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

Optimal. Leaf size=401 \[ \frac {a^3 (c+d x)^3}{3 d}+\frac {3 a^2 b d (c+d x) \text {Li}_2\left (e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {a^2 b (c+d x)^3}{d}-\frac {3 a^2 b d^2 \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^3}+\frac {6 a b^2 d (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^2}-\frac {3 a b^2 (c+d x)^2 \coth (e+f x)}{f}-\frac {3 a b^2 (c+d x)^2}{f}+\frac {a b^2 (c+d x)^3}{d}+\frac {3 a b^2 d^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{f^3}+\frac {b^3 d (c+d x) \text {Li}_2\left (e^{2 (e+f x)}\right )}{f^2}-\frac {b^3 d (c+d x) \coth (e+f x)}{f^2}+\frac {b^3 (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {b^3 (c+d x)^2 \coth ^2(e+f x)}{2 f}+\frac {b^3 c d x}{f}-\frac {b^3 (c+d x)^3}{3 d}-\frac {b^3 d^2 \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^3}+\frac {b^3 d^2 \log (\sinh (e+f x))}{f^3}+\frac {b^3 d^2 x^2}{2 f} \]

[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

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Rubi [A]  time = 0.70, antiderivative size = 401, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 11, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {3722, 3716, 2190, 2531, 2282, 6589, 3720, 2279, 2391, 32, 3475} \[ \frac {3 a^2 b d (c+d x) \text {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}-\frac {3 a^2 b d^2 \text {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3}+\frac {3 a b^2 d^2 \text {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^3}+\frac {b^3 d (c+d x) \text {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}-\frac {b^3 d^2 \text {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3}+\frac {3 a^2 b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {a^2 b (c+d x)^3}{d}+\frac {a^3 (c+d x)^3}{3 d}+\frac {6 a b^2 d (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^2}-\frac {3 a b^2 (c+d x)^2 \coth (e+f x)}{f}-\frac {3 a b^2 (c+d x)^2}{f}+\frac {a b^2 (c+d x)^3}{d}-\frac {b^3 d (c+d x) \coth (e+f x)}{f^2}+\frac {b^3 (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {b^3 (c+d x)^2 \coth ^2(e+f x)}{2 f}+\frac {b^3 c d x}{f}-\frac {b^3 (c+d x)^3}{3 d}+\frac {b^3 d^2 \log (\sinh (e+f x))}{f^3}+\frac {b^3 d^2 x^2}{2 f} \]

Antiderivative was successfully verified.

