\(\int \frac {(c+d x)^3}{(a+b \tanh (e+f x))^2} \, dx\) [73]

Optimal result

Integrand size = 20, antiderivative size = 642 \[ \int \frac {(c+d x)^3}{(a+b \tanh (e+f x))^2} \, dx=-\frac {2 b^2 (c+d x)^3}{\left (a^2-b^2\right )^2 f}+\frac {2 b^2 (c+d x)^3}{(a-b) (a+b)^2 \left (a-b+(a+b) e^{2 e+2 f x}\right ) f}+\frac {(c+d x)^4}{4 (a-b)^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}-\frac {2 b (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f}+\frac {2 b^2 (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f}+\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^3}-\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^2}+\frac {3 b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}-\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 \left (a^2-b^2\right )^2 f^4}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^3}-\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^3}-\frac {3 b d^3 \operatorname {PolyLog}\left (4,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 (a-b)^2 (a+b) f^4}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (4,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 \left (a^2-b^2\right )^2 f^4} \]

[Out]

-2*b^2*(d*x+c)^3/(a^2-b^2)^2/f+2*b^2*(d*x+c)^3/(a-b)/(a+b)^2/(a-b+(a+b)*exp(2*f*x+2*e))/f+1/4*(d*x+c)^4/(a-b)^
2/d+3*b^2*d*(d*x+c)^2*ln(1+(a+b)*exp(2*f*x+2*e)/(a-b))/(a^2-b^2)^2/f^2-2*b*(d*x+c)^3*ln(1+(a+b)*exp(2*f*x+2*e)
/(a-b))/(a-b)^2/(a+b)/f+2*b^2*(d*x+c)^3*ln(1+(a+b)*exp(2*f*x+2*e)/(a-b))/(a^2-b^2)^2/f+3*b^2*d^2*(d*x+c)*polyl
og(2,-(a+b)*exp(2*f*x+2*e)/(a-b))/(a^2-b^2)^2/f^3-3*b*d*(d*x+c)^2*polylog(2,-(a+b)*exp(2*f*x+2*e)/(a-b))/(a-b)
^2/(a+b)/f^2+3*b^2*d*(d*x+c)^2*polylog(2,-(a+b)*exp(2*f*x+2*e)/(a-b))/(a^2-b^2)^2/f^2-3/2*b^2*d^3*polylog(3,-(
a+b)*exp(2*f*x+2*e)/(a-b))/(a^2-b^2)^2/f^4+3*b*d^2*(d*x+c)*polylog(3,-(a+b)*exp(2*f*x+2*e)/(a-b))/(a-b)^2/(a+b
)/f^3-3*b^2*d^2*(d*x+c)*polylog(3,-(a+b)*exp(2*f*x+2*e)/(a-b))/(a^2-b^2)^2/f^3-3/2*b*d^3*polylog(4,-(a+b)*exp(
2*f*x+2*e)/(a-b))/(a-b)^2/(a+b)/f^4+3/2*b^2*d^3*polylog(4,-(a+b)*exp(2*f*x+2*e)/(a-b))/(a^2-b^2)^2/f^4

Rubi [A] (verified)

Time = 1.48 (sec) , antiderivative size = 642, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3815, 2221, 2611, 6744, 2320, 6724, 2286, 2216, 2215, 2222} \[ \int \frac {(c+d x)^3}{(a+b \tanh (e+f x))^2} \, dx=\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^3 \left (a^2-b^2\right )^2}-\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^3 \left (a^2-b^2\right )^2}+\frac {3 b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^2 \left (a^2-b^2\right )^2}+\frac {3 b^2 d (c+d x)^2 \log \left (\frac {(a+b) e^{2 e+2 f x}}{a-b}+1\right )}{f^2 \left (a^2-b^2\right )^2}+\frac {2 b^2 (c+d x)^3 \log \left (\frac {(a+b) e^{2 e+2 f x}}{a-b}+1\right )}{f \left (a^2-b^2\right )^2}-\frac {2 b^2 (c+d x)^3}{f \left (a^2-b^2\right )^2}-\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 f^4 \left (a^2-b^2\right )^2}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (4,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 f^4 \left (a^2-b^2\right )^2}+\frac {2 b^2 (c+d x)^3}{f (a-b) (a+b)^2 \left ((a+b) e^{2 e+2 f x}+a-b\right )}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^3 (a-b)^2 (a+b)}-\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^2 (a-b)^2 (a+b)}-\frac {2 b (c+d x)^3 \log \left (\frac {(a+b) e^{2 e+2 f x}}{a-b}+1\right )}{f (a-b)^2 (a+b)}+\frac {(c+d x)^4}{4 d (a-b)^2}-\frac {3 b d^3 \operatorname {PolyLog}\left (4,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 f^4 (a-b)^2 (a+b)} \]

