3.57 \(\int \frac{(c+d x)^3}{(a+b \coth (e+f x))^2} \, dx\)

Optimal. Leaf size=638 \[ \frac{3 b^2 d^2 (c+d x) \text{PolyLog}\left (2,\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^3 \left (a^2-b^2\right )^2}-\frac{3 b^2 d^2 (c+d x) \text{PolyLog}\left (3,\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^3 \left (a^2-b^2\right )^2}+\frac{3 b^2 d (c+d x)^2 \text{PolyLog}\left (2,\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^2 \left (a^2-b^2\right )^2}-\frac{3 b^2 d^3 \text{PolyLog}\left (3,\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 f^4 \left (a^2-b^2\right )^2}+\frac{3 b^2 d^3 \text{PolyLog}\left (4,\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 f^4 \left (a^2-b^2\right )^2}+\frac{3 b d^2 (c+d x) \text{PolyLog}\left (3,\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^3 (a-b)^2 (a+b)}-\frac{3 b d (c+d x)^2 \text{PolyLog}\left (2,\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^2 (a-b)^2 (a+b)}-\frac{3 b d^3 \text{PolyLog}\left (4,\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 f^4 (a-b)^2 (a+b)}+\frac{3 b^2 d (c+d x)^2 \log \left (1-\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^2 \left (a^2-b^2\right )^2}+\frac{2 b^2 (c+d x)^3 \log \left (1-\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f \left (a^2-b^2\right )^2}-\frac{2 b^2 (c+d x)^3}{f \left (a^2-b^2\right )^2}+\frac{2 b^2 (c+d x)^3}{f (a-b) (a+b)^2 \left (-(a+b) e^{2 e+2 f x}+a-b\right )}-\frac{2 b (c+d x)^3 \log \left (1-\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f (a-b)^2 (a+b)}+\frac{(c+d x)^4}{4 d (a-b)^2} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 2.27365, antiderivative size = 638, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3734, 2254, 2185, 2184, 2190, 2531, 6609, 2282, 6589, 2191} \[ \frac{3 b^2 d^2 (c+d x) \text{PolyLog}\left (2,\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^3 \left (a^2-b^2\right )^2}-\frac{3 b^2 d^2 (c+d x) \text{PolyLog}\left (3,\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^3 \left (a^2-b^2\right )^2}+\frac{3 b^2 d (c+d x)^2 \text{PolyLog}\left (2,\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^2 \left (a^2-b^2\right )^2}-\frac{3 b^2 d^3 \text{PolyLog}\left (3,\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 f^4 \left (a^2-b^2\right )^2}+\frac{3 b^2 d^3 \text{PolyLog}\left (4,\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 f^4 \left (a^2-b^2\right )^2}+\frac{3 b d^2 (c+d x) \text{PolyLog}\left (3,\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^3 (a-b)^2 (a+b)}-\frac{3 b d (c+d x)^2 \text{PolyLog}\left (2,\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^2 (a-b)^2 (a+b)}-\frac{3 b d^3 \text{PolyLog}\left (4,\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 f^4 (a-b)^2 (a+b)}+\frac{3 b^2 d (c+d x)^2 \log \left (1-\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^2 \left (a^2-b^2\right )^2}+\frac{2 b^2 (c+d x)^3 \log \left (1-\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f \left (a^2-b^2\right )^2}-\frac{2 b^2 (c+d x)^3}{f \left (a^2-b^2\right )^2}+\frac{2 b^2 (c+d x)^3}{f (a-b) (a+b)^2 \left (-(a+b) e^{2 e+2 f x}+a-b\right )}-\frac{2 b (c+d x)^3 \log \left (1-\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f (a-b)^2 (a+b)}+\frac{(c+d x)^4}{4 d (a-b)^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(a + b*Coth[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 3734

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 2254

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 2185

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 2184

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 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 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 6609

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

Rule 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 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 2191

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]

