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

Optimal. Leaf size=271 \[ -\frac{3 a b d^2 (c+d x) \text{PolyLog}\left (3,e^{2 (e+f x)}\right )}{f^3}+\frac{3 a b d (c+d x)^2 \text{PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}+\frac{3 a b d^3 \text{PolyLog}\left (4,e^{2 (e+f x)}\right )}{2 f^4}+\frac{3 b^2 d^2 (c+d x) \text{PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^3}-\frac{3 b^2 d^3 \text{PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^4}+\frac{a^2 (c+d x)^4}{4 d}+\frac{2 a b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac{a b (c+d x)^4}{2 d}+\frac{3 b^2 d (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f^2}-\frac{b^2 (c+d x)^3 \coth (e+f x)}{f}-\frac{b^2 (c+d x)^3}{f}+\frac{b^2 (c+d x)^4}{4 d} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.546393, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {3722, 3716, 2190, 2531, 6609, 2282, 6589, 3720, 32} \[ -\frac{3 a b d^2 (c+d x) \text{PolyLog}\left (3,e^{2 (e+f x)}\right )}{f^3}+\frac{3 a b d (c+d x)^2 \text{PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}+\frac{3 a b d^3 \text{PolyLog}\left (4,e^{2 (e+f x)}\right )}{2 f^4}+\frac{3 b^2 d^2 (c+d x) \text{PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^3}-\frac{3 b^2 d^3 \text{PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^4}+\frac{a^2 (c+d x)^4}{4 d}+\frac{2 a b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac{a b (c+d x)^4}{2 d}+\frac{3 b^2 d (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f^2}-\frac{b^2 (c+d x)^3 \coth (e+f x)}{f}-\frac{b^2 (c+d x)^3}{f}+\frac{b^2 (c+d x)^4}{4 d} \]

Antiderivative was successfully verified.

[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 3722

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 3716

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)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

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

Rule 2531

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

Rule 6609

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

Rule 2282

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

Rule 6589

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

Rule 3720

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 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]

Rubi steps

\begin{align*} \int (c+d x)^3 (a+b \coth (e+f x))^2 \, dx &=\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 \text{Li}_2\left (e^{2 (e+f x)}\right )}{f^2}-\frac{\left (6 a b d^2\right ) \int (c+d x) \text{Li}_2\left (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) \text{Li}_2\left (e^{2 (e+f x)}\right )}{f^3}+\frac{3 a b d (c+d x)^2 \text{Li}_2\left (e^{2 (e+f x)}\right )}{f^2}-\frac{3 a b d^2 (c+d x) \text{Li}_3\left (e^{2 (e+f x)}\right )}{f^3}+\frac{\left (3 a b d^3\right ) \int \text{Li}_3\left (e^{2 (e+f x)}\right ) \, dx}{f^3}-\frac{\left (3 b^2 d^3\right ) \int \text{Li}_2\left (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) \text{Li}_2\left (e^{2 (e+f x)}\right )}{f^3}+\frac{3 a b d (c+d x)^2 \text{Li}_2\left (e^{2 (e+f x)}\right )}{f^2}-\frac{3 a b d^2 (c+d x) \text{Li}_3\left (e^{2 (e+f x)}\right )}{f^3}+\frac{\left (3 a b d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^4}-\frac{\left (3 b^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_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) \text{Li}_2\left (e^{2 (e+f x)}\right )}{f^3}+\frac{3 a b d (c+d x)^2 \text{Li}_2\left (e^{2 (e+f x)}\right )}{f^2}-\frac{3 b^2 d^3 \text{Li}_3\left (e^{2 (e+f x)}\right )}{2 f^4}-\frac{3 a b d^2 (c+d x) \text{Li}_3\left (e^{2 (e+f x)}\right )}{f^3}+\frac{3 a b d^3 \text{Li}_4\left (e^{2 (e+f x)}\right )}{2 f^4}\\ \end{align*}

