3.63 \(\int (c+d x)^3 (a+b \tanh (e+f x))^3 \, dx\)

Optimal. Leaf size=566 \[ \frac {a^3 (c+d x)^4}{4 d}-\frac {9 a^2 b d^2 (c+d x) \text {Li}_3\left (-e^{2 (e+f x)}\right )}{2 f^3}+\frac {9 a^2 b d (c+d x)^2 \text {Li}_2\left (-e^{2 (e+f x)}\right )}{2 f^2}+\frac {3 a^2 b (c+d x)^3 \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac {3 a^2 b (c+d x)^4}{4 d}+\frac {9 a^2 b d^3 \text {Li}_4\left (-e^{2 (e+f x)}\right )}{4 f^4}+\frac {9 a b^2 d^2 (c+d x) \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^3}+\frac {9 a b^2 d (c+d x)^2 \log \left (e^{2 (e+f x)}+1\right )}{f^2}-\frac {3 a b^2 (c+d x)^3 \tanh (e+f x)}{f}-\frac {3 a b^2 (c+d x)^3}{f}+\frac {3 a b^2 (c+d x)^4}{4 d}-\frac {9 a b^2 d^3 \text {Li}_3\left (-e^{2 (e+f x)}\right )}{2 f^4}-\frac {3 b^3 d^2 (c+d x) \text {Li}_3\left (-e^{2 (e+f x)}\right )}{2 f^3}+\frac {3 b^3 d^2 (c+d x) \log \left (e^{2 (e+f x)}+1\right )}{f^3}+\frac {3 b^3 d (c+d x)^2 \text {Li}_2\left (-e^{2 (e+f x)}\right )}{2 f^2}-\frac {3 b^3 d (c+d x)^2 \tanh (e+f x)}{2 f^2}+\frac {b^3 (c+d x)^3 \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac {b^3 (c+d x)^3 \tanh ^2(e+f x)}{2 f}-\frac {3 b^3 d (c+d x)^2}{2 f^2}+\frac {b^3 (c+d x)^3}{2 f}-\frac {b^3 (c+d x)^4}{4 d}+\frac {3 b^3 d^3 \text {Li}_2\left (-e^{2 (e+f x)}\right )}{2 f^4}+\frac {3 b^3 d^3 \text {Li}_4\left (-e^{2 (e+f x)}\right )}{4 f^4} \]

[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*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*pol
ylog(2,-exp(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^4-3/2*b^3*d*(d*x+c)^2*
tanh(f*x+e)/f^2-3*a*b^2*(d*x+c)^3*tanh(f*x+e)/f-1/2*b^3*(d*x+c)^3*tanh(f*x+e)^2/f

________________________________________________________________________________________

Rubi [A]  time = 1.02, antiderivative size = 566, 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, 3718, 2190, 2531, 6609, 2282, 6589, 3720, 32, 2279, 2391} \[ -\frac {9 a^2 b d^2 (c+d x) \text {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3}+\frac {9 a^2 b d (c+d x)^2 \text {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^2}+\frac {9 a^2 b d^3 \text {PolyLog}\left (4,-e^{2 (e+f x)}\right )}{4 f^4}+\frac {9 a b^2 d^2 (c+d x) \text {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^3}-\frac {9 a b^2 d^3 \text {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^4}-\frac {3 b^3 d^2 (c+d x) \text {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3}+\frac {3 b^3 d (c+d x)^2 \text {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^2}+\frac {3 b^3 d^3 \text {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^4}+\frac {3 b^3 d^3 \text {PolyLog}\left (4,-e^{2 (e+f x)}\right )}{4 f^4}+\frac {3 a^2 b (c+d x)^3 \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac {3 a^2 b (c+d x)^4}{4 d}+\frac {a^3 (c+d x)^4}{4 d}+\frac {9 a b^2 d (c+d x)^2 \log \left (e^{2 (e+f x)}+1\right )}{f^2}-\frac {3 a b^2 (c+d x)^3 \tanh (e+f x)}{f}-\frac {3 a b^2 (c+d x)^3}{f}+\frac {3 a b^2 (c+d x)^4}{4 d}+\frac {3 b^3 d^2 (c+d x) \log \left (e^{2 (e+f x)}+1\right )}{f^3}-\frac {3 b^3 d (c+d x)^2 \tanh (e+f x)}{2 f^2}+\frac {b^3 (c+d x)^3 \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac {b^3 (c+d x)^3 \tanh ^2(e+f x)}{2 f}-\frac {3 b^3 d (c+d x)^2}{2 f^2}+\frac {b^3 (c+d x)^3}{2 f}-\frac {b^3 (c+d x)^4}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3*(a + b*Tanh[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^2*(c + d*x)*Lo
g[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*Po
lyLog[4, -E^(2*(e + f*x))])/(4*f^4) - (3*b^3*d*(c + d*x)^2*Tanh[e + f*x])/(2*f^2) - (3*a*b^2*(c + d*x)^3*Tanh[
e + f*x])/f - (b^3*(c + d*x)^3*Tanh[e + f*x]^2)/(2*f)

