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

Optimal. Leaf size=475 \[ \frac{2 b^2 d (c+d x) \text{PolyLog}\left (2,\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^2 \left (a^2-b^2\right )^2}+\frac{b^2 d^2 \text{PolyLog}\left (2,\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^3 \left (a^2-b^2\right )^2}-\frac{b^2 d^2 \text{PolyLog}\left (3,\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^3 \left (a^2-b^2\right )^2}-\frac{2 b d (c+d x) \text{PolyLog}\left (2,\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^2 (a-b)^2 (a+b)}+\frac{b d^2 \text{PolyLog}\left (3,\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^3 (a-b)^2 (a+b)}+\frac{2 b^2 d (c+d x) \log \left (1-\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^2 \left (a^2-b^2\right )^2}+\frac{2 b^2 (c+d x)^2 \log \left (1-\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f \left (a^2-b^2\right )^2}-\frac{2 b^2 (c+d x)^2}{f \left (a^2-b^2\right )^2}+\frac{2 b^2 (c+d x)^2}{f (a-b) (a+b)^2 \left (-(a+b) e^{2 e+2 f x}+a-b\right )}-\frac{2 b (c+d x)^2 \log \left (1-\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f (a-b)^2 (a+b)}+\frac{(c+d x)^3}{3 d (a-b)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.71445, antiderivative size = 475, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 11, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.55, Rules used = {3734, 2254, 2185, 2184, 2190, 2531, 2282, 6589, 2191, 2279, 2391} \[ \frac{2 b^2 d (c+d x) \text{PolyLog}\left (2,\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^2 \left (a^2-b^2\right )^2}+\frac{b^2 d^2 \text{PolyLog}\left (2,\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^3 \left (a^2-b^2\right )^2}-\frac{b^2 d^2 \text{PolyLog}\left (3,\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^3 \left (a^2-b^2\right )^2}-\frac{2 b d (c+d x) \text{PolyLog}\left (2,\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^2 (a-b)^2 (a+b)}+\frac{b d^2 \text{PolyLog}\left (3,\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^3 (a-b)^2 (a+b)}+\frac{2 b^2 d (c+d x) \log \left (1-\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f^2 \left (a^2-b^2\right )^2}+\frac{2 b^2 (c+d x)^2 \log \left (1-\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f \left (a^2-b^2\right )^2}-\frac{2 b^2 (c+d x)^2}{f \left (a^2-b^2\right )^2}+\frac{2 b^2 (c+d x)^2}{f (a-b) (a+b)^2 \left (-(a+b) e^{2 e+2 f x}+a-b\right )}-\frac{2 b (c+d x)^2 \log \left (1-\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{f (a-b)^2 (a+b)}+\frac{(c+d x)^3}{3 d (a-b)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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]

