Integrand size = 20, antiderivative size = 210 \[ \int \frac {(c+d x)^3}{a+b \coth (e+f x)} \, dx=\frac {(c+d x)^4}{4 (a+b) d}-\frac {b (c+d x)^3 \log \left (1-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right ) f}+\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 \left (a^2-b^2\right ) f^2}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 \left (a^2-b^2\right ) f^3}+\frac {3 b d^3 \operatorname {PolyLog}\left (4,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{4 \left (a^2-b^2\right ) f^4} \]
1/4*(d*x+c)^4/(a+b)/d-b*(d*x+c)^3*ln(1+(-a+b)/(a+b)/exp(2*f*x+2*e))/(a^2-b ^2)/f+3/2*b*d*(d*x+c)^2*polylog(2,(a-b)/(a+b)/exp(2*f*x+2*e))/(a^2-b^2)/f^ 2+3/2*b*d^2*(d*x+c)*polylog(3,(a-b)/(a+b)/exp(2*f*x+2*e))/(a^2-b^2)/f^3+3/ 4*b*d^3*polylog(4,(a-b)/(a+b)/exp(2*f*x+2*e))/(a^2-b^2)/f^4
Time = 1.53 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.18 \[ \int \frac {(c+d x)^3}{a+b \coth (e+f x)} \, dx=\frac {1}{4} \left (\frac {2 b (c+d x)^4}{(a+b) d \left (a \left (-1+e^{2 e}\right )+b \left (1+e^{2 e}\right )\right )}-\frac {4 b (c+d x)^3 \log \left (1+\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )}{(a-b) (a+b) f}+\frac {3 b d \left (2 f^2 (c+d x)^2 \operatorname {PolyLog}\left (2,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )+d \left (2 f (c+d x) \operatorname {PolyLog}\left (3,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )+d \operatorname {PolyLog}\left (4,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )\right )\right )}{(a-b) (a+b) f^4}+\frac {x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \sinh (e)}{b \cosh (e)+a \sinh (e)}\right ) \]
((2*b*(c + d*x)^4)/((a + b)*d*(a*(-1 + E^(2*e)) + b*(1 + E^(2*e)))) - (4*b *(c + d*x)^3*Log[1 + (-a + b)/((a + b)*E^(2*(e + f*x)))])/((a - b)*(a + b) *f) + (3*b*d*(2*f^2*(c + d*x)^2*PolyLog[2, (a - b)/((a + b)*E^(2*(e + f*x) ))] + d*(2*f*(c + d*x)*PolyLog[3, (a - b)/((a + b)*E^(2*(e + f*x)))] + d*P olyLog[4, (a - b)/((a + b)*E^(2*(e + f*x)))])))/((a - b)*(a + b)*f^4) + (x *(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)*Sinh[e])/(b*Cosh[e] + a*Sinh[ e]))/4
Time = 0.97 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.95, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 4214, 25, 2620, 3011, 7163, 2720, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(c+d x)^3}{a+b \coth (e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c+d x)^3}{a-i b \tan \left (i e+i f x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 4214 |
| \(\displaystyle 2 b \int -\frac {e^{-2 (e+f x)} (c+d x)^3}{(a+b)^2-\left (a^2-b^2\right ) e^{-2 (e+f x)}}dx+\frac {(c+d x)^4}{4 d (a+b)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(c+d x)^4}{4 d (a+b)}-2 b \int \frac {e^{-2 (e+f x)} (c+d x)^3}{(a+b)^2-\left (a^2-b^2\right ) e^{-2 (e+f x)}}dx\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {(c+d x)^4}{4 d (a+b)}-2 b \left (\frac {(c+d x)^3 \log \left (1-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 f \left (a^2-b^2\right )}-\frac {3 d \int (c+d x)^2 \log \left (1-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )dx}{2 f \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {(c+d x)^4}{4 d (a+b)}-2 b \left (\frac {(c+d x)^3 \log \left (1-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 f \left (a^2-b^2\right )}-\frac {3 d \left (\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 f}-\frac {d \int (c+d x) \operatorname {PolyLog}\left (2,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )dx}{f}\right )}{2 f \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle \frac {(c+d x)^4}{4 d (a+b)}-2 b \left (\frac {(c+d x)^3 \log \left (1-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 f \left (a^2-b^2\right )}-\frac {3 d \left (\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 f}-\frac {d \left (\frac {d \int \operatorname {PolyLog}\left (3,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )dx}{2 f}-\frac {(c+d x) \operatorname {PolyLog}\left (3,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 f}\right )}{f}\right )}{2 f \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {(c+d x)^4}{4 d (a+b)}-2 b \left (\frac {(c+d x)^3 \log \left (1-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 f \left (a^2-b^2\right )}-\frac {3 d \left (\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 f}-\frac {d \left (-\frac {d \int e^{2 (e+f x)} \operatorname {PolyLog}\left (3,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )de^{-2 (e+f x)}}{4 f^2}-\frac {(c+d x) \operatorname {PolyLog}\left (3,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 f}\right )}{f}\right )}{2 f \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {(c+d x)^4}{4 d (a+b)}-2 b \left (\frac {(c+d x)^3 \log \left (1-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 f \left (a^2-b^2\right )}-\frac {3 d \left (\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 f}-\frac {d \left (-\frac {(c+d x) \operatorname {PolyLog}\left (3,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{2 f}-\frac {d \operatorname {PolyLog}\left (4,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{4 f^2}\right )}{f}\right )}{2 f \left (a^2-b^2\right )}\right )\) |
(c + d*x)^4/(4*(a + b)*d) - 2*b*(((c + d*x)^3*Log[1 - (a - b)/((a + b)*E^( 2*(e + f*x)))])/(2*(a^2 - b^2)*f) - (3*d*(((c + d*x)^2*PolyLog[2, (a - b)/ ((a + b)*E^(2*(e + f*x)))])/(2*f) - (d*(-1/2*((c + d*x)*PolyLog[3, (a - b) /((a + b)*E^(2*(e + f*x)))])/f - (d*PolyLog[4, (a - b)/((a + b)*E^(2*(e + f*x)))])/(4*f^2)))/f))/(2*(a^2 - b^2)*f))
3.1.52.3.1 Defintions of rubi rules used
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] - Si mp[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]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
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] + Simp[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]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*tan[(e_.) + Pi*(k_.) + (f_.)*( x_)]), x_Symbol] :> Simp[(c + d*x)^(m + 1)/(d*(m + 1)*(a + I*b)), x] + Simp [2*I*b Int[(c + d*x)^m*E^(2*I*k*Pi)*(E^Simp[2*I*(e + f*x), x]/((a + I*b)^ 2 + (a^2 + b^2)*E^(2*I*k*Pi)*E^Simp[2*I*(e + f*x), x])), x], x] /; FreeQ[{a , b, c, d, e, f}, x] && IntegerQ[4*k] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> 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]