[In]

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

[Out]

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

Rule 2215

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

Rule 2216

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

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 2222

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

Rule 2286

Int[((a_.) + (b_.)*(F_)^(u_))^(p_.)*((c_.) + (d_.)*(F_)^(v_))^(q_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> W
ith[{w = ExpandIntegrand[(e + f*x)^m, (a + b*F^u)^p*(c + d*F^v)^q, x]}, Int[w, x] /; SumQ[w]] /; FreeQ[{F, a,
b, c, d, e, f, m}, x] && IntegersQ[p, q] && LinearQ[{u, v}, x] && RationalQ[Simplify[u/v]]

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 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 3815

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandIntegrand[(
c + d*x)^m, (1/(a - I*b) - 2*I*(b/(a^2 + b^2 + (a - I*b)^2*E^(2*I*(e + f*x)))))^(-n), x], x] /; FreeQ[{a, b, c
, d, e, f}, x] && NeQ[a^2 + b^2, 0] && ILtQ[n, 0] && IGtQ[m, 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*} \text {integral}& = \int \left (\frac {(c+d x)^3}{(a-b)^2}+\frac {4 b e^{2 e+2 f x} (c+d x)^3}{(a-b)^2 \left (-a \left (1-\frac {b}{a}\right )-a \left (1+\frac {b}{a}\right ) e^{2 e+2 f x}\right )}+\frac {4 b^2 e^{4 e+4 f x} (c+d x)^3}{(a-b)^2 \left (a \left (1-\frac {b}{a}\right )+a \left (1+\frac {b}{a}\right ) e^{2 e+2 f x}\right )^2}\right ) \, dx \\ & = \frac {(c+d x)^4}{4 (a-b)^2 d}+\frac {(4 b) \int \frac {e^{2 e+2 f x} (c+d x)^3}{-a \left (1-\frac {b}{a}\right )-a \left (1+\frac {b}{a}\right ) e^{2 e+2 f x}} \, dx}{(a-b)^2}+\frac {\left (4 b^2\right ) \int \frac {e^{4 e+4 f x} (c+d x)^3}{\left (a \left (1-\frac {b}{a}\right )+a \left (1+\frac {b}{a}\right ) e^{2 e+2 f x}\right )^2} \, dx}{(a-b)^2} \\ & = \frac {(c+d x)^4}{4 (a-b)^2 d}-\frac {2 b (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f}+\frac {\left (4 b^2\right ) \int \left (\frac {(c+d x)^3}{(a+b)^2}+\frac {(a-b)^2 (c+d x)^3}{(a+b)^2 \left (a-b+(a+b) e^{2 e+2 f x}\right )^2}+\frac {2 (-a+b) (c+d x)^3}{(a+b)^2 \left (a-b+(a+b) e^{2 e+2 f x}\right )}\right ) \, dx}{(a-b)^2}+\frac {(6 b d) \int (c+d x)^2 \log \left (1+\frac {\left (1+\frac {b}{a}\right ) e^{2 e+2 f x}}{1-\frac {b}{a}}\right ) \, dx}{(a-b)^2 (a+b) f} \\ & = \frac {(c+d x)^4}{4 (a-b)^2 d}+\frac {b^2 (c+d x)^4}{\left (a^2-b^2\right )^2 d}-\frac {2 b (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f}-\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^2}+\frac {\left (4 b^2\right ) \int \frac {(c+d x)^3}{\left (a-b+(a+b) e^{2 e+2 f x}\right )^2} \, dx}{(a+b)^2}-\frac {\left (8 b^2\right ) \int \frac {(c+d x)^3}{a-b+(a+b) e^{2 e+2 f x}} \, dx}{(a-b) (a+b)^2}+\frac {\left (6 b d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {\left (1+\frac {b}{a}\right ) e^{2 e+2 f x}}{1-\frac {b}{a}}\right ) \, dx}{(a-b)^2 (a+b) f^2} \\ & = \frac {(c+d x)^4}{4 (a-b)^2 d}-\frac {b^2 (c+d x)^4}{\left (a^2-b^2\right )^2 d}-\frac {2 b (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f}-\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^2}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^3}+\frac {\left (4 b^2\right ) \int \frac {(c+d x)^3}{a-b+(a+b) e^{2 e+2 f x}} \, dx}{(a-b) (a+b)^2}+\frac {\left (8 b^2\right ) \int \frac {e^{2 e+2 f x} (c+d x)^3}{a-b+(a+b) e^{2 e+2 f x}} \, dx}{(a-b)^2 (a+b)}-\frac {\left (4 b^2\right ) \int \frac {e^{2 e+2 f x} (c+d x)^3}{\left (a-b+(a+b) e^{2 e+2 f x}\right )^2} \, dx}{a^2-b^2}-\frac {\left (3 b d^3\right ) \int \operatorname {PolyLog}\left (3,-\frac {\left (1+\frac {b}{a}\right ) e^{2 e+2 f x}}{1-\frac {b}{a}}\right ) \, dx}{(a-b)^2 (a+b) f^3} \\ & = \frac {2 b^2 (c+d x)^3}{(a-b) (a+b)^2 \left (a-b+(a+b) e^{2 e+2 f x}\right ) f}+\frac {(c+d x)^4}{4 (a-b)^2 d}-\frac {2 b (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f}+\frac {4 b^2 (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f}-\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^2}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^3}-\frac {\left (4 b^2\right ) \int \frac {e^{2 e+2 f x} (c+d x)^3}{a-b+(a+b) e^{2 e+2 f x}} \, dx}{(a-b)^2 (a+b)}-\frac {\left (3 b d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,-\frac {(a+b) x}{a-b}\right )}{x} \, dx,x,e^{2 e+2 f x}\right )}{2 (a-b)^2 (a+b) f^4}-\frac {\left (6 b^2 d\right ) \int \frac {(c+d x)^2}{a-b+(a+b) e^{2 e+2 f x}} \, dx}{(a-b) (a+b)^2 f}-\frac {\left (12 b^2 d\right ) \int (c+d x)^2 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right ) \, dx}{\left (a^2-b^2\right )^2 f} \\ & = -\frac {2 b^2 (c+d x)^3}{\left (a^2-b^2\right )^2 f}+\frac {2 b^2 (c+d x)^3}{(a-b) (a+b)^2 \left (a-b+(a+b) e^{2 e+2 f x}\right ) f}+\frac {(c+d x)^4}{4 (a-b)^2 d}-\frac {2 b (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f}+\frac {2 b^2 (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f}-\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^2}+\frac {6 b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^3}-\frac {3 b d^3 \operatorname {PolyLog}\left (4,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 (a-b)^2 (a+b) f^4}-\frac {\left (12 b^2 d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right ) \, dx}{\left (a^2-b^2\right )^2 f^2}+\frac {\left (6 b^2 d\right ) \int \frac {e^{2 e+2 f x} (c+d x)^2}{a-b+(a+b) e^{2 e+2 f x}} \, dx}{(a-b)^2 (a+b) f}+\frac {\left (6 b^2 d\right ) \int (c+d x)^2 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right ) \, dx}{\left (a^2-b^2\right )^2 f} \\ & = -\frac {2 b^2 (c+d x)^3}{\left (a^2-b^2\right )^2 f}+\frac {2 b^2 (c+d x)^3}{(a-b) (a+b)^2 \left (a-b+(a+b) e^{2 e+2 f x}\right ) f}+\frac {(c+d x)^4}{4 (a-b)^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}-\frac {2 b (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f}+\frac {2 b^2 (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f}-\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^2}+\frac {3 b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^3}-\frac {6 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^3}-\frac {3 b d^3 \operatorname {PolyLog}\left (4,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 (a-b)^2 (a+b) f^4}+\frac {\left (6 b^2 d^3\right ) \int \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right ) \, dx}{\left (a^2-b^2\right )^2 f^3}-\frac {\left (6 b^2 d^2\right ) \int (c+d x) \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right ) \, dx}{\left (a^2-b^2\right )^2 f^2}+\frac {\left (6 b^2 d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right ) \, dx}{\left (a^2-b^2\right )^2 f^2} \\ & = -\frac {2 b^2 (c+d x)^3}{\left (a^2-b^2\right )^2 f}+\frac {2 b^2 (c+d x)^3}{(a-b) (a+b)^2 \left (a-b+(a+b) e^{2 e+2 f x}\right ) f}+\frac {(c+d x)^4}{4 (a-b)^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}-\frac {2 b (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f}+\frac {2 b^2 (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f}+\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^3}-\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^2}+\frac {3 b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^3}-\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^3}-\frac {3 b d^3 \operatorname {PolyLog}\left (4,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 (a-b)^2 (a+b) f^4}+\frac {\left (3 b^2 d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,-\frac {(a+b) x}{a-b}\right )}{x} \, dx,x,e^{2 e+2 f x}\right )}{\left (a^2-b^2\right )^2 f^4}-\frac {\left (3 b^2 d^3\right ) \int \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right ) \, dx}{\left (a^2-b^2\right )^2 f^3}-\frac {\left (3 b^2 d^3\right ) \int \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right ) \, dx}{\left (a^2-b^2\right )^2 f^3} \\ & = -\frac {2 b^2 (c+d x)^3}{\left (a^2-b^2\right )^2 f}+\frac {2 b^2 (c+d x)^3}{(a-b) (a+b)^2 \left (a-b+(a+b) e^{2 e+2 f x}\right ) f}+\frac {(c+d x)^4}{4 (a-b)^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}-\frac {2 b (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f}+\frac {2 b^2 (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f}+\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^3}-\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^2}+\frac {3 b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^3}-\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^3}-\frac {3 b d^3 \operatorname {PolyLog}\left (4,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 (a-b)^2 (a+b) f^4}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (4,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^4}-\frac {\left (3 b^2 d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {(a+b) x}{a-b}\right )}{x} \, dx,x,e^{2 e+2 f x}\right )}{2 \left (a^2-b^2\right )^2 f^4}-\frac {\left (3 b^2 d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,-\frac {(a+b) x}{a-b}\right )}{x} \, dx,x,e^{2 e+2 f x}\right )}{2 \left (a^2-b^2\right )^2 f^4} \\ & = -\frac {2 b^2 (c+d x)^3}{\left (a^2-b^2\right )^2 f}+\frac {2 b^2 (c+d x)^3}{(a-b) (a+b)^2 \left (a-b+(a+b) e^{2 e+2 f x}\right ) f}+\frac {(c+d x)^4}{4 (a-b)^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}-\frac {2 b (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f}+\frac {2 b^2 (c+d x)^3 \log \left (1+\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f}+\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^3}-\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^2}+\frac {3 b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}-\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 \left (a^2-b^2\right )^2 f^4}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^3}-\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^3}-\frac {3 b d^3 \operatorname {PolyLog}\left (4,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 (a-b)^2 (a+b) f^4}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (4,-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 \left (a^2-b^2\right )^2 f^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.37 (sec) , antiderivative size = 656, normalized size of antiderivative = 1.02 \[ \int \frac {(c+d x)^3}{(a+b \tanh (e+f x))^2} \, dx=\frac {16 b c^2 f^3 (-3 b d+2 a c f) x+\frac {16 (a-b) b^2 f^3 (c+d x)^3}{b \left (-1+e^{2 e}\right )+a \left (1+e^{2 e}\right )}-\frac {8 a (a-b) b f^4 (c+d x)^4}{d \left (b \left (-1+e^{2 e}\right )+a \left (1+e^{2 e}\right )\right )}+48 b c d f^2 (b d-a c f) x \log \left (1+\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )+24 b d^2 f^2 (b d-2 a c f) x^2 \log \left (1+\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )-16 a b d^3 f^3 x^3 \log \left (1+\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )+8 b c^2 f^2 (3 b d-2 a c f) \log \left (a-b+(a+b) e^{2 (e+f x)}\right )+24 b c d f (-b d+a c f) \operatorname {PolyLog}\left (2,\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )-12 b d^2 (b d-2 a c f) \left (2 f x \operatorname {PolyLog}\left (2,\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )+\operatorname {PolyLog}\left (3,\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )\right )+12 a b d^3 \left (2 f^2 x^2 \operatorname {PolyLog}\left (2,\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )+2 f x \operatorname {PolyLog}\left (3,\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )+\operatorname {PolyLog}\left (4,\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )\right )+\frac {(a-b) (a+b) f^3 \left (\left (a^2+b^2\right ) f x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \cosh (f x)+\left (a^2-b^2\right ) f x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \cosh (2 e+f x)+2 b \left (-4 b (c+d x)^3+a f x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )\right ) \sinh (f x)\right )}{(a \cosh (e)+b \sinh (e)) (a \cosh (e+f x)+b \sinh (e+f x))}}{8 (a-b)^2 (a+b)^2 f^4} \]