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 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 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 \tanh (e+f x))^3 \, dx &=\int \left (a^3 (c+d x)^3+3 a^2 b (c+d x)^3 \tanh (e+f x)+3 a b^2 (c+d x)^3 \tanh ^2(e+f x)+b^3 (c+d x)^3 \tanh ^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 \tanh (e+f x) \, dx+\left (3 a b^2\right ) \int (c+d x)^3 \tanh ^2(e+f x) \, dx+b^3 \int (c+d x)^3 \tanh ^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 \tanh (e+f x)}{f}-\frac {b^3 (c+d x)^3 \tanh ^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 \tanh (e+f x) \, dx+\frac {\left (9 a b^2 d\right ) \int (c+d x)^2 \tanh (e+f x) \, dx}{f}+\frac {\left (3 b^3 d\right ) \int (c+d x)^2 \tanh ^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 a^2 b (c+d x)^3 \log \left (1+e^{2 (e+f x)}\right )}{f}-\frac {3 b^3 d (c+d x)^2 \tanh (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x)^3 \tanh (e+f x)}{f}-\frac {b^3 (c+d x)^3 \tanh ^2(e+f x)}{2 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) \tanh (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 {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 {3 b^3 d (c+d x)^2 \tanh (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x)^3 \tanh (e+f x)}{f}-\frac {b^3 (c+d x)^3 \tanh ^2(e+f x)}{2 f}-\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^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 (c+d x)^2 \tanh (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x)^3 \tanh (e+f x)}{f}-\frac {b^3 (c+d x)^3 \tanh ^2(e+f x)}{2 f}+\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^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 {3 b^3 d (c+d x)^2 \tanh (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x)^3 \tanh (e+f x)}{f}-\frac {b^3 (c+d x)^3 \tanh ^2(e+f x)}{2 f}+\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^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 (c+d x)^2 \tanh (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x)^3 \tanh (e+f x)}{f}-\frac {b^3 (c+d x)^3 \tanh ^2(e+f x)}{2 f}+\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^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}-\frac {3 b^3 d (c+d x)^2 \tanh (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x)^3 \tanh (e+f x)}{f}-\frac {b^3 (c+d x)^3 \tanh ^2(e+f x)}{2 f}\\ \end {align*}

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Mathematica [B]  time = 14.61, size = 2010, normalized size = 3.55 \[ \text {Result too large to show} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^3*(a + b*Tanh[e + f*x])^3,x]

[Out]

(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*Lo
g[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))] + (72*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*Log[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*Log[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) + ((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)*Sec
h[e + f*x]^2)/(2*f) + (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*Si
nh[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*Co
sh[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] + C
osh[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*Sech[e]*Sech[e + f*x]*(b^3*c^2*d*Sinh[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 = 1.20, size = 12909, normalized size = 22.81 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*(a+b*tanh(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(I*cosh(f*x + e) + I*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)*c
osh(f*x + e))*sinh(f*x + e))*dilog(-I*cosh(f*x + e) - I*sinh(f*x + e)) + 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)*c
osh(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) + I) + 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)*sin
h(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) - I) + 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(I*cosh(f*x + e) + I*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(-I*cosh(f*x + e) - I*sinh(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, I*cosh(f*x + e) + I*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*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*co
sh(f*x + e))*sinh(f*x + e))*polylog(4, -I*cosh(f*x + e) - I*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, I*cosh(f*x +
e) + I*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, -I*cosh(f*x + e) - I*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 \[ \int {\left (d x + c\right )}^{3} {\left (b \tanh \left (f x + e\right ) + a\right )}^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*(a+b*tanh(f*x+e))^3,x, algorithm="giac")

