3.47 \(\int (c+d x)^3 (a+b \coth (e+f x))^3 \, dx\)
Optimal. Leaf size=556 \[ \frac {a^3 (c+d x)^4}{4 d}-\frac {9 a^2 b d^2 (c+d x) \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^3}+\frac {9 a^2 b d (c+d x)^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}+\frac {3 a^2 b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {3 a^2 b (c+d x)^4}{4 d}+\frac {9 a^2 b d^3 \text {Li}_4\left (e^{2 (e+f x)}\right )}{4 f^4}+\frac {9 a b^2 d^2 (c+d x) \text {Li}_2\left (e^{2 (e+f x)}\right )}{f^3}+\frac {9 a b^2 d (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f^2}-\frac {3 a b^2 (c+d x)^3 \coth (e+f x)}{f}-\frac {3 a b^2 (c+d x)^3}{f}+\frac {3 a b^2 (c+d x)^4}{4 d}-\frac {9 a b^2 d^3 \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^4}-\frac {3 b^3 d^2 (c+d x) \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^3}+\frac {3 b^3 d^2 (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^3}+\frac {3 b^3 d (c+d x)^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}-\frac {3 b^3 d (c+d x)^2 \coth (e+f x)}{2 f^2}+\frac {b^3 (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {b^3 (c+d x)^3 \coth ^2(e+f x)}{2 f}-\frac {3 b^3 d (c+d x)^2}{2 f^2}+\frac {b^3 (c+d x)^3}{2 f}-\frac {b^3 (c+d x)^4}{4 d}+\frac {3 b^3 d^3 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^4}+\frac {3 b^3 d^3 \text {Li}_4\left (e^{2 (e+f x)}\right )}{4 f^4} \]
[Out]
-3/2*b^3*d*(d*x+c)^2/f^2-3*a*b^2*(d*x+c)^3/f+1/2*b^3*(d*x+c)^3/f+1/4*a^3*(d*x+c)^4/d-3/4*a^2*b*(d*x+c)^4/d+3/4
*a*b^2*(d*x+c)^4/d-1/4*b^3*(d*x+c)^4/d-3/2*b^3*d*(d*x+c)^2*coth(f*x+e)/f^2-3*a*b^2*(d*x+c)^3*coth(f*x+e)/f-1/2
*b^3*(d*x+c)^3*coth(f*x+e)^2/f+3*b^3*d^2*(d*x+c)*ln(1-exp(2*f*x+2*e))/f^3+9*a*b^2*d*(d*x+c)^2*ln(1-exp(2*f*x+2
*e))/f^2+3*a^2*b*(d*x+c)^3*ln(1-exp(2*f*x+2*e))/f+b^3*(d*x+c)^3*ln(1-exp(2*f*x+2*e))/f+3/2*b^3*d^3*polylog(2,e
xp(2*f*x+2*e))/f^4+9*a*b^2*d^2*(d*x+c)*polylog(2,exp(2*f*x+2*e))/f^3+9/2*a^2*b*d*(d*x+c)^2*polylog(2,exp(2*f*x
+2*e))/f^2+3/2*b^3*d*(d*x+c)^2*polylog(2,exp(2*f*x+2*e))/f^2-9/2*a*b^2*d^3*polylog(3,exp(2*f*x+2*e))/f^4-9/2*a
^2*b*d^2*(d*x+c)*polylog(3,exp(2*f*x+2*e))/f^3-3/2*b^3*d^2*(d*x+c)*polylog(3,exp(2*f*x+2*e))/f^3+9/4*a^2*b*d^3
*polylog(4,exp(2*f*x+2*e))/f^4+3/4*b^3*d^3*polylog(4,exp(2*f*x+2*e))/f^4
________________________________________________________________________________________
Rubi [A] time = 1.05, antiderivative size = 556, normalized size of antiderivative =
1.00, number of steps used = 28, number of rules used = 11, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) =
0.550, Rules used = {3722, 3716, 2190, 2531, 6609, 2282, 6589, 3720, 32, 2279, 2391}
\[ -\frac {9 a^2 b d^2 (c+d x) \text {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3}+\frac {9 a^2 b d (c+d x)^2 \text {PolyLog}\left (2,e^{2 (e+f x)}\right )}{2 f^2}+\frac {9 a^2 b d^3 \text {PolyLog}\left (4,e^{2 (e+f x)}\right )}{4 f^4}+\frac {9 a b^2 d^2 (c+d x) \text {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^3}-\frac {9 a b^2 d^3 \text {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^4}-\frac {3 b^3 d^2 (c+d x) \text {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3}+\frac {3 b^3 d (c+d x)^2 \text {PolyLog}\left (2,e^{2 (e+f x)}\right )}{2 f^2}+\frac {3 b^3 d^3 \text {PolyLog}\left (2,e^{2 (e+f x)}\right )}{2 f^4}+\frac {3 b^3 d^3 \text {PolyLog}\left (4,e^{2 (e+f x)}\right )}{4 f^4}+\frac {3 a^2 b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {3 a^2 b (c+d x)^4}{4 d}+\frac {a^3 (c+d x)^4}{4 d}+\frac {9 a b^2 d (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f^2}-\frac {3 a b^2 (c+d x)^3 \coth (e+f x)}{f}-\frac {3 a b^2 (c+d x)^3}{f}+\frac {3 a b^2 (c+d x)^4}{4 d}+\frac {3 b^3 d^2 (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^3}-\frac {3 b^3 d (c+d x)^2 \coth (e+f x)}{2 f^2}+\frac {b^3 (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {b^3 (c+d x)^3 \coth ^2(e+f x)}{2 f}-\frac {3 b^3 d (c+d x)^2}{2 f^2}+\frac {b^3 (c+d x)^3}{2 f}-\frac {b^3 (c+d x)^4}{4 d} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*x)^3*(a + b*Coth[e + f*x])^3,x]
[Out]
(-3*b^3*d*(c + d*x)^2)/(2*f^2) - (3*a*b^2*(c + d*x)^3)/f + (b^3*(c + d*x)^3)/(2*f) + (a^3*(c + d*x)^4)/(4*d) -
(3*a^2*b*(c + d*x)^4)/(4*d) + (3*a*b^2*(c + d*x)^4)/(4*d) - (b^3*(c + d*x)^4)/(4*d) - (3*b^3*d*(c + d*x)^2*Co
th[e + f*x])/(2*f^2) - (3*a*b^2*(c + d*x)^3*Coth[e + f*x])/f - (b^3*(c + d*x)^3*Coth[e + f*x]^2)/(2*f) + (3*b^
3*d^2*(c + d*x)*Log[1 - E^(2*(e + f*x))])/f^3 + (9*a*b^2*d*(c + d*x)^2*Log[1 - E^(2*(e + f*x))])/f^2 + (3*a^2*
b*(c + d*x)^3*Log[1 - E^(2*(e + f*x))])/f + (b^3*(c + d*x)^3*Log[1 - E^(2*(e + f*x))])/f + (3*b^3*d^3*PolyLog[
2, E^(2*(e + f*x))])/(2*f^4) + (9*a*b^2*d^2*(c + d*x)*PolyLog[2, E^(2*(e + f*x))])/f^3 + (9*a^2*b*d*(c + d*x)^
2*PolyLog[2, E^(2*(e + f*x))])/(2*f^2) + (3*b^3*d*(c + d*x)^2*PolyLog[2, E^(2*(e + f*x))])/(2*f^2) - (9*a*b^2*
d^3*PolyLog[3, E^(2*(e + f*x))])/(2*f^4) - (9*a^2*b*d^2*(c + d*x)*PolyLog[3, E^(2*(e + f*x))])/(2*f^3) - (3*b^
3*d^2*(c + d*x)*PolyLog[3, E^(2*(e + f*x))])/(2*f^3) + (9*a^2*b*d^3*PolyLog[4, E^(2*(e + f*x))])/(4*f^4) + (3*
b^3*d^3*PolyLog[4, E^(2*(e + f*x))])/(4*f^4)
Rule 32
Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]
Rule 2190
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Rule 2279
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Rule 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 2391
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]
Rule 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 3716
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c
+ d*x)^(m + 1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]
Rule 3720
Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(c + d*x)^m*(b*Tan[e
+ f*x])^(n - 1))/(f*(n - 1)), x] + (-Dist[(b*d*m)/(f*(n - 1)), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]
Rule 3722
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 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]
Rubi steps