[In]

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

[Out]

(16*b*c^2*f^3*(-3*b*d + 2*a*c*f)*x + (16*(a - b)*b^2*f^3*(c + d*x)^3)/(b*(-1 + E^(2*e)) + a*(1 + E^(2*e))) - (
8*a*(a - b)*b*f^4*(c + d*x)^4)/(d*(b*(-1 + E^(2*e)) + a*(1 + E^(2*e)))) + 48*b*c*d*f^2*(b*d - a*c*f)*x*Log[1 +
(a - b)/((a + b)*E^(2*(e + f*x)))] + 24*b*d^2*f^2*(b*d - 2*a*c*f)*x^2*Log[1 + (a - b)/((a + b)*E^(2*(e + f*x)
))] - 16*a*b*d^3*f^3*x^3*Log[1 + (a - b)/((a + b)*E^(2*(e + f*x)))] + 8*b*c^2*f^2*(3*b*d - 2*a*c*f)*Log[a - b
+ (a + b)*E^(2*(e + f*x))] + 24*b*c*d*f*(-(b*d) + a*c*f)*PolyLog[2, (-a + b)/((a + b)*E^(2*(e + f*x)))] - 12*b
*d^2*(b*d - 2*a*c*f)*(2*f*x*PolyLog[2, (-a + b)/((a + b)*E^(2*(e + f*x)))] + PolyLog[3, (-a + b)/((a + b)*E^(2
*(e + f*x)))]) + 12*a*b*d^3*(2*f^2*x^2*PolyLog[2, (-a + b)/((a + b)*E^(2*(e + f*x)))] + 2*f*x*PolyLog[3, (-a +
b)/((a + b)*E^(2*(e + f*x)))] + PolyLog[4, (-a + b)/((a + b)*E^(2*(e + f*x)))]) + ((a - b)*(a + b)*f^3*((a^2
+ b^2)*f*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)*Cosh[f*x] + (a^2 - b^2)*f*x*(4*c^3 + 6*c^2*d*x + 4*c*d^
2*x^2 + d^3*x^3)*Cosh[2*e + f*x] + 2*b*(-4*b*(c + d*x)^3 + a*f*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3))*
Sinh[f*x]))/((a*Cosh[e] + b*Sinh[e])*(a*Cosh[e + f*x] + b*Sinh[e + f*x])))/(8*(a - b)^2*(a + b)^2*f^4)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2682\) vs. \(2(622)=1244\).

Time = 0.38 (sec) , antiderivative size = 2683, normalized size of antiderivative = 4.18

method	result	size
risch	\(\text {Expression too large to display}\)	\(2683\)

[In]