[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 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 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 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 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]

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 3475

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

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 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 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 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]

Rubi steps

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

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Mathematica [C]  time = 13.39, size = 2029, normalized size = 5.06 \[ \text {Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/2*(a^2*b*d^2*E^e*Csch[e]*((2*f^3*x^3)/E^(2*e) - 3*(1 - E^(-2*e))*f^2*x^2*Log[1 - E^(-e - f*x)] - 3*(1 - E^(
-2*e))*f^2*x^2*Log[1 + E^(-e - f*x)] + 6*(1 - E^(-2*e))*(f*x*PolyLog[2, -E^(-e - f*x)] + PolyLog[3, -E^(-e - f
*x)]) + 6*(1 - E^(-2*e))*(f*x*PolyLog[2, E^(-e - f*x)] + PolyLog[3, E^(-e - f*x)])))/f^3 - (b^3*d^2*E^e*Csch[e
]*((2*f^3*x^3)/E^(2*e) - 3*(1 - E^(-2*e))*f^2*x^2*Log[1 - E^(-e - f*x)] - 3*(1 - E^(-2*e))*f^2*x^2*Log[1 + E^(
-e - f*x)] + 6*(1 - E^(-2*e))*(f*x*PolyLog[2, -E^(-e - f*x)] + PolyLog[3, -E^(-e - f*x)]) + 6*(1 - E^(-2*e))*(
f*x*PolyLog[2, E^(-e - f*x)] + PolyLog[3, E^(-e - f*x)])))/(6*f^3) - (b^3*d^2*Csch[e]*(-(f*x*Cosh[e]) + Log[Co
sh[f*x]*Sinh[e] + Cosh[e]*Sinh[f*x]]*Sinh[e]))/(f^3*(-Cosh[e]^2 + Sinh[e]^2)) - (6*a*b^2*c*d*Csch[e]*(-(f*x*Co
sh[e]) + Log[Cosh[f*x]*Sinh[e] + Cosh[e]*Sinh[f*x]]*Sinh[e]))/(f^2*(-Cosh[e]^2 + Sinh[e]^2)) - (3*a^2*b*c^2*Cs
ch[e]*(-(f*x*Cosh[e]) + Log[Cosh[f*x]*Sinh[e] + Cosh[e]*Sinh[f*x]]*Sinh[e]))/(f*(-Cosh[e]^2 + Sinh[e]^2)) - (b
^3*c^2*Csch[e]*(-(f*x*Cosh[e]) + Log[Cosh[f*x]*Sinh[e] + Cosh[e]*Sinh[f*x]]*Sinh[e]))/(f*(-Cosh[e]^2 + Sinh[e]
^2)) + (Csch[e]*Csch[e + f*x]^2*(-6*b^3*c*d*Cosh[e] - 18*a*b^2*c^2*f*Cosh[e] - 6*b^3*d^2*x*Cosh[e] - 36*a*b^2*
c*d*f*x*Cosh[e] - 18*a^2*b*c^2*f^2*x*Cosh[e] - 6*b^3*c^2*f^2*x*Cosh[e] - 18*a*b^2*d^2*f*x^2*Cosh[e] - 18*a^2*b
*c*d*f^2*x^2*Cosh[e] - 6*b^3*c*d*f^2*x^2*Cosh[e] - 6*a^2*b*d^2*f^2*x^3*Cosh[e] - 2*b^3*d^2*f^2*x^3*Cosh[e] + 6
