2*e^2*f + 3*a^2*b*c*d^2*e^2*f + 3*I*a^3*c^2*d*e*f^2 - 3*a^2*b*c^2*d*e*f^2 - I
*a^3*c^3*f^3 + a^2*b*c^3*f^3)*sinh(d*x + c)^2)*log(-I*cosh(d*x + c) - I*sinh(d*x + c) + 1) - ((a^2*b + b^3)*d^
3*f^3*x^3 + 3*(a^2*b + b^3)*c*d^2*e^2*f - 3*((a^2*b + b^3)*c^2 + 2*(a^3 + a*b^2)*c)*d*e*f^2 + ((a^2*b + b^3)*c
^3 + 3*(a^3 + a*b^2)*c^2)*f^3 + 3*((a^2*b + b^3)*d^3*e*f^2 - (a^3 + a*b^2)*d^2*f^3)*x^2 - ((a^2*b + b^3)*d^3*f
^3*x^3 + 3*(a^2*b + b^3)*c*d^2*e^2*f - 3*((a^2*b + b^3)*c^2 + 2*(a^3 + a*b^2)*c)*d*e*f^2 + ((a^2*b + b^3)*c^3
+ 3*(a^3 + a*b^2)*c^2)*f^3 + 3*((a^2*b + b^3)*d^3*e*f^2 - (a^3 + a*b^2)*d^2*f^3)*x^2 + 3*((a^2*b + b^3)*d^3*e^
2*f - 2*(a^3 + a*b^2)*d^2*e*f^2)*x)*cosh(d*x + c)^2 - 2*((a^2*b + b^3)*d^3*f^3*x^3 + 3*(a^2*b + b^3)*c*d^2*e^2
*f - 3*((a^2*b + b^3)*c^2 + 2*(a^3 + a*b^2)*c)*d*e*f^2 + ((a^2*b + b^3)*c^3 + 3*(a^3 + a*b^2)*c^2)*f^3 + 3*((a
^2*b + b^3)*d^3*e*f^2 - (a^3 + a*b^2)*d^2*f^3)*x^2 + 3*((a^2*b + b^3)*d^3*e^2*f - 2*(a^3 + a*b^2)*d^2*e*f^2)*x
)*cosh(d*x + c)*sinh(d*x + c) - ((a^2*b + b^3)*d^3*f^3*x^3 + 3*(a^2*b + b^3)*c*d^2*e^2*f - 3*((a^2*b + b^3)*c^
2 + 2*(a^3 + a*b^2)*c)*d*e*f^2 + ((a^2*b + b^3)*c^3 + 3*(a^3 + a*b^2)*c^2)*f^3 + 3*((a^2*b + b^3)*d^3*e*f^2 -
(a^3 + a*b^2)*d^2*f^3)*x^2 + 3*((a^2*b + b^3)*d^3*e^2*f - 2*(a^3 + a*b^2)*d^2*e*f^2)*x)*sinh(d*x + c)^2 + 3*((
a^2*b + b^3)*d^3*e^2*f - 2*(a^3 + a*b^2)*d^2*e*f^2)*x)*log(-cosh(d*x + c) - sinh(d*x + c) + 1) - 6*(b^3*f^3*co
sh(d*x + c)^2 + 2*b^3*f^3*cosh(d*x + c)*sinh(d*x + c) + b^3*f^3*sinh(d*x + c)^2 - b^3*f^3)*polylog(4, (a*cosh(
d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) - 6*(b^3*f^3*cosh(d
*x + c)^2 + 2*b^3*f^3*cosh(d*x + c)*sinh(d*x + c) + b^3*f^3*sinh(d*x + c)^2 - b^3*f^3)*polylog(4, (a*cosh(d*x
+ c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) + 6*((a^2*b + b^3)*f^3*
cosh(d*x + c)^2 + 2*(a^2*b + b^3)*f^3*cosh(d*x + c)*sinh(d*x + c) + (a^2*b + b^3)*f^3*sinh(d*x + c)^2 - (a^2*b
 + b^3)*f^3)*polylog(4, cosh(d*x + c) + sinh(d*x + c)) - (6*I*a^3*f^3 - 6*a^2*b*f^3 - 6*(I*a^3*f^3 - a^2*b*f^3
)*cosh(d*x + c)^2 - 12*(I*a^3*f^3 - a^2*b*f^3)*cosh(d*x + c)*sinh(d*x + c) - 6*(I*a^3*f^3 - a^2*b*f^3)*sinh(d*
x + c)^2)*polylog(4, I*cosh(d*x + c) + I*sinh(d*x + c)) - (-6*I*a^3*f^3 - 6*a^2*b*f^3 - 6*(-I*a^3*f^3 - a^2*b*
f^3)*cosh(d*x + c)^2 - 12*(-I*a^3*f^3 - a^2*b*f^3)*cosh(d*x + c)*sinh(d*x + c) - 6*(-I*a^3*f^3 - a^2*b*f^3)*si
nh(d*x + c)^2)*polylog(4, -I*cosh(d*x + c) - I*sinh(d*x + c)) + 6*((a^2*b + b^3)*f^3*cosh(d*x + c)^2 + 2*(a^2*