\begin {align*} \int (c+d x)^3 (a+b \coth (e+f x))^3 \, dx &=\int \left (a^3 (c+d x)^3+3 a^2 b (c+d x)^3 \coth (e+f x)+3 a b^2 (c+d x)^3 \coth ^2(e+f x)+b^3 (c+d x)^3 \coth ^3(e+f x)\right ) \, dx\\ &=\frac {a^3 (c+d x)^4}{4 d}+\left (3 a^2 b\right ) \int (c+d x)^3 \coth (e+f x) \, dx+\left (3 a b^2\right ) \int (c+d x)^3 \coth ^2(e+f x) \, dx+b^3 \int (c+d x)^3 \coth ^3(e+f x) \, dx\\ &=\frac {a^3 (c+d x)^4}{4 d}-\frac {3 a^2 b (c+d x)^4}{4 d}-\frac {3 a b^2 (c+d x)^3 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^3 \coth ^2(e+f x)}{2 f}-\left (6 a^2 b\right ) \int \frac {e^{2 (e+f x)} (c+d x)^3}{1-e^{2 (e+f x)}} \, dx+\left (3 a b^2\right ) \int (c+d x)^3 \, dx+b^3 \int (c+d x)^3 \coth (e+f x) \, dx+\frac {\left (9 a b^2 d\right ) \int (c+d x)^2 \coth (e+f x) \, dx}{f}+\frac {\left (3 b^3 d\right ) \int (c+d x)^2 \coth ^2(e+f x) \, dx}{2 f}\\ &=-\frac {3 a b^2 (c+d x)^3}{f}+\frac {a^3 (c+d x)^4}{4 d}-\frac {3 a^2 b (c+d x)^4}{4 d}+\frac {3 a b^2 (c+d x)^4}{4 d}-\frac {b^3 (c+d x)^4}{4 d}-\frac {3 b^3 d (c+d x)^2 \coth (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x)^3 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^3 \coth ^2(e+f x)}{2 f}+\frac {3 a^2 b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}-\left (2 b^3\right ) \int \frac {e^{2 (e+f x)} (c+d x)^3}{1-e^{2 (e+f x)}} \, dx+\frac {\left (3 b^3 d^2\right ) \int (c+d x) \coth (e+f x) \, dx}{f^2}-\frac {\left (9 a^2 b d\right ) \int (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right ) \, dx}{f}-\frac {\left (18 a b^2 d\right ) \int \frac {e^{2 (e+f x)} (c+d x)^2}{1-e^{2 (e+f x)}} \, dx}{f}+\frac {\left (3 b^3 d\right ) \int (c+d x)^2 \, dx}{2 f}\\ &=-\frac {3 b^3 d (c+d x)^2}{2 f^2}-\frac {3 a b^2 (c+d x)^3}{f}+\frac {b^3 (c+d x)^3}{2 f}+\frac {a^3 (c+d x)^4}{4 d}-\frac {3 a^2 b (c+d x)^4}{4 d}+\frac {3 a b^2 (c+d x)^4}{4 d}-\frac {b^3 (c+d x)^4}{4 d}-\frac {3 b^3 d (c+d x)^2 \coth (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x)^3 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^3 \coth ^2(e+f x)}{2 f}+\frac {9 a b^2 d (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {9 a^2 b d (c+d x)^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}-\frac {\left (9 a^2 b d^2\right ) \int (c+d x) \text {Li}_2\left (e^{2 (e+f x)}\right ) \, dx}{f^2}-\frac {\left (18 a b^2 d^2\right ) \int (c+d x) \log \left (1-e^{2 (e+f x)}\right ) \, dx}{f^2}-\frac {\left (6 b^3 d^2\right ) \int \frac {e^{2 (e+f x)} (c+d x)}{1-e^{2 (e+f x)}} \, dx}{f^2}-\frac {\left (3 b^3 d\right ) \int (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right ) \, dx}{f}\\ &=-\frac {3 b^3 d (c+d x)^2}{2 f^2}-\frac {3 a b^2 (c+d x)^3}{f}+\frac {b^3 (c+d x)^3}{2 f}+\frac {a^3 (c+d x)^4}{4 d}-\frac {3 a^2 b (c+d x)^4}{4 d}+\frac {3 a b^2 (c+d x)^4}{4 d}-\frac {b^3 (c+d x)^4}{4 d}-\frac {3 b^3 d (c+d x)^2 \coth (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x)^3 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^3 \coth ^2(e+f x)}{2 f}+\frac {3 b^3 d^2 (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^3}+\frac {9 a b^2 d (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {9 a b^2 d^2 (c+d x) \text {Li}_2\left (e^{2 (e+f x)}\right )}{f^3}+\frac {9 a^2 b d (c+d x)^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}+\frac {3 b^3 d (c+d x)^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}-\frac {9 a^2 b d^2 (c+d x) \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^3}+\frac {\left (9 a^2 b d^3\right ) \int \text {Li}_3\left (e^{2 (e+f x)}\right ) \, dx}{2 f^3}-\frac {\left (9 a b^2 d^3\right ) \int \text {Li}_2\left (e^{2 (e+f x)}\right ) \, dx}{f^3}-\frac {\left (3 b^3 d^3\right ) \int \log \left (1-e^{2 (e+f x)}\right ) \, dx}{f^3}-\frac {\left (3 b^3 d^2\right ) \int (c+d x) \text {Li}_2\left (e^{2 (e+f x)}\right ) \, dx}{f^2}\\ &=-\frac {3 b^3 d (c+d x)^2}{2 f^2}-\frac {3 a b^2 (c+d x)^3}{f}+\frac {b^3 (c+d x)^3}{2 f}+\frac {a^3 (c+d x)^4}{4 d}-\frac {3 a^2 b (c+d x)^4}{4 d}+\frac {3 a b^2 (c+d x)^4}{4 d}-\frac {b^3 (c+d x)^4}{4 d}-\frac {3 b^3 d (c+d x)^2 \coth (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x)^3 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^3 \coth ^2(e+f x)}{2 f}+\frac {3 b^3 d^2 (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^3}+\frac {9 a b^2 d (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {9 a b^2 d^2 (c+d x) \text {Li}_2\left (e^{2 (e+f x)}\right )}{f^3}+\frac {9 a^2 b d (c+d x)^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}+\frac {3 b^3 d (c+d x)^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}-\frac {9 a^2 b d^2 (c+d x) \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^3}-\frac {3 b^3 d^2 (c+d x) \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^3}+\frac {\left (9 a^2 b d^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{4 f^4}-\frac {\left (9 a b^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^4}-\frac {\left (3 b^3 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^4}+\frac {\left (3 b^3 d^3\right ) \int \text {Li}_3\left (e^{2 (e+f x)}\right ) \, dx}{2 f^3}\\ &=-\frac {3 b^3 d (c+d x)^2}{2 f^2}-\frac {3 a b^2 (c+d x)^3}{f}+\frac {b^3 (c+d x)^3}{2 f}+\frac {a^3 (c+d x)^4}{4 d}-\frac {3 a^2 b (c+d x)^4}{4 d}+\frac {3 a b^2 (c+d x)^4}{4 d}-\frac {b^3 (c+d x)^4}{4 d}-\frac {3 b^3 d (c+d x)^2 \coth (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x)^3 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^3 \coth ^2(e+f x)}{2 f}+\frac {3 b^3 d^2 (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^3}+\frac {9 a b^2 d (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {3 b^3 d^3 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^4}+\frac {9 a b^2 d^2 (c+d x) \text {Li}_2\left (e^{2 (e+f x)}\right )}{f^3}+\frac {9 a^2 b d (c+d x)^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}+\frac {3 b^3 d (c+d x)^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}-\frac {9 a b^2 d^3 \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^4}-\frac {9 a^2 b d^2 (c+d x) \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^3}-\frac {3 b^3 d^2 (c+d x) \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^3}+\frac {9 a^2 b d^3 \text {Li}_4\left (e^{2 (e+f x)}\right )}{4 f^4}+\frac {\left (3 b^3 d^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{4 f^4}\\ &=-\frac {3 b^3 d (c+d x)^2}{2 f^2}-\frac {3 a b^2 (c+d x)^3}{f}+\frac {b^3 (c+d x)^3}{2 f}+\frac {a^3 (c+d x)^4}{4 d}-\frac {3 a^2 b (c+d x)^4}{4 d}+\frac {3 a b^2 (c+d x)^4}{4 d}-\frac {b^3 (c+d x)^4}{4 d}-\frac {3 b^3 d (c+d x)^2 \coth (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x)^3 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^3 \coth ^2(e+f x)}{2 f}+\frac {3 b^3 d^2 (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^3}+\frac {9 a b^2 d (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {3 b^3 d^3 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^4}+\frac {9 a b^2 d^2 (c+d x) \text {Li}_2\left (e^{2 (e+f x)}\right )}{f^3}+\frac {9 a^2 b d (c+d x)^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}+\frac {3 b^3 d (c+d x)^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}-\frac {9 a b^2 d^3 \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^4}-\frac {9 a^2 b d^2 (c+d x) \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^3}-\frac {3 b^3 d^2 (c+d x) \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^3}+\frac {9 a^2 b d^3 \text {Li}_4\left (e^{2 (e+f x)}\right )}{4 f^4}+\frac {3 b^3 d^3 \text {Li}_4\left (e^{2 (e+f x)}\right )}{4 f^4}\\ \end {align*}
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Mathematica [B] time = 14.52, size = 2043, normalized size = 3.67 \[ \text {Result too large to show} \]
Antiderivative was successfully verified.