Rule 2279

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

Rule 2391

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

Rubi steps

\begin{align*} \int \frac{(c+d x)^2}{(a+b \coth (e+f x))^2} \, dx &=\int \left (\frac{(c+d x)^2}{(a-b)^2}+\frac{4 b^2 e^{4 e+4 f x} (c+d x)^2}{(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)^2}{(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)^3}{3 (a-b)^2 d}+\frac{(4 b) \int \frac{e^{2 e+2 f x} (c+d x)^2}{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)^2}{\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)^3}{3 (a-b)^2 d}-\frac{2 b (c+d x)^2 \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)^2}{(a+b)^2}+\frac{(a-b)^2 (c+d x)^2}{(a+b)^2 \left (a-b-(a+b) e^{2 e+2 f x}\right )^2}+\frac{2 (-a+b) (c+d x)^2}{(a+b)^2 \left (a-b-(a+b) e^{2 e+2 f x}\right )}\right ) \, dx}{(a-b)^2}+\frac{(4 b d) \int (c+d x) \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)^3}{3 (a-b)^2 d}+\frac{4 b^2 (c+d x)^3}{3 \left (a^2-b^2\right )^2 d}-\frac{2 b (c+d x)^2 \log \left (1-\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f}-\frac{2 b d (c+d x) \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)^2}{\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)^2}{a-b+(-a-b) e^{2 e+2 f x}} \, dx}{(a-b) (a+b)^2}+\frac{\left (2 b d^2\right ) \int \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)^3}{3 (a-b)^2 d}-\frac{4 b^2 (c+d x)^3}{3 \left (a^2-b^2\right )^2 d}-\frac{2 b (c+d x)^2 \log \left (1-\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f}-\frac{2 b d (c+d x) \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)^2}{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)^2}{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)^2}{\left (a-b+(-a-b) e^{2 e+2 f x}\right )^2} \, dx}{a^2-b^2}+\frac{\left (b d^2\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 )}{(a-b)^2 (a+b) f^3}\\ &=\frac{2 b^2 (c+d x)^2}{(a-b) (a+b)^2 \left (a-b-(a+b) e^{2 e+2 f x}\right ) f}+\frac{(c+d x)^3}{3 (a-b)^2 d}-\frac{2 b (c+d x)^2 \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)^2 \log \left (1-\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f}-\frac{2 b d (c+d x) \text{Li}_2\left (\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^2}+\frac{b d^2 \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)^2}{a-b+(-a-b) e^{2 e+2 f x}} \, dx}{(a-b)^2 (a+b)}-\frac{\left (4 b^2 d\right ) \int \frac{c+d x}{a-b+(-a-b) e^{2 e+2 f x}} \, dx}{(a-b) (a+b)^2 f}-\frac{\left (8 b^2 d\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}\\ &=-\frac{2 b^2 (c+d x)^2}{\left (a^2-b^2\right )^2 f}+\frac{2 b^2 (c+d x)^2}{(a-b) (a+b)^2 \left (a-b-(a+b) e^{2 e+2 f x}\right ) f}+\frac{(c+d x)^3}{3 (a-b)^2 d}-\frac{2 b (c+d x)^2 \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)^2 \log \left (1-\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f}-\frac{2 b d (c+d x) \text{Li}_2\left (\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^2}+\frac{4 b^2 d (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^2}+\frac{b d^2 \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 d^2\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^2}-\frac{\left (4 b^2 d\right ) \int \frac{e^{2 e+2 f x} (c+d x)}{a-b+(-a-b) e^{2 e+2 f x}} \, dx}{(a-b)^2 (a+b) f}+\frac{\left (4 b^2 d\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}\\ &=-\frac{2 b^2 (c+d x)^2}{\left (a^2-b^2\right )^2 f}+\frac{2 b^2 (c+d x)^2}{(a-b) (a+b)^2 \left (a-b-(a+b) e^{2 e+2 f x}\right ) f}+\frac{(c+d x)^3}{3 (a-b)^2 d}+\frac{2 b^2 d (c+d x) \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)^2 \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)^2 \log \left (1-\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f}-\frac{2 b d (c+d x) \text{Li}_2\left (\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^2}+\frac{2 b^2 d (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^2}+\frac{b d^2 \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 (2 b^2 d^2\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 )}{\left (a^2-b^2\right )^2 f^3}-\frac{\left (2 b^2 d^2\right ) \int \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 (2 b^2 d^2\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^2}\\ &=-\frac{2 b^2 (c+d x)^2}{\left (a^2-b^2\right )^2 f}+\frac{2 b^2 (c+d x)^2}{(a-b) (a+b)^2 \left (a-b-(a+b) e^{2 e+2 f x}\right ) f}+\frac{(c+d x)^3}{3 (a-b)^2 d}+\frac{2 b^2 d (c+d x) \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)^2 \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)^2 \log \left (1-\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f}-\frac{2 b d (c+d x) \text{Li}_2\left (\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^2}+\frac{2 b^2 d (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^2}+\frac{b d^2 \text{Li}_3\left (\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^3}-\frac{2 b^2 d^2 \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{\left (b^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\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^3}+\frac{\left (b^2 d^2\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 )}{\left (a^2-b^2\right )^2 f^3}\\ &=-\frac{2 b^2 (c+d x)^2}{\left (a^2-b^2\right )^2 f}+\frac{2 b^2 (c+d x)^2}{(a-b) (a+b)^2 \left (a-b-(a+b) e^{2 e+2 f x}\right ) f}+\frac{(c+d x)^3}{3 (a-b)^2 d}+\frac{2 b^2 d (c+d x) \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)^2 \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)^2 \log \left (1-\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f}+\frac{b^2 d^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^3}-\frac{2 b d (c+d x) \text{Li}_2\left (\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^2}+\frac{2 b^2 d (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^2}+\frac{b d^2 \text{Li}_3\left (\frac{(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^3}-\frac{b^2 d^2 \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}\\ \end{align*}