[Out]

integrate((d*x + c)^3*(b*tanh(f*x + e) + a)^3, x)

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maple [B]  time = 0.67, size = 1788, normalized size = 3.16 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3*(a+b*tanh(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^3-36/f^2*b^2*a*c*d^2*e*x+18/f^2*
b*a^2*c*d^2*e^2*x+4/f^3*b^3*c*d^2*e^3-2/f^3*b^3*e^3*d^3*x-9/2/f^4*b*e^4*a^2*d^3-6/f*b^2*a*d^3*x^3+12/f^4*b^2*a
*d^3*e^3-6/f^3*b^3*d^3*e*x-3/f^2*b^3*c^2*d*e^2+3/2/f^2*b^3*c^2*d*polylog(2,-exp(2*f*x+2*e))+2/f^4*b^3*d^3*e^3*
ln(exp(f*x+e))+6/f^4*b^3*d^3*e*ln(exp(f*x+e))+3/f^3*b^3*d^3*ln(exp(2*f*x+2*e)+1)*x+1/f*b^3*d^3*ln(exp(2*f*x+2*
e)+1)*x^3+3/2/f^2*b^3*d^3*polylog(2,-exp(2*f*x+2*e))*x^2-3/2/f^3*b^3*d^3*polylog(3,-exp(2*f*x+2*e))*x+3/f^3*b^
3*c*d^2*ln(exp(2*f*x+2*e)+1)+3/f*b*a^2*c^3*ln(exp(2*f*x+2*e)+1)-6/f*b*a^2*c^3*ln(exp(f*x+e))-6/f^3*b^3*c*d^2*l
n(exp(f*x+e))-3/2/f^3*b^3*c*d^2*polylog(3,-exp(2*f*x+2*e))-3/f^4*b^3*e^2*d^3-3/f^2*b^3*d^3*x^2-3/2/f^4*b^3*e^4
*d^3-9/2*a*b^2*d^3*polylog(3,-exp(2*f*x+2*e))/f^4+3/2*b^3*d^3*polylog(2,-exp(2*f*x+2*e))/f^4+3/4*b^3*d^3*polyl
og(4,-exp(2*f*x+2*e))/f^4-3*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^2-18/f*b*a
^2*c^2*d*e*x+1/f*b^3*c^3*ln(exp(2*f*x+2*e)+1)-2/f*b^3*c^3*ln(exp(f*x+e))+9/4*a^2*b*d^3*polylog(4,-exp(2*f*x+2*
e))/f^4+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*e
xp(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)^2-3/
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+12/f^3*b*a^2
*c*d^2*e^3-9/f^2*b*a^2*c^2*d*e^2+18/f^3*b^2*a*d^3*e^2*x-18/f^3*b^2*a*c*d^2*e^2-6/f^3*b*e^3*a^2*d^3*x+6/f^2*b^3
*c*d^2*e^2*x-18/f*b^2*a*c*d^2*x^2-6/f*b^3*c^2*d*e*x+9/f^2*b^2*a*d^3*ln(exp(2*f*x+2*e)+1)*x^2+9/f^3*b^2*a*c*d^2
*polylog(2,-exp(2*f*x+2*e))+6/f^4*b*a^2*d^3*e^3*ln(exp(f*x+e))+9/2/f^2*b*a^2*c^2*d*polylog(2,-exp(2*f*x+2*e))-
6/f^3*b^3*c*d^2*e^2*ln(exp(f*x+e))-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(2*f*x+2*e)+1)+3/f*b*a^2*d^3*ln(exp(2*f*x+2*e)+1)*x^3+9/2/
f^2*b*a^2*d^3*polylog(2,-exp(2*f*x+2*e))*x^2-9/2/f^3*b*a^2*d^3*polylog(3,-exp(2*f*x+2*e))*x-9/2/f^3*b*a^2*c*d^
2*polylog(3,-exp(2*f*x+2*e))+9/f^3*b^2*a*d^3*polylog(2,-exp(2*f*x+2*e))*x+3/f*b^3*ln(exp(2*f*x+2*e)+1)*c*d^2*x
^2+3/f^2*b^3*polylog(2,-exp(2*f*x+2*e))*c*d^2*x+3/f*b^3*ln(exp(2*f*x+2*e)+1)*c^2*d*x+18/f^2*b*a^2*c^2*d*e*ln(e
xp(f*x+e))+36/f^3*b^2*a*c*d^2*e*ln(exp(f*x+e))+9/f*b*ln(exp(2*f*x+2*e)+1)*a^2*c*d^2*x^2+9/f^2*b*polylog(2,-exp
(2*f*x+2*e))*a^2*c*d^2*x-18/f^3*b*a^2*c*d^2*e^2*ln(exp(f*x+e))+9/f*b*ln(exp(2*f*x+2*e)+1)*a^2*c^2*d*x+18/f^2*b
^2*ln(exp(2*f*x+2*e)+1)*a*c*d^2*x