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] - Simp[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]
Leaf count of result is larger than twice the leaf count of optimal. \(1157\) vs. \(2(209)=418\).
Time = 0.45 (sec) , antiderivative size = 1158, normalized size of antiderivative = 5.51
1/4/(a+b)*d^3*x^4+1/4/(a+b)/d*c^4+1/2*b/(a+b)/(a-b)*d^3*x^4+3/2/f^4*b/(a+b )/(a-b)*d^3*e^4-3/4/f^4*b/(a+b)/(a-b)*d^3*polylog(4,(a+b)*exp(2*f*x+2*e)/( a-b))-1/f*b/(a+b)*c^3/(a-b)*ln(exp(2*f*x+2*e)*a+b*exp(2*f*x+2*e)-a+b)+1/(a +b)*d^2*c*x^3+3/2/(a+b)*d*c^2*x^2+1/(a+b)*c^3*x+2/f^3*b/(a+b)/(a-b)*d^3*e^ 3*x+2*b/(a+b)/(a-b)*d^2*c*x^3-4/f^3*b/(a+b)/(a-b)*d^2*c*e^3-1/f*b/(a+b)/(a -b)*d^3*ln(1-(a+b)*exp(2*f*x+2*e)/(a-b))*x^3-3/2/f^2*b/(a+b)/(a-b)*d^3*pol ylog(2,(a+b)*exp(2*f*x+2*e)/(a-b))*x^2+3/2/f^3*b/(a+b)/(a-b)*d^2*c*polylog (3,(a+b)*exp(2*f*x+2*e)/(a-b))+1/f^4*b/(a+b)*d^3*e^3/(a-b)*ln(exp(2*f*x+2* e)*a+b*exp(2*f*x+2*e)-a+b)-3/2/f^2*b/(a+b)/(a-b)*d*c^2*polylog(2,(a+b)*exp (2*f*x+2*e)/(a-b))-1/f^4*b/(a+b)/(a-b)*d^3*ln(1-(a+b)*exp(2*f*x+2*e)/(a-b) )*e^3+3/2/f^3*b/(a+b)/(a-b)*d^3*polylog(3,(a+b)*exp(2*f*x+2*e)/(a-b))*x-3/ f*b/(a+b)/(a-b)*d^2*c*ln(1-(a+b)*exp(2*f*x+2*e)/(a-b))*x^2+3/f^3*b/(a+b)/( a-b)*d^2*c*ln(1-(a+b)*exp(2*f*x+2*e)/(a-b))*e^2-2/f^4*b/(a+b)*d^3*e^3/(a-b )*ln(exp(f*x+e))+3*b/(a+b)/(a-b)*d*c^2*x^2+3/f^2*b/(a+b)/(a-b)*d*c^2*e^2-6 /f^2*b/(a+b)*d*c^2*e/(a-b)*ln(exp(f*x+e))+6/f^3*b/(a+b)*d^2*c*e^2/(a-b)*ln (exp(f*x+e))+6/f*b/(a+b)/(a-b)*d*c^2*e*x-6/f^2*b/(a+b)/(a-b)*d^2*c*e^2*x-3 /f^2*b/(a+b)/(a-b)*d^2*c*polylog(2,(a+b)*exp(2*f*x+2*e)/(a-b))*x+3/f^2*b/( a+b)*d*c^2*e/(a-b)*ln(exp(2*f*x+2*e)*a+b*exp(2*f*x+2*e)-a+b)-3/f*b/(a+b)/( a-b)*d*c^2*ln(1-(a+b)*exp(2*f*x+2*e)/(a-b))*x-3/f^2*b/(a+b)/(a-b)*d*c^2*ln (1-(a+b)*exp(2*f*x+2*e)/(a-b))*e-3/f^3*b/(a+b)*d^2*c*e^2/(a-b)*ln(exp(2...
Leaf count of result is larger than twice the leaf count of optimal. 730 vs. \(2 (201) = 402\).
Time = 0.28 (sec) , antiderivative size = 730, normalized size of antiderivative = 3.48 \[ \int \frac {(c+d x)^3}{a+b \coth (e+f x)} \, dx=\frac {{\left (a + b\right )} d^{3} f^{4} x^{4} + 4 \, {\left (a + b\right )} c d^{2} f^{4} x^{3} + 6 \, {\left (a + b\right )} c^{2} d f^{4} x^{2} + 4 \, {\left (a + b\right )} c^{3} f^{4} x - 24 \, b d^{3} {\rm polylog}\left (4, \sqrt {\frac {a + b}{a - b}} {\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )}\right ) - 24 \, b d^{3} {\rm polylog}\left (4, -\sqrt {\frac {a + b}{a - b}} {\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )}\right ) - 12 \, {\left (b d^{3} f^{2} x^{2} + 2 \, b c d^{2} f^{2} x + b c^{2} d f^{2}\right )} {\rm Li}_2\left (\sqrt {\frac {a + b}{a - b}} {\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )}\right ) - 12 \, {\left (b d^{3} f^{2} x^{2} + 2 \, b c d^{2} f^{2} x + b c^{2} d f^{2}\right )} {\rm Li}_2\left (-\sqrt {\frac {a + b}{a - b}} {\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )}\right ) + 4 \, {\left (b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + 3 \, b c^{2} d e