*b^3*c*d*Cosh[e + 2*f*x] + 18*a*b^2*c^2*f*Cosh[e + 2*f*x] + 6*b^3*d^2*x*Cosh[e + 2*f*x] + 36*a*b^2*c*d*f*x*Cos
h[e + 2*f*x] + 9*a^2*b*c^2*f^2*x*Cosh[e + 2*f*x] + 3*b^3*c^2*f^2*x*Cosh[e + 2*f*x] + 18*a*b^2*d^2*f*x^2*Cosh[e
+ 2*f*x] + 9*a^2*b*c*d*f^2*x^2*Cosh[e + 2*f*x] + 3*b^3*c*d*f^2*x^2*Cosh[e + 2*f*x] + 3*a^2*b*d^2*f^2*x^3*Cosh
[e + 2*f*x] + b^3*d^2*f^2*x^3*Cosh[e + 2*f*x] + 9*a^2*b*c^2*f^2*x*Cosh[3*e + 2*f*x] + 3*b^3*c^2*f^2*x*Cosh[3*e
+ 2*f*x] + 9*a^2*b*c*d*f^2*x^2*Cosh[3*e + 2*f*x] + 3*b^3*c*d*f^2*x^2*Cosh[3*e + 2*f*x] + 3*a^2*b*d^2*f^2*x^3*
Cosh[3*e + 2*f*x] + b^3*d^2*f^2*x^3*Cosh[3*e + 2*f*x] - 6*b^3*c^2*f*Sinh[e] - 12*b^3*c*d*f*x*Sinh[e] - 6*a^3*c
^2*f^2*x*Sinh[e] - 18*a*b^2*c^2*f^2*x*Sinh[e] - 6*b^3*d^2*f*x^2*Sinh[e] - 6*a^3*c*d*f^2*x^2*Sinh[e] - 18*a*b^2
*c*d*f^2*x^2*Sinh[e] - 2*a^3*d^2*f^2*x^3*Sinh[e] - 6*a*b^2*d^2*f^2*x^3*Sinh[e] - 3*a^3*c^2*f^2*x*Sinh[e + 2*f*
x] - 9*a*b^2*c^2*f^2*x*Sinh[e + 2*f*x] - 3*a^3*c*d*f^2*x^2*Sinh[e + 2*f*x] - 9*a*b^2*c*d*f^2*x^2*Sinh[e + 2*f*
x] - a^3*d^2*f^2*x^3*Sinh[e + 2*f*x] - 3*a*b^2*d^2*f^2*x^3*Sinh[e + 2*f*x] + 3*a^3*c^2*f^2*x*Sinh[3*e + 2*f*x]
+ 9*a*b^2*c^2*f^2*x*Sinh[3*e + 2*f*x] + 3*a^3*c*d*f^2*x^2*Sinh[3*e + 2*f*x] + 9*a*b^2*c*d*f^2*x^2*Sinh[3*e +
2*f*x] + a^3*d^2*f^2*x^3*Sinh[3*e + 2*f*x] + 3*a*b^2*d^2*f^2*x^3*Sinh[3*e + 2*f*x]))/(12*f^2) - (3*a*b^2*d^2*C
sch[e]*Sech[e]*((f^2*x^2)/E^ArcTanh[Tanh[e]] - (I*(-(f*x*(-Pi + (2*I)*ArcTanh[Tanh[e]])) - Pi*Log[1 + E^(2*f*x
)] - 2*(I*f*x + I*ArcTanh[Tanh[e]])*Log[1 - E^((2*I)*(I*f*x + I*ArcTanh[Tanh[e]]))] + Pi*Log[Cosh[f*x]] + (2*I
)*ArcTanh[Tanh[e]]*Log[I*Sinh[f*x + ArcTanh[Tanh[e]]]] + I*PolyLog[2, E^((2*I)*(I*f*x + I*ArcTanh[Tanh[e]]))])
*Tanh[e])/Sqrt[1 - Tanh[e]^2]))/(f^3*Sqrt[Sech[e]^2*(Cosh[e]^2 - Sinh[e]^2)]) - (3*a^2*b*c*d*Csch[e]*Sech[e]*(
(f^2*x^2)/E^ArcTanh[Tanh[e]] - (I*(-(f*x*(-Pi + (2*I)*ArcTanh[Tanh[e]])) - Pi*Log[1 + E^(2*f*x)] - 2*(I*f*x +
I*ArcTanh[Tanh[e]])*Log[1 - E^((2*I)*(I*f*x + I*ArcTanh[Tanh[e]]))] + Pi*Log[Cosh[f*x]] + (2*I)*ArcTanh[Tanh[e
]]*Log[I*Sinh[f*x + ArcTanh[Tanh[e]]]] + I*PolyLog[2, E^((2*I)*(I*f*x + I*ArcTanh[Tanh[e]]))])*Tanh[e])/Sqrt[1
- Tanh[e]^2]))/(f^2*Sqrt[Sech[e]^2*(Cosh[e]^2 - Sinh[e]^2)]) - (b^3*c*d*Csch[e]*Sech[e]*((f^2*x^2)/E^ArcTanh[
