\(\int (c+d x)^3 (a+b \coth (e+f x))^2 \, dx\) [42]

Optimal result

Integrand size = 20, antiderivative size = 271 \[ \int (c+d x)^3 (a+b \coth (e+f x))^2 \, dx=-\frac {b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}-\frac {a b (c+d x)^4}{2 d}+\frac {b^2 (c+d x)^4}{4 d}-\frac {b^2 (c+d x)^3 \coth (e+f x)}{f}+\frac {3 b^2 d (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {2 a b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^3}+\frac {3 a b d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}-\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^4}-\frac {3 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{f^3}+\frac {3 a b d^3 \operatorname {PolyLog}\left (4,e^{2 (e+f x)}\right )}{2 f^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {3803, 3797, 2221, 2611, 6744, 2320, 6724, 3801, 32} \[ \int (c+d x)^3 (a+b \coth (e+f x))^2 \, dx=\frac {a^2 (c+d x)^4}{4 d}-\frac {3 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{f^3}+\frac {3 a b d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}+\frac {2 a b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {a b (c+d x)^4}{2 d}+\frac {3 a b d^3 \operatorname {PolyLog}\left (4,e^{2 (e+f x)}\right )}{2 f^4}+\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^3}+\frac {3 b^2 d (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f^2}-\frac {b^2 (c+d x)^3 \coth (e+f x)}{f}-\frac {b^2 (c+d x)^3}{f}+\frac {b^2 (c+d x)^4}{4 d}-\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^4} \]

[In]

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

[Out]

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

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 2320

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

Rule 2611

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

Rule 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]

Rule 6744

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

Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 (c+d x)^3+2 a b (c+d x)^3 \coth (e+f x)+b^2 (c+d x)^3 \coth ^2(e+f x)\right ) \, dx \\ & = \frac {a^2 (c+d x)^4}{4 d}+(2 a b) \int (c+d x)^3 \coth (e+f x) \, dx+b^2 \int (c+d x)^3 \coth ^2(e+f x) \, dx \\ & = \frac {a^2 (c+d x)^4}{4 d}-\frac {a b (c+d x)^4}{2 d}-\frac {b^2 (c+d x)^3 \coth (e+f x)}{f}-(4 a b) \int \frac {e^{2 (e+f x)} (c+d x)^3}{1-e^{2 (e+f x)}} \, dx+b^2 \int (c+d x)^3 \, dx+\frac {\left (3 b^2 d\right ) \int (c+d x)^2 \coth (e+f x) \, dx}{f} \\ & = -\frac {b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}-\frac {a b (c+d x)^4}{2 d}+\frac {b^2 (c+d x)^4}{4 d}-\frac {b^2 (c+d x)^3 \coth (e+f x)}{f}+\frac {2 a b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {(6 a b d) \int (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right ) \, dx}{f}-\frac {\left (6 b^2 d\right ) \int \frac {e^{2 (e+f x)} (c+d x)^2}{1-e^{2 (e+f x)}} \, dx}{f} \\ & = -\frac {b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}-\frac {a b (c+d x)^4}{2 d}+\frac {b^2 (c+d x)^4}{4 d}-\frac {b^2 (c+d x)^3 \coth (e+f x)}{f}+\frac {3 b^2 d (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {2 a b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {3 a b d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}-\frac {\left (6 a b d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right ) \, dx}{f^2}-\frac {\left (6 b^2 d^2\right ) \int (c+d x) \log \left (1-e^{2 (e+f x)}\right ) \, dx}{f^2} \\ & = -\frac {b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}-\frac {a b (c+d x)^4}{2 d}+\frac {b^2 (c+d x)^4}{4 d}-\frac {b^2 (c+d x)^3 \coth (e+f x)}{f}+\frac {3 b^2 d (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {2 a b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^3}+\frac {3 a b d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}-\frac {3 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{f^3}+\frac {\left (3 a b d^3\right ) \int \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right ) \, dx}{f^3}-\frac {\left (3 b^2 d^3\right ) \int \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right ) \, dx}{f^3} \\ & = -\frac {b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}-\frac {a b (c+d x)^4}{2 d}+\frac {b^2 (c+d x)^4}{4 d}-\frac {b^2 (c+d x)^3 \coth (e+f x)}{f}+\frac {3 b^2 d (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {2 a b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^3}+\frac {3 a b d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}-\frac {3 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{f^3}+\frac {\left (3 a b d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^4}-\frac {\left (3 b^2 d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^4} \\ & = -\frac {b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}-\frac {a b (c+d x)^4}{2 d}+\frac {b^2 (c+d x)^4}{4 d}-\frac {b^2 (c+d x)^3 \coth (e+f x)}{f}+\frac {3 b^2 d (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {2 a b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {3 b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^3}+\frac {3 a b d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}-\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^4}-\frac {3 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{f^3}+\frac {3 a b d^3 \operatorname {PolyLog}\left (4,e^{2 (e+f x)}\right )}{2 f^4} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(843\) vs. \(2(271)=542\).