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maxima [B]  time = 1.36, size = 1297, normalized size = 2.29 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*(a+b*tanh(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 + b^3*c^3*(x + e/f + log(e^(-2*f*x - 2*e) + 1)/f + 2*e^(-2
*f*x - 2*e)/(f*(2*e^(-2*f*x - 2*e) + e^(-4*f*x - 4*e) + 1))) + a^3*c^3*x + 3*a^2*b*c^3*log(cosh(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 + 12*(2*b^3*c*d^2 + (c^3*f^2 + 6*c^2*d*f)*a*b^2)*x + (12*a*b^2*c^3*f^2*x*e^
(4*e) + (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)*e^(4*f*x) + 2*(12*a*b^2*c^3*f*e^(2*e) + 6*b^3*c^2*d*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 + 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 + 12*((c^3*f^2*e^(2*e) + 3*c^2*d*f*e^(2*e))*a*b^2 + (c^2*d*f*e^(2*e) + 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) - 6*(3*a*b^2*c^2*d*f + b^3*c*d^2)*x/f^2 + 3*(3*a*b
^2*c^2*d*f + b^3*c*d^2)*log(e^(2*f*x + 2*e) + 1)/f^3 + 1/3*(4*f^3*x^3*log(e^(2*f*x + 2*e) + 1) + 6*f^2*x^2*dil
og(-e^(2*f*x + 2*e)) - 6*f*x*polylog(3, -e^(2*f*x + 2*e)) + 3*polylog(4, -e^(2*f*x + 2*e)))*(3*a^2*b*d^3 + b^3
*d^3)/f^4 + 3/2*(3*a^2*b*c*d^2*f + b^3*c*d^2*f + 3*a*b^2*d^3)*(2*f^2*x^2*log(e^(2*f*x + 2*e) + 1) + 2*f*x*dilo
g(-e^(2*f*x + 2*e)) - polylog(3, -e^(2*f*x + 2*e)))/f^4 + 3/2*(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)*(2*f*x*log(e^(2*f*x + 2*e) + 1) + dilog(-e^(2*f*x + 2*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 \[ \int {\left (a+b\,\mathrm {tanh}\left (e+f\,x\right )\right )}^3\,{\left (c+d\,x\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*tanh(e + f*x))^3*(c + d*x)^3,x)

[Out]

int((a + b*tanh(e + f*x))^3*(c + d*x)^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \tanh {\left (e + f x \right )}\right )^{3} \left (c + d x\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**3*(a+b*tanh(f*x+e))**3,x)

[Out]

Integral((a + b*tanh(e + f*x))**3*(c + d*x)**3, x)

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