\(\int (c+d x)^2 (a+b \coth (e+f x))^3 \, dx\) [48]

Optimal result

Integrand size = 20, antiderivative size = 401 \[ \int (c+d x)^2 (a+b \coth (e+f x))^3 \, dx=\frac {b^3 c d x}{f}+\frac {b^3 d^2 x^2}{2 f}-\frac {3 a b^2 (c+d x)^2}{f}+\frac {a^3 (c+d x)^3}{3 d}-\frac {a^2 b (c+d x)^3}{d}+\frac {a b^2 (c+d x)^3}{d}-\frac {b^3 (c+d x)^3}{3 d}-\frac {b^3 d (c+d x) \coth (e+f x)}{f^2}-\frac {3 a b^2 (c+d x)^2 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^2 \coth ^2(e+f x)}{2 f}+\frac {6 a b^2 d (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 d^2 \log (\sinh (e+f x))}{f^3}+\frac {3 a b^2 d^2 \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^3}+\frac {3 a^2 b d (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}+\frac {b^3 d (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}-\frac {3 a^2 b d^2 \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3}-\frac {b^3 d^2 \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3} \]

[Out]

b^3*c*d*x/f+1/2*b^3*d^2*x^2/f-3*a*b^2*(d*x+c)^2/f+1/3*a^3*(d*x+c)^3/d-a^2*b*(d*x+c)^3/d+a*b^2*(d*x+c)^3/d-1/3*
b^3*(d*x+c)^3/d-b^3*d*(d*x+c)*coth(f*x+e)/f^2-3*a*b^2*(d*x+c)^2*coth(f*x+e)/f-1/2*b^3*(d*x+c)^2*coth(f*x+e)^2/
f+6*a*b^2*d*(d*x+c)*ln(1-exp(2*f*x+2*e))/f^2+3*a^2*b*(d*x+c)^2*ln(1-exp(2*f*x+2*e))/f+b^3*(d*x+c)^2*ln(1-exp(2
*f*x+2*e))/f+b^3*d^2*ln(sinh(f*x+e))/f^3+3*a*b^2*d^2*polylog(2,exp(2*f*x+2*e))/f^3+3*a^2*b*d*(d*x+c)*polylog(2
,exp(2*f*x+2*e))/f^2+b^3*d*(d*x+c)*polylog(2,exp(2*f*x+2*e))/f^2-3/2*a^2*b*d^2*polylog(3,exp(2*f*x+2*e))/f^3-1
/2*b^3*d^2*polylog(3,exp(2*f*x+2*e))/f^3

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {3803, 3797, 2221, 2611, 2320, 6724, 3801, 2317, 2438, 32, 3556} \[ \int (c+d x)^2 (a+b \coth (e+f x))^3 \, dx=\frac {a^3 (c+d x)^3}{3 d}+\frac {3 a^2 b d (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {a^2 b (c+d x)^3}{d}-\frac {3 a^2 b d^2 \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3}+\frac {6 a b^2 d (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^2}-\frac {3 a b^2 (c+d x)^2 \coth (e+f x)}{f}-\frac {3 a b^2 (c+d x)^2}{f}+\frac {a b^2 (c+d x)^3}{d}+\frac {3 a b^2 d^2 \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^3}+\frac {b^3 d (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}-\frac {b^3 d (c+d x) \coth (e+f x)}{f^2}+\frac {b^3 (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {b^3 (c+d x)^2 \coth ^2(e+f x)}{2 f}+\frac {b^3 c d x}{f}-\frac {b^3 (c+d x)^3}{3 d}-\frac {b^3 d^2 \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3}+\frac {b^3 d^2 \log (\sinh (e+f x))}{f^3}+\frac {b^3 d^2 x^2}{2 f} \]

[In]

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

[Out]

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2221

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

Rule 2317

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

Rule 2320

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

Rule 2438

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

Rule 2611

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

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 3801

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(c + d*x)^m*((b*Tan[e
+ f*x])^(n - 1)/(f*(n - 1))), x] + (-Dist[b*d*(m/(f*(n - 1))), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]