Tanh[e]] - (I*(-(f*x*(-Pi + (2*I)*ArcTanh[Tanh[e]])) - Pi*Log[1 + E^(2*f*x)] - 2*(I*f*x + I*ArcTanh[Tanh[e]])*
Log[1 - E^((2*I)*(I*f*x + I*ArcTanh[Tanh[e]]))] + Pi*Log[Cosh[f*x]] + (2*I)*ArcTanh[Tanh[e]]*Log[I*Sinh[f*x +
ArcTanh[Tanh[e]]]] + I*PolyLog[2, E^((2*I)*(I*f*x + I*ArcTanh[Tanh[e]]))])*Tanh[e])/Sqrt[1 - Tanh[e]^2]))/(f^2
*Sqrt[Sech[e]^2*(Cosh[e]^2 - Sinh[e]^2)])

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fricas [C]  time = 0.62, size = 6356, normalized size = 15.85 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*f^3*x^3 + 3*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*d*f^3*x^2 + 18*a*b^2*d^
2*e^2 + 3*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c^2*f^3*x - 6*b^3*d^2*e - 2*(3*a^2*b + b^3)*d^2*e^3 + ((a^3 - 3*a^2*
b + 3*a*b^2 - b^3)*d^2*f^3*x^3 + 18*a*b^2*d^2*e^2 - 6*b^3*d^2*e - 2*(3*a^2*b + b^3)*d^2*e^3 - 6*(3*a^2*b + b^3
)*c^2*e*f^2 - 3*(6*a*b^2*d^2*f^2 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*d*f^3)*x^2 - 6*(6*a*b^2*c*d*e - (3*a^2*b
+ b^3)*c*d*e^2)*f - 3*(12*a*b^2*c*d*f^2 + 2*b^3*d^2*f - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c^2*f^3)*x)*cosh(f*x +
e)^4 + 4*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*f^3*x^3 + 18*a*b^2*d^2*e^2 - 6*b^3*d^2*e - 2*(3*a^2*b + b^3)*d^
2*e^3 - 6*(3*a^2*b + b^3)*c^2*e*f^2 - 3*(6*a*b^2*d^2*f^2 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*d*f^3)*x^2 - 6*(6
*a*b^2*c*d*e - (3*a^2*b + b^3)*c*d*e^2)*f - 3*(12*a*b^2*c*d*f^2 + 2*b^3*d^2*f - (a^3 - 3*a^2*b + 3*a*b^2 - b^3
)*c^2*f^3)*x)*cosh(f*x + e)*sinh(f*x + e)^3 + ((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*f^3*x^3 + 18*a*b^2*d^2*e^2
- 6*b^3*d^2*e - 2*(3*a^2*b + b^3)*d^2*e^3 - 6*(3*a^2*b + b^3)*c^2*e*f^2 - 3*(6*a*b^2*d^2*f^2 - (a^3 - 3*a^2*b
+ 3*a*b^2 - b^3)*c*d*f^3)*x^2 - 6*(6*a*b^2*c*d*e - (3*a^2*b + b^3)*c*d*e^2)*f - 3*(12*a*b^2*c*d*f^2 + 2*b^3*d^
2*f - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c^2*f^3)*x)*sinh(f*x + e)^4 + 6*(3*a*b^2*c^2 - (3*a^2*b + b^3)*c^2*e)*f^
2 - 2*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*f^3*x^3 + 18*a*b^2*d^2*e^2 - 6*b^3*d^2*e - 2*(3*a^2*b + b^3)*d^2*e^
3 - 3*(2*(3*a^2*b + b^3)*c^2*e - (3*a*b^2 + b^3)*c^2)*f^2 + 3*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*d*f^3 - (3*a*
b^2 - b^3)*d^2*f^2)*x^2 - 3*(12*a*b^2*c*d*e - b^3*c*d - 2*(3*a^2*b + b^3)*c*d*e^2)*f - 3*(b^3*d^2*f - (a^3 - 3