f^{2} - b c^{3} f^{3}\right )} \log \left (2 \, {\left (a + b\right )} \cosh \left (f x + e\right ) + 2 \, {\left (a + b\right )} \sinh \left (f x + e\right ) + 2 \, {\left (a - b\right )} \sqrt {\frac {a + b}{a - b}}\right ) + 4 \, {\left (b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + 3 \, b c^{2} d e f^{2} - b c^{3} f^{3}\right )} \log \left (2 \, {\left (a + b\right )} \cosh \left (f x + e\right ) + 2 \, {\left (a + b\right )} \sinh \left (f x + e\right ) - 2 \, {\left (a - b\right )} \sqrt {\frac {a + b}{a - b}}\right ) - 4 \, {\left (b d^{3} f^{3} x^{3} + 3 \, b c d^{2} f^{3} x^{2} + 3 \, b c^{2} d f^{3} x + b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + 3 \, b c^{2} d e f^{2}\right )} \log \left (\sqrt {\frac {a + b}{a - b}} {\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )} + 1\right ) - 4 \, {\left (b d^{3} f^{3} x^{3} + 3 \, b c d^{2} f^{3} x^{2} + 3 \, b c^{2} d f^{3} x + b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + 3 \, b c^{2} d e f^{2}\right )} \log \left (-\sqrt {\frac {a + b}{a - b}} {\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )} + 1\right ) + 24 \, {\left (b d^{3} f x + b c d^{2} f\right )} {\rm polylog}\left (3, \sqrt {\frac {a + b}{a - b}} {\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )}\right ) + 24 \, {\left (b d^{3} f x + b c d^{2} f\right )} {\rm polylog}\left (3, -\sqrt {\frac {a + b}{a - b}} {\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )}\right )}{4 \, {\left (a^{2} - b^{2}\right )} f^{4}} \]
1/4*((a + b)*d^3*f^4*x^4 + 4*(a + b)*c*d^2*f^4*x^3 + 6*(a + b)*c^2*d*f^4*x ^2 + 4*(a + b)*c^3*f^4*x - 24*b*d^3*polylog(4, sqrt((a + b)/(a - b))*(cosh (f*x + e) + sinh(f*x + e))) - 24*b*d^3*polylog(4, -sqrt((a + b)/(a - b))*( cosh(f*x + e) + sinh(f*x + e))) - 12*(b*d^3*f^2*x^2 + 2*b*c*d^2*f^2*x + b* c^2*d*f^2)*dilog(sqrt((a + b)/(a - b))*(cosh(f*x + e) + sinh(f*x + e))) - 12*(b*d^3*f^2*x^2 + 2*b*c*d^2*f^2*x + b*c^2*d*f^2)*dilog(-sqrt((a + b)/(a - b))*(cosh(f*x + e) + sinh(f*x + e))) + 4*(b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^3)*log(2*(a + b)*cosh(f*x + e) + 2*(a + b)*sinh( f*x + e) + 2*(a - b)*sqrt((a + b)/(a - b))) + 4*(b*d^3*e^3 - 3*b*c*d^2*e^2 *f + 3*b*c^2*d*e*f^2 - b*c^3*f^3)*log(2*(a + b)*cosh(f*x + e) + 2*(a + b)* sinh(f*x + e) - 2*(a - b)*sqrt((a + b)/(a - b))) - 4*(b*d^3*f^3*x^3 + 3*b* c*d^2*f^3*x^2 + 3*b*c^2*d*f^3*x + b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d* e*f^2)*log(sqrt((a + b)/(a - b))*(cosh(f*x + e) + sinh(f*x + e)) + 1) - 4* (b*d^3*f^3*x^3 + 3*b*c*d^2*f^3*x^2 + 3*b*c^2*d*f^3*x + b*d^3*e^3 - 3*b*c*d ^2*e^2*f + 3*b*c^2*d*e*f^2)*log(-sqrt((a + b)/(a - b))*(cosh(f*x + e) + si nh(f*x + e)) + 1) + 24*(b*d^3*f*x + b*c*d^2*f)*polylog(3, sqrt((a + b)/(a - b))*(cosh(f*x + e) + sinh(f*x + e))) + 24*(b*d^3*f*x + b*c*d^2*f)*polylo g(3, -sqrt((a + b)/(a - b))*(cosh(f*x + e) + sinh(f*x + e))))/((a^2 - b^2) *f^4)
\[ \int \frac {(c+d x)^3}{a+b \coth (e+f x)} \, dx=\int \frac {\left (c + d x\right )^{3}}{a + b \coth {\left (e + f x \right )}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 521 vs. \(2 (201) = 402\).