Mathematica [A]  time = 8.29476, size = 509, normalized size = 1.07 \[ \frac{-6 b d (b d-2 a c f) \text{PolyLog}\left (2,\frac{(a-b) e^{-2 (e+f x)}}{a+b}\right )+6 a b d^2 \left (2 f x \text{PolyLog}\left (2,\frac{(a-b) e^{-2 (e+f x)}}{a+b}\right )+\text{PolyLog}\left (3,\frac{(a-b) e^{-2 (e+f x)}}{a+b}\right )\right )-\frac{f^2 (a-b) (a+b) \left (-f x \left (a^2-b^2\right ) \left (3 c^2+3 c d x+d^2 x^2\right ) \cosh (2 e+f x)+f x \left (a^2+b^2\right ) \left (3 c^2+3 c d x+d^2 x^2\right ) \cosh (f x)+2 b \sinh (f x) \left (a f x \left (3 c^2+3 c d x+d^2 x^2\right )-3 b (c+d x)^2\right )\right )}{(a \sinh (e)+b \cosh (e)) (a \sinh (e+f x)+b \cosh (e+f x))}+\frac{12 b d f^2 x^2 (a-b) (2 a c f-b d)}{a \left (e^{2 e}-1\right )+b \left (e^{2 e}+1\right )}+\frac{24 b c f^2 x (a-b) (a c f-b d)}{a \left (e^{2 e}-1\right )+b \left (e^{2 e}+1\right )}+12 b d f x (b d-2 a c f) \log \left (\frac{(b-a) e^{-2 (e+f x)}}{a+b}+1\right )+12 b c f (a c f-b d) \left (2 f x-\log \left (-(a+b) e^{2 (e+f x)}+a-b\right )\right )+\frac{8 a b d^2 f^3 x^3 (a-b)}{a \left (e^{2 e}-1\right )+b \left (e^{2 e}+1\right )}-12 a b d^2 f^2 x^2 \log \left (\frac{(b-a) e^{-2 (e+f x)}}{a+b}+1\right )}{6 f^3 (a-b)^2 (a+b)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.263, size = 1217, normalized size = 2.6 \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)^2/(a+b*coth(f*x+e))^2,x)

[Out]

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

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Maxima [A]  time = 2.79082, size = 1014, normalized size = 2.13 \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)^2/(a+b*coth(f*x+e))^2,x, algorithm="maxima")

[Out]

-4*b^2*c*d*f*x/(a^4*f^2 - 2*a^2*b^2*f^2 + b^4*f^2) - (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*b*d^2/(a^4*f^3 - 2*a^2*b^2*f^3 + b^4*f^3) + 2*b^2*c*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^2*(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)) - (2*a*b*c*d*f - b^2*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) + 2/3*(2*a*b*d^2*f^3*x^3
+ 3*(2*a*b*c*d*f - b^2*d^2)*f^2*x^2)/(a^4*f^3 - 2*a^2*b^2*f^3 + b^4*f^3) + 1/3*(12*b^2*c*d*x + (a^2*d^2*f - 2
*a*b*d^2*f + b^2*d^2*f)*x^3 + 3*(a^2*c*d*f - 2*a*b*c*d*f + (c*d*f + 2*d^2)*b^2)*x^2 - ((a^2*d^2*f*e^(2*e) - b^
2*d^2*f*e^(2*e))*x^3 + 3*(a^2*c*d*f*e^(2*e) - b^2*c*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 = 2.90485, size = 7876, normalized size = 16.58 \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)^2/(a+b*coth(f*x+e))^2,x, algorithm="fricas")