Rule 3803

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 6724

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

Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 (c+d x)^2+3 a^2 b (c+d x)^2 \coth (e+f x)+3 a b^2 (c+d x)^2 \coth ^2(e+f x)+b^3 (c+d x)^2 \coth ^3(e+f x)\right ) \, dx \\ & = \frac {a^3 (c+d x)^3}{3 d}+\left (3 a^2 b\right ) \int (c+d x)^2 \coth (e+f x) \, dx+\left (3 a b^2\right ) \int (c+d x)^2 \coth ^2(e+f x) \, dx+b^3 \int (c+d x)^2 \coth ^3(e+f x) \, dx \\ & = \frac {a^3 (c+d x)^3}{3 d}-\frac {a^2 b (c+d x)^3}{d}-\frac {3 a b^2 (c+d x)^2 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^2 \coth ^2(e+f x)}{2 f}-\left (6 a^2 b\right ) \int \frac {e^{2 (e+f x)} (c+d x)^2}{1-e^{2 (e+f x)}} \, dx+\left (3 a b^2\right ) \int (c+d x)^2 \, dx+b^3 \int (c+d x)^2 \coth (e+f x) \, dx+\frac {\left (6 a b^2 d\right ) \int (c+d x) \coth (e+f x) \, dx}{f}+\frac {\left (b^3 d\right ) \int (c+d x) \coth ^2(e+f x) \, dx}{f} \\ & = -\frac {3 a b^2 (c+d x)^2}{f}+\frac {a^3 (c+d x)^3}{3 d}-\frac {a^2 b (c+d x)^3}{d}+\frac {a b^2 (c+d x)^3}{d}-\frac {b^3 (c+d x)^3}{3 d}-\frac {b^3 d (c+d x) \coth (e+f x)}{f^2}-\frac {3 a b^2 (c+d x)^2 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^2 \coth ^2(e+f x)}{2 f}+\frac {3 a^2 b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}-\left (2 b^3\right ) \int \frac {e^{2 (e+f x)} (c+d x)^2}{1-e^{2 (e+f x)}} \, dx+\frac {\left (b^3 d^2\right ) \int \coth (e+f x) \, dx}{f^2}-\frac {\left (6 a^2 b d\right ) \int (c+d x) \log \left (1-e^{2 (e+f x)}\right ) \, dx}{f}-\frac {\left (12 a b^2 d\right ) \int \frac {e^{2 (e+f x)} (c+d x)}{1-e^{2 (e+f x)}} \, dx}{f}+\frac {\left (b^3 d\right ) \int (c+d x) \, dx}{f} \\ & = \frac {b^3 c d x}{f}+\frac {b^3 d^2 x^2}{2 f}-\frac {3 a b^2 (c+d x)^2}{f}+\frac {a^3 (c+d x)^3}{3 d}-\frac {a^2 b (c+d x)^3}{d}+\frac {a b^2 (c+d x)^3}{d}-\frac {b^3 (c+d x)^3}{3 d}-\frac {b^3 d (c+d x) \coth (e+f x)}{f^2}-\frac {3 a b^2 (c+d x)^2 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^2 \coth ^2(e+f x)}{2 f}+\frac {6 a b^2 d (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 d^2 \log (\sinh (e+f x))}{f^3}+\frac {3 a^2 b d (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}-\frac {\left (3 a^2 b d^2\right ) \int \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right ) \, dx}{f^2}-\frac {\left (6 a b^2 d^2\right ) \int \log \left (1-e^{2 (e+f x)}\right ) \, dx}{f^2}-\frac {\left (2 b^3 d\right ) \int (c+d x) \log \left (1-e^{2 (e+f x)}\right ) \, dx}{f} \\ & = \frac {b^3 c d x}{f}+\frac {b^3 d^2 x^2}{2 f}-\frac {3 a b^2 (c+d x)^2}{f}+\frac {a^3 (c+d x)^3}{3 d}-\frac {a^2 b (c+d x)^3}{d}+\frac {a b^2 (c+d x)^3}{d}-\frac {b^3 (c+d x)^3}{3 d}-\frac {b^3 d (c+d x) \coth (e+f x)}{f^2}-\frac {3 a b^2 (c+d x)^2 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^2 \coth ^2(e+f x)}{2 f}+\frac {6 a b^2 d (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 d^2 \log (\sinh (e+f x))}{f^3}+\frac {3 a^2 b d (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}+\frac {b^3 d (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}-\frac {\left (3 a^2 b d^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^3}-\frac {\left (3 a b^2 d^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{f^3}-\frac {\left (b^3 d^2\right ) \int \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right ) \, dx}{f^2} \\ & = \frac {b^3 c d x}{f}+\frac {b^3 d^2 x^2}{2 f}-\frac {3 a b^2 (c+d x)^2}{f}+\frac {a^3 (c+d x)^3}{3 d}-\frac {a^2 b (c+d x)^3}{d}+\frac {a b^2 (c+d x)^3}{d}-\frac {b^3 (c+d x)^3}{3 d}-\frac {b^3 d (c+d x) \coth (e+f x)}{f^2}-\frac {3 a b^2 (c+d x)^2 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^2 \coth ^2(e+f x)}{2 f}+\frac {6 a b^2 d (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 d^2 \log (\sinh (e+f x))}{f^3}+\frac {3 a b^2 d^2 \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^3}+\frac {3 a^2 b d (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}+\frac {b^3 d (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}-\frac {3 a^2 b d^2 \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3}-\frac {\left (b^3 d^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^3} \\ & = \frac {b^3 c d x}{f}+\frac {b^3 d^2 x^2}{2 f}-\frac {3 a b^2 (c+d x)^2}{f}+\frac {a^3 (c+d x)^3}{3 d}-\frac {a^2 b (c+d x)^3}{d}+\frac {a b^2 (c+d x)^3}{d}-\frac {b^3 (c+d x)^3}{3 d}-\frac {b^3 d (c+d x) \coth (e+f x)}{f^2}-\frac {3 a b^2 (c+d x)^2 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^2 \coth ^2(e+f x)}{2 f}+\frac {6 a b^2 d (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 d^2 \log (\sinh (e+f x))}{f^3}+\frac {3 a b^2 d^2 \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^3}+\frac {3 a^2 b d (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}+\frac {b^3 d (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}-\frac {3 a^2 b d^2 \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3}-\frac {b^3 d^2 \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 7.47 (sec) , antiderivative size = 585, normalized size of antiderivative = 1.46 \[ \int (c+d x)^2 (a+b \coth (e+f x))^3 \, dx=\frac {-\frac {8 b e^{2 e} f x \left (9 a b d f (2 c+d x)+3 a^2 f^2 \left (3 c^2+3 c d x+d^2 x^2\right )+b^2 \left (3 c^2 f^2+3 c d f^2 x+d^2 \left (3+f^2 x^2\right )\right )\right )}{-1+e^{2 e}}+12 b \left (6 a b d f (c+d x)+3 a^2 f^2 (c+d x)^2+b^2 \left (c^2 f^2+2 c d f^2 x+d^2 \left (1+f^2 x^2\right )\right )\right ) \log \left (1-e^{2 (e+f x)}\right )+12 b d \left (3 a b d+3 a^2 f (c+d x)+b^2 f (c+d x)\right ) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )-6 b \left (3 a^2+b^2\right ) d^2 \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )+f \text {csch}(e) \text {csch}^2(e+f x) \left (-2 b \left (9 a b f (c+d x)^2+3 a^2 f^2 x \left (3 c^2+3 c d x+d^2 x^2\right )+b^2 \left (3 c^2 f^2 x+d^2 x \left (3+f^2 x^2\right )+3 c \left (d+d f^2 x^2\right )\right )\right ) \cosh (e)+b \left (18 a b f (c+d x)^2+3 a^2 f^2 x \left (3 c^2+3 c d x+d^2 x^2\right )+b^2 \left (3 c^2 f^2 x+3 c d \left (2+f^2 x^2\right )+d^2 x \left (6+f^2 x^2\right )\right )\right ) \cosh (e+2 f x)+f \left (b \left (3 a^2+b^2\right ) f x \left (3 c^2+3 c d x+d^2 x^2\right ) \cosh (3 e+2 f x)-2 \left (3 b^3 (c+d x)^2+a^3 f x \left (3 c^2+3 c d x+d^2 x^2\right )+3 a b^2 f x \left (3 c^2+3 c d x+d^2 x^2\right )-a \left (a^2+3 b^2\right ) f x \left (3 c^2+3 c d x+d^2 x^2\right ) \cosh (2 (e+f x))\right ) \sinh (e)\right )\right )}{12 f^3} \]

