3.1.47 \(\int (c+d x)^3 (a+b \coth (e+f x))^3 \, dx\) [47]

Optimal. Leaf size=556 \[ -\frac {3 b^3 d (c+d x)^2}{2 f^2}-\frac {3 a b^2 (c+d x)^3}{f}+\frac {b^3 (c+d x)^3}{2 f}+\frac {a^3 (c+d x)^4}{4 d}-\frac {3 a^2 b (c+d x)^4}{4 d}+\frac {3 a b^2 (c+d x)^4}{4 d}-\frac {b^3 (c+d x)^4}{4 d}-\frac {3 b^3 d (c+d x)^2 \coth (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x)^3 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^3 \coth ^2(e+f x)}{2 f}+\frac {3 b^3 d^2 (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^3}+\frac {9 a b^2 d (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {3 b^3 d^3 \text {PolyLog}\left (2,e^{2 (e+f x)}\right )}{2 f^4}+\frac {9 a b^2 d^2 (c+d x) \text {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^3}+\frac {9 a^2 b d (c+d x)^2 \text {PolyLog}\left (2,e^{2 (e+f x)}\right )}{2 f^2}+\frac {3 b^3 d (c+d x)^2 \text {PolyLog}\left (2,e^{2 (e+f x)}\right )}{2 f^2}-\frac {9 a b^2 d^3 \text {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^4}-\frac {9 a^2 b d^2 (c+d x) \text {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3}-\frac {3 b^3 d^2 (c+d x) \text {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3}+\frac {9 a^2 b d^3 \text {PolyLog}\left (4,e^{2 (e+f x)}\right )}{4 f^4}+\frac {3 b^3 d^3 \text {PolyLog}\left (4,e^{2 (e+f x)}\right )}{4 f^4} \]

[Out]

-3/2*b^3*d*(d*x+c)^2/f^2-3*a*b^2*(d*x+c)^3/f+1/2*b^3*(d*x+c)^3/f+1/4*a^3*(d*x+c)^4/d-3/4*a^2*b*(d*x+c)^4/d+3/4
*a*b^2*(d*x+c)^4/d-1/4*b^3*(d*x+c)^4/d-3/2*b^3*d*(d*x+c)^2*coth(f*x+e)/f^2-3*a*b^2*(d*x+c)^3*coth(f*x+e)/f-1/2
*b^3*(d*x+c)^3*coth(f*x+e)^2/f+3*b^3*d^2*(d*x+c)*ln(1-exp(2*f*x+2*e))/f^3+9*a*b^2*d*(d*x+c)^2*ln(1-exp(2*f*x+2
*e))/f^2+3*a^2*b*(d*x+c)^3*ln(1-exp(2*f*x+2*e))/f+b^3*(d*x+c)^3*ln(1-exp(2*f*x+2*e))/f+3/2*b^3*d^3*polylog(2,e
xp(2*f*x+2*e))/f^4+9*a*b^2*d^2*(d*x+c)*polylog(2,exp(2*f*x+2*e))/f^3+9/2*a^2*b*d*(d*x+c)^2*polylog(2,exp(2*f*x
+2*e))/f^2+3/2*b^3*d*(d*x+c)^2*polylog(2,exp(2*f*x+2*e))/f^2-9/2*a*b^2*d^3*polylog(3,exp(2*f*x+2*e))/f^4-9/2*a
^2*b*d^2*(d*x+c)*polylog(3,exp(2*f*x+2*e))/f^3-3/2*b^3*d^2*(d*x+c)*polylog(3,exp(2*f*x+2*e))/f^3+9/4*a^2*b*d^3
*polylog(4,exp(2*f*x+2*e))/f^4+3/4*b^3*d^3*polylog(4,exp(2*f*x+2*e))/f^4

________________________________________________________________________________________

Rubi [A]
time = 0.72, antiderivative size = 556, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 11, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {3803, 3797, 2221, 2611, 6744, 2320, 6724, 3801, 32, 2317, 2438} \begin {gather*} \frac {a^3 (c+d x)^4}{4 d}-\frac {9 a^2 b d^2 (c+d x) \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^3}+\frac {9 a^2 b d (c+d x)^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}+\frac {3 a^2 b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {3 a^2 b (c+d x)^4}{4 d}+\frac {9 a^2 b d^3 \text {Li}_4\left (e^{2 (e+f x)}\right )}{4 f^4}+\frac {9 a b^2 d^2 (c+d x) \text {Li}_2\left (e^{2 (e+f x)}\right )}{f^3}+\frac {9 a b^2 d (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f^2}-\frac {3 a b^2 (c+d x)^3 \coth (e+f x)}{f}-\frac {3 a b^2 (c+d x)^3}{f}+\frac {3 a b^2 (c+d x)^4}{4 d}-\frac {9 a b^2 d^3 \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^4}-\frac {3 b^3 d^2 (c+d x) \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^3}+\frac {3 b^3 d^2 (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^3}+\frac {3 b^3 d (c+d x)^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}-\frac {3 b^3 d (c+d x)^2 \coth (e+f x)}{2 f^2}+\frac {b^3 (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {b^3 (c+d x)^3 \coth ^2(e+f x)}{2 f}-\frac {3 b^3 d (c+d x)^2}{2 f^2}+\frac {b^3 (c+d x)^3}{2 f}-\frac {b^3 (c+d x)^4}{4 d}+\frac {3 b^3 d^3 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^4}+\frac {3 b^3 d^3 \text {Li}_4\left (e^{2 (e+f x)}\right )}{4 f^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b^3*d*(c + d*x)^2)/(2*f^2) - (3*a*b^2*(c + d*x)^3)/f + (b^3*(c + d*x)^3)/(2*f) + (a^3*(c + d*x)^4)/(4*d) -
 (3*a^2*b*(c + d*x)^4)/(4*d) + (3*a*b^2*(c + d*x)^4)/(4*d) - (b^3*(c + d*x)^4)/(4*d) - (3*b^3*d*(c + d*x)^2*Co