Time = 4.91 (sec) , antiderivative size = 843, normalized size of antiderivative = 3.11 \[ \int (c+d x)^3 (a+b \coth (e+f x))^2 \, dx=-\frac {2 b c^2 (3 b d+2 a c f) x}{f}-\frac {2 b^2 (c+d x)^3}{\left (-1+e^{2 e}\right ) f}-\frac {a b (c+d x)^4}{d \left (-1+e^{2 e}\right )}+\frac {6 b c d (b d+a c f) x \log \left (1-e^{-e-f x}\right )}{f^2}+\frac {3 b d^2 (b d+2 a c f) x^2 \log \left (1-e^{-e-f x}\right )}{f^2}+\frac {2 a b d^3 x^3 \log \left (1-e^{-e-f x}\right )}{f}+\frac {6 b c d (b d+a c f) x \log \left (1+e^{-e-f x}\right )}{f^2}+\frac {3 b d^2 (b d+2 a c f) x^2 \log \left (1+e^{-e-f x}\right )}{f^2}+\frac {2 a b d^3 x^3 \log \left (1+e^{-e-f x}\right )}{f}+\frac {b c^2 (3 b d+2 a c f) \log \left (1-e^{e+f x}\right )}{f^2}+\frac {b c^2 (3 b d+2 a c f) \log \left (1+e^{e+f x}\right )}{f^2}-\frac {6 b c d (b d+a c f) \operatorname {PolyLog}\left (2,-e^{-e-f x}\right )}{f^3}-\frac {6 b d^2 (b d+2 a c f) x \operatorname {PolyLog}\left (2,-e^{-e-f x}\right )}{f^3}-\frac {6 a b d^3 x^2 \operatorname {PolyLog}\left (2,-e^{-e-f x}\right )}{f^2}-\frac {6 b c d (b d+a c f) \operatorname {PolyLog}\left (2,e^{-e-f x}\right )}{f^3}-\frac {6 b d^2 (b d+2 a c f) x \operatorname {PolyLog}\left (2,e^{-e-f x}\right )}{f^3}-\frac {6 a b d^3 x^2 \operatorname {PolyLog}\left (2,e^{-e-f x}\right )}{f^2}-\frac {6 b d^2 (b d+2 a c f) \operatorname {PolyLog}\left (3,-e^{-e-f x}\right )}{f^4}-\frac {12 a b d^3 x \operatorname {PolyLog}\left (3,-e^{-e-f x}\right )}{f^3}-\frac {6 b d^2 (b d+2 a c f) \operatorname {PolyLog}\left (3,e^{-e-f x}\right )}{f^4}-\frac {12 a b d^3 x \operatorname {PolyLog}\left (3,e^{-e-f x}\right )}{f^3}-\frac {12 a b d^3 \operatorname {PolyLog}\left (4,-e^{-e-f x}\right )}{f^4}-\frac {12 a b d^3 \operatorname {PolyLog}\left (4,e^{-e-f x}\right )}{f^4}+\frac {\text {csch}(e) \text {csch}(e+f x) \left (-\left (\left (a^2+b^2\right ) f x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \cosh (f x)\right )+\left (a^2+b^2\right ) f x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \cosh (2 e+f x)+2 b \left (\left (4 b (c+d x)^3+a f x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )\right ) \sinh (f x)+a f x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \sinh (2 e+f x)\right )\right )}{8 f} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1424\) vs. \(2(261)=522\).