[In]
Integrate[(c + d*x)^3*(a + b*Coth[e + f*x])^3,x]
[Out]
((-(b^3*c^3) - 3*b^3*c^2*d*x - 3*b^3*c*d^2*x^2 - b^3*d^3*x^3)*Csch[e + f*x]^2)/(2*f) - (b*E^(2*e)*(24*b^2*c*d^
2*x + 72*a*b*c^2*d*f*x + 24*a^2*c^3*f^2*x + 8*b^2*c^3*f^2*x + 12*b^2*d^3*x^2 + 72*a*b*c*d^2*f*x^2 + 36*a^2*c^2
*d*f^2*x^2 + 12*b^2*c^2*d*f^2*x^2 + 24*a*b*d^3*f*x^3 + 24*a^2*c*d^2*f^2*x^3 + 8*b^2*c*d^2*f^2*x^3 + 6*a^2*d^3*
f^2*x^4 + 2*b^2*d^3*f^2*x^4 - 36*a*b*c^2*d*Log[1 - E^(2*(e + f*x))] + (36*a*b*c^2*d*Log[1 - E^(2*(e + f*x))])/
E^(2*e) - (12*b^2*c*d^2*Log[1 - E^(2*(e + f*x))])/f + (12*b^2*c*d^2*Log[1 - E^(2*(e + f*x))])/(E^(2*e)*f) - 12
*a^2*c^3*f*Log[1 - E^(2*(e + f*x))] - 4*b^2*c^3*f*Log[1 - E^(2*(e + f*x))] + (12*a^2*c^3*f*Log[1 - E^(2*(e + f
*x))])/E^(2*e) + (4*b^2*c^3*f*Log[1 - E^(2*(e + f*x))])/E^(2*e) - 72*a*b*c*d^2*x*Log[1 - E^(2*(e + f*x))] + (7
2*a*b*c*d^2*x*Log[1 - E^(2*(e + f*x))])/E^(2*e) - (12*b^2*d^3*x*Log[1 - E^(2*(e + f*x))])/f + (12*b^2*d^3*x*Lo
g[1 - E^(2*(e + f*x))])/(E^(2*e)*f) - 36*a^2*c^2*d*f*x*Log[1 - E^(2*(e + f*x))] - 12*b^2*c^2*d*f*x*Log[1 - E^(
2*(e + f*x))] + (36*a^2*c^2*d*f*x*Log[1 - E^(2*(e + f*x))])/E^(2*e) + (12*b^2*c^2*d*f*x*Log[1 - E^(2*(e + f*x)
)])/E^(2*e) - 36*a*b*d^3*x^2*Log[1 - E^(2*(e + f*x))] + (36*a*b*d^3*x^2*Log[1 - E^(2*(e + f*x))])/E^(2*e) - 36
*a^2*c*d^2*f*x^2*Log[1 - E^(2*(e + f*x))] - 12*b^2*c*d^2*f*x^2*Log[1 - E^(2*(e + f*x))] + (36*a^2*c*d^2*f*x^2*
Log[1 - E^(2*(e + f*x))])/E^(2*e) + (12*b^2*c*d^2*f*x^2*Log[1 - E^(2*(e + f*x))])/E^(2*e) - 12*a^2*d^3*f*x^3*L
og[1 - E^(2*(e + f*x))] - 4*b^2*d^3*f*x^3*Log[1 - E^(2*(e + f*x))] + (12*a^2*d^3*f*x^3*Log[1 - E^(2*(e + f*x))
])/E^(2*e) + (4*b^2*d^3*f*x^3*Log[1 - E^(2*(e + f*x))])/E^(2*e) - (6*d*(-1 + E^(2*e))*(6*a*b*d*f*(c + d*x) + 3
*a^2*f^2*(c + d*x)^2 + b^2*(d^2 + c^2*f^2 + 2*c*d*f^2*x + d^2*f^2*x^2))*PolyLog[2, E^(2*(e + f*x))])/(E^(2*e)*
f^2) + (6*d^2*(-1 + E^(2*e))*(3*a*b*d + 3*a^2*f*(c + d*x) + b^2*f*(c + d*x))*PolyLog[3, E^(2*(e + f*x))])/(E^(
2*e)*f^2) - (9*a^2*d^3*PolyLog[4, E^(2*(e + f*x))])/f^2 - (3*b^2*d^3*PolyLog[4, E^(2*(e + f*x))])/f^2 + (9*a^2
*d^3*PolyLog[4, E^(2*(e + f*x))])/(E^(2*e)*f^2) + (3*b^2*d^3*PolyLog[4, E^(2*(e + f*x))])/(E^(2*e)*f^2)))/(4*(
-1 + E^(2*e))*f^2) + (3*x^2*(-(a^3*c^2*d) + 3*a^2*b*c^2*d - 3*a*b^2*c^2*d + b^3*c^2*d + a^3*c^2*d*Cosh[2*e] +
3*a^2*b*c^2*d*Cosh[2*e] + 3*a*b^2*c^2*d*Cosh[2*e] + b^3*c^2*d*Cosh[2*e] + a^3*c^2*d*Sinh[2*e] + 3*a^2*b*c^2*d*
Sinh[2*e] + 3*a*b^2*c^2*d*Sinh[2*e] + b^3*c^2*d*Sinh[2*e]))/(2*(-1 + Cosh[2*e] + Sinh[2*e])) + (x^3*(-(a^3*c*d
^2) + 3*a^2*b*c*d^2 - 3*a*b^2*c*d^2 + b^3*c*d^2 + a^3*c*d^2*Cosh[2*e] + 3*a^2*b*c*d^2*Cosh[2*e] + 3*a*b^2*c*d^
2*Cosh[2*e] + b^3*c*d^2*Cosh[2*e] + a^3*c*d^2*Sinh[2*e] + 3*a^2*b*c*d^2*Sinh[2*e] + 3*a*b^2*c*d^2*Sinh[2*e] +
b^3*c*d^2*Sinh[2*e]))/(-1 + Cosh[2*e] + Sinh[2*e]) + (x^4*(-(a^3*d^3) + 3*a^2*b*d^3 - 3*a*b^2*d^3 + b^3*d^3 +
a^3*d^3*Cosh[2*e] + 3*a^2*b*d^3*Cosh[2*e] + 3*a*b^2*d^3*Cosh[2*e] + b^3*d^3*Cosh[2*e] + a^3*d^3*Sinh[2*e] + 3*
a^2*b*d^3*Sinh[2*e] + 3*a*b^2*d^3*Sinh[2*e] + b^3*d^3*Sinh[2*e]))/(4*(-1 + Cosh[2*e] + Sinh[2*e])) + x*(a^3*c^
3 + 3*a*b^2*c^3 + (3*a^2*b*c^3)/(-1 + Cosh[2*e] + Sinh[2*e]) + (3*a^2*b*c^3*Cosh[2*e] + 3*a^2*b*c^3*Sinh[2*e])
/(-1 + Cosh[2*e] + Sinh[2*e]) + (2*b^3*c^3*Cosh[2*e] + 2*b^3*c^3*Sinh[2*e])/((-1 + Cosh[2*e] + Sinh[2*e])*(1 +
Cosh[2*e] + Cosh[4*e] + Sinh[2*e] + Sinh[4*e])) + (2*b^3*c^3*Cosh[4*e] + 2*b^3*c^3*Sinh[4*e])/((-1 + Cosh[2*e
] + Sinh[2*e])*(1 + Cosh[2*e] + Cosh[4*e] + Sinh[2*e] + Sinh[4*e])) + (b^3*c^3)/(-1 + Cosh[6*e] + Sinh[6*e]) +
(b^3*c^3*Cosh[6*e] + b^3*c^3*Sinh[6*e])/(-1 + Cosh[6*e] + Sinh[6*e])) + (3*Csch[e]*Csch[e + f*x]*(b^3*c^2*d*S
inh[f*x] + 2*a*b^2*c^3*f*Sinh[f*x] + 2*b^3*c*d^2*x*Sinh[f*x] + 6*a*b^2*c^2*d*f*x*Sinh[f*x] + b^3*d^3*x^2*Sinh[
f*x] + 6*a*b^2*c*d^2*f*x^2*Sinh[f*x] + 2*a*b^2*d^3*f*x^3*Sinh[f*x]))/(2*f^2)
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fricas [C] time = 0.66, size = 11137, normalized size = 20.03 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^3*(a+b*coth(f*x+e))^3,x, algorithm="fricas")
[Out]
1/4*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^3*f^4*x^4 + 4*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*d^2*f^4*x^3 + 6*(a^3 -
3*a^2*b + 3*a*b^2 - b^3)*c^2*d*f^4*x^2 - 24*a*b^2*d^3*e^3 + 4*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c^3*f^4*x + 12*b
^3*d^3*e^2 + 2*(3*a^2*b + b^3)*d^3*e^4 + ((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^3*f^4*x^4 - 24*a*b^2*d^3*e^3 + 12*
b^3*d^3*e^2 + 2*(3*a^2*b + b^3)*d^3*e^4 - 8*(3*a^2*b + b^3)*c^3*e*f^3 - 4*(6*a*b^2*d^3*f^3 - (a^3 - 3*a^2*b +
3*a*b^2 - b^3)*c*d^2*f^4)*x^3 - 12*(6*a*b^2*c^2*d*e - (3*a^2*b + b^3)*c^2*d*e^2)*f^2 - 6*(12*a*b^2*c*d^2*f^3 +
2*b^3*d^3*f^2 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c^2*d*f^4)*x^2 + 8*(9*a*b^2*c*d^2*e^2 - 3*b^3*c*d^2*e - (3*a^
2*b + b^3)*c*d^2*e^3)*f - 4*(18*a*b^2*c^2*d*f^3 + 6*b^3*c*d^2*f^2 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c^3*f^4)*x
)*cosh(f*x + e)^4 + 4*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^3*f^4*x^4 - 24*a*b^2*d^3*e^3 + 12*b^3*d^3*e^2 + 2*(3*