[Out]

-1/3*((a^3 + a^2*b - a*b^2 - b^3)*d^2*f^3*x^3 + 3*(a^3 + a^2*b - a*b^2 - b^3)*c*d*f^3*x^2 + 3*(a^3 + a^2*b - a
*b^2 - b^3)*c^2*f^3*x + 4*(a^2*b - a*b^2)*d^2*e^3 + 6*(a*b^2 - b^3)*d^2*e^2 + 6*(2*(a^2*b - a*b^2)*c^2*e + (a*
b^2 - b^3)*c^2)*f^2 - ((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d^2*f^3*x^3 + 4*(a^2*b + a*b^2)*d^2*e^3 + 12*(a^2*b + a
*b^2)*c^2*e*f^2 + 6*(a*b^2 + b^3)*d^2*e^2 + 3*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*c*d*f^3 - 2*(a*b^2 + b^3)*d^2*f
^2)*x^2 - 12*((a^2*b + a*b^2)*c*d*e^2 + (a*b^2 + b^3)*c*d*e)*f + 3*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*c^2*f^3 -
4*(a*b^2 + b^3)*c*d*f^2)*x)*cosh(f*x + e)^2 - 2*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d^2*f^3*x^3 + 4*(a^2*b + a*b^
2)*d^2*e^3 + 12*(a^2*b + a*b^2)*c^2*e*f^2 + 6*(a*b^2 + b^3)*d^2*e^2 + 3*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*c*d*f
^3 - 2*(a*b^2 + b^3)*d^2*f^2)*x^2 - 12*((a^2*b + a*b^2)*c*d*e^2 + (a*b^2 + b^3)*c*d*e)*f + 3*((a^3 + 3*a^2*b +
3*a*b^2 + b^3)*c^2*f^3 - 4*(a*b^2 + b^3)*c*d*f^2)*x)*cosh(f*x + e)*sinh(f*x + e) - ((a^3 + 3*a^2*b + 3*a*b^2
+ b^3)*d^2*f^3*x^3 + 4*(a^2*b + a*b^2)*d^2*e^3 + 12*(a^2*b + a*b^2)*c^2*e*f^2 + 6*(a*b^2 + b^3)*d^2*e^2 + 3*((
a^3 + 3*a^2*b + 3*a*b^2 + b^3)*c*d*f^3 - 2*(a*b^2 + b^3)*d^2*f^2)*x^2 - 12*((a^2*b + a*b^2)*c*d*e^2 + (a*b^2 +
b^3)*c*d*e)*f + 3*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*c^2*f^3 - 4*(a*b^2 + b^3)*c*d*f^2)*x)*sinh(f*x + e)^2 - 12
*((a^2*b - a*b^2)*c*d*e^2 + (a*b^2 - b^3)*c*d*e)*f - 6*(2*(a^2*b - a*b^2)*d^2*f*x + 2*(a^2*b - a*b^2)*c*d*f -
(a*b^2 - b^3)*d^2 - (2*(a^2*b + a*b^2)*d^2*f*x + 2*(a^2*b + a*b^2)*c*d*f - (a*b^2 + b^3)*d^2)*cosh(f*x + e)^2
- 2*(2*(a^2*b + a*b^2)*d^2*f*x + 2*(a^2*b + a*b^2)*c*d*f - (a*b^2 + b^3)*d^2)*cosh(f*x + e)*sinh(f*x + e) - (2
*(a^2*b + a*b^2)*d^2*f*x + 2*(a^2*b + a*b^2)*c*d*f - (a*b^2 + b^3)*d^2)*sinh(f*x + e)^2)*dilog(sqrt((a + b)/(a
- b))*(cosh(f*x + e) + sinh(f*x + e))) - 6*(2*(a^2*b - a*b^2)*d^2*f*x + 2*(a^2*b - a*b^2)*c*d*f - (a*b^2 - b^
