3.1.63 \(\int (c+d x)^3 (a+b \tanh (e+f x))^3 \, dx\) [63]

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

[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 = 0.70, 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 = {3803, 3799, 2221, 2611, 6744, 2320, 6724, 3801, 32, 2317, 2438} \begin {gather*} \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} \end {gather*}

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 2221

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

Rule 2317

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

Rule 2320

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

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2611

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

Rule 3799

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 3801

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 3803

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 6724

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

Rule 6744

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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(2010\) vs. \(2(566)=1132\).
time = 11.22, size = 2010, normalized size = 3.55 \begin {gather*} \text {Result too large to show} \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1833\) vs. \(2(534)=1068\).
time = 3.92, size = 1834, normalized size = 3.24

method	result	size
risch	\(\text {Expression too large to display}\)	\(1834\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-36/f^2*b^2*a*c*d^2*e*x+9/f*b*ln(1+exp(2*f*x+2*e))*a^2*c^2*d*x+9/f*b*ln(1+exp(2*f*x+2*e))*a^2*c*d^2*x^2-18/f^3
*b*a^2*c*d^2*e^2*ln(exp(f*x+e))+18/f^2*b^2*ln(1+exp(2*f*x+2*e))*a*c*d^2*x+36/f^3*b^2*a*c*d^2*e*ln(exp(f*x+e))+
18/f^2*b*a^2*c*d^2*e^2*x-18/f*b*a^2*c^2*d*e*x+18/f^2*b*a^2*c^2*d*e*ln(exp(f*x+e))+9/f^2*b*polylog(2,-exp(2*f*x
+2*e))*a^2*c*d^2*x-9/2*a*b^2*d^3*polylog(3,-exp(2*f*x+2*e))/f^4+9/4*a^2*b*d^3*polylog(4,-exp(2*f*x+2*e))/f^4-3
/f^2*b^3*d^3*x^2-2/f*b^3*c^3*ln(exp(f*x+e))+1/f*b^3*c^3*ln(1+exp(2*f*x+2*e))-6/f^3*b*e^3*a^2*d^3*x-18/f^3*b^2*
a*c*d^2*e^2-9/f^2*b*a^2*c^2*d*e^2+12/f^3*b*a^2*c*d^2*e^3+3/f^2*b^3*polylog(2,-exp(2*f*x+2*e))*c*d^2*x+9/f^3*b^
2*a*c*d^2*polylog(2,-exp(2*f*x+2*e))+9/2/f^2*b*a^2*c^2*d*polylog(2,-exp(2*f*x+2*e))-9/2/f^3*b*a^2*c*d^2*polylo
g(3,-exp(2*f*x+2*e))-6/f^3*b^3*c*d^2*e^2*ln(exp(f*x+e))+6/f^4*b*a^2*d^3*e^3*ln(exp(f*x+e))-9/2/f^3*b*a^2*d^3*p
olylog(3,-exp(2*f*x+2*e))*x+9/f^2*b^2*a*c^2*d*ln(1+exp(2*f*x+2*e))-18/f^4*b^2*a*d^3*e^2*ln(exp(f*x+e))+6/f^2*b
^3*c^2*d*e*ln(exp(f*x+e))-18/f^2*b^2*a*c^2*d*ln(exp(f*x+e))-1/4*d^3*b^3*x^4+1/4/d*a^3*c^4+1/4/d*b^3*c^4+1/4*d^
3*a^3*x^4+3/f*b*a^2*d^3*ln(1+exp(2*f*x+2*e))*x^3+9/2/f^2*b*a^2*d^3*polylog(2,-exp(2*f*x+2*e))*x^2+9/f^2*b^2*a*
d^3*ln(1+exp(2*f*x+2*e))*x^2+9/f^3*b^2*a*d^3*polylog(2,-exp(2*f*x+2*e))*x+3/f*b^3*ln(1+exp(2*f*x+2*e))*c^2*d*x
+3/f*b^3*ln(1+exp(2*f*x+2*e))*c*d^2*x^2+6/f^2*b^3*c*d^2*e^2*x+18/f^3*b^2*e^2*a*d^3*x-18/f*b^2*a*c*d^2*x^2-6/f*
b^3*c^2*d*e*x-6/f^3*b^3*d^3*e*x+12/f^4*b^2*e^3*a*d^3-3/f^2*b^3*c^2*d*e^2-2/f^3*b^3*e^3*d^3*x-6/f*b^2*a*d^3*x^3
-9/2/f^4*b*e^4*a^2*d^3+4/f^3*b^3*c*d^2*e^3+1/f*b^3*d^3*ln(1+exp(2*f*x+2*e))*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*d^3*ln(1+exp(2*f*x+2*e))*x+3/f^3*b^3*c
*d^2*ln(1+exp(2*f*x+2*e))+3/f*b*a^2*c^3*ln(1+exp(2*f*x+2*e))+6/f^4*b^3*d^3*e*ln(exp(f*x+e))+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))-3/2/f^3*b^3*c*d^2*polylog(3,-exp(2*f*x+2*e))-6/f^3
*b^3*c*d^2*ln(exp(f*x+e))-6/f*b*a^2*c^3*ln(exp(f*x+e))+3/2*b^3*d^3*polylog(2,-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*d^2*a^2*b*c*x^3+3*d^2*a*b^2*c*x^3-9/2*d*a^2*b*c^2*x^2+9/2*d*a*b^2*c^2*x^2+3*a
^2*b*c^3*x+3*a*b^2*c^3*x+3/4*d^3*a*b^2*x^4+d^2*a^3*c*x^3-d^2*b^3*c*x^3+3/2*d*a^3*c^2*x^2-3/2*d*b^3*c^2*x^2+a^3
*c^3*x+b^3*c^3*x+3/4/d*a^2*b*c^4+3/4/d*a*b^2*c^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*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/(1+exp(2*f*x+2*e))^2-3/4*d^3*a^2*b*x^4-3/f^4*b^3*e^2*d^3-3/2/f^4*b^3*e^4*d^3