int((d*x+c)^3/(a+b*tanh(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

2/(a^2+2*a*b+b^2)/f*b^2/(a-b)/(-a+b)*d^3*x^3-4/(a^2+2*a*b+b^2)/f^4*b^2/(a-b)/(-a+b)*d^3*e^3+3/2/(a^2+2*a*b+b^2
)/f^4*b^2/(a-b)/(-a+b)*d^3*polylog(3,(a+b)*exp(2*f*x+2*e)/(-a+b))-6/(a^2+2*a*b+b^2)/f^4*b^2/(a-b)^2*e^2*d^3*ln
(exp(f*x+e))+3/(a^2+2*a*b+b^2)/f^4*b^2/(a-b)^2*e^2*d^3*ln(exp(2*f*x+2*e)*a+b*exp(2*f*x+2*e)+a-b)+4/(a^2+2*a*b+
b^2)/f*b/(a-b)^2*a*c^3*ln(exp(f*x+e))-2/(a^2+2*a*b+b^2)/f*b/(a-b)^2*a*c^3*ln(exp(2*f*x+2*e)*a+b*exp(2*f*x+2*e)
+a-b)-6/(a^2+2*a*b+b^2)/f^2*b^2/(a-b)^2*c^2*d*ln(exp(f*x+e))+3/(a^2+2*a*b+b^2)/f^2*b^2/(a-b)^2*c^2*d*ln(exp(2*
f*x+2*e)*a+b*exp(2*f*x+2*e)+a-b)-4/(a^2+2*a*b+b^2)*b/(a-b)/(-a+b)*d^2*c*a*x^3+8/(a^2+2*a*b+b^2)/f^3*b/(a-b)/(-
a+b)*d^2*c*a*e^3-4/(a^2+2*a*b+b^2)/f^3*b/(a-b)/(-a+b)*d^3*a*e^3*x-6/(a^2+2*a*b+b^2)/f^2*b/(a-b)*a*c^2*d/(-a+b)
*e^2+12/(a^2+2*a*b+b^2)/f^2*b^2/(a-b)/(-a+b)*d^2*c*e*x-6/(a^2+2*a*b+b^2)*b/(a-b)*a*c^2*d/(-a+b)*x^2-3/(a^2+2*a
*b+b^2)/f^3*b/(a-b)/(-a+b)*d^2*c*a*polylog(3,(a+b)*exp(2*f*x+2*e)/(-a+b))+2/(a^2+2*a*b+b^2)/f*b/(a-b)/(-a+b)*d
^3*a*ln(1-(a+b)*exp(2*f*x+2*e)/(-a+b))*x^3+3/(a^2+2*a*b+b^2)/f^2*b/(a-b)/(-a+b)*d^3*a*polylog(2,(a+b)*exp(2*f*
x+2*e)/(-a+b))*x^2+2/(a^2+2*a*b+b^2)/f^4*b/(a-b)/(-a+b)*d^3*a*ln(1-(a+b)*exp(2*f*x+2*e)/(-a+b))*e^3-3/(a^2+2*a
*b+b^2)/f^3*b/(a-b)/(-a+b)*d^3*a*polylog(3,(a+b)*exp(2*f*x+2*e)/(-a+b))*x+12/(a^2+2*a*b+b^2)/f^3*b/(a-b)^2*e^2
*d^2*c*a*ln(exp(f*x+e))-6/(a^2+2*a*b+b^2)/f^3*b/(a-b)^2*e^2*d^2*c*a*ln(exp(2*f*x+2*e)*a+b*exp(2*f*x+2*e)+a-b)+
3/(a^2+2*a*b+b^2)/f^2*b/(a-b)*a*c^2*d/(-a+b)*polylog(2,(a+b)*exp(2*f*x+2*e)/(-a+b))-6/(a^2+2*a*b+b^2)/f^2*b^2/
(a-b)/(-a+b)*d^2*c*ln(1-(a+b)*exp(2*f*x+2*e)/(-a+b))*x-6/(a^2+2*a*b+b^2)/f^3*b^2/(a-b)/(-a+b)*d^2*c*ln(1-(a+b)
*exp(2*f*x+2*e)/(-a+b))*e-12/(a^2+2*a*b+b^2)/f^2*b/(a-b)^2*e*a*c^2*d*ln(exp(f*x+e))+6/(a^2+2*a*b+b^2)/f^2*b/(a