th[e + f*x])/(2*f^2) - (3*a*b^2*(c + d*x)^3*Coth[e + f*x])/f - (b^3*(c + d*x)^3*Coth[e + f*x]^2)/(2*f) + (3*b^
3*d^2*(c + d*x)*Log[1 - E^(2*(e + f*x))])/f^3 + (9*a*b^2*d*(c + d*x)^2*Log[1 - E^(2*(e + f*x))])/f^2 + (3*a^2*
b*(c + d*x)^3*Log[1 - E^(2*(e + f*x))])/f + (b^3*(c + d*x)^3*Log[1 - E^(2*(e + f*x))])/f + (3*b^3*d^3*PolyLog[
2, E^(2*(e + f*x))])/(2*f^4) + (9*a*b^2*d^2*(c + d*x)*PolyLog[2, E^(2*(e + f*x))])/f^3 + (9*a^2*b*d*(c + d*x)^
2*PolyLog[2, E^(2*(e + f*x))])/(2*f^2) + (3*b^3*d*(c + d*x)^2*PolyLog[2, E^(2*(e + f*x))])/(2*f^2) - (9*a*b^2*
d^3*PolyLog[3, E^(2*(e + f*x))])/(2*f^4) - (9*a^2*b*d^2*(c + d*x)*PolyLog[3, E^(2*(e + f*x))])/(2*f^3) - (3*b^
3*d^2*(c + d*x)*PolyLog[3, E^(2*(e + f*x))])/(2*f^3) + (9*a^2*b*d^3*PolyLog[4, E^(2*(e + f*x))])/(4*f^4) + (3*
b^3*d^3*PolyLog[4, E^(2*(e + f*x))])/(4*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 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 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*} \int (c+d x)^3 (a+b \coth (e+f x))^3 \, dx &=\int \left (a^3 (c+d x)^3+3 a^2 b (c+d x)^3 \coth (e+f x)+3 a b^2 (c+d x)^3 \coth ^2(e+f x)+b^3 (c+d x)^3 \coth ^3(e+f x)\right ) \, dx\\ &=\frac {a^3 (c+d x)^4}{4 d}+\left (3 a^2 b\right ) \int (c+d x)^3 \coth (e+f x) \, dx+\left (3 a b^2\right ) \int (c+d x)^3 \coth ^2(e+f x) \, dx+b^3 \int (c+d x)^3 \coth ^3(e+f x) \, dx\\ &=\frac {a^3 (c+d x)^4}{4 d}-\frac {3 a^2 b (c+d x)^4}{4 d}-\frac {3 a b^2 (c+d x)^3 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^3 \coth ^2(e+f x)}{2 f}-\left (6 a^2 b\right ) \int \frac {e^{2 (e+f x)} (c+d x)^3}{1-e^{2 (e+f x)}} \, dx+\left (3 a b^2\right ) \int (c+d x)^3 \, dx+b^3 \int (c+d x)^3 \coth (e+f x) \, dx+\frac {\left (9 a b^2 d\right ) \int (c+d x)^2 \coth (e+f x) \, dx}{f}+\frac {\left (3 b^3 d\right ) \int (c+d x)^2 \coth ^2(e+f x) \, dx}{2 f}\\ &=-\frac {3 a b^2 (c+d x)^3}{f}+\frac {a^3 (c+d x)^4}{4 d}-\frac {3 a^2 b (c+d x)^4}{4 d}+\frac {3 a b^2 (c+d x)^4}{4 d}-\frac {b^3 (c+d x)^4}{4 d}-\frac {3 b^3 d (c+d x)^2 \coth (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x)^3 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^3 \coth ^2(e+f x)}{2 f}+\frac {3 a^2 b (c+d x)^3 \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)^3}{1-e^{2 (e+f x)}} \, dx+\frac {\left (3 b^3 d^2\right ) \int (c+d x) \coth (e+f x) \, dx}{f^2}-\frac {\left (9 a^2 b d\right ) \int (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right ) \, dx}{f}-\frac {\left (18 a b^2 d\right ) \int \frac {e^{2 (e+f x)} (c+d x)^2}{1-e^{2 (e+f x)}} \, dx}{f}+\frac {\left (3 b^3 d\right ) \int (c+d x)^2 \, dx}{2 f}\\ &=-\frac {3 b^3 d (c+d x)^2}{2 f^2}-\frac {3 a b^2 (c+d x)^3}{f}+\frac {b^3 (c+d x)^3}{2 f}+\frac {a^3 (c+d x)^4}{4 d}-\frac {3 a^2 b (c+d x)^4}{4 d}+\frac {3 a b^2 (c+d x)^4}{4 d}-\frac {b^3 (c+d x)^4}{4 d}-\frac {3 b^3 d (c+d x)^2 \coth (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x)^3 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^3 \coth ^2(e+f x)}{2 f}+\frac {9 a b^2 d (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {9 a^2 b d (c+d x)^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}-\frac {\left (9 a^2 b d^2\right ) \int (c+d x) \text {Li}_2\left (e^{2 (e+f x)}\right ) \, dx}{f^2}-\frac {\left (18 a b^2 d^2\right ) \int (c+d x) \log \left (1-e^{2 (e+f x)}\right ) \, dx}{f^2}-\frac {\left (6 b^3 d^2\right ) \int \frac {e^{2 (e+f x)} (c+d x)}{1-e^{2 (e+f x)}} \, dx}{f^2}-\frac {\left (3 b^3 d\right ) \int (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right ) \, dx}{f}\\ &=-\frac {3 b^3 d (c+d x)^2}{2 f^2}-\frac {3 a b^2 (c+d x)^3}{f}+\frac {b^3 (c+d x)^3}{2 f}+\frac {a^3 (c+d x)^4}{4 d}-\frac {3 a^2 b (c+d x)^4}{4 d}+\frac {3 a b^2 (c+d x)^4}{4 d}-\frac {b^3 (c+d x)^4}{4 d}-\frac {3 b^3 d (c+d x)^2 \coth (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x)^3 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^3 \coth ^2(e+f x)}{2 f}+\frac {3 b^3 d^2 (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^3}+\frac {9 a b^2 d (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {9 a b^2 d^2 (c+d x) \text {Li}_2\left (e^{2 (e+f x)}\right )}{f^3}+\frac {9 a^2 b d (c+d x)^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}+\frac {3 b^3 d (c+d x)^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}-\frac {9 a^2 b d^2 (c+d x) \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^3}+\frac {\left (9 a^2 b d^3\right ) \int \text {Li}_3\left (e^{2 (e+f x)}\right ) \, dx}{2 f^3}-\frac {\left (9 a b^2 d^3\right ) \int \text {Li}_2\left (e^{2 (e+f x)}\right ) \, dx}{f^3}-\frac {\left (3 b^3 d^3\right ) \int \log \left (1-e^{2 (e+f x)}\right ) \, dx}{f^3}-\frac {\left (3 b^3 d^2\right ) \int (c+d x) \text {Li}_2\left (e^{2 (e+f x)}\right ) \, dx}{f^2}\\ &=-\frac {3 b^3 d (c+d x)^2}{2 f^2}-\frac {3 a b^2 (c+d x)^3}{f}+\frac {b^3 (c+d x)^3}{2 f}+\frac {a^3 (c+d x)^4}{4 d}-\frac {3 a^2 b (c+d x)^4}{4 d}+\frac {3 a b^2 (c+d x)^4}{4 d}-\frac {b^3 (c+d x)^4}{4 d}-\frac {3 b^3 d (c+d x)^2 \coth (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x)^3 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^3 \coth ^2(e+f x)}{2 f}+\frac {3 b^3 d^2 (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^3}+\frac {9 a b^2 d (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {9 a b^2 d^2 (c+d x) \text {Li}_2\left (e^{2 (e+f x)}\right )}{f^3}+\frac {9 a^2 b d (c+d x)^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}+\frac {3 b^3 d (c+d x)^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}-\frac {9 a^2 b d^2 (c+d x) \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^3}-\frac {3 b^3 d^2 (c+d x) \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^3}+\frac {\left (9 a^2 b d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{4 f^4}-\frac {\left (9 a b^2 d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^4}-\frac {\left (3 b^3 d^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^4}+\frac {\left (3 b^3 d^3\right ) \int \text {Li}_3\left (e^{2 (e+f x)}\right ) \, dx}{2 f^3}\\ &=-\frac {3 b^3 d (c+d x)^2}{2 f^2}-\frac {3 a b^2 (c+d x)^3}{f}+\frac {b^3 (c+d x)^3}{2 f}+\frac {a^3 (c+d x)^4}{4 d}-\frac {3 a^2 b (c+d x)^4}{4 d}+\frac {3 a b^2 (c+d x)^4}{4 d}-\frac {b^3 (c+d x)^4}{4 d}-\frac {3 b^3 d (c+d x)^2 \coth (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x)^3 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^3 \coth ^2(e+f x)}{2 f}+\frac {3 b^3 d^2 (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^3}+\frac {9 a b^2 d (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {3 b^3 d^3 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^4}+\frac {9 a b^2 d^2 (c+d x) \text {Li}_2\left (e^{2 (e+f x)}\right )}{f^3}+\frac {9 a^2 b d (c+d x)^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}+\frac {3 b^3 d (c+d x)^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}-\frac {9 a b^2 d^3 \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^4}-\frac {9 a^2 b d^2 (c+d x) \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^3}-\frac {3 b^3 d^2 (c+d x) \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^3}+\frac {9 a^2 b d^3 \text {Li}_4\left (e^{2 (e+f x)}\right )}{4 f^4}+\frac {\left (3 b^3 d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{4 f^4}\\ &=-\frac {3 b^3 d (c+d x)^2}{2 f^2}-\frac {3 a b^2 (c+d x)^3}{f}+\frac {b^3 (c+d x)^3}{2 f}+\frac {a^3 (c+d x)^4}{4 d}-\frac {3 a^2 b (c+d x)^4}{4 d}+\frac {3 a b^2 (c+d x)^4}{4 d}-\frac {b^3 (c+d x)^4}{4 d}-\frac {3 b^3 d (c+d x)^2 \coth (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x)^3 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^3 \coth ^2(e+f x)}{2 f}+\frac {3 b^3 d^2 (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^3}+\frac {9 a b^2 d (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^3 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {3 b^3 d^3 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^4}+\frac {9 a b^2 d^2 (c+d x) \text {Li}_2\left (e^{2 (e+f x)}\right )}{f^3}+\frac {9 a^2 b d (c+d x)^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}+\frac {3 b^3 d (c+d x)^2 \text {Li}_2\left (e^{2 (e+f x)}\right )}{2 f^2}-\frac {9 a b^2 d^3 \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^4}-\frac {9 a^2 b d^2 (c+d x) \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^3}-\frac {3 b^3 d^2 (c+d x) \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^3}+\frac {9 a^2 b d^3 \text {Li}_4\left (e^{2 (e+f x)}\right )}{4 f^4}+\frac {3 b^3 d^3 \text {Li}_4\left (e^{2 (e+f x)}\right )}{4 f^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(2043\) vs. \(2(556)=1112\).