Rubi steps

\begin{align*} \int \frac{(c+d x)^3}{(a+b \coth (e+f x))^2} \, dx &=\int \left (\frac{(c+d x)^3}{(a-b)^2}+\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}+\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 )}\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 \text{Li}_2\left (\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) \text{Li}_2\left (\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 \text{Li}_2\left (\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) \text{Li}_3\left (\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 \text{Li}_3\left (\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 \text{Li}_2\left (\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) \text{Li}_3\left (\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 ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\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 \text{Li}_2\left (\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 \text{Li}_2\left (\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) \text{Li}_3\left (\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^3}-\frac{3 b d^3 \text{Li}_4\left (\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) \text{Li}_2\left (-\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 \text{Li}_2\left (\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 \text{Li}_2\left (\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) \text{Li}_3\left (\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) \text{Li}_3\left (\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 \text{Li}_4\left (\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 \text{Li}_3\left (-\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) \text{Li}_2\left (-\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) \text{Li}_2\left (\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 \text{Li}_2\left (\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 \text{Li}_2\left (\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) \text{Li}_3\left (\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) \text{Li}_3\left (\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 \text{Li}_4\left (\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 ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\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 \text{Li}_2\left (-\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 \text{Li}_3\left (-\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) \text{Li}_2\left (\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 \text{Li}_2\left (\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 \text{Li}_2\left (\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) \text{Li}_3\left (\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) \text{Li}_3\left (\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 \text{Li}_4\left (\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 \text{Li}_4\left (\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 ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\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 ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\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) \text{Li}_2\left (\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 \text{Li}_2\left (\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 \text{Li}_2\left (\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 \text{Li}_3\left (\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) \text{Li}_3\left (\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) \text{Li}_3\left (\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 \text{Li}_4\left (\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 \text{Li}_4\left (\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]  time = 12.2993, size = 646, normalized size = 1.01 \[ \frac{-12 b d^2 (b d-2 a c f) \left (2 f x \text{PolyLog}\left (2,\frac{(a-b) e^{-2 (e+f x)}}{a+b}\right )+\text{PolyLog}\left (3,\frac{(a-b) e^{-2 (e+f x)}}{a+b}\right )\right )+24 b c d f (a c f-b d) \text{PolyLog}\left (2,\frac{(a-b) e^{-2 (e+f x)}}{a+b}\right )+12 a b d^3 \left (2 f^2 x^2 \text{PolyLog}\left (2,\frac{(a-b) e^{-2 (e+f x)}}{a+b}\right )+2 f x \text{PolyLog}\left (3,\frac{(a-b) e^{-2 (e+f x)}}{a+b}\right )+\text{PolyLog}\left (4,\frac{(a-b) e^{-2 (e+f x)}}{a+b}\right )\right )-\frac{f^3 (a-b) (a+b) \left (-f x \left (a^2-b^2\right ) \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right ) \cosh (2 e+f x)+f x \left (a^2+b^2\right ) \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right ) \cosh (f x)+2 b \sinh (f x) \left (a f x \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right )-4 b (c+d x)^3\right )\right )}{(a \sinh (e)+b \cosh (e)) (a \sinh (e+f x)+b \cosh (e+f x))}-\frac{16 b^2 f^3 (a-b) (c+d x)^3}{a \left (e^{2 e}-1\right )+b \left (e^{2 e}+1\right )}+8 b c^2 f^2 (2 a c f-3 b d) \left (2 f x-\log \left (-(a+b) e^{2 (e+f x)}+a-b\right )\right )+24 b d^2 f^2 x^2 (b d-2 a c f) \log \left (\frac{(b-a) e^{-2 (e+f x)}}{a+b}+1\right )+\frac{8 a b f^4 (a-b) (c+d x)^4}{d \left (a \left (e^{2 e}-1\right )+b \left (e^{2 e}+1\right )\right )}+48 b c d f^2 x (b d-a c f) \log \left (\frac{(b-a) e^{-2 (e+f x)}}{a+b}+1\right )-16 a b d^3 f^3 x^3 \log \left (\frac{(b-a) e^{-2 (e+f x)}}{a+b}+1\right )}{8 f^4 (a-b)^2 (a+b)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-16*(a - b)*b^2*f^3*(c + d*x)^3)/(a*(-1 + E^(2*e)) + b*(1 + E^(2*e))) + (8*a*(a - b)*b*f^4*(c + d*x)^4)/(d*(
a*(-1 + E^(2*e)) + b*(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)*(2*f*x - 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]))/((b*Cosh[e] + a
*Sinh[e])*(b*Cosh[e + f*x] + a*Sinh[e + f*x])))/(8*(a - b)^2*(a + b)^2*f^4)