Time = 0.32 (sec) , antiderivative size = 521, normalized size of antiderivative = 2.48 \[ \int \frac {(c+d x)^3}{a+b \coth (e+f x)} \, dx=-\frac {3 \, {\left (2 \, f x \log \left (-\frac {{\left (a e^{\left (2 \, e\right )} + b e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a - b} + 1\right ) + {\rm Li}_2\left (\frac {{\left (a e^{\left (2 \, e\right )} + b e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a - b}\right )\right )} b c^{2} d}{2 \, {\left (a^{2} f^{2} - b^{2} f^{2}\right )}} - \frac {3 \, {\left (2 \, f^{2} x^{2} \log \left (-\frac {{\left (a e^{\left (2 \, e\right )} + b e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a - b} + 1\right ) + 2 \, f x {\rm Li}_2\left (\frac {{\left (a e^{\left (2 \, e\right )} + b e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a - b}\right ) - {\rm Li}_{3}(\frac {{\left (a e^{\left (2 \, e\right )} + b e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a - b})\right )} b c d^{2}}{2 \, {\left (a^{2} f^{3} - b^{2} f^{3}\right )}} - \frac {{\left (4 \, f^{3} x^{3} \log \left (-\frac {{\left (a e^{\left (2 \, e\right )} + b e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a - b} + 1\right ) + 6 \, f^{2} x^{2} {\rm Li}_2\left (\frac {{\left (a e^{\left (2 \, e\right )} + b e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a - b}\right ) - 6 \, f x {\rm Li}_{3}(\frac {{\left (a e^{\left (2 \, e\right )} + b e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a - b}) + 3 \, {\rm Li}_{4}(\frac {{\left (a e^{\left (2 \, e\right )} + b e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a - b})\right )} b d^{3}}{3 \, {\left (a^{2} f^{4} - b^{2} f^{4}\right )}} - c^{3} {\left (\frac {b \log \left (-{\left (a - b\right )} e^{\left (-2 \, f x - 2 \, e\right )} + a + b\right )}{{\left (a^{2} - b^{2}\right )} f} - \frac {f x + e}{{\left (a + b\right )} f}\right )} + \frac {b d^{3} f^{4} x^{4} + 4 \, b c d^{2} f^{4} x^{3} + 6 \, b c^{2} d f^{4} x^{2}}{2 \, {\left (a^{2} f^{4} - b^{2} f^{4}\right )}} + \frac {d^{3} x^{4} + 4 \, c d^{2} x^{3} + 6 \, c^{2} d x^{2}}{4 \, {\left (a + b\right )}} \]
-3/2*(2*f*x*log(-(a*e^(2*e) + b*e^(2*e))*e^(2*f*x)/(a - b) + 1) + dilog((a *e^(2*e) + b*e^(2*e))*e^(2*f*x)/(a - b)))*b*c^2*d/(a^2*f^2 - b^2*f^2) - 3/ 2*(2*f^2*x^2*log(-(a*e^(2*e) + b*e^(2*e))*e^(2*f*x)/(a - b) + 1) + 2*f*x*d ilog((a*e^(2*e) + b*e^(2*e))*e^(2*f*x)/(a - b)) - polylog(3, (a*e^(2*e) + b*e^(2*e))*e^(2*f*x)/(a - b)))*b*c*d^2/(a^2*f^3 - b^2*f^3) - 1/3*(4*f^3*x^ 3*log(-(a*e^(2*e) + b*e^(2*e))*e^(2*f*x)/(a - b) + 1) + 6*f^2*x^2*dilog((a *e^(2*e) + b*e^(2*e))*e^(2*f*x)/(a - b)) - 6*f*x*polylog(3, (a*e^(2*e) + b *e^(2*e))*e^(2*f*x)/(a - b)) + 3*polylog(4, (a*e^(2*e) + b*e^(2*e))*e^(2*f *x)/(a - b)))*b*d^3/(a^2*f^4 - b^2*f^4) - c^3*(b*log(-(a - b)*e^(-2*f*x - 2*e) + a + b)/((a^2 - b^2)*f) - (f*x + e)/((a + b)*f)) + 1/2*(b*d^3*f^4*x^ 4 + 4*b*c*d^2*f^4*x^3 + 6*b*c^2*d*f^4*x^2)/(a^2*f^4 - b^2*f^4) + 1/4*(d^3* x^4 + 4*c*d^2*x^3 + 6*c^2*d*x^2)/(a + b)
\[ \int \frac {(c+d x)^3}{a+b \coth (e+f x)} \, dx=\int { \frac {{\left (d x + c\right )}^{3}}{b \coth \left (f x + e\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(c+d x)^3}{a+b \coth (e+f x)} \, dx=\int \frac {{\left (c+d\,x\right )}^3}{a+b\,\mathrm {coth}\left (e+f\,x\right )} \,d x \]