time = 11.12, size = 2043, normalized size = 3.67 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-(b^3*c^3) - 3*b^3*c^2*d*x - 3*b^3*c*d^2*x^2 - b^3*d^3*x^3)*Csch[e + f*x]^2)/(2*f) - (b*E^(2*e)*(24*b^2*c*d^
2*x + 72*a*b*c^2*d*f*x + 24*a^2*c^3*f^2*x + 8*b^2*c^3*f^2*x + 12*b^2*d^3*x^2 + 72*a*b*c*d^2*f*x^2 + 36*a^2*c^2
*d*f^2*x^2 + 12*b^2*c^2*d*f^2*x^2 + 24*a*b*d^3*f*x^3 + 24*a^2*c*d^2*f^2*x^3 + 8*b^2*c*d^2*f^2*x^3 + 6*a^2*d^3*
f^2*x^4 + 2*b^2*d^3*f^2*x^4 - 36*a*b*c^2*d*Log[1 - E^(2*(e + f*x))] + (36*a*b*c^2*d*Log[1 - E^(2*(e + f*x))])/
E^(2*e) - (12*b^2*c*d^2*Log[1 - E^(2*(e + f*x))])/f + (12*b^2*c*d^2*Log[1 - E^(2*(e + f*x))])/(E^(2*e)*f) - 12
*a^2*c^3*f*Log[1 - E^(2*(e + f*x))] - 4*b^2*c^3*f*Log[1 - E^(2*(e + f*x))] + (12*a^2*c^3*f*Log[1 - E^(2*(e + f
*x))])/E^(2*e) + (4*b^2*c^3*f*Log[1 - E^(2*(e + f*x))])/E^(2*e) - 72*a*b*c*d^2*x*Log[1 - E^(2*(e + f*x))] + (7
2*a*b*c*d^2*x*Log[1 - E^(2*(e + f*x))])/E^(2*e) - (12*b^2*d^3*x*Log[1 - E^(2*(e + f*x))])/f + (12*b^2*d^3*x*Lo
g[1 - E^(2*(e + f*x))])/(E^(2*e)*f) - 36*a^2*c^2*d*f*x*Log[1 - E^(2*(e + f*x))] - 12*b^2*c^2*d*f*x*Log[1 - E^(
2*(e + f*x))] + (36*a^2*c^2*d*f*x*Log[1 - E^(2*(e + f*x))])/E^(2*e) + (12*b^2*c^2*d*f*x*Log[1 - E^(2*(e + f*x)
)])/E^(2*e) - 36*a*b*d^3*x^2*Log[1 - E^(2*(e + f*x))] + (36*a*b*d^3*x^2*Log[1 - E^(2*(e + f*x))])/E^(2*e) - 36
*a^2*c*d^2*f*x^2*Log[1 - E^(2*(e + f*x))] - 12*b^2*c*d^2*f*x^2*Log[1 - E^(2*(e + f*x))] + (36*a^2*c*d^2*f*x^2*
Log[1 - E^(2*(e + f*x))])/E^(2*e) + (12*b^2*c*d^2*f*x^2*Log[1 - E^(2*(e + f*x))])/E^(2*e) - 12*a^2*d^3*f*x^3*L
og[1 - E^(2*(e + f*x))] - 4*b^2*d^3*f*x^3*Log[1 - E^(2*(e + f*x))] + (12*a^2*d^3*f*x^3*Log[1 - E^(2*(e + f*x))
])/E^(2*e) + (4*b^2*d^3*f*x^3*Log[1 - E^(2*(e + f*x))])/E^(2*e) - (6*d*(-1 + E^(2*e))*(6*a*b*d*f*(c + d*x) + 3
*a^2*f^2*(c + d*x)^2 + b^2*(d^2 + c^2*f^2 + 2*c*d*f^2*x + d^2*f^2*x^2))*PolyLog[2, E^(2*(e + f*x))])/(E^(2*e)*
f^2) + (6*d^2*(-1 + E^(2*e))*(3*a*b*d + 3*a^2*f*(c + d*x) + b^2*f*(c + d*x))*PolyLog[3, E^(2*(e + f*x))])/(E^(
2*e)*f^2) - (9*a^2*d^3*PolyLog[4, E^(2*(e + f*x))])/f^2 - (3*b^2*d^3*PolyLog[4, E^(2*(e + f*x))])/f^2 + (9*a^2
*d^3*PolyLog[4, E^(2*(e + f*x))])/(E^(2*e)*f^2) + (3*b^2*d^3*PolyLog[4, E^(2*(e + f*x))])/(E^(2*e)*f^2)))/(4*(
-1 + E^(2*e))*f^2) + (3*x^2*(-(a^3*c^2*d) + 3*a^2*b*c^2*d - 3*a*b^2*c^2*d + b^3*c^2*d + a^3*c^2*d*Cosh[2*e] +
3*a^2*b*c^2*d*Cosh[2*e] + 3*a*b^2*c^2*d*Cosh[2*e] + b^3*c^2*d*Cosh[2*e] + a^3*c^2*d*Sinh[2*e] + 3*a^2*b*c^2*d*
Sinh[2*e] + 3*a*b^2*c^2*d*Sinh[2*e] + b^3*c^2*d*Sinh[2*e]))/(2*(-1 + Cosh[2*e] + Sinh[2*e])) + (x^3*(-(a^3*c*d
^2) + 3*a^2*b*c*d^2 - 3*a*b^2*c*d^2 + b^3*c*d^2 + a^3*c*d^2*Cosh[2*e] + 3*a^2*b*c*d^2*Cosh[2*e] + 3*a*b^2*c*d^
2*Cosh[2*e] + b^3*c*d^2*Cosh[2*e] + a^3*c*d^2*Sinh[2*e] + 3*a^2*b*c*d^2*Sinh[2*e] + 3*a*b^2*c*d^2*Sinh[2*e] +
b^3*c*d^2*Sinh[2*e]))/(-1 + Cosh[2*e] + Sinh[2*e]) + (x^4*(-(a^3*d^3) + 3*a^2*b*d^3 - 3*a*b^2*d^3 + b^3*d^3 +
a^3*d^3*Cosh[2*e] + 3*a^2*b*d^3*Cosh[2*e] + 3*a*b^2*d^3*Cosh[2*e] + b^3*d^3*Cosh[2*e] + a^3*d^3*Sinh[2*e] + 3*
a^2*b*d^3*Sinh[2*e] + 3*a*b^2*d^3*Sinh[2*e] + b^3*d^3*Sinh[2*e]))/(4*(-1 + Cosh[2*e] + Sinh[2*e])) + x*(a^3*c^
3 + 3*a*b^2*c^3 + (3*a^2*b*c^3)/(-1 + Cosh[2*e] + Sinh[2*e]) + (3*a^2*b*c^3*Cosh[2*e] + 3*a^2*b*c^3*Sinh[2*e])
/(-1 + Cosh[2*e] + Sinh[2*e]) + (2*b^3*c^3*Cosh[2*e] + 2*b^3*c^3*Sinh[2*e])/((-1 + Cosh[2*e] + Sinh[2*e])*(1 +
 Cosh[2*e] + Cosh[4*e] + Sinh[2*e] + Sinh[4*e])) + (2*b^3*c^3*Cosh[4*e] + 2*b^3*c^3*Sinh[4*e])/((-1 + Cosh[2*e
] + Sinh[2*e])*(1 + Cosh[2*e] + Cosh[4*e] + Sinh[2*e] + Sinh[4*e])) + (b^3*c^3)/(-1 + Cosh[6*e] + Sinh[6*e]) +
 (b^3*c^3*Cosh[6*e] + b^3*c^3*Sinh[6*e])/(-1 + Cosh[6*e] + Sinh[6*e])) + (3*Csch[e]*Csch[e + f*x]*(b^3*c^2*d*S
inh[f*x] + 2*a*b^2*c^3*f*Sinh[f*x] + 2*b^3*c*d^2*x*Sinh[f*x] + 6*a*b^2*c^2*d*f*x*Sinh[f*x] + b^3*d^3*x^2*Sinh[
f*x] + 6*a*b^2*c*d^2*f*x^2*Sinh[f*x] + 2*a*b^2*d^3*f*x^3*Sinh[f*x]))/(2*f^2)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2822\) vs. \(2(524)=1048\).