________________________________________________________________________________________

Maple [B]  time = 0.301, size = 2039, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 2.77539, size = 1426, normalized size = 2.24 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+b*coth(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) + dilog
((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))

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Fricas [C]  time = 3.55548, size = 12840, normalized size = 20.13 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((a^3 + a^2*b - a*b^2 - b^3)*d^3*f^4*x^4 + 4*(a^3 + a^2*b - a*b^2 - b^3)*c*d^2*f^4*x^3 + 6*(a^3 + a^2*b -
a*b^2 - b^3)*c^2*d*f^4*x^2 + 4*(a^3 + a^2*b - a*b^2 - b^3)*c^3*f^4*x - 4*(a^2*b - a*b^2)*d^3*e^4 - 8*(a*b^2 -
b^3)*d^3*e^3 + 8*(2*(a^2*b - a*b^2)*c^3*e + (a*b^2 - b^3)*c^3)*f^3 - 24*((a^2*b - a*b^2)*c^2*d*e^2 + (a*b^2 -
b^3)*c^2*d*e)*f^2 - ((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d^3*f^4*x^4 - 4*(a^2*b + a*b^2)*d^3*e^4 + 16*(a^2*b + a*
b^2)*c^3*e*f^3 - 8*(a*b^2 + b^3)*d^3*e^3 + 4*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*c*d^2*f^4 - 2*(a*b^2 + b^3)*d^3*
f^3)*x^3 - 24*((a^2*b + a*b^2)*c^2*d*e^2 + (a*b^2 + b^3)*c^2*d*e)*f^2 + 6*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*c^2
*d*f^4 - 4*(a*b^2 + b^3)*c*d^2*f^3)*x^2 + 8*(2*(a^2*b + a*b^2)*c*d^2*e^3 + 3*(a*b^2 + b^3)*c*d^2*e^2)*f + 4*((
a^3 + 3*a^2*b + 3*a*b^2 + b^3)*c^3*f^4 - 6*(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 - 4*(a^2*b + a*b^2)*d^3*e^4 + 16*(a^2*b + a*b^2)*c^3*e*f^3 - 8*(a*b^2 + b^3)*d^3*e^3
+ 4*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*c*d^2*f^4 - 2*(a*b^2 + b^3)*d^3*f^3)*x^3 - 24*((a^2*b + a*b^2)*c^2*d*e^2
+ (a*b^2 + b^3)*c^2*d*e)*f^2 + 6*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*c^2*d*f^4 - 4*(a*b^2 + b^3)*c*d^2*f^3)*x^2
+ 8*(2*(a^2*b + a*b^2)*c*d^2*e^3 + 3*(a*b^2 + b^3)*c*d^2*e^2)*f + 4*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*c^3*f^4 -
6*(a*b^2 + b^3)*c^2*d*f^3)*x)*cosh(f*x + e)*sinh(f*x + e) - ((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d^3*f^4*x^4 - 4*