Time = 0.52 (sec) , antiderivative size = 1425, normalized size of antiderivative = 5.26

method	result	size
risch	\(\text {Expression too large to display}\)	\(1425\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3239 vs. \(2 (259) = 518\).

Time = 0.32 (sec) , antiderivative size = 3239, normalized size of antiderivative = 11.95 \[ \int (c+d x)^3 (a+b \coth (e+f x))^2 \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/4*((a^2 - 2*a*b + b^2)*d^3*f^4*x^4 + 4*(a^2 - 2*a*b + b^2)*c*d^2*f^4*x^3 + 6*(a^2 - 2*a*b + b^2)*c^2*d*f^4*
x^2 + 4*a*b*d^3*e^4 + 4*(a^2 - 2*a*b + b^2)*c^3*f^4*x - 8*b^2*d^3*e^3 - 8*(2*a*b*c^3*e - b^2*c^3)*f^3 + 24*(a*
b*c^2*d*e^2 - b^2*c^2*d*e)*f^2 - ((a^2 - 2*a*b + b^2)*d^3*f^4*x^4 + 4*a*b*d^3*e^4 - 16*a*b*c^3*e*f^3 - 8*b^2*d
^3*e^3 - 4*(2*b^2*d^3*f^3 - (a^2 - 2*a*b + b^2)*c*d^2*f^4)*x^3 + 24*(a*b*c^2*d*e^2 - b^2*c^2*d*e)*f^2 - 6*(4*b
^2*c*d^2*f^3 - (a^2 - 2*a*b + b^2)*c^2*d*f^4)*x^2 - 8*(2*a*b*c*d^2*e^3 - 3*b^2*c*d^2*e^2)*f - 4*(6*b^2*c^2*d*f
^3 - (a^2 - 2*a*b + b^2)*c^3*f^4)*x)*cosh(f*x + e)^2 - 2*((a^2 - 2*a*b + b^2)*d^3*f^4*x^4 + 4*a*b*d^3*e^4 - 16
*a*b*c^3*e*f^3 - 8*b^2*d^3*e^3 - 4*(2*b^2*d^3*f^3 - (a^2 - 2*a*b + b^2)*c*d^2*f^4)*x^3 + 24*(a*b*c^2*d*e^2 - b
^2*c^2*d*e)*f^2 - 6*(4*b^2*c*d^2*f^3 - (a^2 - 2*a*b + b^2)*c^2*d*f^4)*x^2 - 8*(2*a*b*c*d^2*e^3 - 3*b^2*c*d^2*e
^2)*f - 4*(6*b^2*c^2*d*f^3 - (a^2 - 2*a*b + b^2)*c^3*f^4)*x)*cosh(f*x + e)*sinh(f*x + e) - ((a^2 - 2*a*b + b^2
)*d^3*f^4*x^4 + 4*a*b*d^3*e^4 - 16*a*b*c^3*e*f^3 - 8*b^2*d^3*e^3 - 4*(2*b^2*d^3*f^3 - (a^2 - 2*a*b + b^2)*c*d^
2*f^4)*x^3 + 24*(a*b*c^2*d*e^2 - b^2*c^2*d*e)*f^2 - 6*(4*b^2*c*d^2*f^3 - (a^2 - 2*a*b + b^2)*c^2*d*f^4)*x^2 -
8*(2*a*b*c*d^2*e^3 - 3*b^2*c*d^2*e^2)*f - 4*(6*b^2*c^2*d*f^3 - (a^2 - 2*a*b + b^2)*c^3*f^4)*x)*sinh(f*x + e)^2
- 8*(2*a*b*c*d^2*e^3 - 3*b^2*c*d^2*e^2)*f + 24*(a*b*d^3*f^2*x^2 + a*b*c^2*d*f^2 + b^2*c*d^2*f - (a*b*d^3*f^2*
x^2 + a*b*c^2*d*f^2 + b^2*c*d^2*f + (2*a*b*c*d^2*f^2 + b^2*d^3*f)*x)*cosh(f*x + e)^2 - 2*(a*b*d^3*f^2*x^2 + a*