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1241 vs. \(2 (546) = 1092\).
time = 0.60, size = 1241, normalized size = 2.19 \begin {gather*} \frac {1}{4} \, a^{3} d^{3} x^{4} + a^{3} c d^{2} x^{3} + \frac {3}{2} \, a^{3} c^{2} d x^{2} + b^{3} c^{3} {\left (x + \frac {e}{f} + \frac {\log \left (e^{\left (-2 \, f x - 2 \, e\right )} + 1\right )}{f} + \frac {2 \, e^{\left (-2 \, f x - 2 \, e\right )}}{f {\left (2 \, e^{\left (-2 \, f x - 2 \, e\right )} + e^{\left (-4 \, f x - 4 \, e\right )} + 1\right )}}\right )} + a^{3} c^{3} x + \frac {3 \, a^{2} b c^{3} \log \left (\cosh \left (f x + e\right )\right )}{f} + \frac {24 \, a b^{2} c^{3} f + 12 \, b^{3} c^{2} d + {\left (3 \, a^{2} b d^{3} f^{2} + 3 \, a b^{2} d^{3} f^{2} + b^{3} d^{3} f^{2}\right )} x^{4} + 4 \, {\left (3 \, a^{2} b c d^{2} f^{2} + b^{3} c d^{2} f^{2} + 3 \, {\left (c d^{2} f^{2} + 2 \, d^{3} f\right )} a b^{2}\right )} x^{3} + 6 \, {\left (3 \, a^{2} b c^{2} d f^{2} + 3 \, {\left (c^{2} d f^{2} + 4 \, c d^{2} f\right )} a b^{2} + {\left (c^{2} d f^{2} + 2 \, d^{3}\right )} b^{3}\right )} x^{2} + 12 \, {\left (2 \, b^{3} c d^{2} + {\left (c^{3} f^{2} + 6 \, c^{2} d f\right )} a b^{2}\right )} x + {\left (12 \, a b^{2} c^{3} f^{2} x e^{\left (4 \, e\right )} + {\left (3 \, a^{2} b d^{3} f^{2} + 3 \, a b^{2} d^{3} f^{2} + b^{3} d^{3} f^{2}\right )} x^{4} e^{\left (4 \, e\right )} + 4 \, {\left (3 \, a^{2} b c d^{2} f^{2} + 3 \, a b^{2} c d^{2} f^{2} + b^{3} c d^{2} f^{2}\right )} x^{3} e^{\left (4 \, e\right )} + 6 \, {\left (3 \, a^{2} b c^{2} d f^{2} + 3 \, a b^{2} c^{2} d f^{2} + b^{3} c^{2} d f^{2}\right )} x^{2} e^{\left (4 \, e\right )}\right )} e^{\left (4 \, f x\right )} + 2 \, {\left ({\left (3 \, a^{2} b d^{3} f^{2} + 3 \, a b^{2} d^{3} f^{2} + b^{3} d^{3} f^{2}\right )} x^{4} e^{\left (2 \, e\right )} + 4 \, {\left (3 \, a^{2} b c d^{2} f^{2} + 3 \, {\left (c d^{2} f^{2} + d^{3} f\right )} a b^{2} + {\left (c d^{2} f^{2} + d^{3} f\right )} b^{3}\right )} x^{3} e^{\left (2 \, e\right )} + 6 \, {\left (3 \, a^{2} b c^{2} d f^{2} + 3 \, {\left (c^{2} d f^{2} + 2 \, c d^{2} f\right )} a b^{2} + {\left (c^{2} d f^{2} + 2 \, c d^{2} f + d^{3}\right )} b^{3}\right )} x^{2} e^{\left (2 \, e\right )} + 12 \, {\left ({\left (c^{3} f^{2} + 3 \, c^{2} d f\right )} a b^{2} + {\left (c^{2} d f + c d^{2}\right )} b^{3}\right )} x e^{\left (2 \, e\right )} + 6 \, {\left (2 \, a b^{2} c^{3} f + b^{3} c^{2} d\right )} e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{4 \, {\left (f^{2} e^{\left (4 \, f x + 4 \, e\right )} + 2 \, f^{2} e^{\left (2 \, f x + 2 \, e\right )} + f^{2}\right )}} - \frac {6 \, {\left (3 \, a b^{2} c^{2} d f + b^{3} c d^{2}\right )} x}{f^{2}} + \frac {3 \, {\left (3 \, a b^{2} c^{2} d f + b^{3} c d^{2}\right )} \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right )}{f^{3}} + \frac {{\left (4 \, f^{3} x^{3} \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right ) + 6 \, f^{2} x^{2} {\rm Li}_2\left (-e^{\left (2 \, f x + 2 \, e\right )}\right ) - 6 \, f x {\rm Li}_{3}(-e^{\left (2 \, f x + 2 \, e\right )}) + 3 \, {\rm Li}_{4}(-e^{\left (2 \, f x + 2 \, e\right )})\right )} {\left (3 \, a^{2} b d^{3} + b^{3} d^{3}\right )}}{3 \, f^{4}} + \frac {3 \, {\left (3 \, a^{2} b c d^{2} f + b^{3} c d^{2} f + 3 \, a b^{2} d^{3}\right )} {\left (2 \, f^{2} x^{2} \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right ) + 2 \, f x {\rm Li}_2\left (-e^{\left (2 \, f x + 2 \, e\right )}\right ) - {\rm Li}_{3}(-e^{\left (2 \, f x + 2 \, e\right )})\right )}}{2 \, f^{4}} + \frac {3 \, {\left (3 \, a^{2} b c^{2} d f^{2} + 6 \, a b^{2} c d^{2} f + {\left (c^{2} d f^{2} + d^{3}\right )} b^{3}\right )} {\left (2 \, f x \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (2 \, f x + 2 \, e\right )}\right )\right )}}{2 \, f^{4}} - \frac {{\left (3 \, a^{2} b d^{3} + b^{3} d^{3}\right )} f^{4} x^{4} + 4 \, {\left (3 \, a^{2} b c d^{2} f + b^{3} c d^{2} f + 3 \, a b^{2} d^{3}\right )} f^{3} x^{3} + 6 \, {\left (3 \, a^{2} b c^{2} d f^{2} + 6 \, a b^{2} c d^{2} f + {\left (c^{2} d f^{2} + d^{3}\right )} b^{3}\right )} f^{2} x^{2}}{2 \, f^{4}} \end {gather*}