(a^2*b + a*b^2)*d^3*e^4 + 16*(a^2*b + a*b^2)*c^3*e*f^3 - 8*(a*b^2 + b^3)*d^3*e^3 + 4*((a^3 + 3*a^2*b + 3*a*b^2
+ b^3)*c*d^2*f^4 - 2*(a*b^2 + b^3)*d^3*f^3)*x^3 - 24*((a^2*b + a*b^2)*c^2*d*e^2 + (a*b^2 + b^3)*c^2*d*e)*f^2
+ 6*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*c^2*d*f^4 - 4*(a*b^2 + b^3)*c*d^2*f^3)*x^2 + 8*(2*(a^2*b + a*b^2)*c*d^2*e
^3 + 3*(a*b^2 + b^3)*c*d^2*e^2)*f + 4*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*c^3*f^4 - 6*(a*b^2 + b^3)*c^2*d*f^3)*x)
*sinh(f*x + e)^2 + 8*(2*(a^2*b - a*b^2)*c*d^2*e^3 + 3*(a*b^2 - b^3)*c*d^2*e^2)*f - 24*((a^2*b - a*b^2)*d^3*f^2
*x^2 + (a^2*b - a*b^2)*c^2*d*f^2 - (a*b^2 - b^3)*c*d^2*f - ((a^2*b + a*b^2)*d^3*f^2*x^2 + (a^2*b + a*b^2)*c^2*
d*f^2 - (a*b^2 + b^3)*c*d^2*f + (2*(a^2*b + a*b^2)*c*d^2*f^2 - (a*b^2 + b^3)*d^3*f)*x)*cosh(f*x + e)^2 - 2*((a
^2*b + a*b^2)*d^3*f^2*x^2 + (a^2*b + a*b^2)*c^2*d*f^2 - (a*b^2 + b^3)*c*d^2*f + (2*(a^2*b + a*b^2)*c*d^2*f^2 -
(a*b^2 + b^3)*d^3*f)*x)*cosh(f*x + e)*sinh(f*x + e) - ((a^2*b + a*b^2)*d^3*f^2*x^2 + (a^2*b + a*b^2)*c^2*d*f^
2 - (a*b^2 + b^3)*c*d^2*f + (2*(a^2*b + a*b^2)*c*d^2*f^2 - (a*b^2 + b^3)*d^3*f)*x)*sinh(f*x + e)^2 + (2*(a^2*b
- a*b^2)*c*d^2*f^2 - (a*b^2 - b^3)*d^3*f)*x)*dilog(sqrt((a + b)/(a - b))*(cosh(f*x + e) + sinh(f*x + e))) - 2
4*((a^2*b - a*b^2)*d^3*f^2*x^2 + (a^2*b - a*b^2)*c^2*d*f^2 - (a*b^2 - b^3)*c*d^2*f - ((a^2*b + a*b^2)*d^3*f^2*
x^2 + (a^2*b + a*b^2)*c^2*d*f^2 - (a*b^2 + b^3)*c*d^2*f + (2*(a^2*b + a*b^2)*c*d^2*f^2 - (a*b^2 + b^3)*d^3*f)*
x)*cosh(f*x + e)^2 - 2*((a^2*b + a*b^2)*d^3*f^2*x^2 + (a^2*b + a*b^2)*c^2*d*f^2 - (a*b^2 + b^3)*c*d^2*f + (2*(
a^2*b + a*b^2)*c*d^2*f^2 - (a*b^2 + b^3)*d^3*f)*x)*cosh(f*x + e)*sinh(f*x + e) - ((a^2*b + a*b^2)*d^3*f^2*x^2
+ (a^2*b + a*b^2)*c^2*d*f^2 - (a*b^2 + b^3)*c*d^2*f + (2*(a^2*b + a*b^2)*c*d^2*f^2 - (a*b^2 + b^3)*d^3*f)*x)*s
inh(f*x + e)^2 + (2*(a^2*b - a*b^2)*c*d^2*f^2 - (a*b^2 - b^3)*d^3*f)*x)*dilog(-sqrt((a + b)/(a - b))*(cosh(f*x
+ e) + sinh(f*x + e))) + 4*(2*(a^2*b - a*b^2)*d^3*e^3 - 2*(a^2*b - a*b^2)*c^3*f^3 + 3*(a*b^2 - b^3)*d^3*e^2 +