b*c^2*d*f^2 + b^2*c*d^2*f + (2*a*b*c*d^2*f^2 + b^2*d^3*f)*x)*cosh(f*x + e)*sinh(f*x + e) - (a*b*d^3*f^2*x^2 +
a*b*c^2*d*f^2 + b^2*c*d^2*f + (2*a*b*c*d^2*f^2 + b^2*d^3*f)*x)*sinh(f*x + e)^2 + (2*a*b*c*d^2*f^2 + b^2*d^3*f)
*x)*dilog(cosh(f*x + e) + sinh(f*x + e)) + 24*(a*b*d^3*f^2*x^2 + a*b*c^2*d*f^2 + b^2*c*d^2*f - (a*b*d^3*f^2*x^
2 + a*b*c^2*d*f^2 + b^2*c*d^2*f + (2*a*b*c*d^2*f^2 + b^2*d^3*f)*x)*cosh(f*x + e)^2 - 2*(a*b*d^3*f^2*x^2 + a*b*
c^2*d*f^2 + b^2*c*d^2*f + (2*a*b*c*d^2*f^2 + b^2*d^3*f)*x)*cosh(f*x + e)*sinh(f*x + e) - (a*b*d^3*f^2*x^2 + a*
b*c^2*d*f^2 + b^2*c*d^2*f + (2*a*b*c*d^2*f^2 + b^2*d^3*f)*x)*sinh(f*x + e)^2 + (2*a*b*c*d^2*f^2 + b^2*d^3*f)*x
)*dilog(-cosh(f*x + e) - sinh(f*x + e)) + 4*(2*a*b*d^3*f^3*x^3 + 2*a*b*c^3*f^3 + 3*b^2*c^2*d*f^2 + 3*(2*a*b*c*
d^2*f^3 + b^2*d^3*f^2)*x^2 - (2*a*b*d^3*f^3*x^3 + 2*a*b*c^3*f^3 + 3*b^2*c^2*d*f^2 + 3*(2*a*b*c*d^2*f^3 + b^2*d
^3*f^2)*x^2 + 6*(a*b*c^2*d*f^3 + b^2*c*d^2*f^2)*x)*cosh(f*x + e)^2 - 2*(2*a*b*d^3*f^3*x^3 + 2*a*b*c^3*f^3 + 3*
b^2*c^2*d*f^2 + 3*(2*a*b*c*d^2*f^3 + b^2*d^3*f^2)*x^2 + 6*(a*b*c^2*d*f^3 + b^2*c*d^2*f^2)*x)*cosh(f*x + e)*sin
h(f*x + e) - (2*a*b*d^3*f^3*x^3 + 2*a*b*c^3*f^3 + 3*b^2*c^2*d*f^2 + 3*(2*a*b*c*d^2*f^3 + b^2*d^3*f^2)*x^2 + 6*
(a*b*c^2*d*f^3 + b^2*c*d^2*f^2)*x)*sinh(f*x + e)^2 + 6*(a*b*c^2*d*f^3 + b^2*c*d^2*f^2)*x)*log(cosh(f*x + e) +
sinh(f*x + e) + 1) - 4*(2*a*b*d^3*e^3 - 2*a*b*c^3*f^3 - 3*b^2*d^3*e^2 + 3*(2*a*b*c^2*d*e - b^2*c^2*d)*f^2 - (2
*a*b*d^3*e^3 - 2*a*b*c^3*f^3 - 3*b^2*d^3*e^2 + 3*(2*a*b*c^2*d*e - b^2*c^2*d)*f^2 - 6*(a*b*c*d^2*e^2 - b^2*c*d^
2*e)*f)*cosh(f*x + e)^2 - 2*(2*a*b*d^3*e^3 - 2*a*b*c^3*f^3 - 3*b^2*d^3*e^2 + 3*(2*a*b*c^2*d*e - b^2*c^2*d)*f^2
- 6*(a*b*c*d^2*e^2 - b^2*c*d^2*e)*f)*cosh(f*x + e)*sinh(f*x + e) - (2*a*b*d^3*e^3 - 2*a*b*c^3*f^3 - 3*b^2*d^3
*e^2 + 3*(2*a*b*c^2*d*e - b^2*c^2*d)*f^2 - 6*(a*b*c*d^2*e^2 - b^2*c*d^2*e)*f)*sinh(f*x + e)^2 - 6*(a*b*c*d^2*e