a^2*b + b^3)*d^3*e^4 - 8*(3*a^2*b + b^3)*c^3*e*f^3 - 4*(6*a*b^2*d^3*f^3 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*d^
2*f^4)*x^3 - 12*(6*a*b^2*c^2*d*e - (3*a^2*b + b^3)*c^2*d*e^2)*f^2 - 6*(12*a*b^2*c*d^2*f^3 + 2*b^3*d^3*f^2 - (a
^3 - 3*a^2*b + 3*a*b^2 - b^3)*c^2*d*f^4)*x^2 + 8*(9*a*b^2*c*d^2*e^2 - 3*b^3*c*d^2*e - (3*a^2*b + b^3)*c*d^2*e^
3)*f - 4*(18*a*b^2*c^2*d*f^3 + 6*b^3*c*d^2*f^2 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c^3*f^4)*x)*cosh(f*x + e)*sin
h(f*x + e)^3 + ((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^3*f^4*x^4 - 24*a*b^2*d^3*e^3 + 12*b^3*d^3*e^2 + 2*(3*a^2*b +
b^3)*d^3*e^4 - 8*(3*a^2*b + b^3)*c^3*e*f^3 - 4*(6*a*b^2*d^3*f^3 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*d^2*f^4)*
x^3 - 12*(6*a*b^2*c^2*d*e - (3*a^2*b + b^3)*c^2*d*e^2)*f^2 - 6*(12*a*b^2*c*d^2*f^3 + 2*b^3*d^3*f^2 - (a^3 - 3*
a^2*b + 3*a*b^2 - b^3)*c^2*d*f^4)*x^2 + 8*(9*a*b^2*c*d^2*e^2 - 3*b^3*c*d^2*e - (3*a^2*b + b^3)*c*d^2*e^3)*f -
4*(18*a*b^2*c^2*d*f^3 + 6*b^3*c*d^2*f^2 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c^3*f^4)*x)*sinh(f*x + e)^4 + 8*(3*a
*b^2*c^3 - (3*a^2*b + b^3)*c^3*e)*f^3 - 12*(6*a*b^2*c^2*d*e - b^3*c^2*d - (3*a^2*b + b^3)*c^2*d*e^2)*f^2 - 2*(
(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^3*f^4*x^4 - 24*a*b^2*d^3*e^3 + 12*b^3*d^3*e^2 + 2*(3*a^2*b + b^3)*d^3*e^4 -
4*(2*(3*a^2*b + b^3)*c^3*e - (3*a*b^2 + b^3)*c^3)*f^3 + 4*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*d^2*f^4 - (3*a*b^
2 - b^3)*d^3*f^3)*x^3 - 6*(12*a*b^2*c^2*d*e - b^3*c^2*d - 2*(3*a^2*b + b^3)*c^2*d*e^2)*f^2 - 6*(b^3*d^3*f^2 -
(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c^2*d*f^4 + 2*(3*a*b^2 - b^3)*c*d^2*f^3)*x^2 + 8*(9*a*b^2*c*d^2*e^2 - 3*b^3*c*
d^2*e - (3*a^2*b + b^3)*c*d^2*e^3)*f - 4*(3*b^3*c*d^2*f^2 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c^3*f^4 + 3*(3*a*b
^2 - b^3)*c^2*d*f^3)*x)*cosh(f*x + e)^2 - 2*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^3*f^4*x^4 - 24*a*b^2*d^3*e^3 +
12*b^3*d^3*e^2 + 2*(3*a^2*b + b^3)*d^3*e^4 - 4*(2*(3*a^2*b + b^3)*c^3*e - (3*a*b^2 + b^3)*c^3)*f^3 + 4*((a^3 -
3*a^2*b + 3*a*b^2 - b^3)*c*d^2*f^4 - (3*a*b^2 - b^3)*d^3*f^3)*x^3 - 6*(12*a*b^2*c^2*d*e - b^3*c^2*d - 2*(3*a^
2*b + b^3)*c^2*d*e^2)*f^2 - 6*(b^3*d^3*f^2 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c^2*d*f^4 + 2*(3*a*b^2 - b^3)*c*d
^2*f^3)*x^2 - 3*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^3*f^4*x^4 - 24*a*b^2*d^3*e^3 + 12*b^3*d^3*e^2 + 2*(3*a^2*b
+ b^3)*d^3*e^4 - 8*(3*a^2*b + b^3)*c^3*e*f^3 - 4*(6*a*b^2*d^3*f^3 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*d^2*f^4)
*x^3 - 12*(6*a*b^2*c^2*d*e - (3*a^2*b + b^3)*c^2*d*e^2)*f^2 - 6*(12*a*b^2*c*d^2*f^3 + 2*b^3*d^3*f^2 - (a^3 - 3
*a^2*b + 3*a*b^2 - b^3)*c^2*d*f^4)*x^2 + 8*(9*a*b^2*c*d^2*e^2 - 3*b^3*c*d^2*e - (3*a^2*b + b^3)*c*d^2*e^3)*f -
4*(18*a*b^2*c^2*d*f^3 + 6*b^3*c*d^2*f^2 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c^3*f^4)*x)*cosh(f*x + e)^2 + 8*(9*
a*b^2*c*d^2*e^2 - 3*b^3*c*d^2*e - (3*a^2*b + b^3)*c*d^2*e^3)*f - 4*(3*b^3*c*d^2*f^2 - (a^3 - 3*a^2*b + 3*a*b^2
- b^3)*c^3*f^4 + 3*(3*a*b^2 - b^3)*c^2*d*f^3)*x)*sinh(f*x + e)^2 + 8*(9*a*b^2*c*d^2*e^2 - 3*b^3*c*d^2*e - (3*
a^2*b + b^3)*c*d^2*e^3)*f + 12*((3*a^2*b + b^3)*d^3*f^2*x^2 + 6*a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*
d*f^2 + ((3*a^2*b + b^3)*d^3*f^2*x^2 + 6*a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2 + 2*(3*a*b^2*d^3*
f + (3*a^2*b + b^3)*c*d^2*f^2)*x)*cosh(f*x + e)^4 + 4*((3*a^2*b + b^3)*d^3*f^2*x^2 + 6*a*b^2*c*d^2*f + b^3*d^3
+ (3*a^2*b + b^3)*c^2*d*f^2 + 2*(3*a*b^2*d^3*f + (3*a^2*b + b^3)*c*d^2*f^2)*x)*cosh(f*x + e)*sinh(f*x + e)^3
+ ((3*a^2*b + b^3)*d^3*f^2*x^2 + 6*a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2 + 2*(3*a*b^2*d^3*f + (3
*a^2*b + b^3)*c*d^2*f^2)*x)*sinh(f*x + e)^4 - 2*((3*a^2*b + b^3)*d^3*f^2*x^2 + 6*a*b^2*c*d^2*f + b^3*d^3 + (3*
a^2*b + b^3)*c^2*d*f^2 + 2*(3*a*b^2*d^3*f + (3*a^2*b + b^3)*c*d^2*f^2)*x)*cosh(f*x + e)^2 - 2*((3*a^2*b + b^3)
*d^3*f^2*x^2 + 6*a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2 - 3*((3*a^2*b + b^3)*d^3*f^2*x^2 + 6*a*b^
2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2 + 2*(3*a*b^2*d^3*f + (3*a^2*b + b^3)*c*d^2*f^2)*x)*cosh(f*x +
e)^2 + 2*(3*a*b^2*d^3*f + (3*a^2*b + b^3)*c*d^2*f^2)*x)*sinh(f*x + e)^2 + 2*(3*a*b^2*d^3*f + (3*a^2*b + b^3)*c
*d^2*f^2)*x + 4*(((3*a^2*b + b^3)*d^3*f^2*x^2 + 6*a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2 + 2*(3*a
*b^2*d^3*f + (3*a^2*b + b^3)*c*d^2*f^2)*x)*cosh(f*x + e)^3 - ((3*a^2*b + b^3)*d^3*f^2*x^2 + 6*a*b^2*c*d^2*f +
b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2 + 2*(3*a*b^2*d^3*f + (3*a^2*b + b^3)*c*d^2*f^2)*x)*cosh(f*x + e))*sinh(f*x
+ e))*dilog(cosh(f*x + e) + sinh(f*x + e)) + 12*((3*a^2*b + b^3)*d^3*f^2*x^2 + 6*a*b^2*c*d^2*f + b^3*d^3 + (3
*a^2*b + b^3)*c^2*d*f^2 + ((3*a^2*b + b^3)*d^3*f^2*x^2 + 6*a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2
+ 2*(3*a*b^2*d^3*f + (3*a^2*b + b^3)*c*d^2*f^2)*x)*cosh(f*x + e)^4 + 4*((3*a^2*b + b^3)*d^3*f^2*x^2 + 6*a*b^2
*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2 + 2*(3*a*b^2*d^3*f + (3*a^2*b + b^3)*c*d^2*f^2)*x)*cosh(f*x + e