3*(2*(a^2*b - a*b^2)*c^2*d*e + (a*b^2 - b^3)*c^2*d)*f^2 - (2*(a^2*b + a*b^2)*d^3*e^3 - 2*(a^2*b + a*b^2)*c^3*
f^3 + 3*(a*b^2 + b^3)*d^3*e^2 + 3*(2*(a^2*b + a*b^2)*c^2*d*e + (a*b^2 + b^3)*c^2*d)*f^2 - 6*((a^2*b + a*b^2)*c
*d^2*e^2 + (a*b^2 + b^3)*c*d^2*e)*f)*cosh(f*x + e)^2 - 2*(2*(a^2*b + a*b^2)*d^3*e^3 - 2*(a^2*b + a*b^2)*c^3*f^
3 + 3*(a*b^2 + b^3)*d^3*e^2 + 3*(2*(a^2*b + a*b^2)*c^2*d*e + (a*b^2 + b^3)*c^2*d)*f^2 - 6*((a^2*b + a*b^2)*c*d
^2*e^2 + (a*b^2 + b^3)*c*d^2*e)*f)*cosh(f*x + e)*sinh(f*x + e) - (2*(a^2*b + a*b^2)*d^3*e^3 - 2*(a^2*b + a*b^2
)*c^3*f^3 + 3*(a*b^2 + b^3)*d^3*e^2 + 3*(2*(a^2*b + a*b^2)*c^2*d*e + (a*b^2 + b^3)*c^2*d)*f^2 - 6*((a^2*b + a*
b^2)*c*d^2*e^2 + (a*b^2 + b^3)*c*d^2*e)*f)*sinh(f*x + e)^2 - 6*((a^2*b - a*b^2)*c*d^2*e^2 + (a*b^2 - b^3)*c*d^
2*e)*f)*log(2*(a + b)*cosh(f*x + e) + 2*(a + b)*sinh(f*x + e) + 2*(a - b)*sqrt((a + b)/(a - b))) + 4*(2*(a^2*b
- a*b^2)*d^3*e^3 - 2*(a^2*b - a*b^2)*c^3*f^3 + 3*(a*b^2 - b^3)*d^3*e^2 + 3*(2*(a^2*b - a*b^2)*c^2*d*e + (a*b^
2 - b^3)*c^2*d)*f^2 - (2*(a^2*b + a*b^2)*d^3*e^3 - 2*(a^2*b + a*b^2)*c^3*f^3 + 3*(a*b^2 + b^3)*d^3*e^2 + 3*(2*
(a^2*b + a*b^2)*c^2*d*e + (a*b^2 + b^3)*c^2*d)*f^2 - 6*((a^2*b + a*b^2)*c*d^2*e^2 + (a*b^2 + b^3)*c*d^2*e)*f)*
cosh(f*x + e)^2 - 2*(2*(a^2*b + a*b^2)*d^3*e^3 - 2*(a^2*b + a*b^2)*c^3*f^3 + 3*(a*b^2 + b^3)*d^3*e^2 + 3*(2*(a
^2*b + a*b^2)*c^2*d*e + (a*b^2 + b^3)*c^2*d)*f^2 - 6*((a^2*b + a*b^2)*c*d^2*e^2 + (a*b^2 + b^3)*c*d^2*e)*f)*co
sh(f*x + e)*sinh(f*x + e) - (2*(a^2*b + a*b^2)*d^3*e^3 - 2*(a^2*b + a*b^2)*c^3*f^3 + 3*(a*b^2 + b^3)*d^3*e^2 +
3*(2*(a^2*b + a*b^2)*c^2*d*e + (a*b^2 + b^3)*c^2*d)*f^2 - 6*((a^2*b + a*b^2)*c*d^2*e^2 + (a*b^2 + b^3)*c*d^2*
e)*f)*sinh(f*x + e)^2 - 6*((a^2*b - a*b^2)*c*d^2*e^2 + (a*b^2 - b^3)*c*d^2*e)*f)*log(2*(a + b)*cosh(f*x + e) +
2*(a + b)*sinh(f*x + e) - 2*(a - b)*sqrt((a + b)/(a - b))) - 4*(2*(a^2*b - a*b^2)*d^3*f^3*x^3 + 2*(a^2*b - a*
b^2)*d^3*e^3 + 6*(a^2*b - a*b^2)*c^2*d*e*f^2 + 3*(a*b^2 - b^3)*d^3*e^2 + 3*(2*(a^2*b - a*b^2)*c*d^2*f^3 - (a*b
^2 - b^3)*d^3*f^2)*x^2 - (2*(a^2*b + a*b^2)*d^3*f^3*x^3 + 2*(a^2*b + a*b^2)*d^3*e^3 + 6*(a^2*b + a*b^2)*c^2*d*
