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

Optimal. Leaf size=638 \[ \frac {3 b^2 d^2 (c+d x) \text {Li}_2\left (\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^3 \left (a^2-b^2\right )^2}-\frac {3 b^2 d^2 (c+d x) \text {Li}_3\left (\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^3 \left (a^2-b^2\right )^2}+\frac {3 b^2 d (c+d x)^2 \text {Li}_2\left (\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^2 \left (a^2-b^2\right )^2}+\frac {3 b^2 d (c+d x)^2 \log \left (1-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^2 \left (a^2-b^2\right )^2}+\frac {2 b^2 (c+d x)^3 \log \left (1-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{f \left (a^2-b^2\right )^2}-\frac {2 b^2 (c+d x)^3}{f \left (a^2-b^2\right )^2}-\frac {3 b^2 d^3 \text {Li}_3\left (\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 f^4 \left (a^2-b^2\right )^2}+\frac {3 b^2 d^3 \text {Li}_4\left (\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 f^4 \left (a^2-b^2\right )^2}+\frac {2 b^2 (c+d x)^3}{f (a-b) (a+b)^2 \left (-(a+b) e^{2 e+2 f x}+a-b\right )}+\frac {3 b d^2 (c+d x) \text {Li}_3\left (\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^3 (a-b)^2 (a+b)}-\frac {3 b d (c+d x)^2 \text {Li}_2\left (\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^2 (a-b)^2 (a+b)}-\frac {2 b (c+d x)^3 \log \left (1-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{f (a-b)^2 (a+b)}+\frac {(c+d x)^4}{4 d (a-b)^2}-\frac {3 b d^3 \text {Li}_4\left (\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{2 f^4 (a-b)^2 (a+b)} \]

[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]  time = 2.27, antiderivative size = 638, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 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 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 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 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]

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 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 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 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 6589

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

Rule 6609

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

Rubi steps

\begin {align*} \int \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*}

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Mathematica [A]  time = 12.71, size = 1022, normalized size = 1.60 \[ \frac {b \left (-4 a \left (a \left (-1+e^{2 e}\right )+b \left (1+e^{2 e}\right )\right ) f^3 x^3 \log \left (\frac {e^{-2 (e+f x)} (b-a)}{a+b}+1\right ) d^4+3 a \left (a \left (-1+e^{2 e}\right )+b \left (1+e^{2 e}\right )\right ) \left (2 f^2 \text {Li}_2\left (\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right ) x^2+2 f \text {Li}_3\left (\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right ) x+\text {Li}_4\left (\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )\right ) d^4+6 \left (a \left (-1+e^{2 e}\right )+b \left (1+e^{2 e}\right )\right ) f^2 (b d-2 a c f) x^2 \log \left (\frac {e^{-2 (e+f x)} (b-a)}{a+b}+1\right ) d^3-3 \left (a \left (-1+e^{2 e}\right )+b \left (1+e^{2 e}\right )\right ) (b d-2 a c f) \left (2 f x \text {Li}_2\left (\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )+\text {Li}_3\left (\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )\right ) d^3-12 c \left (a \left (-1+e^{2 e}\right )+b \left (1+e^{2 e}\right )\right ) f^2 (a c f-b d) x \log \left (\frac {e^{-2 (e+f x)} (b-a)}{a+b}+1\right ) d^2+6 c \left (a \left (-1+e^{2 e}\right )+b \left (1+e^{2 e}\right )\right ) f (a c f-b d) \text {Li}_2\left (\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right ) d^2-4 (a-b) b f^3 (c+d x)^3 d+2 c^2 \left (a \left (-1+e^{2 e}\right )+b \left (1+e^{2 e}\right )\right ) f^2 (2 a c f-3 b d) \left (2 f x-\log \left (a-(a+b) e^{2 (e+f x)}-b\right )\right ) d+2 a (a-b) f^4 (c+d x)^4\right )}{2 (a-b)^2 (a+b)^2 d \left (a \left (-1+e^{2 e}\right )+b \left (1+e^{2 e}\right )\right ) f^4}+\frac {-a^2 d^3 f \cosh (f x) x^4-b^2 d^3 f \cosh (f x) x^4+a^2 d^3 f \cosh (2 e+f x) x^4-b^2 d^3 f \cosh (2 e+f x) x^4-2 a b d^3 f \sinh (f x) x^4-4 a^2 c d^2 f \cosh (f x) x^3-4 b^2 c d^2 f \cosh (f x) x^3+4 a^2 c d^2 f \cosh (2 e+f x) x^3-4 b^2 c d^2 f \cosh (2 e+f x) x^3+8 b^2 d^3 \sinh (f x) x^3-8 a b c d^2 f \sinh (f x) x^3-6 a^2 c^2 d f \cosh (f x) x^2-6 b^2 c^2 d f \cosh (f x) x^2+6 a^2 c^2 d f \cosh (2 e+f x) x^2-6 b^2 c^2 d f \cosh (2 e+f x) x^2+24 b^2 c d^2 \sinh (f x) x^2-12 a b c^2 d f \sinh (f x) x^2-4 a^2 c^3 f \cosh (f x) x-4 b^2 c^3 f \cosh (f x) x+4 a^2 c^3 f \cosh (2 e+f x) x-4 b^2 c^3 f \cosh (2 e+f x) x+24 b^2 c^2 d \sinh (f x) x-8 a b c^3 f \sinh (f x) x+8 b^2 c^3 \sinh (f x)}{8 (a-b) (a+b) f (b \cosh (e)+a \sinh (e)) (b \cosh (e+f x)+a \sinh (e+f x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [C]  time = 0.57, size = 6171, normalized size = 9.67 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+b*coth(f*x+e))^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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x + c\right )}^{3}}{{\left (b \coth \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]

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)

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maple [B]  time = 1.07, size = 2423, normalized size = 3.80 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.03, size = 1056, normalized size = 1.66 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+b*coth(f*x+e))^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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c+d\,x\right )}^3}{{\left (a+b\,\mathrm {coth}\left (e+f\,x\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c + d x\right )^{3}}{\left (a + b \coth {\left (e + f x \right )}\right )^{2}}\, dx \]

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]

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

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