time = 4.37, size = 2823, normalized size = 5.08

method result size
risch \(\text {Expression too large to display}\) \(2823\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6/f*b^2*a*d^3*x^3-2/f^3*b^3*e^3*d^3*x+18/f^4*b*a^2*d^3*polylog(4,exp(f*x+e))+3/f*b*a^2*c^3*ln(exp(f*x+e)+1)+3
/f*b*a^2*c^3*ln(exp(f*x+e)-1)+18/f^4*b*a^2*d^3*polylog(4,-exp(f*x+e))+3/f^3*b^3*d^3*ln(exp(f*x+e)+1)*x+3/f^3*b
^3*d^3*ln(1-exp(f*x+e))*x-6/f^3*b^3*d^3*polylog(3,-exp(f*x+e))*x+1/f^4*b^3*d^3*e^3*ln(1-exp(f*x+e))+6/f^4*b^3*
d^3*e*ln(exp(f*x+e))+2/f^4*b^3*d^3*e^3*ln(exp(f*x+e))-18/f^4*b^2*a*d^3*polylog(3,-exp(f*x+e))-18/f^4*b^2*a*d^3
*polylog(3,exp(f*x+e))-3/f^4*b^3*d^3*e*ln(exp(f*x+e)-1)-1/f^4*b^3*d^3*e^3*ln(exp(f*x+e)-1)+3/f^2*b^3*c^2*d*pol
ylog(2,-exp(f*x+e))+3/f^2*b^3*c^2*d*polylog(2,exp(f*x+e))+1/f*b^3*d^3*ln(exp(f*x+e)+1)*x^3+3/f^2*b^3*d^3*polyl
og(2,-exp(f*x+e))*x^2+1/f*b^3*d^3*ln(1-exp(f*x+e))*x^3+3/f^2*b^3*d^3*polylog(2,exp(f*x+e))*x^2-6/f^3*b^3*d^3*p
olylog(3,exp(f*x+e))*x+3/f^4*b^3*d^3*e*ln(1-exp(f*x+e))+12/f^3*b*a^2*c*d^2*e^3-6/f*b^3*c^2*d*e*x-6/f^3*b*e^3*a
^2*d^3*x+18/f^3*b^2*e^2*a*d^3*x-18/f*b^2*a*c*d^2*x^2+6/f^2*b^3*c*d^2*e^2*x-18/f*b*a^2*c^2*d*e*x+18/f^2*b*a^2*c
*d^2*e^2*x-3*d^2*a^2*b*c*x^3+3*d^2*a*b^2*c*x^3-9/2*d*a^2*b*c^2*x^2+9/2*d*a*b^2*c^2*x^2+3*a^2*b*c^3*x+3*a*b^2*c
^3*x-18/f^3*b*a^2*c*d^2*e^2*ln(exp(f*x+e))+9/f^3*b*a^2*c*d^2*e^2*ln(exp(f*x+e)-1)+9/f*b*ln(1-exp(f*x+e))*a^2*c
^2*d*x+9/f^2*b*ln(1-exp(f*x+e))*a^2*c^2*d*e+18/f^3*b^2*ln(1-exp(f*x+e))*a*c*d^2*e+9/f*b*ln(exp(f*x+e)+1)*a^2*c
^2*d*x+9/f*b*ln(1-exp(f*x+e))*a^2*c*d^2*x^2+18/f^2*b*polylog(2,exp(f*x+e))*a^2*c*d^2*x+9/f*b*ln(exp(f*x+e)+1)*
a^2*c*d^2*x^2+1/4*d^3*a^3*x^4-1/4*d^3*b^3*x^4+1/4/d*a^3*c^4+1/4/d*b^3*c^4+1/f*b^3*c^3*ln(exp(f*x+e)+1)+1/f*b^3
*c^3*ln(exp(f*x+e)-1)-2/f*b^3*c^3*ln(exp(f*x+e))+6/f^4*b^3*d^3*polylog(4,-exp(f*x+e))+6/f^4*b^3*d^3*polylog(4,
exp(f*x+e))+3/f^4*b^3*d^3*polylog(2,-exp(f*x+e))+3/f^4*b^3*d^3*polylog(2,exp(f*x+e))-36/f^2*b^2*a*c*d^2*e*x-3/
4*d^3*a^2*b*x^4+3/4*d^3*a*b^2*x^4+d^2*a^3*c*x^3-d^2*b^3*c*x^3+3/2*d*a^3*c^2*x^2-3/2*d*b^3*c^2*x^2+a^3*c^3*x+b^
3*c^3*x+3/4/d*a^2*b*c^4+3/4/d*a*b^2*c^4+18/f^2*b*polylog(2,-exp(f*x+e))*a^2*c*d^2*x-9/f^3*b*ln(1-exp(f*x+e))*a
^2*c*d^2*e^2+36/f^3*b^2*a*c*d^2*e*ln(exp(f*x+e))+18/f^2*b*a^2*c^2*d*e*ln(exp(f*x+e))-18/f^3*b^2*a*c*d^2*e*ln(e
xp(f*x+e)-1)-9/f^2*b*a^2*c^2*d*e*ln(exp(f*x+e)-1)+18/f^2*b^2*ln(exp(f*x+e)+1)*a*c*d^2*x+18/f^2*b^2*ln(1-exp(f*
x+e))*a*c*d^2*x-6/f^3*b^3*c*d^2*polylog(3,-exp(f*x+e))-6/f^3*b^3*c*d^2*polylog(3,exp(f*x+e))-6/f^3*b^3*c*d^2*l
n(exp(f*x+e))-6/f*b*a^2*c^3*ln(exp(f*x+e))+3/f^3*b^3*c*d^2*ln(exp(f*x+e)+1)+3/f^3*b^3*c*d^2*ln(exp(f*x+e)-1)+3
/f^4*b*a^2*d^3*e^3*ln(1-exp(f*x+e))-6/f^3*b^3*c*d^2*e^2*ln(exp(f*x+e))+6/f^4*b*a^2*d^3*e^3*ln(exp(f*x+e))+18/f
^3*b^2*a*c*d^2*polylog(2,-exp(f*x+e))+18/f^3*b^2*a*c*d^2*polylog(2,exp(f*x+e))+3/f^3*b^3*c*d^2*e^2*ln(exp(f*x+
e)-1)-3/f^4*b*a^2*d^3*e^3*ln(exp(f*x+e)-1)+3/f*b*a^2*d^3*ln(exp(f*x+e)+1)*x^3+9/f^2*b*a^2*d^3*polylog(2,-exp(f
*x+e))*x^2-18/f^3*b*a^2*d^3*polylog(3,-exp(f*x+e))*x+9/f^2*b*a^2*c^2*d*polylog(2,-exp(f*x+e))+9/f^2*b*a^2*c^2*
d*polylog(2,exp(f*x+e))+9/f^2*b^2*a*d^3*ln(exp(f*x+e)+1)*x^2+18/f^3*b^2*a*d^3*polylog(2,-exp(f*x+e))*x+9/f^2*b
^2*a*d^3*ln(1-exp(f*x+e))*x^2-9/f^4*b^2*a*d^3*ln(1-exp(f*x+e))*e^2+18/f^3*b^2*a*d^3*polylog(2,exp(f*x+e))*x+3/
f*b^3*ln(exp(f*x+e)+1)*c^2*d*x+3/f*b^3*ln(1-exp(f*x+e))*c^2*d*x+3/f^2*b^3*ln(1-exp(f*x+e))*c^2*d*e+6/f^2*b^3*p
olylog(2,-exp(f*x+e))*c*d^2*x+3/f*b^3*ln(1-exp(f*x+e))*c*d^2*x^2+6/f^2*b^3*polylog(2,exp(f*x+e))*c*d^2*x+3/f*b
^3*ln(exp(f*x+e)+1)*c*d^2*x^2-18/f^3*b*a^2*c*d^2*polylog(3,-exp(f*x+e))-18/f^3*b*a^2*c*d^2*polylog(3,exp(f*x+e
))-3/f^2*b^3*c^2*d*e*ln(exp(f*x+e)-1)+3/f*b*a^2*d^3*ln(1-exp(f*x+e))*x^3+9/f^2*b*a^2*d^3*polylog(2,exp(f*x+e))
*x^2-18/f^3*b*a^2*d^3*polylog(3,exp(f*x+e))*x-3/f^3*b^3*ln(1-exp(f*x+e))*c*d^2*e^2-18/f^4*b^2*a*d^3*e^2*ln(exp
(f*x+e))+6/f^2*b^3*c^2*d*e*ln(exp(f*x+e))-18/f^2*b^2*a*c^2*d*ln(exp(f*x+e))+9/f^4*b^2*a*d^3*e^2*ln(exp(f*x+e)-
1)+9/f^2*b^2*a*c^2*d*ln(exp(f*x+e)+1)+9/f^2*b^2*a*c^2*d*ln(exp(f*x+e)-1)-3/f^4*b^3*e^2*d^3-3/f^2*b^3*d^3*x^2-3
/2/f^4*b^3*e^4*d^3+4/f^3*b^3*c*d^2*e^3-3/f^2*b^3*c^2*d*e^2-6/f^3*b^3*d^3*e*x+12/f^4*b^2*e^3*a*d^3-9/2/f^4*b*e^
4*a^2*d^3-9/f^2*b*a^2*c^2*d*e^2-18/f^3*b^2*a*c*d^2*e^2-b^2*(6*a*d^3*f*x^3*exp(2*f*x+2*e)+2*b*d^3*f*x^3*exp(2*f
*x+2*e)+18*a*c*d^2*f*x^2*exp(2*f*x+2*e)+6*b*c*d^2*f*x^2*exp(2*f*x+2*e)+18*a*c^2*d*f*x*exp(2*f*x+2*e)-6*a*d^3*f
*x^3+6*b*c^2*d*f*x*exp(2*f*x+2*e)+3*b*d^3*x^2*exp(2*f*x+2*e)+6*a*c^3*f*exp(2*f*x+2*e)-18*a*c*d^2*f*x^2+2*b*c^3
*f*exp(2*f*x+2*e)+6*b*c*d^2*x*exp(2*f*x+2*e)-18*a*c^2*d*f*x+3*b*c^2*d*exp(2*f*x+2*e)-3*b*d^3*x^2-6*a*c^3*f-6*b
*c*d^2*x-3*b*c^2*d)/f^2/(exp(2*f*x+2*e)-1)^2

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1468 vs. \(2 (536) = 1072\).