b + b^3)*f^3*cosh(d*x + c)*sinh(d*x + c) + (a^2*b + b^3)*f^3*sinh(d*x + c)^2 - (a^2*b + b^3)*f^3)*polylog(4, -
cosh(d*x + c) - sinh(d*x + c)) - 6*(b^3*d*f^3*x + b^3*d*e*f^2 - (b^3*d*f^3*x + b^3*d*e*f^2)*cosh(d*x + c)^2 -
2*(b^3*d*f^3*x + b^3*d*e*f^2)*cosh(d*x + c)*sinh(d*x + c) - (b^3*d*f^3*x + b^3*d*e*f^2)*sinh(d*x + c)^2)*polyl
og(3, (a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) - 6*(
b^3*d*f^3*x + b^3*d*e*f^2 - (b^3*d*f^3*x + b^3*d*e*f^2)*cosh(d*x + c)^2 - 2*(b^3*d*f^3*x + b^3*d*e*f^2)*cosh(d
*x + c)*sinh(d*x + c) - (b^3*d*f^3*x + b^3*d*e*f^2)*sinh(d*x + c)^2)*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x
+ c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) + 6*((a^2*b + b^3)*d*f^3*x + (a^2*b + b^3
)*d*e*f^2 - (a^3 + a*b^2)*f^3 - ((a^2*b + b^3)*d*f^3*x + (a^2*b + b^3)*d*e*f^2 - (a^3 + a*b^2)*f^3)*cosh(d*x +
 c)^2 - 2*((a^2*b + b^3)*d*f^3*x + (a^2*b + b^3)*d*e*f^2 - (a^3 + a*b^2)*f^3)*cosh(d*x + c)*sinh(d*x + c) - ((
a^2*b + b^3)*d*f^3*x + (a^2*b + b^3)*d*e*f^2 - (a^3 + a*b^2)*f^3)*sinh(d*x + c)^2)*polylog(3, cosh(d*x + c) +
sinh(d*x + c)) - (-6*I*a^3*d*f^3*x + 6*a^2*b*d*f^3*x - 6*I*a^3*d*e*f^2 + 6*a^2*b*d*e*f^2 + (6*I*a^3*d*f^3*x -
6*a^2*b*d*f^3*x + 6*I*a^3*d*e*f^2 - 6*a^2*b*d*e*f^2)*cosh(d*x + c)^2 + (12*I*a^3*d*f^3*x - 12*a^2*b*d*f^3*x +
12*I*a^3*d*e*f^2 - 12*a^2*b*d*e*f^2)*cosh(d*x + c)*sinh(d*x + c) + (6*I*a^3*d*f^3*x - 6*a^2*b*d*f^3*x + 6*I*a^
3*d*e*f^2 - 6*a^2*b*d*e*f^2)*sinh(d*x + c)^2)*polylog(3, I*cosh(d*x + c) + I*sinh(d*x + c)) - (6*I*a^3*d*f^3*x
 + 6*a^2*b*d*f^3*x + 6*I*a^3*d*e*f^2 + 6*a^2*b*d*e*f^2 + (-6*I*a^3*d*f^3*x - 6*a^2*b*d*f^3*x - 6*I*a^3*d*e*f^2
 - 6*a^2*b*d*e*f^2)*cosh(d*x + c)^2 + (-12*I*a^3*d*f^3*x - 12*a^2*b*d*f^3*x - 12*I*a^3*d*e*f^2 - 12*a^2*b*d*e*
f^2)*cosh(d*x + c)*sinh(d*x + c) + (-6*I*a^3*d*f^3*x - 6*a^2*b*d*f^3*x - 6*I*a^3*d*e*f^2 - 6*a^2*b*d*e*f^2)*si
nh(d*x + c)^2)*polylog(3, -I*cosh(d*x + c) - I*sinh(d*x + c)) + 6*((a^2*b + b^3)*d*f^3*x + (a^2*b + b^3)*d*e*f
^2 + (a^3 + a*b^2)*f^3 - ((a^2*b + b^3)*d*f^3*x + (a^2*b + b^3)*d*e*f^2 + (a^3 + a*b^2)*f^3)*cosh(d*x + c)^2 -
 2*((a^2*b + b^3)*d*f^3*x + (a^2*b + b^3)*d*e*f^2 + (a^3 + a*b^2)*f^3)*cosh(d*x + c)*sinh(d*x + c) - ((a^2*b +
 b^3)*d*f^3*x + (a^2*b + b^3)*d*e*f^2 + (a^3 + a*b^2)*f^3)*sinh(d*x + c)^2)*polylog(3, -cosh(d*x + c) - sinh(d
*x + c)) + 2*((a^3 + a*b^2)*d^3*f^3*x^3 + 3*(a^3 + a*b^2)*d^3*e*f^2*x^2 + 3*(a^3 + a*b^2)*d^3*e^2*f*x + (a^3 +