-b)^2*e*a*c^2*d*ln(exp(2*f*x+2*e)*a+b*exp(2*f*x+2*e)+a-b)+2/(a-b)/f/(a^2+2*a*b+b^2)*(d^3*x^3+3*c*d^2*x^2+3*c^2
*d*x+c^3)*b^2/(exp(2*f*x+2*e)*a+b*exp(2*f*x+2*e)+a-b)+1/4*d^3/(a^2+2*a*b+b^2)*x^4+1/4/d/(a^2+2*a*b+b^2)*c^4+2/
(a^2+2*a*b+b^2)/f^4*b/(a-b)^2*e^3*d^3*a*ln(exp(2*f*x+2*e)*a+b*exp(2*f*x+2*e)+a-b)-6/(a^2+2*a*b+b^2)/f^3*b^2/(a
-b)^2*e*d^2*c*ln(exp(2*f*x+2*e)*a+b*exp(2*f*x+2*e)+a-b)+d^2/(a^2+2*a*b+b^2)*c*x^3+3/2*d/(a^2+2*a*b+b^2)*c^2*x^
2+1/(a^2+2*a*b+b^2)*c^3*x-6/(a^2+2*a*b+b^2)/f^3*b^2/(a-b)/(-a+b)*d^3*e^2*x+12/(a^2+2*a*b+b^2)/f^2*b/(a-b)/(-a+
b)*d^2*c*a*e^2*x-12/(a^2+2*a*b+b^2)/f*b/(a-b)*a*c^2*d/(-a+b)*e*x+6/(a^2+2*a*b+b^2)/f*b/(a-b)/(-a+b)*d^2*c*a*ln
(1-(a+b)*exp(2*f*x+2*e)/(-a+b))*x^2-6/(a^2+2*a*b+b^2)/f^3*b/(a-b)/(-a+b)*d^2*c*a*ln(1-(a+b)*exp(2*f*x+2*e)/(-a
+b))*e^2+6/(a^2+2*a*b+b^2)/f^2*b/(a-b)/(-a+b)*d^2*c*a*polylog(2,(a+b)*exp(2*f*x+2*e)/(-a+b))*x+6/(a^2+2*a*b+b^
2)/f*b/(a-b)*a*c^2*d/(-a+b)*ln(1-(a+b)*exp(2*f*x+2*e)/(-a+b))*x+6/(a^2+2*a*b+b^2)/f^2*b/(a-b)*a*c^2*d/(-a+b)*l
n(1-(a+b)*exp(2*f*x+2*e)/(-a+b))*e+6/(a^2+2*a*b+b^2)/f*b^2/(a-b)/(-a+b)*d^2*c*x^2+6/(a^2+2*a*b+b^2)/f^3*b^2/(a
-b)/(-a+b)*d^2*c*e^2-1/(a^2+2*a*b+b^2)*b/(a-b)/(-a+b)*d^3*a*x^4-3/(a^2+2*a*b+b^2)/f^4*b/(a-b)/(-a+b)*d^3*a*e^4
-3/(a^2+2*a*b+b^2)/f^2*b^2/(a-b)/(-a+b)*d^3*ln(1-(a+b)*exp(2*f*x+2*e)/(-a+b))*x^2+3/(a^2+2*a*b+b^2)/f^4*b^2/(a
-b)/(-a+b)*d^3*ln(1-(a+b)*exp(2*f*x+2*e)/(-a+b))*e^2-3/(a^2+2*a*b+b^2)/f^3*b^2/(a-b)/(-a+b)*d^3*polylog(2,(a+b
)*exp(2*f*x+2*e)/(-a+b))*x-3/(a^2+2*a*b+b^2)/f^3*b^2/(a-b)/(-a+b)*d^2*c*polylog(2,(a+b)*exp(2*f*x+2*e)/(-a+b))
+12/(a^2+2*a*b+b^2)/f^3*b^2/(a-b)^2*e*d^2*c*ln(exp(f*x+e))+3/2/(a^2+2*a*b+b^2)/f^4*b/(a-b)/(-a+b)*d^3*a*polylo
g(4,(a+b)*exp(2*f*x+2*e)/(-a+b))-4/(a^2+2*a*b+b^2)/f^4*b/(a-b)^2*e^3*d^3*a*ln(exp(f*x+e))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6160 vs. \(2 (619) = 1238\).