time = 0.39, size = 1468, normalized size = 2.64 \begin {gather*} \frac {1}{4} \, a^{3} d^{3} x^{4} + a^{3} c d^{2} x^{3} + \frac {3}{2} \, a^{3} c^{2} d x^{2} + a^{3} c^{3} x + \frac {3 \, a^{2} b c^{3} \log \left (\sinh \left (f x + e\right )\right )}{f} + \frac {24 \, a b^{2} c^{3} f + 12 \, b^{3} c^{2} d + {\left (3 \, a^{2} b d^{3} f^{2} + 3 \, a b^{2} d^{3} f^{2} + b^{3} d^{3} f^{2}\right )} x^{4} + 4 \, {\left (3 \, a^{2} b c d^{2} f^{2} + b^{3} c d^{2} f^{2} + 3 \, {\left (c d^{2} f^{2} + 2 \, d^{3} f\right )} a b^{2}\right )} x^{3} + 6 \, {\left (3 \, a^{2} b c^{2} d f^{2} + 3 \, {\left (c^{2} d f^{2} + 4 \, c d^{2} f\right )} a b^{2} + {\left (c^{2} d f^{2} + 2 \, d^{3}\right )} b^{3}\right )} x^{2} + 4 \, {\left (3 \, {\left (c^{3} f^{2} + 6 \, c^{2} d f\right )} a b^{2} + {\left (c^{3} f^{2} + 6 \, c d^{2}\right )} b^{3}\right )} x + {\left ({\left (3 \, a^{2} b d^{3} f^{2} + 3 \, a b^{2} d^{3} f^{2} + b^{3} d^{3} f^{2}\right )} x^{4} e^{\left (4 \, e\right )} + 4 \, {\left (3 \, a^{2} b c d^{2} f^{2} + 3 \, a b^{2} c d^{2} f^{2} + b^{3} c d^{2} f^{2}\right )} x^{3} e^{\left (4 \, e\right )} + 6 \, {\left (3 \, a^{2} b c^{2} d f^{2} + 3 \, a b^{2} c^{2} d f^{2} + b^{3} c^{2} d f^{2}\right )} x^{2} e^{\left (4 \, e\right )} + 4 \, {\left (3 \, a b^{2} c^{3} f^{2} + b^{3} c^{3} f^{2}\right )} x e^{\left (4 \, e\right )}\right )} e^{\left (4 \, f x\right )} - 2 \, {\left ({\left (3 \, a^{2} b d^{3} f^{2} + 3 \, a b^{2} d^{3} f^{2} + b^{3} d^{3} f^{2}\right )} x^{4} e^{\left (2 \, e\right )} + 4 \, {\left (3 \, a^{2} b c d^{2} f^{2} + 3 \, {\left (c d^{2} f^{2} + d^{3} f\right )} a b^{2} + {\left (c d^{2} f^{2} + d^{3} f\right )} b^{3}\right )} x^{3} e^{\left (2 \, e\right )} + 6 \, {\left (3 \, a^{2} b c^{2} d f^{2} + 3 \, {\left (c^{2} d f^{2} + 2 \, c d^{2} f\right )} a b^{2} + {\left (c^{2} d f^{2} + 2 \, c d^{2} f + d^{3}\right )} b^{3}\right )} x^{2} e^{\left (2 \, e\right )} + 4 \, {\left (3 \, {\left (c^{3} f^{2} + 3 \, c^{2} d f\right )} a b^{2} + {\left (c^{3} f^{2} + 3 \, c^{2} d f + 3 \, c d^{2}\right )} b^{3}\right )} x e^{\left (2 \, e\right )} + 2 \, {\left (6 \, a b^{2} c^{3} f + {\left (2 \, c^{3} f + 3 \, c^{2} d\right )} b^{3}\right )} e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{4 \, {\left (f^{2} e^{\left (4 \, f x + 4 \, e\right )} - 2 \, f^{2} e^{\left (2 \, f x + 2 \, e\right )} + f^{2}\right )}} - \frac {2 \, {\left (9 \, a b^{2} c^{2} d f + {\left (c^{3} f^{2} + 3 \, c d^{2}\right )} b^{3}\right )} x}{f^{2}} + \frac {{\left (9 \, a b^{2} c^{2} d f + {\left (c^{3} f^{2} + 3 \, c d^{2}\right )} b^{3}\right )} \log \left (e^{\left (f x + e\right )} + 1\right )}{f^{3}} + \frac {{\left (9 \, a b^{2} c^{2} d f + {\left (c^{3} f^{2} + 3 \, c d^{2}\right )} b^{3}\right )} \log \left (e^{\left (f x + e\right )} - 1\right )}{f^{3}} + \frac {{\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 )} {\left (3 \, a^{2} b d^{3} + b^{3} d^{3}\right )}}{f^{4}} + \frac {{\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 )} {\left (3 \, a^{2} b d^{3} + b^{3} d^{3}\right )}}{f^{4}} + \frac {3 \, {\left (3 \, a^{2} b c d^{2} f + b^{3} c d^{2} f + 3 \, a 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 (3 \, a^{2} b c d^{2} f + b^{3} c d^{2} f + 3 \, a 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 (3 \, a^{2} b c^{2} d f^{2} + 6 \, a b^{2} c d^{2} f + {\left (c^{2} d f^{2} + d^{3}\right )} b^{3}\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^{4}} + \frac {3 \, {\left (3 \, a^{2} b c^{2} d f^{2} + 6 \, a b^{2} c d^{2} f + {\left (c^{2} d f^{2} + d^{3}\right )} b^{3}\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^{4}} - \frac {{\left (3 \, a^{2} b d^{3} + b^{3} d^{3}\right )} f^{4} x^{4} + 4 \, {\left (3 \, a^{2} b c d^{2} f + b^{3} c d^{2} f + 3 \, a b^{2} d^{3}\right )} f^{3} x^{3} + 6 \, {\left (3 \, a^{2} b c^{2} d f^{2} + 6 \, a b^{2} c d^{2} f + {\left (c^{2} d f^{2} + d^{3}\right )} b^{3}\right )} f^{2} x^{2}}{2 \, f^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a^3*d^3*x^4 + a^3*c*d^2*x^3 + 3/2*a^3*c^2*d*x^2 + a^3*c^3*x + 3*a^2*b*c^3*log(sinh(f*x + e))/f + 1/4*(24*a
*b^2*c^3*f + 12*b^3*c^2*d + (3*a^2*b*d^3*f^2 + 3*a*b^2*d^3*f^2 + b^3*d^3*f^2)*x^4 + 4*(3*a^2*b*c*d^2*f^2 + b^3
*c*d^2*f^2 + 3*(c*d^2*f^2 + 2*d^3*f)*a*b^2)*x^3 + 6*(3*a^2*b*c^2*d*f^2 + 3*(c^2*d*f^2 + 4*c*d^2*f)*a*b^2 + (c^
2*d*f^2 + 2*d^3)*b^3)*x^2 + 4*(3*(c^3*f^2 + 6*c^2*d*f)*a*b^2 + (c^3*f^2 + 6*c*d^2)*b^3)*x + ((3*a^2*b*d^3*f^2
+ 3*a*b^2*d^3*f^2 + b^3*d^3*f^2)*x^4*e^(4*e) + 4*(3*a^2*b*c*d^2*f^2 + 3*a*b^2*c*d^2*f^2 + b^3*c*d^2*f^2)*x^3*e
^(4*e) + 6*(3*a^2*b*c^2*d*f^2 + 3*a*b^2*c^2*d*f^2 + b^3*c^2*d*f^2)*x^2*e^(4*e) + 4*(3*a*b^2*c^3*f^2 + b^3*c^3*
f^2)*x*e^(4*e))*e^(4*f*x) - 2*((3*a^2*b*d^3*f^2 + 3*a*b^2*d^3*f^2 + b^3*d^3*f^2)*x^4*e^(2*e) + 4*(3*a^2*b*c*d^
2*f^2 + 3*(c*d^2*f^2 + d^3*f)*a*b^2 + (c*d^2*f^2 + d^3*f)*b^3)*x^3*e^(2*e) + 6*(3*a^2*b*c^2*d*f^2 + 3*(c^2*d*f
^2 + 2*c*d^2*f)*a*b^2 + (c^2*d*f^2 + 2*c*d^2*f + d^3)*b^3)*x^2*e^(2*e) + 4*(3*(c^3*f^2 + 3*c^2*d*f)*a*b^2 + (c
^3*f^2 + 3*c^2*d*f + 3*c*d^2)*b^3)*x*e^(2*e) + 2*(6*a*b^2*c^3*f + (2*c^3*f + 3*c^2*d)*b^3)*e^(2*e))*e^(2*f*x))
/(f^2*e^(4*f*x + 4*e) - 2*f^2*e^(2*f*x + 2*e) + f^2) - 2*(9*a*b^2*c^2*d*f + (c^3*f^2 + 3*c*d^2)*b^3)*x/f^2 + (
9*a*b^2*c^2*d*f + (c^3*f^2 + 3*c*d^2)*b^3)*log(e^(f*x + e) + 1)/f^3 + (9*a*b^2*c^2*d*f + (c^3*f^2 + 3*c*d^2)*b
^3)*log(e^(f*x + e) - 1)/f^3 + (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)))*(3*a^2*b*d^3 + b^3*d^3)/f^4 + (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)))*(3*a^2*b*d^3 + b^3*d^
3)/f^4 + 3*(3*a^2*b*c*d^2*f + b^3*c*d^2*f + 3*a*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 + 3*(3*a^2*b*c*d^2*f + b^3*c*d^2*f + 3*a*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 + 3*(3*a^2*b*c^2*d*f^2 + 6*a*b^2*c*d^2*f
 + (c^2*d*f^2 + d^3)*b^3)*(f*x*log(e^(f*x + e) + 1) + dilog(-e^(f*x + e)))/f^4 + 3*(3*a^2*b*c^2*d*f^2 + 6*a*b^
2*c*d^2*f + (c^2*d*f^2 + d^3)*b^3)*(f*x*log(-e^(f*x + e) + 1) + dilog(e^(f*x + e)))/f^4 - 1/2*((3*a^2*b*d^3 +
b^3*d^3)*f^4*x^4 + 4*(3*a^2*b*c*d^2*f + b^3*c*d^2*f + 3*a*b^2*d^3)*f^3*x^3 + 6*(3*a^2*b*c^2*d*f^2 + 6*a*b^2*c*
d^2*f + (c^2*d*f^2 + d^3)*b^3)*f^2*x^2)/f^4

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 16978 vs. \(2 (536) = 1072\).