*a^2*b + 3*a*b^2 - b^3)*c^2*f^3 + 2*(3*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 + 18*a*b^2*d^2*e^2 - 6*b^3*d^2*e - 2*(3*a^2*b + b^3)*d^2*e^3 - 3*(2*(3*a^2*b + b^3)*c^2*e -
(3*a*b^2 + b^3)*c^2)*f^2 + 3*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*d*f^3 - (3*a*b^2 - b^3)*d^2*f^2)*x^2 - 3*((a^
3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*f^3*x^3 + 18*a*b^2*d^2*e^2 - 6*b^3*d^2*e - 2*(3*a^2*b + b^3)*d^2*e^3 - 6*(3*a
^2*b + b^3)*c^2*e*f^2 - 3*(6*a*b^2*d^2*f^2 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*d*f^3)*x^2 - 6*(6*a*b^2*c*d*e -
(3*a^2*b + b^3)*c*d*e^2)*f - 3*(12*a*b^2*c*d*f^2 + 2*b^3*d^2*f - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c^2*f^3)*x)*
cosh(f*x + e)^2 - 3*(12*a*b^2*c*d*e - b^3*c*d - 2*(3*a^2*b + b^3)*c*d*e^2)*f - 3*(b^3*d^2*f - (a^3 - 3*a^2*b +
3*a*b^2 - b^3)*c^2*f^3 + 2*(3*a*b^2 - b^3)*c*d*f^2)*x)*sinh(f*x + e)^2 - 6*(6*a*b^2*c*d*e - b^3*c*d - (3*a^2*
b + b^3)*c*d*e^2)*f + 6*(3*a*b^2*d^2 + (3*a^2*b + b^3)*d^2*f*x + (3*a*b^2*d^2 + (3*a^2*b + b^3)*d^2*f*x + (3*a
^2*b + b^3)*c*d*f)*cosh(f*x + e)^4 + 4*(3*a*b^2*d^2 + (3*a^2*b + b^3)*d^2*f*x + (3*a^2*b + b^3)*c*d*f)*cosh(f*
x + e)*sinh(f*x + e)^3 + (3*a*b^2*d^2 + (3*a^2*b + b^3)*d^2*f*x + (3*a^2*b + b^3)*c*d*f)*sinh(f*x + e)^4 + (3*
a^2*b + b^3)*c*d*f - 2*(3*a*b^2*d^2 + (3*a^2*b + b^3)*d^2*f*x + (3*a^2*b + b^3)*c*d*f)*cosh(f*x + e)^2 - 2*(3*
a*b^2*d^2 + (3*a^2*b + b^3)*d^2*f*x + (3*a^2*b + b^3)*c*d*f - 3*(3*a*b^2*d^2 + (3*a^2*b + b^3)*d^2*f*x + (3*a^
2*b + b^3)*c*d*f)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 4*((3*a*b^2*d^2 + (3*a^2*b + b^3)*d^2*f*x + (3*a^2*b + b^
3)*c*d*f)*cosh(f*x + e)^3 - (3*a*b^2*d^2 + (3*a^2*b + b^3)*d^2*f*x + (3*a^2*b + b^3)*c*d*f)*cosh(f*x + e))*sin
h(f*x + e))*dilog(cosh(f*x + e) + sinh(f*x + e)) + 6*(3*a*b^2*d^2 + (3*a^2*b + b^3)*d^2*f*x + (3*a*b^2*d^2 + (
3*a^2*b + b^3)*d^2*f*x + (3*a^2*b + b^3)*c*d*f)*cosh(f*x + e)^4 + 4*(3*a*b^2*d^2 + (3*a^2*b + b^3)*d^2*f*x + (
3*a^2*b + b^3)*c*d*f)*cosh(f*x + e)*sinh(f*x + e)^3 + (3*a*b^2*d^2 + (3*a^2*b + b^3)*d^2*f*x + (3*a^2*b + b^3)
*c*d*f)*sinh(f*x + e)^4 + (3*a^2*b + b^3)*c*d*f - 2*(3*a*b^2*d^2 + (3*a^2*b + b^3)*d^2*f*x + (3*a^2*b + b^3)*c
*d*f)*cosh(f*x + e)^2 - 2*(3*a*b^2*d^2 + (3*a^2*b + b^3)*d^2*f*x + (3*a^2*b + b^3)*c*d*f - 3*(3*a*b^2*d^2 + (3