)*sinh(f*x + e)^3 + ((3*a^2*b + b^3)*d^3*f^2*x^2 + 6*a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2 + 2*(
3*a*b^2*d^3*f + (3*a^2*b + b^3)*c*d^2*f^2)*x)*sinh(f*x + e)^4 - 2*((3*a^2*b + b^3)*d^3*f^2*x^2 + 6*a*b^2*c*d^2
*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2 + 2*(3*a*b^2*d^3*f + (3*a^2*b + b^3)*c*d^2*f^2)*x)*cosh(f*x + e)^2 -
2*((3*a^2*b + b^3)*d^3*f^2*x^2 + 6*a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2 - 3*((3*a^2*b + b^3)*d^
3*f^2*x^2 + 6*a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2 + 2*(3*a*b^2*d^3*f + (3*a^2*b + b^3)*c*d^2*f
^2)*x)*cosh(f*x + e)^2 + 2*(3*a*b^2*d^3*f + (3*a^2*b + b^3)*c*d^2*f^2)*x)*sinh(f*x + e)^2 + 2*(3*a*b^2*d^3*f +
(3*a^2*b + b^3)*c*d^2*f^2)*x + 4*(((3*a^2*b + b^3)*d^3*f^2*x^2 + 6*a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*
c^2*d*f^2 + 2*(3*a*b^2*d^3*f + (3*a^2*b + b^3)*c*d^2*f^2)*x)*cosh(f*x + e)^3 - ((3*a^2*b + b^3)*d^3*f^2*x^2 +
6*a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2 + 2*(3*a*b^2*d^3*f + (3*a^2*b + b^3)*c*d^2*f^2)*x)*cosh(
f*x + e))*sinh(f*x + e))*dilog(-cosh(f*x + e) - sinh(f*x + e)) + 4*((3*a^2*b + b^3)*d^3*f^3*x^3 + 9*a*b^2*c^2*
d*f^2 + 3*b^3*c*d^2*f + (3*a^2*b + b^3)*c^3*f^3 + ((3*a^2*b + b^3)*d^3*f^3*x^3 + 9*a*b^2*c^2*d*f^2 + 3*b^3*c*d
^2*f + (3*a^2*b + b^3)*c^3*f^3 + 3*(3*a*b^2*d^3*f^2 + (3*a^2*b + b^3)*c*d^2*f^3)*x^2 + 3*(6*a*b^2*c*d^2*f^2 +
b^3*d^3*f + (3*a^2*b + b^3)*c^2*d*f^3)*x)*cosh(f*x + e)^4 + 4*((3*a^2*b + b^3)*d^3*f^3*x^3 + 9*a*b^2*c^2*d*f^2
+ 3*b^3*c*d^2*f + (3*a^2*b + b^3)*c^3*f^3 + 3*(3*a*b^2*d^3*f^2 + (3*a^2*b + b^3)*c*d^2*f^3)*x^2 + 3*(6*a*b^2*
c*d^2*f^2 + b^3*d^3*f + (3*a^2*b + b^3)*c^2*d*f^3)*x)*cosh(f*x + e)*sinh(f*x + e)^3 + ((3*a^2*b + b^3)*d^3*f^3
*x^3 + 9*a*b^2*c^2*d*f^2 + 3*b^3*c*d^2*f + (3*a^2*b + b^3)*c^3*f^3 + 3*(3*a*b^2*d^3*f^2 + (3*a^2*b + b^3)*c*d^
2*f^3)*x^2 + 3*(6*a*b^2*c*d^2*f^2 + b^3*d^3*f + (3*a^2*b + b^3)*c^2*d*f^3)*x)*sinh(f*x + e)^4 + 3*(3*a*b^2*d^3
*f^2 + (3*a^2*b + b^3)*c*d^2*f^3)*x^2 - 2*((3*a^2*b + b^3)*d^3*f^3*x^3 + 9*a*b^2*c^2*d*f^2 + 3*b^3*c*d^2*f + (
3*a^2*b + b^3)*c^3*f^3 + 3*(3*a*b^2*d^3*f^2 + (3*a^2*b + b^3)*c*d^2*f^3)*x^2 + 3*(6*a*b^2*c*d^2*f^2 + b^3*d^3*
f + (3*a^2*b + b^3)*c^2*d*f^3)*x)*cosh(f*x + e)^2 - 2*((3*a^2*b + b^3)*d^3*f^3*x^3 + 9*a*b^2*c^2*d*f^2 + 3*b^3
*c*d^2*f + (3*a^2*b + b^3)*c^3*f^3 + 3*(3*a*b^2*d^3*f^2 + (3*a^2*b + b^3)*c*d^2*f^3)*x^2 - 3*((3*a^2*b + b^3)*
d^3*f^3*x^3 + 9*a*b^2*c^2*d*f^2 + 3*b^3*c*d^2*f + (3*a^2*b + b^3)*c^3*f^3 + 3*(3*a*b^2*d^3*f^2 + (3*a^2*b + b^
3)*c*d^2*f^3)*x^2 + 3*(6*a*b^2*c*d^2*f^2 + b^3*d^3*f + (3*a^2*b + b^3)*c^2*d*f^3)*x)*cosh(f*x + e)^2 + 3*(6*a*
b^2*c*d^2*f^2 + b^3*d^3*f + (3*a^2*b + b^3)*c^2*d*f^3)*x)*sinh(f*x + e)^2 + 3*(6*a*b^2*c*d^2*f^2 + b^3*d^3*f +
(3*a^2*b + b^3)*c^2*d*f^3)*x + 4*(((3*a^2*b + b^3)*d^3*f^3*x^3 + 9*a*b^2*c^2*d*f^2 + 3*b^3*c*d^2*f + (3*a^2*b
+ b^3)*c^3*f^3 + 3*(3*a*b^2*d^3*f^2 + (3*a^2*b + b^3)*c*d^2*f^3)*x^2 + 3*(6*a*b^2*c*d^2*f^2 + b^3*d^3*f + (3*
a^2*b + b^3)*c^2*d*f^3)*x)*cosh(f*x + e)^3 - ((3*a^2*b + b^3)*d^3*f^3*x^3 + 9*a*b^2*c^2*d*f^2 + 3*b^3*c*d^2*f
+ (3*a^2*b + b^3)*c^3*f^3 + 3*(3*a*b^2*d^3*f^2 + (3*a^2*b + b^3)*c*d^2*f^3)*x^2 + 3*(6*a*b^2*c*d^2*f^2 + b^3*d
^3*f + (3*a^2*b + b^3)*c^2*d*f^3)*x)*cosh(f*x + e))*sinh(f*x + e))*log(cosh(f*x + e) + sinh(f*x + e) + 1) + 4*
(9*a*b^2*d^3*e^2 - 3*b^3*d^3*e - (3*a^2*b + b^3)*d^3*e^3 + (3*a^2*b + b^3)*c^3*f^3 + (9*a*b^2*d^3*e^2 - 3*b^3*
d^3*e - (3*a^2*b + b^3)*d^3*e^3 + (3*a^2*b + b^3)*c^3*f^3 + 3*(3*a*b^2*c^2*d - (3*a^2*b + b^3)*c^2*d*e)*f^2 -
3*(6*a*b^2*c*d^2*e - b^3*c*d^2 - (3*a^2*b + b^3)*c*d^2*e^2)*f)*cosh(f*x + e)^4 + 4*(9*a*b^2*d^3*e^2 - 3*b^3*d^
3*e - (3*a^2*b + b^3)*d^3*e^3 + (3*a^2*b + b^3)*c^3*f^3 + 3*(3*a*b^2*c^2*d - (3*a^2*b + b^3)*c^2*d*e)*f^2 - 3*
(6*a*b^2*c*d^2*e - b^3*c*d^2 - (3*a^2*b + b^3)*c*d^2*e^2)*f)*cosh(f*x + e)*sinh(f*x + e)^3 + (9*a*b^2*d^3*e^2
- 3*b^3*d^3*e - (3*a^2*b + b^3)*d^3*e^3 + (3*a^2*b + b^3)*c^3*f^3 + 3*(3*a*b^2*c^2*d - (3*a^2*b + b^3)*c^2*d*e
)*f^2 - 3*(6*a*b^2*c*d^2*e - b^3*c*d^2 - (3*a^2*b + b^3)*c*d^2*e^2)*f)*sinh(f*x + e)^4 + 3*(3*a*b^2*c^2*d - (3
*a^2*b + b^3)*c^2*d*e)*f^2 - 2*(9*a*b^2*d^3*e^2 - 3*b^3*d^3*e - (3*a^2*b + b^3)*d^3*e^3 + (3*a^2*b + b^3)*c^3*
f^3 + 3*(3*a*b^2*c^2*d - (3*a^2*b + b^3)*c^2*d*e)*f^2 - 3*(6*a*b^2*c*d^2*e - b^3*c*d^2 - (3*a^2*b + b^3)*c*d^2
*e^2)*f)*cosh(f*x + e)^2 - 2*(9*a*b^2*d^3*e^2 - 3*b^3*d^3*e - (3*a^2*b + b^3)*d^3*e^3 + (3*a^2*b + b^3)*c^3*f^
3 + 3*(3*a*b^2*c^2*d - (3*a^2*b + b^3)*c^2*d*e)*f^2 - 3*(9*a*b^2*d^3*e^2 - 3*b^3*d^3*e - (3*a^2*b + b^3)*d^3*e
^3 + (3*a^2*b + b^3)*c^3*f^3 + 3*(3*a*b^2*c^2*d - (3*a^2*b + b^3)*c^2*d*e)*f^2 - 3*(6*a*b^2*c*d^2*e - b^3*c*d^
2 - (3*a^2*b + b^3)*c*d^2*e^2)*f)*cosh(f*x + e)^2 - 3*(6*a*b^2*c*d^2*e - b^3*c*d^2 - (3*a^2*b + b^3)*c*d^2*e^2
)*f)*sinh(f*x + e)^2 - 3*(6*a*b^2*c*d^2*e - b^3*c*d^2 - (3*a^2*b + b^3)*c*d^2*e^2)*f + 4*((9*a*b^2*d^3*e^2 - 3
*b^3*d^3*e - (3*a^2*b + b^3)*d^3*e^3 + (3*a^2*b + b^3)*c^3*f^3 + 3*(3*a*b^2*c^2*d - (3*a^2*b + b^3)*c^2*d*e)*f
^2 - 3*(6*a*b^2*c*d^2*e - b^3*c*d^2 - (3*a^2*b + b^3)*c*d^2*e^2)*f)*cosh(f*x + e)^3 - (9*a*b^2*d^3*e^2 - 3*b^3
*d^3*e - (3*a^2*b + b^3)*d^3*e^3 + (3*a^2*b + b^3)*c^3*f^3 + 3*(3*a*b^2*c^2*d - (3*a^2*b + b^3)*c^2*d*e)*f^2 -
