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

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

[Out]

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

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Rubi [A]
time = 0.50, 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 = {3803, 3797, 2221, 2611, 2320, 6724, 3801, 2317, 2438, 32, 3556} \begin {gather*} \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} \end {gather*}

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 2221

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

Rule 2317

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

Rule 2320

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

Rule 2438

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

Rule 2611

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

Rule 3556

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

Rule 3797

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

Rule 3801

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

Rule 3803

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

Rule 6724

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

Rubi steps

\begin {align*} \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 ) \text {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 ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{f^3}-\frac {\left (b^3 d^2\right ) \int \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 ) \text {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] Result contains complex when optimal does not.
time = 10.57, size = 1885, normalized size = 4.70 \begin {gather*} -\frac {a^2 b d^2 e^{-e} \text {csch}(e) \left (2 f^2 x^2 \left (2 e^{2 e} f x-3 \left (-1+e^{2 e}\right ) \log \left (1-e^{2 (e+f x)}\right )\right )-6 \left (-1+e^{2 e}\right ) f x \text {PolyLog}\left (2,e^{2 (e+f x)}\right )+3 \left (-1+e^{2 e}\right ) \text {PolyLog}\left (3,e^{2 (e+f x)}\right )\right )}{4 f^3}-\frac {b^3 d^2 e^{-e} \text {csch}(e) \left (2 f^2 x^2 \left (2 e^{2 e} f x-3 \left (-1+e^{2 e}\right ) \log \left (1-e^{2 (e+f x)}\right )\right )-6 \left (-1+e^{2 e}\right ) f x \text {PolyLog}\left (2,e^{2 (e+f x)}\right )+3 \left (-1+e^{2 e}\right ) \text {PolyLog}\left (3,e^{2 (e+f x)}\right )\right )}{12 f^3}-\frac {b^3 d^2 \text {csch}(e) (-f x \cosh (e)+\log (\cosh (f x) \sinh (e)+\cosh (e) \sinh (f x)) \sinh (e))}{f^3 \left (-\cosh ^2(e)+\sinh ^2(e)\right )}-\frac {6 a b^2 c d \text {csch}(e) (-f x \cosh (e)+\log (\cosh (f x) \sinh (e)+\cosh (e) \sinh (f x)) \sinh (e))}{f^2 \left (-\cosh ^2(e)+\sinh ^2(e)\right )}-\frac {3 a^2 b c^2 \text {csch}(e) (-f x \cosh (e)+\log (\cosh (f x) \sinh (e)+\cosh (e) \sinh (f x)) \sinh (e))}{f \left (-\cosh ^2(e)+\sinh ^2(e)\right )}-\frac {b^3 c^2 \text {csch}(e) (-f x \cosh (e)+\log (\cosh (f x) \sinh (e)+\cosh (e) \sinh (f x)) \sinh (e))}{f \left (-\cosh ^2(e)+\sinh ^2(e)\right )}+\frac {\text {csch}(e) \text {csch}^2(e+f x) \left (-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 \cosh (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)\right )}{12 f^2}-\frac {3 a b^2 d^2 \text {csch}(e) \text {sech}(e) \left (e^{-\tanh ^{-1}(\tanh (e))} f^2 x^2-\frac {i \left (-f x \left (-\pi +2 i \tanh ^{-1}(\tanh (e))\right )-\pi \log \left (1+e^{2 f x}\right )-2 \left (i f x+i \tanh ^{-1}(\tanh (e))\right ) \log \left (1-e^{2 i \left (i f x+i \tanh ^{-1}(\tanh (e))\right )}\right )+\pi \log (\cosh (f x))+2 i \tanh ^{-1}(\tanh (e)) \log \left (i \sinh \left (f x+\tanh ^{-1}(\tanh (e))\right )\right )+i \text {PolyLog}\left (2,e^{2 i \left (i f x+i \tanh ^{-1}(\tanh (e))\right )}\right )\right ) \tanh (e)}{\sqrt {1-\tanh ^2(e)}}\right )}{f^3 \sqrt {\text {sech}^2(e) \left (\cosh ^2(e)-\sinh ^2(e)\right )}}-\frac {3 a^2 b c d \text {csch}(e) \text {sech}(e) \left (e^{-\tanh ^{-1}(\tanh (e))} f^2 x^2-\frac {i \left (-f x \left (-\pi +2 i \tanh ^{-1}(\tanh (e))\right )-\pi \log \left (1+e^{2 f x}\right )-2 \left (i f x+i \tanh ^{-1}(\tanh (e))\right ) \log \left (1-e^{2 i \left (i f x+i \tanh ^{-1}(\tanh (e))\right )}\right )+\pi \log (\cosh (f x))+2 i \tanh ^{-1}(\tanh (e)) \log \left (i \sinh \left (f x+\tanh ^{-1}(\tanh (e))\right )\right )+i \text {PolyLog}\left (2,e^{2 i \left (i f x+i \tanh ^{-1}(\tanh (e))\right )}\right )\right ) \tanh (e)}{\sqrt {1-\tanh ^2(e)}}\right )}{f^2 \sqrt {\text {sech}^2(e) \left (\cosh ^2(e)-\sinh ^2(e)\right )}}-\frac {b^3 c d \text {csch}(e) \text {sech}(e) \left (e^{-\tanh ^{-1}(\tanh (e))} f^2 x^2-\frac {i \left (-f x \left (-\pi +2 i \tanh ^{-1}(\tanh (e))\right )-\pi \log \left (1+e^{2 f x}\right )-2 \left (i f x+i \tanh ^{-1}(\tanh (e))\right ) \log \left (1-e^{2 i \left (i f x+i \tanh ^{-1}(\tanh (e))\right )}\right )+\pi \log (\cosh (f x))+2 i \tanh ^{-1}(\tanh (e)) \log \left (i \sinh \left (f x+\tanh ^{-1}(\tanh (e))\right )\right )+i \text {PolyLog}\left (2,e^{2 i \left (i f x+i \tanh ^{-1}(\tanh (e))\right )}\right )\right ) \tanh (e)}{\sqrt {1-\tanh ^2(e)}}\right )}{f^2 \sqrt {\text {sech}^2(e) \left (\cosh ^2(e)-\sinh ^2(e)\right )}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/4*(a^2*b*d^2*Csch[e]*(2*f^2*x^2*(2*E^(2*e)*f*x - 3*(-1 + E^(2*e))*Log[1 - E^(2*(e + f*x))]) - 6*(-1 + E^(2*
e))*f*x*PolyLog[2, E^(2*(e + f*x))] + 3*(-1 + E^(2*e))*PolyLog[3, E^(2*(e + f*x))]))/(E^e*f^3) - (b^3*d^2*Csch
[e]*(2*f^2*x^2*(2*E^(2*e)*f*x - 3*(-1 + E^(2*e))*Log[1 - E^(2*(e + f*x))]) - 6*(-1 + E^(2*e))*f*x*PolyLog[2, E
^(2*(e + f*x))] + 3*(-1 + E^(2*e))*PolyLog[3, E^(2*(e + f*x))]))/(12*E^e*f^3) - (b^3*d^2*Csch[e]*(-(f*x*Cosh[e
]) + Log[Cosh[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*Cosh[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*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)) - (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*Cosh[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*Si
nh[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*Si
nh[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*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[T
anh[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)*Arc
Tanh[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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1585\) vs. \(2(389)=778\).
time = 3.69, size = 1586, normalized size = 3.96