3)*d^2 - (2*(a^2*b + a*b^2)*d^2*f*x + 2*(a^2*b + a*b^2)*c*d*f - (a*b^2 + b^3)*d^2)*cosh(f*x + e)^2 - 2*(2*(a^2
*b + a*b^2)*d^2*f*x + 2*(a^2*b + a*b^2)*c*d*f - (a*b^2 + b^3)*d^2)*cosh(f*x + e)*sinh(f*x + e) - (2*(a^2*b + a
*b^2)*d^2*f*x + 2*(a^2*b + a*b^2)*c*d*f - (a*b^2 + b^3)*d^2)*sinh(f*x + e)^2)*dilog(-sqrt((a + b)/(a - b))*(co
sh(f*x + e) + sinh(f*x + e))) - 6*((a^2*b - a*b^2)*d^2*e^2 + (a^2*b - a*b^2)*c^2*f^2 + (a*b^2 - b^3)*d^2*e - (
(a^2*b + a*b^2)*d^2*e^2 + (a^2*b + a*b^2)*c^2*f^2 + (a*b^2 + b^3)*d^2*e - (2*(a^2*b + a*b^2)*c*d*e + (a*b^2 +
b^3)*c*d)*f)*cosh(f*x + e)^2 - 2*((a^2*b + a*b^2)*d^2*e^2 + (a^2*b + a*b^2)*c^2*f^2 + (a*b^2 + b^3)*d^2*e - (2
*(a^2*b + a*b^2)*c*d*e + (a*b^2 + b^3)*c*d)*f)*cosh(f*x + e)*sinh(f*x + e) - ((a^2*b + a*b^2)*d^2*e^2 + (a^2*b
+ a*b^2)*c^2*f^2 + (a*b^2 + b^3)*d^2*e - (2*(a^2*b + a*b^2)*c*d*e + (a*b^2 + b^3)*c*d)*f)*sinh(f*x + e)^2 - (
2*(a^2*b - a*b^2)*c*d*e + (a*b^2 - b^3)*c*d)*f)*log(2*(a + b)*cosh(f*x + e) + 2*(a + b)*sinh(f*x + e) + 2*(a -
b)*sqrt((a + b)/(a - b))) - 6*((a^2*b - a*b^2)*d^2*e^2 + (a^2*b - a*b^2)*c^2*f^2 + (a*b^2 - b^3)*d^2*e - ((a^
2*b + a*b^2)*d^2*e^2 + (a^2*b + a*b^2)*c^2*f^2 + (a*b^2 + b^3)*d^2*e - (2*(a^2*b + a*b^2)*c*d*e + (a*b^2 + b^3
)*c*d)*f)*cosh(f*x + e)^2 - 2*((a^2*b + a*b^2)*d^2*e^2 + (a^2*b + a*b^2)*c^2*f^2 + (a*b^2 + b^3)*d^2*e - (2*(a
^2*b + a*b^2)*c*d*e + (a*b^2 + b^3)*c*d)*f)*cosh(f*x + e)*sinh(f*x + e) - ((a^2*b + a*b^2)*d^2*e^2 + (a^2*b +
a*b^2)*c^2*f^2 + (a*b^2 + b^3)*d^2*e - (2*(a^2*b + a*b^2)*c*d*e + (a*b^2 + b^3)*c*d)*f)*sinh(f*x + e)^2 - (2*(
a^2*b - a*b^2)*c*d*e + (a*b^2 - b^3)*c*d)*f)*log(2*(a + b)*cosh(f*x + e) + 2*(a + b)*sinh(f*x + e) - 2*(a - b)
*sqrt((a + b)/(a - b))) - 6*((a^2*b - a*b^2)*d^2*f^2*x^2 - (a^2*b - a*b^2)*d^2*e^2 + 2*(a^2*b - a*b^2)*c*d*e*f
- (a*b^2 - b^3)*d^2*e - ((a^2*b + a*b^2)*d^2*f^2*x^2 - (a^2*b + a*b^2)*d^2*e^2 + 2*(a^2*b + a*b^2)*c*d*e*f -
(a*b^2 + b^3)*d^2*e + (2*(a^2*b + a*b^2)*c*d*f^2 - (a*b^2 + b^3)*d^2*f)*x)*cosh(f*x + e)^2 - 2*((a^2*b + a*b^2
)*d^2*f^2*x^2 - (a^2*b + a*b^2)*d^2*e^2 + 2*(a^2*b + a*b^2)*c*d*e*f - (a*b^2 + b^3)*d^2*e + (2*(a^2*b + a*b^2)