*a^2*b + b^3)*d^2*f*x + (3*a^2*b + b^3)*c*d*f)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 4*((3*a*b^2*d^2 + (3*a^2*b +
b^3)*d^2*f*x + (3*a^2*b + b^3)*c*d*f)*cosh(f*x + e)^3 - (3*a*b^2*d^2 + (3*a^2*b + b^3)*d^2*f*x + (3*a^2*b + b
^3)*c*d*f)*cosh(f*x + e))*sinh(f*x + e))*dilog(-cosh(f*x + e) - sinh(f*x + e)) + 3*((3*a^2*b + b^3)*d^2*f^2*x^
2 + 6*a*b^2*c*d*f + b^3*d^2 + (3*a^2*b + b^3)*c^2*f^2 + ((3*a^2*b + b^3)*d^2*f^2*x^2 + 6*a*b^2*c*d*f + b^3*d^2
+ (3*a^2*b + b^3)*c^2*f^2 + 2*(3*a*b^2*d^2*f + (3*a^2*b + b^3)*c*d*f^2)*x)*cosh(f*x + e)^4 + 4*((3*a^2*b + b^
3)*d^2*f^2*x^2 + 6*a*b^2*c*d*f + b^3*d^2 + (3*a^2*b + b^3)*c^2*f^2 + 2*(3*a*b^2*d^2*f + (3*a^2*b + b^3)*c*d*f^
2)*x)*cosh(f*x + e)*sinh(f*x + e)^3 + ((3*a^2*b + b^3)*d^2*f^2*x^2 + 6*a*b^2*c*d*f + b^3*d^2 + (3*a^2*b + b^3)
*c^2*f^2 + 2*(3*a*b^2*d^2*f + (3*a^2*b + b^3)*c*d*f^2)*x)*sinh(f*x + e)^4 - 2*((3*a^2*b + b^3)*d^2*f^2*x^2 + 6
*a*b^2*c*d*f + b^3*d^2 + (3*a^2*b + b^3)*c^2*f^2 + 2*(3*a*b^2*d^2*f + (3*a^2*b + b^3)*c*d*f^2)*x)*cosh(f*x + e
)^2 - 2*((3*a^2*b + b^3)*d^2*f^2*x^2 + 6*a*b^2*c*d*f + b^3*d^2 + (3*a^2*b + b^3)*c^2*f^2 - 3*((3*a^2*b + b^3)*
d^2*f^2*x^2 + 6*a*b^2*c*d*f + b^3*d^2 + (3*a^2*b + b^3)*c^2*f^2 + 2*(3*a*b^2*d^2*f + (3*a^2*b + b^3)*c*d*f^2)*
x)*cosh(f*x + e)^2 + 2*(3*a*b^2*d^2*f + (3*a^2*b + b^3)*c*d*f^2)*x)*sinh(f*x + e)^2 + 2*(3*a*b^2*d^2*f + (3*a^
2*b + b^3)*c*d*f^2)*x + 4*(((3*a^2*b + b^3)*d^2*f^2*x^2 + 6*a*b^2*c*d*f + b^3*d^2 + (3*a^2*b + b^3)*c^2*f^2 +
2*(3*a*b^2*d^2*f + (3*a^2*b + b^3)*c*d*f^2)*x)*cosh(f*x + e)^3 - ((3*a^2*b + b^3)*d^2*f^2*x^2 + 6*a*b^2*c*d*f
+ b^3*d^2 + (3*a^2*b + b^3)*c^2*f^2 + 2*(3*a*b^2*d^2*f + (3*a^2*b + b^3)*c*d*f^2)*x)*cosh(f*x + e))*sinh(f*x +
e))*log(cosh(f*x + e) + sinh(f*x + e) + 1) - 3*(6*a*b^2*d^2*e - b^3*d^2 - (3*a^2*b + b^3)*d^2*e^2 - (3*a^2*b
+ b^3)*c^2*f^2 + (6*a*b^2*d^2*e - b^3*d^2 - (3*a^2*b + b^3)*d^2*e^2 - (3*a^2*b + b^3)*c^2*f^2 - 2*(3*a*b^2*c*d
- (3*a^2*b + b^3)*c*d*e)*f)*cosh(f*x + e)^4 + 4*(6*a*b^2*d^2*e - b^3*d^2 - (3*a^2*b + b^3)*d^2*e^2 - (3*a^2*b
+ b^3)*c^2*f^2 - 2*(3*a*b^2*c*d - (3*a^2*b + b^3)*c*d*e)*f)*cosh(f*x + e)*sinh(f*x + e)^3 + (6*a*b^2*d^2*e -
b^3*d^2 - (3*a^2*b + b^3)*d^2*e^2 - (3*a^2*b + b^3)*c^2*f^2 - 2*(3*a*b^2*c*d - (3*a^2*b + b^3)*c*d*e)*f)*sinh(