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6/f^3*b^2*a*d^2*e^2+1/f^3*b^3*d^2*e^2*ln(exp(f*x+e)-1)+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)-2/f^3*b^3*d^2*e^2*ln(exp(f*x+e))-6/f*b*a^2*c^2*ln(exp(f*x+e))-6/f^3*b*a^2*d^2*polylog(3,-exp(f*x+e))-6
/f^3*b*a^2*d^2*polylog(3,exp(f*x+e))+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+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+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))+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^3*b^3*ln(1-exp(f*x+e))*d^2*e^2-12/f*b*a^2*c*d*e*x+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-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^3*b^2*a*d^2*e*ln(exp(f*x+e)-1)+12/f^3*b^2*a*d^2*e*ln(exp(f*x+e))+3/f*b*ln(exp(f*x+e)+1)*a^2*d^2*x^2+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+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))-3/f^3*b*ln(1-exp(f*x+e))*a^2*d^2*e^2+3/f^3*b*a^2*d^2*e^2*ln(exp(f*x+e)-1)-6/f^3*b*a^2*d^2*e^2*ln(exp(f*x
+e))+6/f^2*b^2*a*c*d*ln(exp(f*x+e)+1)+6/f^2*b^2*a*c*d*ln(exp(f*x+e)-1)-2/f^2*b^3*c*d*e*ln(exp(f*x+e)-1)-12/f^2
*b^2*a*c*d*ln(exp(f*x+e))-12/f^2*b^2*a*d^2*e*x-4/f*b^3*c*d*e*x+4/f^2*b^3*c*d*e*ln(exp(f*x+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(1-exp(f*x+e))*a*d^
2*x+6/f^2*b^2*ln(exp(f*x+e)+1)*a*d^2*x+6/f^2*b*a^2*d^2*e^2*x-6/f^2*b*a^2*c*d*e^2-d^2*a^2*b*x^3+d^2*a*b^2*x^3+d
*a^3*c*x^2-d*b^3*c*x^2+a^3*c^2*x+b^3*c^2*x+1/d*a^2*b*c^3+1/d*a*b^2*c^3+2/f^2*b^3*d^2*e^2*x-6/f*b^2*a*d^2*x^2-2
/f^2*b^3*c*d*e^2+4/f^3*b*a^2*d^2*e^3+1/f*b^3*c^2*ln(exp(f*x+e)+1)-2/f*b^3*c^2*ln(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^3*b^3*d^2*polylog(3,-exp(f*x+e))
-2/f^3*b^3*d^2*polylog(3,exp(f*x+e))+1/3/d*b^3*c^3+1/3*d^2*a^3*x^3-1/3*d^2*b^3*x^3+1/3/d*a^3*c^3+3*a*b^2*c^2*x
-3*d*a^2*b*c*x^2+3*d*a*b^2*c*x^2+3*a^2*b*c^2*x+4/3/f^3*b^3*d^2*e^3+1/f*b^3*c^2*ln(exp(f*x+e)-1)-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-b*c*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. 952 vs. \(2 (398) = 796\).
time = 0.38, size = 952, normalized size = 2.37 \begin {gather*} \frac {1}{3} \, a^{3} d^{2} x^{3} + a^{3} c d x^{2} + a^{3} c^{2} x + \frac {3 \, a^{2} b c^{2} \log \left (\sinh \left (f x + e\right )\right )}{f} + \frac {18 \, a b^{2} c^{2} f + 6 \, b^{3} c d + {\left (3 \, a^{2} b d^{2} f^{2} + 3 \, a b^{2} d^{2} f^{2} + b^{3} d^{2} f^{2}\right )} x^{3} + 3 \, {\left (3 \, a^{2} b c d f^{2} + b^{3} c d f^{2} + 3 \, {\left (c d f^{2} + 2 \, d^{2} f\right )} a b^{2}\right )} x^{2} + 3 \, {\left (3 \, {\left (c^{2} f^{2} + 4 \, c d f\right )} a b^{2} + {\left (c^{2} f^{2} + 2 \, d^{2}\right )} b^{3}\right )} x + {\left ({\left (3 \, a^{2} b d^{2} f^{2} + 3 \, a b^{2} d^{2} f^{2} + b^{3} d^{2} f^{2}\right )} x^{3} e^{\left (4 \, e\right )} + 3 \, {\left (3 \, a^{2} b c d f^{2} + 3 \, a b^{2} c d f^{2} + b^{3} c d f^{2}\right )} x^{2} e^{\left (4 \, e\right )} + 3 \, {\left (3 \, a b^{2} c^{2} f^{2} + b^{3} c^{2} f^{2}\right )} x e^{\left (4 \, e\right )}\right )} e^{\left (4 \, f x\right )} - 2 \, {\left ({\left (3 \, a^{2} b d^{2} f^{2} + 3 \, a b^{2} d^{2} f^{2} + b^{3} d^{2} f^{2}\right )} x^{3} e^{\left (2 \, e\right )} + 3 \, {\left (3 \, a^{2} b c d f^{2} + 3 \, {\left (c d f^{2} + d^{2} f\right )} a b^{2} + {\left (c d f^{2} + d^{2} f\right )} b^{3}\right )} x^{2} e^{\left (2 \, e\right )} + 3 \, {\left (3 \, {\left (c^{2} f^{2} + 2 \, c d f\right )} a b^{2} + {\left (c^{2} f^{2} + 2 \, c d f + d^{2}\right )} b^{3}\right )} x e^{\left (2 \, e\right )} + 3 \, {\left (3 \, a b^{2} c^{2} f + {\left (c^{2} f + c d\right )} b^{3}\right )} e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{3 \, {\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 (6 \, a b^{2} c d f + {\left (c^{2} f^{2} + d^{2}\right )} b^{3}\right )} x}{f^{2}} + \frac {{\left (3 \, a^{2} b d^{2} + b^{3} d^{2}\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^{3}} + \frac {{\left (3 \, a^{2} b d^{2} + b^{3} d^{2}\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^{3}} + \frac {2 \, {\left (3 \, a^{2} b c d f + b^{3} c d f + 3 \, a b^{2} d^{2}\right )} {\left (f x \log \left (e^{\left (f x + e\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (f x + e\right )}\right )\right )}}{f^{3}} + \frac {2 \, {\left (3 \, a^{2} b c d f + b^{3} c d f + 3 \, a b^{2} d^{2}\right )} {\left (f x \log \left (-e^{\left (f x + e\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (f x + e\right )}\right )\right )}}{f^{3}} + \frac {{\left (6 \, a b^{2} c d f + {\left (c^{2} f^{2} + d^{2}\right )} b^{3}\right )} \log \left (e^{\left (f x + e\right )} + 1\right )}{f^{3}} + \frac {{\left (6 \, a b^{2} c d f + {\left (c^{2} f^{2} + d^{2}\right )} b^{3}\right )} \log \left (e^{\left (f x + e\right )} - 1\right )}{f^{3}} - \frac {2 \, {\left ({\left (3 \, a^{2} b d^{2} + b^{3} d^{2}\right )} f^{3} x^{3} + 3 \, {\left (3 \, a^{2} b c d f + b^{3} c d f + 3 \, a b^{2} d^{2}\right )} f^{2} x^{2}\right )}}{3 \, f^{3}} \end {gather*}