^2 - b^2*c*d^2*e)*f)*log(cosh(f*x + e) + sinh(f*x + e) - 1) + 4*(2*a*b*d^3*f^3*x^3 + 2*a*b*d^3*e^3 + 6*a*b*c^2
*d*e*f^2 - 3*b^2*d^3*e^2 + 3*(2*a*b*c*d^2*f^3 + b^2*d^3*f^2)*x^2 - (2*a*b*d^3*f^3*x^3 + 2*a*b*d^3*e^3 + 6*a*b*
c^2*d*e*f^2 - 3*b^2*d^3*e^2 + 3*(2*a*b*c*d^2*f^3 + b^2*d^3*f^2)*x^2 - 6*(a*b*c*d^2*e^2 - b^2*c*d^2*e)*f + 6*(a
*b*c^2*d*f^3 + b^2*c*d^2*f^2)*x)*cosh(f*x + e)^2 - 2*(2*a*b*d^3*f^3*x^3 + 2*a*b*d^3*e^3 + 6*a*b*c^2*d*e*f^2 -
3*b^2*d^3*e^2 + 3*(2*a*b*c*d^2*f^3 + b^2*d^3*f^2)*x^2 - 6*(a*b*c*d^2*e^2 - b^2*c*d^2*e)*f + 6*(a*b*c^2*d*f^3 +
b^2*c*d^2*f^2)*x)*cosh(f*x + e)*sinh(f*x + e) - (2*a*b*d^3*f^3*x^3 + 2*a*b*d^3*e^3 + 6*a*b*c^2*d*e*f^2 - 3*b^
2*d^3*e^2 + 3*(2*a*b*c*d^2*f^3 + b^2*d^3*f^2)*x^2 - 6*(a*b*c*d^2*e^2 - b^2*c*d^2*e)*f + 6*(a*b*c^2*d*f^3 + b^2
*c*d^2*f^2)*x)*sinh(f*x + e)^2 - 6*(a*b*c*d^2*e^2 - b^2*c*d^2*e)*f + 6*(a*b*c^2*d*f^3 + b^2*c*d^2*f^2)*x)*log(
-cosh(f*x + e) - sinh(f*x + e) + 1) - 48*(a*b*d^3*cosh(f*x + e)^2 + 2*a*b*d^3*cosh(f*x + e)*sinh(f*x + e) + a*
b*d^3*sinh(f*x + e)^2 - a*b*d^3)*polylog(4, cosh(f*x + e) + sinh(f*x + e)) - 48*(a*b*d^3*cosh(f*x + e)^2 + 2*a
*b*d^3*cosh(f*x + e)*sinh(f*x + e) + a*b*d^3*sinh(f*x + e)^2 - a*b*d^3)*polylog(4, -cosh(f*x + e) - sinh(f*x +
e)) - 24*(2*a*b*d^3*f*x + 2*a*b*c*d^2*f + b^2*d^3 - (2*a*b*d^3*f*x + 2*a*b*c*d^2*f + b^2*d^3)*cosh(f*x + e)^2
- 2*(2*a*b*d^3*f*x + 2*a*b*c*d^2*f + b^2*d^3)*cosh(f*x + e)*sinh(f*x + e) - (2*a*b*d^3*f*x + 2*a*b*c*d^2*f +
b^2*d^3)*sinh(f*x + e)^2)*polylog(3, cosh(f*x + e) + sinh(f*x + e)) - 24*(2*a*b*d^3*f*x + 2*a*b*c*d^2*f + b^2*
d^3 - (2*a*b*d^3*f*x + 2*a*b*c*d^2*f + b^2*d^3)*cosh(f*x + e)^2 - 2*(2*a*b*d^3*f*x + 2*a*b*c*d^2*f + b^2*d^3)*
cosh(f*x + e)*sinh(f*x + e) - (2*a*b*d^3*f*x + 2*a*b*c*d^2*f + b^2*d^3)*sinh(f*x + e)^2)*polylog(3, -cosh(f*x
+ e) - sinh(f*x + e)))/(f^4*cosh(f*x + e)^2 + 2*f^4*cosh(f*x + e)*sinh(f*x + e) + f^4*sinh(f*x + e)^2 - f^4)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 781 vs. \(2 (259) = 518\).