e*f^2 + 3*(a*b^2 + b^3)*d^3*e^2 + 3*(2*(a^2*b + a*b^2)*c*d^2*f^3 - (a*b^2 + b^3)*d^3*f^2)*x^2 - 6*((a^2*b + a*
b^2)*c*d^2*e^2 + (a*b^2 + b^3)*c*d^2*e)*f + 6*((a^2*b + a*b^2)*c^2*d*f^3 - (a*b^2 + b^3)*c*d^2*f^2)*x)*cosh(f*
x + e)^2 - 2*(2*(a^2*b + a*b^2)*d^3*f^3*x^3 + 2*(a^2*b + a*b^2)*d^3*e^3 + 6*(a^2*b + a*b^2)*c^2*d*e*f^2 + 3*(a
*b^2 + b^3)*d^3*e^2 + 3*(2*(a^2*b + a*b^2)*c*d^2*f^3 - (a*b^2 + b^3)*d^3*f^2)*x^2 - 6*((a^2*b + a*b^2)*c*d^2*e
^2 + (a*b^2 + b^3)*c*d^2*e)*f + 6*((a^2*b + a*b^2)*c^2*d*f^3 - (a*b^2 + b^3)*c*d^2*f^2)*x)*cosh(f*x + e)*sinh(
f*x + e) - (2*(a^2*b + a*b^2)*d^3*f^3*x^3 + 2*(a^2*b + a*b^2)*d^3*e^3 + 6*(a^2*b + a*b^2)*c^2*d*e*f^2 + 3*(a*b
^2 + b^3)*d^3*e^2 + 3*(2*(a^2*b + a*b^2)*c*d^2*f^3 - (a*b^2 + b^3)*d^3*f^2)*x^2 - 6*((a^2*b + a*b^2)*c*d^2*e^2
+ (a*b^2 + b^3)*c*d^2*e)*f + 6*((a^2*b + a*b^2)*c^2*d*f^3 - (a*b^2 + b^3)*c*d^2*f^2)*x)*sinh(f*x + e)^2 - 6*(
(a^2*b - a*b^2)*c*d^2*e^2 + (a*b^2 - b^3)*c*d^2*e)*f + 6*((a^2*b - a*b^2)*c^2*d*f^3 - (a*b^2 - b^3)*c*d^2*f^2)
*x)*log(sqrt((a + b)/(a - b))*(cosh(f*x + e) + sinh(f*x + e)) + 1) - 4*(2*(a^2*b - a*b^2)*d^3*f^3*x^3 + 2*(a^2
*b - a*b^2)*d^3*e^3 + 6*(a^2*b - a*b^2)*c^2*d*e*f^2 + 3*(a*b^2 - b^3)*d^3*e^2 + 3*(2*(a^2*b - a*b^2)*c*d^2*f^3
- (a*b^2 - b^3)*d^3*f^2)*x^2 - (2*(a^2*b + a*b^2)*d^3*f^3*x^3 + 2*(a^2*b + a*b^2)*d^3*e^3 + 6*(a^2*b + a*b^2)
*c^2*d*e*f^2 + 3*(a*b^2 + b^3)*d^3*e^2 + 3*(2*(a^2*b + a*b^2)*c*d^2*f^3 - (a*b^2 + b^3)*d^3*f^2)*x^2 - 6*((a^2
*b + a*b^2)*c*d^2*e^2 + (a*b^2 + b^3)*c*d^2*e)*f + 6*((a^2*b + a*b^2)*c^2*d*f^3 - (a*b^2 + b^3)*c*d^2*f^2)*x)*
cosh(f*x + e)^2 - 2*(2*(a^2*b + a*b^2)*d^3*f^3*x^3 + 2*(a^2*b + a*b^2)*d^3*e^3 + 6*(a^2*b + a*b^2)*c^2*d*e*f^2
+ 3*(a*b^2 + b^3)*d^3*e^2 + 3*(2*(a^2*b + a*b^2)*c*d^2*f^3 - (a*b^2 + b^3)*d^3*f^2)*x^2 - 6*((a^2*b + a*b^2)*
c*d^2*e^2 + (a*b^2 + b^3)*c*d^2*e)*f + 6*((a^2*b + a*b^2)*c^2*d*f^3 - (a*b^2 + b^3)*c*d^2*f^2)*x)*cosh(f*x + e
)*sinh(f*x + e) - (2*(a^2*b + a*b^2)*d^3*f^3*x^3 + 2*(a^2*b + a*b^2)*d^3*e^3 + 6*(a^2*b + a*b^2)*c^2*d*e*f^2 +
3*(a*b^2 + b^3)*d^3*e^2 + 3*(2*(a^2*b + a*b^2)*c*d^2*f^3 - (a*b^2 + b^3)*d^3*f^2)*x^2 - 6*((a^2*b + a*b^2)*c*