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 + 3*a*b^2*d^2
*f^2 + b^3*d^2*f^2)*x^3*e^(4*e) + 3*(3*a^2*b*c*d*f^2 + 3*a*b^2*c*d*f^2 + b^3*c*d*f^2)*x^2*e^(4*e) + 3*(3*a*b^2
*c^2*f^2 + b^3*c^2*f^2)*x*e^(4*e))*e^(4*f*x) - 2*((3*a^2*b*d^2*f^2 + 3*a*b^2*d^2*f^2 + b^3*d^2*f^2)*x^3*e^(2*e
) + 3*(3*a^2*b*c*d*f^2 + 3*(c*d*f^2 + d^2*f)*a*b^2 + (c*d*f^2 + d^2*f)*b^3)*x^2*e^(2*e) + 3*(3*(c^2*f^2 + 2*c*
d*f)*a*b^2 + (c^2*f^2 + 2*c*d*f + d^2)*b^3)*x*e^(2*e) + 3*(3*a*b^2*c^2*f + (c^2*f + c*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*(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*polylog(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) + dilog(-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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 9165 vs. \(2 (398) = 796\).
time = 0.50, size = 9165, normalized size = 22.86 \begin {gather*} \text {Too large to display} \end {gather*}

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*c^
2*f^2 + 3*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*c^2*f^3*x + 6*b^3*c*d*f - 2*(3*a^2*b + b^3)*d^2*cosh(1)^3 - 2*(3*a^2
*b + b^3)*d^2*sinh(1)^3 + ((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*f^3*x^3 - 2*(3*a^2*b + b^3)*d^2*cosh(1)^3 - 2*(
3*a^2*b + b^3)*d^2*sinh(1)^3 - 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*(3*a*b^2*
d^2 + (3*a^2*b + b^3)*c*d*f)*cosh(1)^2 + 6*(3*a*b^2*d^2 + (3*a^2*b + b^3)*c*d*f - (3*a^2*b + b^3)*d^2*cosh(1))
*sinh(1)^2 - 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 - 6*(6*a*b^2*c*d*f
+ b^3*d^2 + (3*a^2*b + b^3)*c^2*f^2)*cosh(1) - 6*(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*cosh(1)^2 - 2*(3*a*b^2*d^2 + (3*a^2*b + b^3)*c*d*f)*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^2*f^3*x^3 - 2*(3*a^2*b + b^3)*d^2*cosh(1)^3 - 2*(3*a^2*b + b^3)*d^
2*sinh(1)^3 - 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*(3*a*b^2*d^2 + (3*a^2*b +
b^3)*c*d*f)*cosh(1)^2 + 6*(3*a*b^2*d^2 + (3*a^2*b + b^3)*c*d*f - (3*a^2*b + b^3)*d^2*cosh(1))*sinh(1)^2 - 3*(1
2*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 - 6*(6*a*b^2*c*d*f + b^3*d^2 + (3*a
^2*b + b^3)*c^2*f^2)*cosh(1) - 6*(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*cosh
(1)^2 - 2*(3*a*b^2*d^2 + (3*a^2*b + b^3)*c*d*f)*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cos
h(1) + sinh(1))^3 + ((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*f^3*x^3 - 2*(3*a^2*b + b^3)*d^2*cosh(1)^3 - 2*(3*a^2*
b + b^3)*d^2*sinh(1)^3 - 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*(3*a*b^2*d^2 +
(3*a^2*b + b^3)*c*d*f)*cosh(1)^2 + 6*(3*a*b^2*d^2 + (3*a^2*b + b^3)*c*d*f - (3*a^2*b + b^3)*d^2*cosh(1))*sinh(
1)^2 - 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 - 6*(6*a*b^2*c*d*f + b^3
*d^2 + (3*a^2*b + b^3)*c^2*f^2)*cosh(1) - 6*(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*cosh(1)^2 - 2*(3*a*b^2*d^2 + (3*a^2*b + b^3)*c*d*f)*cosh(1))*sinh(1))*sinh(f*x + cosh(1) + sinh(1))^4 +
6*(3*a*b^2*d^2 + (3*a^2*b + b^3)*c*d*f)*cosh(1)^2 - 2*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*f^3*x^3 + 3*b^3*c*