Time = 0.28 (sec) , antiderivative size = 781, normalized size of antiderivative = 2.88 \[ \int (c+d x)^3 (a+b \coth (e+f x))^2 \, dx=\frac {1}{4} \, a^{2} d^{3} x^{4} + a^{2} c d^{2} x^{3} + \frac {3}{2} \, a^{2} c^{2} d x^{2} + a^{2} c^{3} x - \frac {6 \, b^{2} c^{2} d x}{f} + \frac {2 \, a b c^{3} \log \left (\sinh \left (f x + e\right )\right )}{f} + \frac {3 \, b^{2} c^{2} d \log \left (e^{\left (f x + e\right )} + 1\right )}{f^{2}} + \frac {3 \, b^{2} c^{2} d \log \left (e^{\left (f x + e\right )} - 1\right )}{f^{2}} + \frac {2 \, {\left (f^{3} x^{3} \log \left (e^{\left (f x + e\right )} + 1\right ) + 3 \, f^{2} x^{2} {\rm Li}_2\left (-e^{\left (f x + e\right )}\right ) - 6 \, f x {\rm Li}_{3}(-e^{\left (f x + e\right )}) + 6 \, {\rm Li}_{4}(-e^{\left (f x + e\right )})\right )} a b d^{3}}{f^{4}} + \frac {2 \, {\left (f^{3} x^{3} \log \left (-e^{\left (f x + e\right )} + 1\right ) + 3 \, f^{2} x^{2} {\rm Li}_2\left (e^{\left (f x + e\right )}\right ) - 6 \, f x {\rm Li}_{3}(e^{\left (f x + e\right )}) + 6 \, {\rm Li}_{4}(e^{\left (f x + e\right )})\right )} a b d^{3}}{f^{4}} - \frac {8 \, b^{2} c^{3} + {\left (2 \, a b d^{3} f + b^{2} d^{3} f\right )} x^{4} + 4 \, {\left (c^{3} f + 6 \, c^{2} d\right )} b^{2} x + 4 \, {\left (2 \, a b c d^{2} f + {\left (c d^{2} f + 2 \, d^{3}\right )} b^{2}\right )} x^{3} + 6 \, {\left (2 \, a b c^{2} d f + {\left (c^{2} d f + 4 \, c d^{2}\right )} b^{2}\right )} x^{2} - {\left (4 \, b^{2} c^{3} f x e^{\left (2 \, e\right )} + {\left (2 \, a b d^{3} f e^{\left (2 \, e\right )} + b^{2} d^{3} f e^{\left (2 \, e\right )}\right )} x^{4} + 4 \, {\left (2 \, a b c d^{2} f e^{\left (2 \, e\right )} + b^{2} c d^{2} f e^{\left (2 \, e\right )}\right )} x^{3} + 6 \, {\left (2 \, a b c^{2} d f e^{\left (2 \, e\right )} + b^{2} c^{2} d f e^{\left (2 \, e\right )}\right )} x^{2}\right )} e^{\left (2 \, f x\right )}}{4 \, {\left (f e^{\left (2 \, f x + 2 \, e\right )} - f\right )}} + \frac {6 \, {\left (a b c^{2} d f + b^{2} c d^{2}\right )} {\left (f x \log \left (e^{\left (f x + e\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (f x + e\right )}\right )\right )}}{f^{3}} + \frac {6 \, {\left (a b c^{2} d f + b^{2} c d^{2}\right )} {\left (f x \log \left (-e^{\left (f x + e\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (f x + e\right )}\right )\right )}}{f^{3}} + \frac {3 \, {\left (2 \, a b c d^{2} f + b^{2} d^{3}\right )} {\left (f^{2} x^{2} \log \left (e^{\left (f x + e\right )} + 1\right ) + 2 \, f x {\rm Li}_2\left (-e^{\left (f x + e\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (f x + e\right )})\right )}}{f^{4}} + \frac {3 \, {\left (2 \, a b c d^{2} f + b^{2} d^{3}\right )} {\left (f^{2} x^{2} \log \left (-e^{\left (f x + e\right )} + 1\right ) + 2 \, f x {\rm Li}_2\left (e^{\left (f x + e\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (f x + e\right )})\right )}}{f^{4}} - \frac {a b d^{3} f^{4} x^{4} + 2 \, {\left (2 \, a b c d^{2} f + b^{2} d^{3}\right )} f^{3} x^{3} + 6 \, {\left (a b c^{2} d f^{2} + b^{2} c d^{2} f\right )} f^{2} x^{2}}{f^{4}} \]