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 + 3*a*b^2*d^3*f^2 + b^3*d^3*f^2)*x^4*e^(4*e) + 4*(3*a^2*b*c*d^2*f^2 + 3*a*b^2*c*d^2*f
^2 + b^3*c*d^2*f^2)*x^3*e^(4*e) + 6*(3*a^2*b*c^2*d*f^2 + 3*a*b^2*c^2*d*f^2 + b^3*c^2*d*f^2)*x^2*e^(4*e))*e^(4*
f*x) + 2*((3*a^2*b*d^3*f^2 + 3*a*b^2*d^3*f^2 + b^3*d^3*f^2)*x^4*e^(2*e) + 4*(3*a^2*b*c*d^2*f^2 + 3*(c*d^2*f^2
+ d^3*f)*a*b^2 + (c*d^2*f^2 + d^3*f)*b^3)*x^3*e^(2*e) + 6*(3*a^2*b*c^2*d*f^2 + 3*(c^2*d*f^2 + 2*c*d^2*f)*a*b^2
+ (c^2*d*f^2 + 2*c*d^2*f + d^3)*b^3)*x^2*e^(2*e) + 12*((c^3*f^2 + 3*c^2*d*f)*a*b^2 + (c^2*d*f + c*d^2)*b^3)*x
*e^(2*e) + 6*(2*a*b^2*c^3*f + b^3*c^2*d)*e^(2*e))*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*dilog(-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*dilog(-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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Fricas [C] Result contains complex when optimal does not.
time = 0.70, size = 21594, normalized size = 38.15 \begin {gather*} \text {Too large to display} \end {gather*}