d*f - 2*(3*a^2*b + b^3)*d^2*cosh(1)^3 - 2*(3*a^2*b + b^3)*d^2*sinh(1)^3 + 3*(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 + 6*(3*a*b^2*d^2 + (3*a^2*b + b^3)*c*d*f)*co
sh(1)^2 + 6*(3*a*b^2*d^2 + (3*a^2*b + b^3)*c*d*f - (3*a^2*b + b^3)*d^2*cosh(1))*sinh(1)^2 - 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 - 6*(6*a*b^2*c*d*f + b^3*d^2 + (3*a^2*b +
b^3)*c^2*f^2)*cosh(1) - 6*(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*cosh(1)^2 -
2*(3*a*b^2*d^2 + (3*a^2*b + b^3)*c*d*f)*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^2 + 6*(3*a*b^2*d^2 +
(3*a^2*b + b^3)*c*d*f - (3*a^2*b + b^3)*d^2*cosh(1))*sinh(1)^2 - 2*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*f^3*x^
3 + 3*b^3*c*d*f - 2*(3*a^2*b + b^3)*d^2*cosh(1)^3 - 2*(3*a^2*b + b^3)*d^2*sinh(1)^3 + 3*(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 + 6*(3*a*b^2*d^2 + (3*a^2*b + b^
3)*c*d*f)*cosh(1)^2 - 3*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d^2*f^3*x^3 - 2*(3*a^2*b + b^3)*d^2*cosh(1)^3 - 2*(3*
a^2*b + b^3)*d^2*sinh(1)^3 - 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*(3*a*b^2*d^
2 + (3*a^2*b + b^3)*c*d*f)*cosh(1)^2 + 6*(3*a*b^2*d^2 + (3*a^2*b + b^3)*c*d*f - (3*a^2*b + b^3)*d^2*cosh(1))*s
inh(1)^2 - 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 - 6*(6*a*b^2*c*d*f +
b^3*d^2 + (3*a^2*b + b^3)*c^2*f^2)*cosh(1) - 6*(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*cosh(1)^2 - 2*(3*a*b^2*d^2 + (3*a^2*b + b^3)*c*d*f)*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1))
^2 + 6*(3*a*b^2*d^2 + (3*a^2*b + b^3)*c*d*f - (3*a^2*b + b^3)*d^2*cosh(1))*sinh(1)^2 - 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 - 6*(6*a*b^2*c*d*f + b^3*d^2 + (3*a^2*b + b^3)*
c^2*f^2)*cosh(1) - 6*(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*cosh(1)^2 - 2*(3
*a*b^2*d^2 + (3*a^2*b + b^3)*c*d*f)*cosh(1))*sinh(1))*sinh(f*x + cosh(1) + sinh(1))^2 - 6*(6*a*b^2*c*d*f + b^3
*d^2 + (3*a^2*b + b^3)*c^2*f^2)*cosh(1) + 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 + cosh(1) + sinh(1))^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 + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1))^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 + cosh(1) + sinh(1))^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 + cosh(1) + sinh(1))^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...

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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 )^{2}\, dx \end {gather*}

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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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)^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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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 )}^2 \,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)^2,x)

[Out]

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

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