3*(6*a*b^2*c*d^2*e - b^3*c*d^2 - (3*a^2*b + b^3)*c*d^2*e^2)*f)*cosh(f*x + e))*sinh(f*x + e))*log(cosh(f*x + e
) + sinh(f*x + e) - 1) + 4*((3*a^2*b + b^3)*d^3*f^3*x^3 - 9*a*b^2*d^3*e^2 + 3*b^3*d^3*e + (3*a^2*b + b^3)*d^3*
e^3 + 3*(3*a^2*b + b^3)*c^2*d*e*f^2 + ((3*a^2*b + b^3)*d^3*f^3*x^3 - 9*a*b^2*d^3*e^2 + 3*b^3*d^3*e + (3*a^2*b
+ b^3)*d^3*e^3 + 3*(3*a^2*b + b^3)*c^2*d*e*f^2 + 3*(3*a*b^2*d^3*f^2 + (3*a^2*b + b^3)*c*d^2*f^3)*x^2 + 3*(6*a*
b^2*c*d^2*e - (3*a^2*b + b^3)*c*d^2*e^2)*f + 3*(6*a*b^2*c*d^2*f^2 + b^3*d^3*f + (3*a^2*b + b^3)*c^2*d*f^3)*x)*
cosh(f*x + e)^4 + 4*((3*a^2*b + b^3)*d^3*f^3*x^3 - 9*a*b^2*d^3*e^2 + 3*b^3*d^3*e + (3*a^2*b + b^3)*d^3*e^3 + 3
*(3*a^2*b + b^3)*c^2*d*e*f^2 + 3*(3*a*b^2*d^3*f^2 + (3*a^2*b + b^3)*c*d^2*f^3)*x^2 + 3*(6*a*b^2*c*d^2*e - (3*a
^2*b + b^3)*c*d^2*e^2)*f + 3*(6*a*b^2*c*d^2*f^2 + b^3*d^3*f + (3*a^2*b + b^3)*c^2*d*f^3)*x)*cosh(f*x + e)*sinh
(f*x + e)^3 + ((3*a^2*b + b^3)*d^3*f^3*x^3 - 9*a*b^2*d^3*e^2 + 3*b^3*d^3*e + (3*a^2*b + b^3)*d^3*e^3 + 3*(3*a^
2*b + b^3)*c^2*d*e*f^2 + 3*(3*a*b^2*d^3*f^2 + (3*a^2*b + b^3)*c*d^2*f^3)*x^2 + 3*(6*a*b^2*c*d^2*e - (3*a^2*b +
b^3)*c*d^2*e^2)*f + 3*(6*a*b^2*c*d^2*f^2 + b^3*d^3*f + (3*a^2*b + b^3)*c^2*d*f^3)*x)*sinh(f*x + e)^4 + 3*(3*a
*b^2*d^3*f^2 + (3*a^2*b + b^3)*c*d^2*f^3)*x^2 - 2*((3*a^2*b + b^3)*d^3*f^3*x^3 - 9*a*b^2*d^3*e^2 + 3*b^3*d^3*e
+ (3*a^2*b + b^3)*d^3*e^3 + 3*(3*a^2*b + b^3)*c^2*d*e*f^2 + 3*(3*a*b^2*d^3*f^2 + (3*a^2*b + b^3)*c*d^2*f^3)*x
^2 + 3*(6*a*b^2*c*d^2*e - (3*a^2*b + b^3)*c*d^2*e^2)*f + 3*(6*a*b^2*c*d^2*f^2 + b^3*d^3*f + (3*a^2*b + b^3)*c^
2*d*f^3)*x)*cosh(f*x + e)^2 - 2*((3*a^2*b + b^3)*d^3*f^3*x^3 - 9*a*b^2*d^3*e^2 + 3*b^3*d^3*e + (3*a^2*b + b^3)
*d^3*e^3 + 3*(3*a^2*b + b^3)*c^2*d*e*f^2 + 3*(3*a*b^2*d^3*f^2 + (3*a^2*b + b^3)*c*d^2*f^3)*x^2 - 3*((3*a^2*b +
b^3)*d^3*f^3*x^3 - 9*a*b^2*d^3*e^2 + 3*b^3*d^3*e + (3*a^2*b + b^3)*d^3*e^3 + 3*(3*a^2*b + b^3)*c^2*d*e*f^2 +
3*(3*a*b^2*d^3*f^2 + (3*a^2*b + b^3)*c*d^2*f^3)*x^2 + 3*(6*a*b^2*c*d^2*e - (3*a^2*b + b^3)*c*d^2*e^2)*f + 3*(6
*a*b^2*c*d^2*f^2 + b^3*d^3*f + (3*a^2*b + b^3)*c^2*d*f^3)*x)*cosh(f*x + e)^2 + 3*(6*a*b^2*c*d^2*e - (3*a^2*b +
b^3)*c*d^2*e^2)*f + 3*(6*a*b^2*c*d^2*f^2 + b^3*d^3*f + (3*a^2*b + b^3)*c^2*d*f^3)*x)*sinh(f*x + e)^2 + 3*(6*a
*b^2*c*d^2*e - (3*a^2*b + b^3)*c*d^2*e^2)*f + 3*(6*a*b^2*c*d^2*f^2 + b^3*d^3*f + (3*a^2*b + b^3)*c^2*d*f^3)*x
+ 4*(((3*a^2*b + b^3)*d^3*f^3*x^3 - 9*a*b^2*d^3*e^2 + 3*b^3*d^3*e + (3*a^2*b + b^3)*d^3*e^3 + 3*(3*a^2*b + b^3
)*c^2*d*e*f^2 + 3*(3*a*b^2*d^3*f^2 + (3*a^2*b + b^3)*c*d^2*f^3)*x^2 + 3*(6*a*b^2*c*d^2*e - (3*a^2*b + b^3)*c*d
^2*e^2)*f + 3*(6*a*b^2*c*d^2*f^2 + b^3*d^3*f + (3*a^2*b + b^3)*c^2*d*f^3)*x)*cosh(f*x + e)^3 - ((3*a^2*b + b^3
)*d^3*f^3*x^3 - 9*a*b^2*d^3*e^2 + 3*b^3*d^3*e + (3*a^2*b + b^3)*d^3*e^3 + 3*(3*a^2*b + b^3)*c^2*d*e*f^2 + 3*(3
*a*b^2*d^3*f^2 + (3*a^2*b + b^3)*c*d^2*f^3)*x^2 + 3*(6*a*b^2*c*d^2*e - (3*a^2*b + b^3)*c*d^2*e^2)*f + 3*(6*a*b
^2*c*d^2*f^2 + b^3*d^3*f + (3*a^2*b + b^3)*c^2*d*f^3)*x)*cosh(f*x + e))*sinh(f*x + e))*log(-cosh(f*x + e) - si
nh(f*x + e) + 1) + 24*((3*a^2*b + b^3)*d^3*cosh(f*x + e)^4 + 4*(3*a^2*b + b^3)*d^3*cosh(f*x + e)*sinh(f*x + e)
^3 + (3*a^2*b + b^3)*d^3*sinh(f*x + e)^4 - 2*(3*a^2*b + b^3)*d^3*cosh(f*x + e)^2 + (3*a^2*b + b^3)*d^3 + 2*(3*
(3*a^2*b + b^3)*d^3*cosh(f*x + e)^2 - (3*a^2*b + b^3)*d^3)*sinh(f*x + e)^2 + 4*((3*a^2*b + b^3)*d^3*cosh(f*x +
e)^3 - (3*a^2*b + b^3)*d^3*cosh(f*x + e))*sinh(f*x + e))*polylog(4, cosh(f*x + e) + sinh(f*x + e)) + 24*((3*a
^2*b + b^3)*d^3*cosh(f*x + e)^4 + 4*(3*a^2*b + b^3)*d^3*cosh(f*x + e)*sinh(f*x + e)^3 + (3*a^2*b + b^3)*d^3*si
nh(f*x + e)^4 - 2*(3*a^2*b + b^3)*d^3*cosh(f*x + e)^2 + (3*a^2*b + b^3)*d^3 + 2*(3*(3*a^2*b + b^3)*d^3*cosh(f*
x + e)^2 - (3*a^2*b + b^3)*d^3)*sinh(f*x + e)^2 + 4*((3*a^2*b + b^3)*d^3*cosh(f*x + e)^3 - (3*a^2*b + b^3)*d^3
*cosh(f*x + e))*sinh(f*x + e))*polylog(4, -cosh(f*x + e) - sinh(f*x + e)) - 24*(3*a*b^2*d^3 + (3*a^2*b + b^3)*
d^3*f*x + (3*a^2*b + b^3)*c*d^2*f + (3*a*b^2*d^3 + (3*a^2*b + b^3)*d^3*f*x + (3*a^2*b + b^3)*c*d^2*f)*cosh(f*x
+ e)^4 + 4*(3*a*b^2*d^3 + (3*a^2*b + b^3)*d^3*f*x + (3*a^2*b + b^3)*c*d^2*f)*cosh(f*x + e)*sinh(f*x + e)^3 +
(3*a*b^2*d^3 + (3*a^2*b + b^3)*d^3*f*x + (3*a^2*b + b^3)*c*d^2*f)*sinh(f*x + e)^4 - 2*(3*a*b^2*d^3 + (3*a^2*b
+ b^3)*d^3*f*x + (3*a^2*b + b^3)*c*d^2*f)*cosh(f*x + e)^2 - 2*(3*a*b^2*d^3 + (3*a^2*b + b^3)*d^3*f*x + (3*a^2*
b + b^3)*c*d^2*f - 3*(3*a*b^2*d^3 + (3*a^2*b + b^3)*d^3*f*x + (3*a^2*b + b^3)*c*d^2*f)*cosh(f*x + e)^2)*sinh(f
*x + e)^2 + 4*((3*a*b^2*d^3 + (3*a^2*b + b^3)*d^3*f*x + (3*a^2*b + b^3)*c*d^2*f)*cosh(f*x + e)^3 - (3*a*b^2*d^
3 + (3*a^2*b + b^3)*d^3*f*x + (3*a^2*b + b^3)*c*d^2*f)*cosh(f*x + e))*sinh(f*x + e))*polylog(3, cosh(f*x + e)
+ sinh(f*x + e)) - 24*(3*a*b^2*d^3 + (3*a^2*b + b^3)*d^3*f*x + (3*a^2*b + b^3)*c*d^2*f + (3*a*b^2*d^3 + (3*a^2
*b + b^3)*d^3*f*x + (3*a^2*b + b^3)*c*d^2*f)*cosh(f*x + e)^4 + 4*(3*a*b^2*d^3 + (3*a^2*b + b^3)*d^3*f*x + (3*a
^2*b + b^3)*c*d^2*f)*cosh(f*x + e)*sinh(f*x + e)^3 + (3*a*b^2*d^3 + (3*a^2*b + b^3)*d^3*f*x + (3*a^2*b + b^3)*
c*d^2*f)*sinh(f*x + e)^4 - 2*(3*a*b^2*d^3 + (3*a^2*b + b^3)*d^3*f*x + (3*a^2*b + b^3)*c*d^2*f)*cosh(f*x + e)^2