time = 0.56, size = 16978, normalized size = 30.54 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^3*f^4*x^4 + 4*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*d^2*f^4*x^3 + 6*(a^3 -
3*a^2*b + 3*a*b^2 - b^3)*c^2*d*f^4*x^2 + 24*a*b^2*c^3*f^3 + 4*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c^3*f^4*x + 12*b
^3*c^2*d*f^2 + 2*(3*a^2*b + b^3)*d^3*cosh(1)^4 + 2*(3*a^2*b + b^3)*d^3*sinh(1)^4 + ((a^3 - 3*a^2*b + 3*a*b^2 -
 b^3)*d^3*f^4*x^4 + 2*(3*a^2*b + b^3)*d^3*cosh(1)^4 + 2*(3*a^2*b + b^3)*d^3*sinh(1)^4 - 4*(6*a*b^2*d^3*f^3 - (
a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*d^2*f^4)*x^3 - 8*(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f)*cosh(1)^3 - 8*(3*a*b
^2*d^3 + (3*a^2*b + b^3)*c*d^2*f - (3*a^2*b + b^3)*d^3*cosh(1))*sinh(1)^3 - 6*(12*a*b^2*c*d^2*f^3 + 2*b^3*d^3*
f^2 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c^2*d*f^4)*x^2 + 12*(6*a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f
^2)*cosh(1)^2 + 12*(6*a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2 + (3*a^2*b + b^3)*d^3*cosh(1)^2 - 2*
(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f)*cosh(1))*sinh(1)^2 - 4*(18*a*b^2*c^2*d*f^3 + 6*b^3*c*d^2*f^2 - (a^3 -
3*a^2*b + 3*a*b^2 - b^3)*c^3*f^4)*x - 8*(9*a*b^2*c^2*d*f^2 + 3*b^3*c*d^2*f + (3*a^2*b + b^3)*c^3*f^3)*cosh(1)
- 8*(9*a*b^2*c^2*d*f^2 + 3*b^3*c*d^2*f + (3*a^2*b + b^3)*c^3*f^3 - (3*a^2*b + b^3)*d^3*cosh(1)^3 + 3*(3*a*b^2*
d^3 + (3*a^2*b + b^3)*c*d^2*f)*cosh(1)^2 - 3*(6*a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2)*cosh(1))*
sinh(1))*cosh(f*x + cosh(1) + sinh(1))^4 + 4*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^3*f^4*x^4 + 2*(3*a^2*b + b^3)*
d^3*cosh(1)^4 + 2*(3*a^2*b + b^3)*d^3*sinh(1)^4 - 4*(6*a*b^2*d^3*f^3 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*d^2*f
^4)*x^3 - 8*(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f)*cosh(1)^3 - 8*(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f - (3*
a^2*b + b^3)*d^3*cosh(1))*sinh(1)^3 - 6*(12*a*b^2*c*d^2*f^3 + 2*b^3*d^3*f^2 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*
c^2*d*f^4)*x^2 + 12*(6*a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2)*cosh(1)^2 + 12*(6*a*b^2*c*d^2*f +
b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2 + (3*a^2*b + b^3)*d^3*cosh(1)^2 - 2*(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f
)*cosh(1))*sinh(1)^2 - 4*(18*a*b^2*c^2*d*f^3 + 6*b^3*c*d^2*f^2 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c^3*f^4)*x -
8*(9*a*b^2*c^2*d*f^2 + 3*b^3*c*d^2*f + (3*a^2*b + b^3)*c^3*f^3)*cosh(1) - 8*(9*a*b^2*c^2*d*f^2 + 3*b^3*c*d^2*f
 + (3*a^2*b + b^3)*c^3*f^3 - (3*a^2*b + b^3)*d^3*cosh(1)^3 + 3*(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f)*cosh(1)
^2 - 3*(6*a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2)*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1))
*sinh(f*x + cosh(1) + sinh(1))^3 + ((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^3*f^4*x^4 + 2*(3*a^2*b + b^3)*d^3*cosh(1
)^4 + 2*(3*a^2*b + b^3)*d^3*sinh(1)^4 - 4*(6*a*b^2*d^3*f^3 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*d^2*f^4)*x^3 -
8*(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f)*cosh(1)^3 - 8*(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f - (3*a^2*b + b^
3)*d^3*cosh(1))*sinh(1)^3 - 6*(12*a*b^2*c*d^2*f^3 + 2*b^3*d^3*f^2 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c^2*d*f^4)
*x^2 + 12*(6*a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2)*cosh(1)^2 + 12*(6*a*b^2*c*d^2*f + b^3*d^3 +
(3*a^2*b + b^3)*c^2*d*f^2 + (3*a^2*b + b^3)*d^3*cosh(1)^2 - 2*(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f)*cosh(1))
*sinh(1)^2 - 4*(18*a*b^2*c^2*d*f^3 + 6*b^3*c*d^2*f^2 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c^3*f^4)*x - 8*(9*a*b^2
*c^2*d*f^2 + 3*b^3*c*d^2*f + (3*a^2*b + b^3)*c^3*f^3)*cosh(1) - 8*(9*a*b^2*c^2*d*f^2 + 3*b^3*c*d^2*f + (3*a^2*
b + b^3)*c^3*f^3 - (3*a^2*b + b^3)*d^3*cosh(1)^3 + 3*(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f)*cosh(1)^2 - 3*(6*
a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2)*cosh(1))*sinh(1))*sinh(f*x + cosh(1) + sinh(1))^4 - 8*(3*
a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f)*cosh(1)^3 - 8*(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f - (3*a^2*b + b^3)*d^
3*cosh(1))*sinh(1)^3 + 12*(6*a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2)*cosh(1)^2 - 2*((a^3 - 3*a^2*
b + 3*a*b^2 - b^3)*d^3*f^4*x^4 + 6*b^3*c^2*d*f^2 + 2*(3*a^2*b + b^3)*d^3*cosh(1)^4 + 2*(3*a^2*b + b^3)*d^3*sin
h(1)^4 + 4*(3*a*b^2 + b^3)*c^3*f^3 + 4*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c*d^2*f^4 - (3*a*b^2 - b^3)*d^3*f^3)*x
^3 - 8*(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f)*cosh(1)^3 - 8*(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f - (3*a^2*b
 + b^3)*d^3*cosh(1))*sinh(1)^3 - 6*(b^3*d^3*f^2 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c^2*d*f^4 + 2*(3*a*b^2 - b^3
)*c*d^2*f^3)*x^2 + 12*(6*a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2)*cosh(1)^2 + 12*(6*a*b^2*c*d^2*f
+ b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2 + (3*a^2*b + b^3)*d^3*cosh(1)^2 - 2*(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2
*f)*cosh(1))*sinh(1)^2 - 4*(3*b^3*c*d^2*f^2 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c^3*f^4 + 3*(3*a*b^2 - b^3)*c^2*
d*f^3)*x - 8*(9*a*b^2*c^2*d*f^2 + 3*b^3*c*d^2*f + (3*a^2*b + b^3)*c^3*f^3)*cosh(1) - 8*(9*a*b^2*c^2*d*f^2 + 3*
b^3*c*d^2*f + (3*a^2*b + b^3)*c^3*f^3 - (3*a^2*b + b^3)*d^3*cosh(1)^3 + 3*(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2
*f)*cosh(1)^2 - 3*(6*a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2)*cosh(1))*sinh(1))*cosh(f*x + cosh(1)
 + sinh(1))^2 + 12*(6*a*b^2*c*d^2*f + b^3*d^3 + (3*a^2*b + b^3)*c^2*d*f^2 + (3*a^2*b + b^3)*d^3*cosh(1)^2 - 2*
(3*a*b^2*d^3 + (3*a^2*b + b^3)*c*d^2*f)*cosh(1)...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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