d^2*e^2 + (a*b^2 + b^3)*c*d^2*e)*f + 6*((a^2*b + a*b^2)*c^2*d*f^3 - (a*b^2 + b^3)*c*d^2*f^2)*x)*sinh(f*x + e)^
2 - 6*((a^2*b - a*b^2)*c*d^2*e^2 + (a*b^2 - b^3)*c*d^2*e)*f + 6*((a^2*b - a*b^2)*c^2*d*f^3 - (a*b^2 - b^3)*c*d
^2*f^2)*x)*log(-sqrt((a + b)/(a - b))*(cosh(f*x + e) + sinh(f*x + e)) + 1) + 48*((a^2*b + a*b^2)*d^3*cosh(f*x
+ e)^2 + 2*(a^2*b + a*b^2)*d^3*cosh(f*x + e)*sinh(f*x + e) + (a^2*b + a*b^2)*d^3*sinh(f*x + e)^2 - (a^2*b - a*
b^2)*d^3)*polylog(4, sqrt((a + b)/(a - b))*(cosh(f*x + e) + sinh(f*x + e))) + 48*((a^2*b + a*b^2)*d^3*cosh(f*x
+ e)^2 + 2*(a^2*b + a*b^2)*d^3*cosh(f*x + e)*sinh(f*x + e) + (a^2*b + a*b^2)*d^3*sinh(f*x + e)^2 - (a^2*b - a
*b^2)*d^3)*polylog(4, -sqrt((a + b)/(a - b))*(cosh(f*x + e) + sinh(f*x + e))) + 24*(2*(a^2*b - a*b^2)*d^3*f*x
+ 2*(a^2*b - a*b^2)*c*d^2*f - (a*b^2 - b^3)*d^3 - (2*(a^2*b + a*b^2)*d^3*f*x + 2*(a^2*b + a*b^2)*c*d^2*f - (a*
b^2 + b^3)*d^3)*cosh(f*x + e)^2 - 2*(2*(a^2*b + a*b^2)*d^3*f*x + 2*(a^2*b + a*b^2)*c*d^2*f - (a*b^2 + b^3)*d^3
)*cosh(f*x + e)*sinh(f*x + e) - (2*(a^2*b + a*b^2)*d^3*f*x + 2*(a^2*b + a*b^2)*c*d^2*f - (a*b^2 + b^3)*d^3)*si
nh(f*x + e)^2)*polylog(3, sqrt((a + b)/(a - b))*(cosh(f*x + e) + sinh(f*x + e))) + 24*(2*(a^2*b - a*b^2)*d^3*f
*x + 2*(a^2*b - a*b^2)*c*d^2*f - (a*b^2 - b^3)*d^3 - (2*(a^2*b + a*b^2)*d^3*f*x + 2*(a^2*b + a*b^2)*c*d^2*f -
(a*b^2 + b^3)*d^3)*cosh(f*x + e)^2 - 2*(2*(a^2*b + a*b^2)*d^3*f*x + 2*(a^2*b + a*b^2)*c*d^2*f - (a*b^2 + b^3)*
d^3)*cosh(f*x + e)*sinh(f*x + e) - (2*(a^2*b + a*b^2)*d^3*f*x + 2*(a^2*b + a*b^2)*c*d^2*f - (a*b^2 + b^3)*d^3)
*sinh(f*x + e)^2)*polylog(3, -sqrt((a + b)/(a - b))*(cosh(f*x + e) + sinh(f*x + e))))/((a^5 + a^4*b - 2*a^3*b^
2 - 2*a^2*b^3 + a*b^4 + b^5)*f^4*cosh(f*x + e)^2 + 2*(a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5)*f^4*c
osh(f*x + e)*sinh(f*x + e) + (a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5)*f^4*sinh(f*x + e)^2 - (a^5 -
a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*f^4)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{3}}{{\left (b \coth \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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