- 2*(3*a*b^2*d^3 + (3*a^2*b + b^3)*d^3*f*x + (3*a^2*b + b^3)*c*d^2*f - 3*(3*a*b^2*d^3 + (3*a^2*b + b^3)*d^3*f
*x + (3*a^2*b + b^3)*c*d^2*f)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 4*((3*a*b^2*d^3 + (3*a^2*b + b^3)*d^3*f*x + (
3*a^2*b + b^3)*c*d^2*f)*cosh(f*x + e)^3 - (3*a*b^2*d^3 + (3*a^2*b + b^3)*d^3*f*x + (3*a^2*b + b^3)*c*d^2*f)*co
sh(f*x + e))*sinh(f*x + e))*polylog(3, -cosh(f*x + e) - sinh(f*x + e)) + 4*(((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d
^3*f^4*x^4 - 24*a*b^2*d^3*e^3 + 12*b^3*d^3*e^2 + 2*(3*a^2*b + b^3)*d^3*e^4 - 8*(3*a^2*b + b^3)*c^3*e*f^3 - 4*(
6*a*b^2*d^3*f^3 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*d^2*f^4)*x^3 - 12*(6*a*b^2*c^2*d*e - (3*a^2*b + b^3)*c^2*d
*e^2)*f^2 - 6*(12*a*b^2*c*d^2*f^3 + 2*b^3*d^3*f^2 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c^2*d*f^4)*x^2 + 8*(9*a*b^
2*c*d^2*e^2 - 3*b^3*c*d^2*e - (3*a^2*b + b^3)*c*d^2*e^3)*f - 4*(18*a*b^2*c^2*d*f^3 + 6*b^3*c*d^2*f^2 - (a^3 -
3*a^2*b + 3*a*b^2 - b^3)*c^3*f^4)*x)*cosh(f*x + e)^3 - ((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^3*f^4*x^4 - 24*a*b^2
*d^3*e^3 + 12*b^3*d^3*e^2 + 2*(3*a^2*b + b^3)*d^3*e^4 - 4*(2*(3*a^2*b + b^3)*c^3*e - (3*a*b^2 + b^3)*c^3)*f^3
+ 4*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*d^2*f^4 - (3*a*b^2 - b^3)*d^3*f^3)*x^3 - 6*(12*a*b^2*c^2*d*e - b^3*c^2*
d - 2*(3*a^2*b + b^3)*c^2*d*e^2)*f^2 - 6*(b^3*d^3*f^2 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c^2*d*f^4 + 2*(3*a*b^2
- b^3)*c*d^2*f^3)*x^2 + 8*(9*a*b^2*c*d^2*e^2 - 3*b^3*c*d^2*e - (3*a^2*b + b^3)*c*d^2*e^3)*f - 4*(3*b^3*c*d^2*
f^2 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c^3*f^4 + 3*(3*a*b^2 - b^3)*c^2*d*f^3)*x)*cosh(f*x + e))*sinh(f*x + e))/
(f^4*cosh(f*x + e)^4 + 4*f^4*cosh(f*x + e)*sinh(f*x + e)^3 + f^4*sinh(f*x + e)^4 - 2*f^4*cosh(f*x + e)^2 + f^4
+ 2*(3*f^4*cosh(f*x + e)^2 - f^4)*sinh(f*x + e)^2 + 4*(f^4*cosh(f*x + e)^3 - f^4*cosh(f*x + e))*sinh(f*x + e)
)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{3} {\left (b \coth \left (f x + e\right ) + a\right )}^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^3*(a+b*coth(f*x+e))^3,x, algorithm="giac")
[Out]
integrate((d*x + c)^3*(b*coth(f*x + e) + a)^3, x)
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maple [B] time = 0.88, size = 2777, normalized size = 4.99 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x+c)^3*(a+b*coth(f*x+e))^3,x)
[Out]
1/4*a^3*d^3*x^4-1/4*b^3*d^3*x^4+c^3*a^3*x+c^3*b^3*x+a^3*c*d^2*x^3-b^3*c*d^2*x^3-3*a^2*b*c*d^2*x^3+3*a*b^2*c*d^
2*x^3-9/2*a^2*b*c^2*d*x^2+9/2*a*b^2*c^2*d*x^2+1/f*b^3*d^3*ln(exp(f*x+e)+1)*x^3+3/f^2*b^3*d^3*polylog(2,-exp(f*
x+e))*x^2-6/f^3*b^3*d^3*polylog(3,-exp(f*x+e))*x+1/f*b^3*d^3*ln(1-exp(f*x+e))*x^3+1/f^4*b^3*d^3*ln(1-exp(f*x+e
))*e^3+3/f^2*b^3*d^3*polylog(2,exp(f*x+e))*x^2-6/f^3*b^3*d^3*polylog(3,exp(f*x+e))*x-3/f^4*b^3*d^3*e*ln(exp(f*
x+e)-1)-18/f^4*b^2*a*d^3*polylog(3,-exp(f*x+e))-18/f^4*b^2*a*d^3*polylog(3,exp(f*x+e))+18/f^4*b*a^2*d^3*polylo
g(4,exp(f*x+e))-6/f^3*b^3*c*d^2*polylog(3,-exp(f*x+e))-6/f^3*b^3*c*d^2*polylog(3,exp(f*x+e))+3/f*b*a^2*c^3*ln(
exp(f*x+e)+1)+3/f*b*a^2*c^3*ln(exp(f*x+e)-1)-3/4*a^2*b*d^3*x^4+3/4*a*b^2*d^3*x^4+3/2*a^3*c^2*d*x^2-3/2*b^3*c^2
*d*x^2+3*c^3*a^2*b*x+3*c^3*a*b^2*x-9/2/f^4*b*e^4*a^2*d^3+4/f^3*b^3*c*d^2*e^3-3/f^2*b^3*c^2*d*e^2-6/f*b^2*a*d^3
*x^3+12/f^4*b^2*a*d^3*e^3-2/f^3*b^3*e^3*d^3*x+3/f^3*b^3*c*d^2*ln(exp(f*x+e)+1)+3/f^3*b^3*c*d^2*ln(exp(f*x+e)-1
)+18/f^4*b*a^2*d^3*polylog(4,-exp(f*x+e))-6/f*b*a^2*c^3*ln(exp(f*x+e))-6/f^3*b^3*c*d^2*ln(exp(f*x+e))+3/f^2*b^
3*c^2*d*polylog(2,-exp(f*x+e))+3/f^2*b^3*c^2*d*polylog(2,exp(f*x+e))-1/f^4*b^3*d^3*e^3*ln(exp(f*x+e)-1)+6/f^4*
b^3*d^3*e*ln(exp(f*x+e))+2/f^4*b^3*d^3*e^3*ln(exp(f*x+e))+3/f^3*b^3*d^3*ln(exp(f*x+e)+1)*x+3/f^3*b^3*d^3*ln(1-
exp(f*x+e))*x+3/f^4*b^3*d^3*ln(1-exp(f*x+e))*e-3/f^4*b^3*e^2*d^3-3/2/f^4*b^3*e^4*d^3-3/f^2*b^3*d^3*x^2+6/f^4*b
^3*d^3*polylog(4,-exp(f*x+e))+6/f^4*b^3*d^3*polylog(4,exp(f*x+e))+1/f*b^3*c^3*ln(exp(f*x+e)+1)+1/f*b^3*c^3*ln(
exp(f*x+e)-1)+3/f^4*b^3*d^3*polylog(2,-exp(f*x+e))+3/f^4*b^3*d^3*polylog(2,exp(f*x+e))-2/f*b^3*c^3*ln(exp(f*x+
e))-b^2*(6*a*d^3*f*x^3*exp(2*f*x+2*e)+2*b*d^3*f*x^3*exp(2*f*x+2*e)+18*a*c*d^2*f*x^2*exp(2*f*x+2*e)+6*b*c*d^2*f
*x^2*exp(2*f*x+2*e)+18*a*c^2*d*f*x*exp(2*f*x+2*e)-6*a*d^3*f*x^3+6*b*c^2*d*f*x*exp(2*f*x+2*e)+3*b*d^3*x^2*exp(2
*f*x+2*e)+6*a*c^3*f*exp(2*f*x+2*e)-18*a*c*d^2*f*x^2+2*b*c^3*f*exp(2*f*x+2*e)+6*b*c*d^2*x*exp(2*f*x+2*e)-18*a*c
^2*d*f*x+3*b*c^2*d*exp(2*f*x+2*e)-3*b*d^3*x^2-6*a*c^3*f-6*b*c*d^2*x-3*b*c^2*d)/f^2/(exp(2*f*x+2*e)-1)^2-36/f^2
*b^2*a*c*d^2*e*x-18/f*b*a^2*c^2*d*e*x+18/f^2*b*a^2*c*d^2*e^2*x-18/f^3*b*a^2*c*d^2*e^2*ln(exp(f*x+e))+9/f^3*b*a
^2*c*d^2*e^2*ln(exp(f*x+e)-1)+18/f^2*b^2*ln(1-exp(f*x+e))*a*c*d^2*x+18/f^2*b^2*ln(exp(f*x+e)+1)*a*c*d^2*x-9/f^
3*b*ln(1-exp(f*x+e))*a^2*c*d^2*e^2+18/f^2*b*a^2*c^2*d*e*ln(exp(f*x+e))+36/f^3*b^2*a*c*d^2*e*ln(exp(f*x+e))-9/f
^2*b*a^2*c^2*d*e*ln(exp(f*x+e)-1)-18/f^3*b^2*a*c*d^2*e*ln(exp(f*x+e)-1)+18/f^3*b^2*ln(1-exp(f*x+e))*a*c*d^2*e+
9/f*b*ln(1-exp(f*x+e))*a^2*c^2*d*x+9/f^2*b*ln(1-exp(f*x+e))*a^2*c^2*d*e+9/f*b*ln(exp(f*x+e)+1)*a^2*c*d^2*x^2+1