f*x + e)^4 - 2*(6*a*b^2*d^2*e - b^3*d^2 - (3*a^2*b + b^3)*d^2*e^2 - (3*a^2*b + b^3)*c^2*f^2 - 2*(3*a*b^2*c*d -
(3*a^2*b + b^3)*c*d*e)*f)*cosh(f*x + e)^2 - 2*(6*a*b^2*d^2*e - b^3*d^2 - (3*a^2*b + b^3)*d^2*e^2 - (3*a^2*b +
b^3)*c^2*f^2 - 3*(6*a*b^2*d^2*e - b^3*d^2 - (3*a^2*b + b^3)*d^2*e^2 - (3*a^2*b + b^3)*c^2*f^2 - 2*(3*a*b^2*c*
d - (3*a^2*b + b^3)*c*d*e)*f)*cosh(f*x + e)^2 - 2*(3*a*b^2*c*d - (3*a^2*b + b^3)*c*d*e)*f)*sinh(f*x + e)^2 - 2
*(3*a*b^2*c*d - (3*a^2*b + b^3)*c*d*e)*f + 4*((6*a*b^2*d^2*e - b^3*d^2 - (3*a^2*b + b^3)*d^2*e^2 - (3*a^2*b +
b^3)*c^2*f^2 - 2*(3*a*b^2*c*d - (3*a^2*b + b^3)*c*d*e)*f)*cosh(f*x + e)^3 - (6*a*b^2*d^2*e - b^3*d^2 - (3*a^2*
b + b^3)*d^2*e^2 - (3*a^2*b + b^3)*c^2*f^2 - 2*(3*a*b^2*c*d - (3*a^2*b + b^3)*c*d*e)*f)*cosh(f*x + e))*sinh(f*
x + e))*log(cosh(f*x + e) + sinh(f*x + e) - 1) + 3*((3*a^2*b + b^3)*d^2*f^2*x^2 + 6*a*b^2*d^2*e - (3*a^2*b + b
^3)*d^2*e^2 + 2*(3*a^2*b + b^3)*c*d*e*f + ((3*a^2*b + b^3)*d^2*f^2*x^2 + 6*a*b^2*d^2*e - (3*a^2*b + b^3)*d^2*e
^2 + 2*(3*a^2*b + b^3)*c*d*e*f + 2*(3*a*b^2*d^2*f + (3*a^2*b + b^3)*c*d*f^2)*x)*cosh(f*x + e)^4 + 4*((3*a^2*b
+ b^3)*d^2*f^2*x^2 + 6*a*b^2*d^2*e - (3*a^2*b + b^3)*d^2*e^2 + 2*(3*a^2*b + b^3)*c*d*e*f + 2*(3*a*b^2*d^2*f +
(3*a^2*b + b^3)*c*d*f^2)*x)*cosh(f*x + e)*sinh(f*x + e)^3 + ((3*a^2*b + b^3)*d^2*f^2*x^2 + 6*a*b^2*d^2*e - (3*
a^2*b + b^3)*d^2*e^2 + 2*(3*a^2*b + b^3)*c*d*e*f + 2*(3*a*b^2*d^2*f + (3*a^2*b + b^3)*c*d*f^2)*x)*sinh(f*x + e
)^4 - 2*((3*a^2*b + b^3)*d^2*f^2*x^2 + 6*a*b^2*d^2*e - (3*a^2*b + b^3)*d^2*e^2 + 2*(3*a^2*b + b^3)*c*d*e*f + 2
*(3*a*b^2*d^2*f + (3*a^2*b + b^3)*c*d*f^2)*x)*cosh(f*x + e)^2 - 2*((3*a^2*b + b^3)*d^2*f^2*x^2 + 6*a*b^2*d^2*e
- (3*a^2*b + b^3)*d^2*e^2 + 2*(3*a^2*b + b^3)*c*d*e*f - 3*((3*a^2*b + b^3)*d^2*f^2*x^2 + 6*a*b^2*d^2*e - (3*a
^2*b + b^3)*d^2*e^2 + 2*(3*a^2*b + b^3)*c*d*e*f + 2*(3*a*b^2*d^2*f + (3*a^2*b + b^3)*c*d*f^2)*x)*cosh(f*x + e)
^2 + 2*(3*a*b^2*d^2*f + (3*a^2*b + b^3)*c*d*f^2)*x)*sinh(f*x + e)^2 + 2*(3*a*b^2*d^2*f + (3*a^2*b + b^3)*c*d*f
^2)*x + 4*(((3*a^2*b + b^3)*d^2*f^2*x^2 + 6*a*b^2*d^2*e - (3*a^2*b + b^3)*d^2*e^2 + 2*(3*a^2*b + b^3)*c*d*e*f
+ 2*(3*a*b^2*d^2*f + (3*a^2*b + b^3)*c*d*f^2)*x)*cosh(f*x + e)^3 - ((3*a^2*b + b^3)*d^2*f^2*x^2 + 6*a*b^2*d^2*
e - (3*a^2*b + b^3)*d^2*e^2 + 2*(3*a^2*b + b^3)*c*d*e*f + 2*(3*a*b^2*d^2*f + (3*a^2*b + b^3)*c*d*f^2)*x)*cosh(