 a*b^2)*d^3*e^3)*sinh(d*x + c))/((a^4 + a^2*b^2)*d^4*cosh(d*x + c)^2 + 2*(a^4 + a^2*b^2)*d^4*cosh(d*x + c)*sin
h(d*x + c) + (a^4 + a^2*b^2)*d^4*sinh(d*x + c)^2 - (a^4 + a^2*b^2)*d^4)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [F]  time = 4.93, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{3} \mathrm {csch}\left (d x +c \right )^{2} \mathrm {sech}\left (d x +c \right )}{a +b \sinh \left (d x +c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^3*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^4 + a^2*b^2)*d) + 2*a*arctan(e^(-d*x - c))/((a^2 + b^
2)*d) + b*log(e^(-2*d*x - 2*c) + 1)/((a^2 + b^2)*d) + 2*e^(-d*x - c)/((a*e^(-2*d*x - 2*c) - a)*d) - b*log(e^(-
d*x - c) + 1)/(a^2*d) - b*log(e^(-d*x - c) - 1)/(a^2*d))*e^3 - 2*(f^3*x^3*e^c + 3*e*f^2*x^2*e^c + 3*e^2*f*x*e^
c)*e^(d*x)/(a*d*e^(2*d*x + 2*c) - a*d) - 3*e^2*f*log(e^(d*x + c) + 1)/(a*d^2) + 3*e^2*f*log(e^(d*x + c) - 1)/(
a*d^2) - (d^3*x^3*log(e^(d*x + c) + 1) + 3*d^2*x^2*dilog(-e^(d*x + c)) - 6*d*x*polylog(3, -e^(d*x + c)) + 6*po
lylog(4, -e^(d*x + c)))*b*f^3/(a^2*d^4) - (d^3*x^3*log(-e^(d*x + c) + 1) + 3*d^2*x^2*dilog(e^(d*x + c)) - 6*d*
x*polylog(3, e^(d*x + c)) + 6*polylog(4, e^(d*x + c)))*b*f^3/(a^2*d^4) - 3*(b*d*e^2*f + 2*a*e*f^2)*(d*x*log(e^
(d*x + c) + 1) + dilog(-e^(d*x + c)))/(a^2*d^3) - 3*(b*d*e^2*f - 2*a*e*f^2)*(d*x*log(-e^(d*x + c) + 1) + dilog
(e^(d*x + c)))/(a^2*d^3) - 3*(b*d*e*f^2 + a*f^3)*(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)))/(a^2*d^4) - 3*(b*d*e*f^2 - a*f^3)*(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)))/(a^2*d^4) + 1/4*(b*d^4*f^3*x^4 + 4*(b*d*e*f^2 + a*f^3)*d^3*x^3 + 6*(b*d^
2*e^2*f + 2*a*d*e*f^2)*d^2*x^2)/(a^2*d^4) + 1/4*(b*d^4*f^3*x^4 + 4*(b*d*e*f^2 - a*f^3)*d^3*x^3 + 6*(b*d^2*e^2*
f - 2*a*d*e*f^2)*d^2*x^2)/(a^2*d^4) - integrate(2*(b^4*f^3*x^3 + 3*b^4*e*f^2*x^2 + 3*b^4*e^2*f*x - (a*b^3*f^3*
x^3*e^c + 3*a*b^3*e*f^2*x^2*e^c + 3*a*b^3*e^2*f*x*e^c)*e^(d*x))/(a^4*b + a^2*b^3 - (a^4*b*e^(2*c) + a^2*b^3*e^
(2*c))*e^(2*d*x) - 2*(a^5*e^c + a^3*b^2*e^c)*e^(d*x)), x) - integrate(2*(b*f^3*x^3 + 3*b*e*f^2*x^2 + 3*b*e^2*f
*x + (a*f^3*x^3*e^c + 3*a*e*f^2*x^2*e^c + 3*a*e^2*f*x*e^c)*e^(d*x))/(a^2 + b^2 + (a^2*e^(2*c) + b^2*e^(2*c))*e
^(2*d*x)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e+f\,x\right )}^3}{\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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