8/f^2*b*polylog(2,-exp(f*x+e))*a^2*c*d^2*x+9/f*b*ln(1-exp(f*x+e))*a^2*c*d^2*x^2+18/f^2*b*polylog(2,exp(f*x+e))
*a^2*c*d^2*x+9/f*b*ln(exp(f*x+e)+1)*a^2*c^2*d*x-6/f^3*b^3*d^3*e*x+12/f^3*b*a^2*c*d^2*e^3-18/f^3*b^2*a*c*d^2*e^
2-6/f^3*b*e^3*a^2*d^3*x+18/f^3*b^2*a*d^3*e^2*x-9/f^2*b*a^2*c^2*d*e^2-6/f*b^3*c^2*d*e*x+6/f^2*b^3*c*d^2*e^2*x-1
8/f*b^2*a*c*d^2*x^2+18/f^3*b^2*a*c*d^2*polylog(2,-exp(f*x+e))+18/f^3*b^2*a*c*d^2*polylog(2,exp(f*x+e))+9/f^4*b
^2*a*d^3*e^2*ln(exp(f*x+e)-1)-18/f^3*b*a^2*c*d^2*polylog(3,-exp(f*x+e))-18/f^3*b*a^2*c*d^2*polylog(3,exp(f*x+e
))+9/f^2*b*a^2*c^2*d*polylog(2,exp(f*x+e))-3/f^4*b*a^2*d^3*e^3*ln(exp(f*x+e)-1)-6/f^3*b^3*c*d^2*e^2*ln(exp(f*x
+e))+3/f^3*b^3*c*d^2*e^2*ln(exp(f*x+e)-1)+6/f^4*b*a^2*d^3*e^3*ln(exp(f*x+e))+9/f^2*b*a^2*c^2*d*polylog(2,-exp(
f*x+e))+3/f*b*a^2*d^3*ln(exp(f*x+e)+1)*x^3+9/f^2*b*a^2*d^3*polylog(2,-exp(f*x+e))*x^2-18/f^3*b*a^2*d^3*polylog
(3,-exp(f*x+e))*x+3/f*b*a^2*d^3*ln(1-exp(f*x+e))*x^3+9/f^2*b*a^2*d^3*polylog(2,exp(f*x+e))*x^2-18/f^3*b*a^2*d^
3*polylog(3,exp(f*x+e))*x-3/f^3*b^3*ln(1-exp(f*x+e))*c*d^2*e^2-18/f^2*b^2*a*c^2*d*ln(exp(f*x+e))+6/f^2*b^3*c^2
*d*e*ln(exp(f*x+e))-18/f^4*b^2*a*d^3*e^2*ln(exp(f*x+e))+9/f^2*b^2*a*c^2*d*ln(exp(f*x+e)+1)+9/f^2*b^2*a*c^2*d*l
n(exp(f*x+e)-1)-3/f^2*b^3*c^2*d*e*ln(exp(f*x+e)-1)+3/f^4*b*a^2*d^3*e^3*ln(1-exp(f*x+e))+3/f*b^3*ln(exp(f*x+e)+
1)*c^2*d*x+3/f*b^3*ln(1-exp(f*x+e))*c^2*d*x+3/f^2*b^3*ln(1-exp(f*x+e))*c^2*d*e+3/f*b^3*ln(exp(f*x+e)+1)*c*d^2*
x^2+6/f^2*b^3*polylog(2,-exp(f*x+e))*c*d^2*x+3/f*b^3*ln(1-exp(f*x+e))*c*d^2*x^2+6/f^2*b^3*polylog(2,exp(f*x+e)
)*c*d^2*x-9/f^4*b^2*a*d^3*e^2*ln(1-exp(f*x+e))+9/f^2*b^2*a*d^3*ln(exp(f*x+e)+1)*x^2+18/f^3*b^2*a*d^3*polylog(2
,-exp(f*x+e))*x+9/f^2*b^2*a*d^3*ln(1-exp(f*x+e))*x^2+18/f^3*b^2*a*d^3*polylog(2,exp(f*x+e))*x
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maxima [B] time = 0.53, size = 1531, normalized size = 2.75 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^3*(a+b*coth(f*x+e))^3,x, algorithm="maxima")
[Out]
1/4*a^3*d^3*x^4 + a^3*c*d^2*x^3 + 3/2*a^3*c^2*d*x^2 + a^3*c^3*x + 3*a^2*b*c^3*log(sinh(f*x + e))/f + 1/4*(24*a
*b^2*c^3*f + 12*b^3*c^2*d + (3*a^2*b*d^3*f^2 + 3*a*b^2*d^3*f^2 + b^3*d^3*f^2)*x^4 + 4*(3*a^2*b*c*d^2*f^2 + b^3
*c*d^2*f^2 + 3*(c*d^2*f^2 + 2*d^3*f)*a*b^2)*x^3 + 6*(3*a^2*b*c^2*d*f^2 + 3*(c^2*d*f^2 + 4*c*d^2*f)*a*b^2 + (c^
2*d*f^2 + 2*d^3)*b^3)*x^2 + 4*(3*(c^3*f^2 + 6*c^2*d*f)*a*b^2 + (c^3*f^2 + 6*c*d^2)*b^3)*x + ((3*a^2*b*d^3*f^2*
e^(4*e) + 3*a*b^2*d^3*f^2*e^(4*e) + b^3*d^3*f^2*e^(4*e))*x^4 + 4*(3*a^2*b*c*d^2*f^2*e^(4*e) + 3*a*b^2*c*d^2*f^
2*e^(4*e) + b^3*c*d^2*f^2*e^(4*e))*x^3 + 6*(3*a^2*b*c^2*d*f^2*e^(4*e) + 3*a*b^2*c^2*d*f^2*e^(4*e) + b^3*c^2*d*
f^2*e^(4*e))*x^2 + 4*(3*a*b^2*c^3*f^2*e^(4*e) + b^3*c^3*f^2*e^(4*e))*x)*e^(4*f*x) - 2*(12*a*b^2*c^3*f*e^(2*e)
+ (3*a^2*b*d^3*f^2*e^(2*e) + 3*a*b^2*d^3*f^2*e^(2*e) + b^3*d^3*f^2*e^(2*e))*x^4 + 2*(2*c^3*f*e^(2*e) + 3*c^2*d
*e^(2*e))*b^3 + 4*(3*a^2*b*c*d^2*f^2*e^(2*e) + 3*(c*d^2*f^2*e^(2*e) + d^3*f*e^(2*e))*a*b^2 + (c*d^2*f^2*e^(2*e
) + d^3*f*e^(2*e))*b^3)*x^3 + 6*(3*a^2*b*c^2*d*f^2*e^(2*e) + 3*(c^2*d*f^2*e^(2*e) + 2*c*d^2*f*e^(2*e))*a*b^2 +
(c^2*d*f^2*e^(2*e) + 2*c*d^2*f*e^(2*e) + d^3*e^(2*e))*b^3)*x^2 + 4*(3*(c^3*f^2*e^(2*e) + 3*c^2*d*f*e^(2*e))*a
*b^2 + (c^3*f^2*e^(2*e) + 3*c^2*d*f*e^(2*e) + 3*c*d^2*e^(2*e))*b^3)*x)*e^(2*f*x))/(f^2*e^(4*f*x + 4*e) - 2*f^2
*e^(2*f*x + 2*e) + f^2) - 2*(9*a*b^2*c^2*d*f + (c^3*f^2 + 3*c*d^2)*b^3)*x/f^2 + (9*a*b^2*c^2*d*f + (c^3*f^2 +
3*c*d^2)*b^3)*log(e^(f*x + e) + 1)/f^3 + (9*a*b^2*c^2*d*f + (c^3*f^2 + 3*c*d^2)*b^3)*log(e^(f*x + e) - 1)/f^3
+ (f^3*x^3*log(e^(f*x + e) + 1) + 3*f^2*x^2*dilog(-e^(f*x + e)) - 6*f*x*polylog(3, -e^(f*x + e)) + 6*polylog(4
, -e^(f*x + e)))*(3*a^2*b*d^3 + b^3*d^3)/f^4 + (f^3*x^3*log(-e^(f*x + e) + 1) + 3*f^2*x^2*dilog(e^(f*x + e)) -
6*f*x*polylog(3, e^(f*x + e)) + 6*polylog(4, e^(f*x + e)))*(3*a^2*b*d^3 + b^3*d^3)/f^4 + 3*(3*a^2*b*c*d^2*f +
b^3*c*d^2*f + 3*a*b^2*d^3)*(f^2*x^2*log(e^(f*x + e) + 1) + 2*f*x*dilog(-e^(f*x + e)) - 2*polylog(3, -e^(f*x +
e)))/f^4 + 3*(3*a^2*b*c*d^2*f + b^3*c*d^2*f + 3*a*b^2*d^3)*(f^2*x^2*log(-e^(f*x + e) + 1) + 2*f*x*dilog(e^(f*
x + e)) - 2*polylog(3, e^(f*x + e)))/f^4 + 3*(3*a^2*b*c^2*d*f^2 + 6*a*b^2*c*d^2*f + (c^2*d*f^2 + d^3)*b^3)*(f*
x*log(e^(f*x + e) + 1) + dilog(-e^(f*x + e)))/f^4 + 3*(3*a^2*b*c^2*d*f^2 + 6*a*b^2*c*d^2*f + (c^2*d*f^2 + d^3)
*b^3)*(f*x*log(-e^(f*x + e) + 1) + dilog(e^(f*x + e)))/f^4 - 1/2*((3*a^2*b*d^3 + b^3*d^3)*f^4*x^4 + 4*(3*a^2*b
*c*d^2*f + b^3*c*d^2*f + 3*a*b^2*d^3)*f^3*x^3 + 6*(3*a^2*b*c^2*d*f^2 + 6*a*b^2*c*d^2*f + (c^2*d*f^2 + d^3)*b^3
)*f^2*x^2)/f^4
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+b\,\mathrm {coth}\left (e+f\,x\right )\right )}^3\,{\left (c+d\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*coth(e + f*x))^3*(c + d*x)^3,x)
[Out]
int((a + b*coth(e + f*x))^3*(c + d*x)^3, x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \coth {\left (e + f x \right )}\right )^{3} \left (c + d x\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)**3*(a+b*coth(f*x+e))**3,x)
[Out]
Integral((a + b*coth(e + f*x))**3*(c + d*x)**3, x)
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