Time = 0.38 (sec) , antiderivative size = 6160, normalized size of antiderivative = 9.60 \[ \int \frac {(c+d x)^3}{(a+b \tanh (e+f x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

\[ \int \frac {(c+d x)^3}{(a+b \tanh (e+f x))^2} \, dx=\int \frac {\left (c + d x\right )^{3}}{\left (a + b \tanh {\left (e + f x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

Integral((c + d*x)**3/(a + b*tanh(e + f*x))**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.56 (sec) , antiderivative size = 1060, normalized size of antiderivative = 1.65 \[ \int \frac {(c+d x)^3}{(a+b \tanh (e+f x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-6*b^2*c^2*d*f*x/(a^4*f^2 - 2*a^2*b^2*f^2 + b^4*f^2) - 2/3*(4*f^3*x^3*log((a*e^(2*e) + b*e^(2*e))*e^(2*f*x)/(a
- b) + 1) + 6*f^2*x^2*dilog(-(a*e^(2*e) + b*e^(2*e))*e^(2*f*x)/(a - b)) - 6*f*x*polylog(3, -(a*e^(2*e) + b*e^
(2*e))*e^(2*f*x)/(a - b)) + 3*polylog(4, -(a*e^(2*e) + b*e^(2*e))*e^(2*f*x)/(a - b)))*a*b*d^3/(a^4*f^4 - 2*a^2
*b^2*f^4 + b^4*f^4) + 3*b^2*c^2*d*log((a*e^(2*e) + b*e^(2*e))*e^(2*f*x) + a - b)/(a^4*f^2 - 2*a^2*b^2*f^2 + b^
4*f^2) - c^3*(2*a*b*log(-(a - b)*e^(-2*f*x - 2*e) - a - b)/((a^4 - 2*a^2*b^2 + b^4)*f) + 2*b^2/((a^4 - 2*a^2*b
^2 + b^4 + (a^4 - 2*a^3*b + 2*a*b^3 - b^4)*e^(-2*f*x - 2*e))*f) - (f*x + e)/((a^2 + 2*a*b + b^2)*f)) - 3/2*(2*
a*b*c*d^2*f - b^2*d^3)*(2*f^2*x^2*log((a*e^(2*e) + b*e^(2*e))*e^(2*f*x)/(a - b) + 1) + 2*f*x*dilog(-(a*e^(2*e)
+ b*e^(2*e))*e^(2*f*x)/(a - b)) - polylog(3, -(a*e^(2*e) + b*e^(2*e))*e^(2*f*x)/(a - b)))/(a^4*f^4 - 2*a^2*b^
2*f^4 + b^4*f^4) - 3*(a*b*c^2*d*f - b^2*c*d^2)*(2*f*x*log((a*e^(2*e) + b*e^(2*e))*e^(2*f*x)/(a - b) + 1) + dil
og(-(a*e^(2*e) + b*e^(2*e))*e^(2*f*x)/(a - b)))/(a^4*f^3 - 2*a^2*b^2*f^3 + b^4*f^3) + (a*b*d^3*f^4*x^4 + 2*(2*
a*b*c*d^2*f - b^2*d^3)*f^3*x^3 + 6*(a*b*c^2*d*f^2 - b^2*c*d^2*f)*f^2*x^2)/(a^4*f^4 - 2*a^2*b^2*f^4 + b^4*f^4)
+ 1/4*(24*b^2*c^2*d*x + (a^2*d^3*f - 2*a*b*d^3*f + b^2*d^3*f)*x^4 + 4*(a^2*c*d^2*f - 2*a*b*c*d^2*f + (c*d^2*f
+ 2*d^3)*b^2)*x^3 + 6*(a^2*c^2*d*f - 2*a*b*c^2*d*f + (c^2*d*f + 4*c*d^2)*b^2)*x^2 + ((a^2*d^3*f*e^(2*e) - b^2*
d^3*f*e^(2*e))*x^4 + 4*(a^2*c*d^2*f*e^(2*e) - b^2*c*d^2*f*e^(2*e))*x^3 + 6*(a^2*c^2*d*f*e^(2*e) - b^2*c^2*d*f*
e^(2*e))*x^2)*e^(2*f*x))/(a^4*f - 2*a^2*b^2*f + b^4*f + (a^4*f*e^(2*e) + 2*a^3*b*f*e^(2*e) - 2*a*b^3*f*e^(2*e)
- b^4*f*e^(2*e))*e^(2*f*x))

Giac [F]

\[ \int \frac {(c+d x)^3}{(a+b \tanh (e+f x))^2} \, dx=\int { \frac {{\left (d x + c\right )}^{3}}{{\left (b \tanh \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

integrate((d*x + c)^3/(b*tanh(f*x + e) + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^3}{(a+b \tanh (e+f x))^2} \, dx=\int \frac {{\left (c+d\,x\right )}^3}{{\left (a+b\,\mathrm {tanh}\left (e+f\,x\right )\right )}^2} \,d x \]

[In]

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

[Out]

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