Mathematica [B]  time = 9.88911, size = 857, normalized size = 3.16 \[ -\frac{a b (\coth (e)-1) (c+d x)^4}{2 d}-\frac{b^2 (\coth (e)-1) (c+d x)^3}{f}-\frac{b c^2 (3 b d+2 a c f) (f x-\log (-\cosh (e+f x)-\sinh (e+f x)+1))}{f^2}+\frac{2 a b d^3 x^3 \log (\cosh (e+f x)-\sinh (e+f x)+1)}{f}+\frac{3 b d^2 (b d+2 a c f) x^2 \log (\cosh (e+f x)-\sinh (e+f x)+1)}{f^2}+\frac{6 b c d (b d+a c f) x \log (\cosh (e+f x)-\sinh (e+f x)+1)}{f^2}+\frac{2 a b d^3 x^3 \log (-\cosh (e+f x)+\sinh (e+f x)+1)}{f}+\frac{3 b d^2 (b d+2 a c f) x^2 \log (-\cosh (e+f x)+\sinh (e+f x)+1)}{f^2}+\frac{6 b c d (b d+a c f) x \log (-\cosh (e+f x)+\sinh (e+f x)+1)}{f^2}-\frac{b c^2 (3 b d+2 a c f) (f x-\log (\cosh (e+f x)+\sinh (e+f x)+1))}{f^2}-\frac{6 b c d (b d+a c f) \text{PolyLog}(2,\cosh (e+f x)-\sinh (e+f x))}{f^3}-\frac{6 b c d (b d+a c f) \text{PolyLog}(2,\sinh (e+f x)-\cosh (e+f x))}{f^3}-\frac{6 b d^2 (b d+2 a c f) (f x \text{PolyLog}(2,\cosh (e+f x)-\sinh (e+f x))+\text{PolyLog}(3,\cosh (e+f x)-\sinh (e+f x)))}{f^4}-\frac{6 b d^2 (b d+2 a c f) (f x \text{PolyLog}(2,\sinh (e+f x)-\cosh (e+f x))+\text{PolyLog}(3,\sinh (e+f x)-\cosh (e+f x)))}{f^4}-\frac{6 a b d^3 \left (f^2 \text{PolyLog}(2,\cosh (e+f x)-\sinh (e+f x)) x^2+2 (f x \text{PolyLog}(3,\cosh (e+f x)-\sinh (e+f x))+\text{PolyLog}(4,\cosh (e+f x)-\sinh (e+f x)))\right )}{f^4}-\frac{6 a b d^3 \left (f^2 \text{PolyLog}(2,\sinh (e+f x)-\cosh (e+f x)) x^2+2 (f x \text{PolyLog}(3,\sinh (e+f x)-\cosh (e+f x))+\text{PolyLog}(4,\sinh (e+f x)-\cosh (e+f x)))\right )}{f^4}+\frac{\text{csch}(e) \text{csch}(e+f x) \left (-\left (a^2+b^2\right ) f x \left (4 c^3+6 d x c^2+4 d^2 x^2 c+d^3 x^3\right ) \cosh (f x)+\left (a^2+b^2\right ) f x \left (4 c^3+6 d x c^2+4 d^2 x^2 c+d^3 x^3\right ) \cosh (2 e+f x)+2 b \left (\left (4 b (c+d x)^3+a f x \left (4 c^3+6 d x c^2+4 d^2 x^2 c+d^3 x^3\right )\right ) \sinh (f x)+a f x \left (4 c^3+6 d x c^2+4 d^2 x^2 c+d^3 x^3\right ) \sinh (2 e+f x)\right )\right )}{8 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b^2*(c + d*x)^3*(-1 + Coth[e]))/f) - (a*b*(c + d*x)^4*(-1 + Coth[e]))/(2*d) - (b*c^2*(3*b*d + 2*a*c*f)*(f*x
- Log[1 - Cosh[e + f*x] - Sinh[e + f*x]]))/f^2 + (6*b*c*d*(b*d + a*c*f)*x*Log[1 + Cosh[e + f*x] - Sinh[e + f*
x]])/f^2 + (3*b*d^2*(b*d + 2*a*c*f)*x^2*Log[1 + Cosh[e + f*x] - Sinh[e + f*x]])/f^2 + (2*a*b*d^3*x^3*Log[1 + C
osh[e + f*x] - Sinh[e + f*x]])/f + (6*b*c*d*(b*d + a*c*f)*x*Log[1 - Cosh[e + f*x] + Sinh[e + f*x]])/f^2 + (3*b
*d^2*(b*d + 2*a*c*f)*x^2*Log[1 - Cosh[e + f*x] + Sinh[e + f*x]])/f^2 + (2*a*b*d^3*x^3*Log[1 - Cosh[e + f*x] +
Sinh[e + f*x]])/f - (b*c^2*(3*b*d + 2*a*c*f)*(f*x - Log[1 + Cosh[e + f*x] + Sinh[e + f*x]]))/f^2 - (6*b*c*d*(b
*d + a*c*f)*PolyLog[2, Cosh[e + f*x] - Sinh[e + f*x]])/f^3 - (6*b*c*d*(b*d + a*c*f)*PolyLog[2, -Cosh[e + f*x]
+ Sinh[e + f*x]])/f^3 - (6*b*d^2*(b*d + 2*a*c*f)*(f*x*PolyLog[2, Cosh[e + f*x] - Sinh[e + f*x]] + PolyLog[3, C
osh[e + f*x] - Sinh[e + f*x]]))/f^4 - (6*b*d^2*(b*d + 2*a*c*f)*(f*x*PolyLog[2, -Cosh[e + f*x] + Sinh[e + f*x]]
+ PolyLog[3, -Cosh[e + f*x] + Sinh[e + f*x]]))/f^4 - (6*a*b*d^3*(f^2*x^2*PolyLog[2, Cosh[e + f*x] - Sinh[e +
f*x]] + 2*(f*x*PolyLog[3, Cosh[e + f*x] - Sinh[e + f*x]] + PolyLog[4, Cosh[e + f*x] - Sinh[e + f*x]])))/f^4 -
(6*a*b*d^3*(f^2*x^2*PolyLog[2, -Cosh[e + f*x] + Sinh[e + f*x]] + 2*(f*x*PolyLog[3, -Cosh[e + f*x] + Sinh[e + f
*x]] + PolyLog[4, -Cosh[e + f*x] + Sinh[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)*Co
sh[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)

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Maple [B]  time = 0.168, size = 1393, normalized size = 5.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.49089, size = 1054, normalized size = 3.89 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C]  time = 2.83648, size = 7128, normalized size = 26.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((a^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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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