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*c^3*f^3 + 4*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c^3*f^4*x + 12*b
^3*c^2*d*f^2 + 2*(3*a^2*b + b^3)*d^3*cosh(1)^4 + 2*(3*a^2*b + b^3)*d^3*sinh(1)^4 + ((a^3 - 3*a^2*b + 3*a*b^2 -
b^3)*d^3*f^4*x^4 + 2*(3*a^2*b + b^3)*d^3*cosh(1)^4 + 2*(3*a^2*b + b^3)*d^3*sinh(1)^4 - 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 - 8*(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f)*cosh(1)^3 - 8*(3*a*b
^2*d^3 + (3*a^2*b + b^3)*c*d^2*f - (3*a^2*b + b^3)*d^3*cosh(1))*sinh(1)^3 - 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 + 12*(6*a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f
^2)*cosh(1)^2 + 12*(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*cosh(1)^2 - 2*
(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f)*cosh(1))*sinh(1)^2 - 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 - 8*(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)*cosh(1)
- 8*(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*cosh(1)^3 + 3*(3*a*b^2*
d^3 + (3*a^2*b + b^3)*c*d^2*f)*cosh(1)^2 - 3*(6*a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2)*cosh(1))*
sinh(1))*cosh(f*x + cosh(1) + sinh(1))^4 + 4*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^3*f^4*x^4 + 2*(3*a^2*b + b^3)*
d^3*cosh(1)^4 + 2*(3*a^2*b + b^3)*d^3*sinh(1)^4 - 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 - 8*(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f)*cosh(1)^3 - 8*(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f - (3*
a^2*b + b^3)*d^3*cosh(1))*sinh(1)^3 - 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 + 12*(6*a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2)*cosh(1)^2 + 12*(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*cosh(1)^2 - 2*(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f
)*cosh(1))*sinh(1)^2 - 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 -
8*(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)*cosh(1) - 8*(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*cosh(1)^3 + 3*(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f)*cosh(1)
^2 - 3*(6*a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2)*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1))
*sinh(f*x + cosh(1) + sinh(1))^3 + ((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^3*f^4*x^4 + 2*(3*a^2*b + b^3)*d^3*cosh(1
)^4 + 2*(3*a^2*b + b^3)*d^3*sinh(1)^4 - 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 -
8*(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f)*cosh(1)^3 - 8*(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f - (3*a^2*b + b^
3)*d^3*cosh(1))*sinh(1)^3 - 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 + 12*(6*a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2)*cosh(1)^2 + 12*(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*cosh(1)^2 - 2*(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f)*cosh(1))
*sinh(1)^2 - 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 - 8*(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)*cosh(1) - 8*(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*cosh(1)^3 + 3*(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f)*cosh(1)^2 - 3*(6*
a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2)*cosh(1))*sinh(1))*sinh(f*x + cosh(1) + sinh(1))^4 - 8*(3*
a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f)*cosh(1)^3 - 8*(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f - (3*a^2*b + b^3)*d^
3*cosh(1))*sinh(1)^3 + 12*(6*a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2)*cosh(1)^2 + 2*((a^3 - 3*a^2*
b + 3*a*b^2 - b^3)*d^3*f^4*x^4 + 6*b^3*c^2*d*f^2 + 2*(3*a^2*b + b^3)*d^3*cosh(1)^4 + 2*(3*a^2*b + b^3)*d^3*sin
h(1)^4 + 4*(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 - 8*(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f)*cosh(1)^3 - 8*(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f - (3*a^2*b
+ b^3)*d^3*cosh(1))*sinh(1)^3 - 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 + 12*(6*a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2)*cosh(1)^2 + 12*(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*cosh(1)^2 - 2*(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2
*f)*cosh(1))*sinh(1)^2 - 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 - 8*(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)*cosh(1) - 8*(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*cosh(1)^3 + 3*(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2
*f)*cosh(1)^2 - 3*(6*a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2)*cosh(1))*sinh(1))*cosh(f*x + cosh(1)
+ sinh(1))^2 + 12*(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*cosh(1)^2 - 2*
(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f)*cosh(1)...

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

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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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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