f*x + e))*sinh(f*x + e))*log(-cosh(f*x + e) - sinh(f*x + e) + 1) - 6*((3*a^2*b + b^3)*d^2*cosh(f*x + e)^4 + 4*
(3*a^2*b + b^3)*d^2*cosh(f*x + e)*sinh(f*x + e)^3 + (3*a^2*b + b^3)*d^2*sinh(f*x + e)^4 - 2*(3*a^2*b + b^3)*d^
2*cosh(f*x + e)^2 + (3*a^2*b + b^3)*d^2 + 2*(3*(3*a^2*b + b^3)*d^2*cosh(f*x + e)^2 - (3*a^2*b + b^3)*d^2)*sinh
(f*x + e)^2 + 4*((3*a^2*b + b^3)*d^2*cosh(f*x + e)^3 - (3*a^2*b + b^3)*d^2*cosh(f*x + e))*sinh(f*x + e))*polyl
og(3, cosh(f*x + e) + sinh(f*x + e)) - 6*((3*a^2*b + b^3)*d^2*cosh(f*x + e)^4 + 4*(3*a^2*b + b^3)*d^2*cosh(f*x
+ e)*sinh(f*x + e)^3 + (3*a^2*b + b^3)*d^2*sinh(f*x + e)^4 - 2*(3*a^2*b + b^3)*d^2*cosh(f*x + e)^2 + (3*a^2*b
+ b^3)*d^2 + 2*(3*(3*a^2*b + b^3)*d^2*cosh(f*x + e)^2 - (3*a^2*b + b^3)*d^2)*sinh(f*x + e)^2 + 4*((3*a^2*b +
b^3)*d^2*cosh(f*x + e)^3 - (3*a^2*b + b^3)*d^2*cosh(f*x + e))*sinh(f*x + e))*polylog(3, -cosh(f*x + e) - sinh(
f*x + e)) + 4*(((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*f^3*x^3 + 18*a*b^2*d^2*e^2 - 6*b^3*d^2*e - 2*(3*a^2*b + b^
3)*d^2*e^3 - 6*(3*a^2*b + b^3)*c^2*e*f^2 - 3*(6*a*b^2*d^2*f^2 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*d*f^3)*x^2 -
6*(6*a*b^2*c*d*e - (3*a^2*b + b^3)*c*d*e^2)*f - 3*(12*a*b^2*c*d*f^2 + 2*b^3*d^2*f - (a^3 - 3*a^2*b + 3*a*b^2
- b^3)*c^2*f^3)*x)*cosh(f*x + e)^3 - ((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*f^3*x^3 + 18*a*b^2*d^2*e^2 - 6*b^3*d
^2*e - 2*(3*a^2*b + b^3)*d^2*e^3 - 3*(2*(3*a^2*b + b^3)*c^2*e - (3*a*b^2 + b^3)*c^2)*f^2 + 3*((a^3 - 3*a^2*b +
3*a*b^2 - b^3)*c*d*f^3 - (3*a*b^2 - b^3)*d^2*f^2)*x^2 - 3*(12*a*b^2*c*d*e - b^3*c*d - 2*(3*a^2*b + b^3)*c*d*e
^2)*f - 3*(b^3*d^2*f - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c^2*f^3 + 2*(3*a*b^2 - b^3)*c*d*f^2)*x)*cosh(f*x + e))*
sinh(f*x + e))/(f^3*cosh(f*x + e)^4 + 4*f^3*cosh(f*x + e)*sinh(f*x + e)^3 + f^3*sinh(f*x + e)^4 - 2*f^3*cosh(f
*x + e)^2 + f^3 + 2*(3*f^3*cosh(f*x + e)^2 - f^3)*sinh(f*x + e)^2 + 4*(f^3*cosh(f*x + e)^3 - f^3*cosh(f*x + e)
)*sinh(f*x + e))

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [B]  time = 0.75, size = 1542, normalized size = 3.85 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.51, size = 997, normalized size = 2.49 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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