[In]

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

[Out]

((-8*b*E^(2*e)*f*x*(9*a*b*d*f*(2*c + d*x) + 3*a^2*f^2*(3*c^2 + 3*c*d*x + d^2*x^2) + b^2*(3*c^2*f^2 + 3*c*d*f^2
*x + d^2*(3 + f^2*x^2))))/(-1 + E^(2*e)) + 12*b*(6*a*b*d*f*(c + d*x) + 3*a^2*f^2*(c + d*x)^2 + b^2*(c^2*f^2 +
2*c*d*f^2*x + d^2*(1 + f^2*x^2)))*Log[1 - E^(2*(e + f*x))] + 12*b*d*(3*a*b*d + 3*a^2*f*(c + d*x) + b^2*f*(c +
d*x))*PolyLog[2, E^(2*(e + f*x))] - 6*b*(3*a^2 + b^2)*d^2*PolyLog[3, E^(2*(e + f*x))] + f*Csch[e]*Csch[e + f*x
]^2*(-2*b*(9*a*b*f*(c + d*x)^2 + 3*a^2*f^2*x*(3*c^2 + 3*c*d*x + d^2*x^2) + b^2*(3*c^2*f^2*x + d^2*x*(3 + f^2*x
^2) + 3*c*(d + d*f^2*x^2)))*Cosh[e] + b*(18*a*b*f*(c + d*x)^2 + 3*a^2*f^2*x*(3*c^2 + 3*c*d*x + d^2*x^2) + b^2*
(3*c^2*f^2*x + 3*c*d*(2 + f^2*x^2) + d^2*x*(6 + f^2*x^2)))*Cosh[e + 2*f*x] + f*(b*(3*a^2 + b^2)*f*x*(3*c^2 + 3
*c*d*x + d^2*x^2)*Cosh[3*e + 2*f*x] - 2*(3*b^3*(c + d*x)^2 + a^3*f*x*(3*c^2 + 3*c*d*x + d^2*x^2) + 3*a*b^2*f*x
*(3*c^2 + 3*c*d*x + d^2*x^2) - a*(a^2 + 3*b^2)*f*x*(3*c^2 + 3*c*d*x + d^2*x^2)*Cosh[2*(e + f*x)])*Sinh[e])))/(
12*f^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1585\) vs. \(2(389)=778\).

Time = 0.63 (sec) , antiderivative size = 1586, normalized size of antiderivative = 3.96

method	result	size
risch	\(\text {Expression too large to display}\)	\(1586\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6356 vs. \(2 (386) = 772\).

Time = 0.34 (sec) , antiderivative size = 6356, normalized size of antiderivative = 15.85 \[ \int (c+d x)^2 (a+b \coth (e+f x))^3 \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 997 vs. \(2 (386) = 772\).