*c*d*f^2 - (a*b^2 + b^3)*d^2*f)*x)*cosh(f*x + e)*sinh(f*x + e) - ((a^2*b + a*b^2)*d^2*f^2*x^2 - (a^2*b + a*b^2
)*d^2*e^2 + 2*(a^2*b + a*b^2)*c*d*e*f - (a*b^2 + b^3)*d^2*e + (2*(a^2*b + a*b^2)*c*d*f^2 - (a*b^2 + b^3)*d^2*f
)*x)*sinh(f*x + e)^2 + (2*(a^2*b - a*b^2)*c*d*f^2 - (a*b^2 - b^3)*d^2*f)*x)*log(sqrt((a + b)/(a - b))*(cosh(f*
x + e) + sinh(f*x + e)) + 1) - 6*((a^2*b - a*b^2)*d^2*f^2*x^2 - (a^2*b - a*b^2)*d^2*e^2 + 2*(a^2*b - a*b^2)*c*
d*e*f - (a*b^2 - b^3)*d^2*e - ((a^2*b + a*b^2)*d^2*f^2*x^2 - (a^2*b + a*b^2)*d^2*e^2 + 2*(a^2*b + a*b^2)*c*d*e
*f - (a*b^2 + b^3)*d^2*e + (2*(a^2*b + a*b^2)*c*d*f^2 - (a*b^2 + b^3)*d^2*f)*x)*cosh(f*x + e)^2 - 2*((a^2*b +
a*b^2)*d^2*f^2*x^2 - (a^2*b + a*b^2)*d^2*e^2 + 2*(a^2*b + a*b^2)*c*d*e*f - (a*b^2 + b^3)*d^2*e + (2*(a^2*b + a
*b^2)*c*d*f^2 - (a*b^2 + b^3)*d^2*f)*x)*cosh(f*x + e)*sinh(f*x + e) - ((a^2*b + a*b^2)*d^2*f^2*x^2 - (a^2*b +
a*b^2)*d^2*e^2 + 2*(a^2*b + a*b^2)*c*d*e*f - (a*b^2 + b^3)*d^2*e + (2*(a^2*b + a*b^2)*c*d*f^2 - (a*b^2 + b^3)*
d^2*f)*x)*sinh(f*x + e)^2 + (2*(a^2*b - a*b^2)*c*d*f^2 - (a*b^2 - b^3)*d^2*f)*x)*log(-sqrt((a + b)/(a - b))*(c
osh(f*x + e) + sinh(f*x + e)) + 1) - 12*((a^2*b + a*b^2)*d^2*cosh(f*x + e)^2 + 2*(a^2*b + a*b^2)*d^2*cosh(f*x
+ e)*sinh(f*x + e) + (a^2*b + a*b^2)*d^2*sinh(f*x + e)^2 - (a^2*b - a*b^2)*d^2)*polylog(3, sqrt((a + b)/(a - b
))*(cosh(f*x + e) + sinh(f*x + e))) - 12*((a^2*b + a*b^2)*d^2*cosh(f*x + e)^2 + 2*(a^2*b + a*b^2)*d^2*cosh(f*x
+ e)*sinh(f*x + e) + (a^2*b + a*b^2)*d^2*sinh(f*x + e)^2 - (a^2*b - a*b^2)*d^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^3*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^3*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^3*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^
3)

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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)**2/(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 )}^{2}}{{\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)^2/(a+b*coth(f*x+e))^2,x, algorithm="giac")

[Out]

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