[In]

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

[Out]

1/4*a^2*d^3*x^4 + a^2*c*d^2*x^3 + 3/2*a^2*c^2*d*x^2 + a^2*c^3*x - 6*b^2*c^2*d*x/f + 2*a*b*c^3*log(sinh(f*x + e
))/f + 3*b^2*c^2*d*log(e^(f*x + e) + 1)/f^2 + 3*b^2*c^2*d*log(e^(f*x + e) - 1)/f^2 + 2*(f^3*x^3*log(e^(f*x + e
) + 1) + 3*f^2*x^2*dilog(-e^(f*x + e)) - 6*f*x*polylog(3, -e^(f*x + e)) + 6*polylog(4, -e^(f*x + e)))*a*b*d^3/
f^4 + 2*(f^3*x^3*log(-e^(f*x + e) + 1) + 3*f^2*x^2*dilog(e^(f*x + e)) - 6*f*x*polylog(3, e^(f*x + e)) + 6*poly
log(4, e^(f*x + e)))*a*b*d^3/f^4 - 1/4*(8*b^2*c^3 + (2*a*b*d^3*f + b^2*d^3*f)*x^4 + 4*(c^3*f + 6*c^2*d)*b^2*x
+ 4*(2*a*b*c*d^2*f + (c*d^2*f + 2*d^3)*b^2)*x^3 + 6*(2*a*b*c^2*d*f + (c^2*d*f + 4*c*d^2)*b^2)*x^2 - (4*b^2*c^3
*f*x*e^(2*e) + (2*a*b*d^3*f*e^(2*e) + b^2*d^3*f*e^(2*e))*x^4 + 4*(2*a*b*c*d^2*f*e^(2*e) + b^2*c*d^2*f*e^(2*e))
*x^3 + 6*(2*a*b*c^2*d*f*e^(2*e) + b^2*c^2*d*f*e^(2*e))*x^2)*e^(2*f*x))/(f*e^(2*f*x + 2*e) - f) + 6*(a*b*c^2*d*
f + b^2*c*d^2)*(f*x*log(e^(f*x + e) + 1) + dilog(-e^(f*x + e)))/f^3 + 6*(a*b*c^2*d*f + b^2*c*d^2)*(f*x*log(-e^
(f*x + e) + 1) + dilog(e^(f*x + e)))/f^3 + 3*(2*a*b*c*d^2*f + b^2*d^3)*(f^2*x^2*log(e^(f*x + e) + 1) + 2*f*x*d
ilog(-e^(f*x + e)) - 2*polylog(3, -e^(f*x + e)))/f^4 + 3*(2*a*b*c*d^2*f + b^2*d^3)*(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^4 - (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)/f^4

Giac [F]

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

[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)

Mupad [F(-1)]

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

[In]

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

[Out]

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