Time = 0.28 (sec) , antiderivative size = 997, normalized size of antiderivative = 2.49 \[ \int (c+d x)^2 (a+b \coth (e+f x))^3 \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/3*a^3*d^2*x^3 + a^3*c*d*x^2 + a^3*c^2*x + 3*a^2*b*c^2*log(sinh(f*x + e))/f + 1/3*(18*a*b^2*c^2*f + 6*b^3*c*d
+ (3*a^2*b*d^2*f^2 + 3*a*b^2*d^2*f^2 + b^3*d^2*f^2)*x^3 + 3*(3*a^2*b*c*d*f^2 + b^3*c*d*f^2 + 3*(c*d*f^2 + 2*d
^2*f)*a*b^2)*x^2 + 3*(3*(c^2*f^2 + 4*c*d*f)*a*b^2 + (c^2*f^2 + 2*d^2)*b^3)*x + ((3*a^2*b*d^2*f^2*e^(4*e) + 3*a
*b^2*d^2*f^2*e^(4*e) + b^3*d^2*f^2*e^(4*e))*x^3 + 3*(3*a^2*b*c*d*f^2*e^(4*e) + 3*a*b^2*c*d*f^2*e^(4*e) + b^3*c
*d*f^2*e^(4*e))*x^2 + 3*(3*a*b^2*c^2*f^2*e^(4*e) + b^3*c^2*f^2*e^(4*e))*x)*e^(4*f*x) - 2*(9*a*b^2*c^2*f*e^(2*e
) + 3*(c^2*f*e^(2*e) + c*d*e^(2*e))*b^3 + (3*a^2*b*d^2*f^2*e^(2*e) + 3*a*b^2*d^2*f^2*e^(2*e) + b^3*d^2*f^2*e^(
2*e))*x^3 + 3*(3*a^2*b*c*d*f^2*e^(2*e) + 3*(c*d*f^2*e^(2*e) + d^2*f*e^(2*e))*a*b^2 + (c*d*f^2*e^(2*e) + d^2*f*
e^(2*e))*b^3)*x^2 + 3*(3*(c^2*f^2*e^(2*e) + 2*c*d*f*e^(2*e))*a*b^2 + (c^2*f^2*e^(2*e) + 2*c*d*f*e^(2*e) + d^2*
e^(2*e))*b^3)*x)*e^(2*f*x))/(f^2*e^(4*f*x + 4*e) - 2*f^2*e^(2*f*x + 2*e) + f^2) - 2*(6*a*b^2*c*d*f + (c^2*f^2
+ d^2)*b^3)*x/f^2 + (3*a^2*b*d^2 + b^3*d^2)*(f^2*x^2*log(e^(f*x + e) + 1) + 2*f*x*dilog(-e^(f*x + e)) - 2*poly
log(3, -e^(f*x + e)))/f^3 + (3*a^2*b*d^2 + b^3*d^2)*(f^2*x^2*log(-e^(f*x + e) + 1) + 2*f*x*dilog(e^(f*x + e))
- 2*polylog(3, e^(f*x + e)))/f^3 + 2*(3*a^2*b*c*d*f + b^3*c*d*f + 3*a*b^2*d^2)*(f*x*log(e^(f*x + e) + 1) + dil
og(-e^(f*x + e)))/f^3 + 2*(3*a^2*b*c*d*f + b^3*c*d*f + 3*a*b^2*d^2)*(f*x*log(-e^(f*x + e) + 1) + dilog(e^(f*x
+ e)))/f^3 + (6*a*b^2*c*d*f + (c^2*f^2 + d^2)*b^3)*log(e^(f*x + e) + 1)/f^3 + (6*a*b^2*c*d*f + (c^2*f^2 + d^2)
*b^3)*log(e^(f*x + e) - 1)/f^3 - 2/3*((3*a^2*b*d^2 + b^3*d^2)*f^3*x^3 + 3*(3*a^2*b*c*d*f + b^3*c*d*f + 3*a*b^2
*d^2)*f^2*x^2)/f^3

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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