Integrand size = 15, antiderivative size = 299 \[ \int (e+f x)^3 \cot ^{-1}(\coth (a+b x)) \, dx=\frac {(e+f x)^4 \cot ^{-1}(\coth (a+b x))}{4 f}-\frac {(e+f x)^4 \arctan \left (e^{2 a+2 b x}\right )}{4 f}+\frac {i (e+f x)^3 \operatorname {PolyLog}\left (2,-i e^{2 a+2 b x}\right )}{4 b}-\frac {i (e+f x)^3 \operatorname {PolyLog}\left (2,i e^{2 a+2 b x}\right )}{4 b}-\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (3,-i e^{2 a+2 b x}\right )}{8 b^2}+\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (3,i e^{2 a+2 b x}\right )}{8 b^2}+\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (4,-i e^{2 a+2 b x}\right )}{8 b^3}-\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (4,i e^{2 a+2 b x}\right )}{8 b^3}-\frac {3 i f^3 \operatorname {PolyLog}\left (5,-i e^{2 a+2 b x}\right )}{16 b^4}+\frac {3 i f^3 \operatorname {PolyLog}\left (5,i e^{2 a+2 b x}\right )}{16 b^4} \]
1/4*(f*x+e)^4*arccot(coth(b*x+a))/f-1/4*(f*x+e)^4*arctan(exp(2*b*x+2*a))/f +1/4*I*(f*x+e)^3*polylog(2,-I*exp(2*b*x+2*a))/b-1/4*I*(f*x+e)^3*polylog(2, I*exp(2*b*x+2*a))/b-3/8*I*f*(f*x+e)^2*polylog(3,-I*exp(2*b*x+2*a))/b^2+3/8 *I*f*(f*x+e)^2*polylog(3,I*exp(2*b*x+2*a))/b^2+3/8*I*f^2*(f*x+e)*polylog(4 ,-I*exp(2*b*x+2*a))/b^3-3/8*I*f^2*(f*x+e)*polylog(4,I*exp(2*b*x+2*a))/b^3- 3/16*I*f^3*polylog(5,-I*exp(2*b*x+2*a))/b^4+3/16*I*f^3*polylog(5,I*exp(2*b *x+2*a))/b^4
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(600\) vs. \(2(299)=598\).
Time = 0.22 (sec) , antiderivative size = 600, normalized size of antiderivative = 2.01 \[ \int (e+f x)^3 \cot ^{-1}(\coth (a+b x)) \, dx=\frac {1}{4} x \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right ) \cot ^{-1}(\coth (a+b x))-\frac {i \left (8 b^4 e^3 x \log \left (1-i e^{2 (a+b x)}\right )+12 b^4 e^2 f x^2 \log \left (1-i e^{2 (a+b x)}\right )+8 b^4 e f^2 x^3 \log \left (1-i e^{2 (a+b x)}\right )+2 b^4 f^3 x^4 \log \left (1-i e^{2 (a+b x)}\right )-8 b^4 e^3 x \log \left (1+i e^{2 (a+b x)}\right )-12 b^4 e^2 f x^2 \log \left (1+i e^{2 (a+b x)}\right )-8 b^4 e f^2 x^3 \log \left (1+i e^{2 (a+b x)}\right )-2 b^4 f^3 x^4 \log \left (1+i e^{2 (a+b x)}\right )-4 b^3 (e+f x)^3 \operatorname {PolyLog}\left (2,-i e^{2 (a+b x)}\right )+4 b^3 (e+f x)^3 \operatorname {PolyLog}\left (2,i e^{2 (a+b x)}\right )+6 b^2 e^2 f \operatorname {PolyLog}\left (3,-i e^{2 (a+b x)}\right )+12 b^2 e f^2 x \operatorname {PolyLog}\left (3,-i e^{2 (a+b x)}\right )+6 b^2 f^3 x^2 \operatorname {PolyLog}\left (3,-i e^{2 (a+b x)}\right )-6 b^2 e^2 f \operatorname {PolyLog}\left (3,i e^{2 (a+b x)}\right )-12 b^2 e f^2 x \operatorname {PolyLog}\left (3,i e^{2 (a+b x)}\right )-6 b^2 f^3 x^2 \operatorname {PolyLog}\left (3,i e^{2 (a+b x)}\right )-6 b e f^2 \operatorname {PolyLog}\left (4,-i e^{2 (a+b x)}\right )-6 b f^3 x \operatorname {PolyLog}\left (4,-i e^{2 (a+b x)}\right )+6 b e f^2 \operatorname {PolyLog}\left (4,i e^{2 (a+b x)}\right )+6 b f^3 x \operatorname {PolyLog}\left (4,i e^{2 (a+b x)}\right )+3 f^3 \operatorname {PolyLog}\left (5,-i e^{2 (a+b x)}\right )-3 f^3 \operatorname {PolyLog}\left (5,i e^{2 (a+b x)}\right )\right )}{16 b^4} \]
(x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3)*ArcCot[Coth[a + b*x]])/4 - ((I/16)*(8*b^4*e^3*x*Log[1 - I*E^(2*(a + b*x))] + 12*b^4*e^2*f*x^2*Log[1 - I*E^(2*(a + b*x))] + 8*b^4*e*f^2*x^3*Log[1 - I*E^(2*(a + b*x))] + 2*b^4*f ^3*x^4*Log[1 - I*E^(2*(a + b*x))] - 8*b^4*e^3*x*Log[1 + I*E^(2*(a + b*x))] - 12*b^4*e^2*f*x^2*Log[1 + I*E^(2*(a + b*x))] - 8*b^4*e*f^2*x^3*Log[1 + I *E^(2*(a + b*x))] - 2*b^4*f^3*x^4*Log[1 + I*E^(2*(a + b*x))] - 4*b^3*(e + f*x)^3*PolyLog[2, (-I)*E^(2*(a + b*x))] + 4*b^3*(e + f*x)^3*PolyLog[2, I*E ^(2*(a + b*x))] + 6*b^2*e^2*f*PolyLog[3, (-I)*E^(2*(a + b*x))] + 12*b^2*e* f^2*x*PolyLog[3, (-I)*E^(2*(a + b*x))] + 6*b^2*f^3*x^2*PolyLog[3, (-I)*E^( 2*(a + b*x))] - 6*b^2*e^2*f*PolyLog[3, I*E^(2*(a + b*x))] - 12*b^2*e*f^2*x *PolyLog[3, I*E^(2*(a + b*x))] - 6*b^2*f^3*x^2*PolyLog[3, I*E^(2*(a + b*x) )] - 6*b*e*f^2*PolyLog[4, (-I)*E^(2*(a + b*x))] - 6*b*f^3*x*PolyLog[4, (-I )*E^(2*(a + b*x))] + 6*b*e*f^2*PolyLog[4, I*E^(2*(a + b*x))] + 6*b*f^3*x*P olyLog[4, I*E^(2*(a + b*x))] + 3*f^3*PolyLog[5, (-I)*E^(2*(a + b*x))] - 3* f^3*PolyLog[5, I*E^(2*(a + b*x))]))/b^4
Time = 1.16 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {5709, 3042, 4668, 3011, 7163, 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.
\(\displaystyle \int (e+f x)^3 \cot ^{-1}(\coth (a+b x)) \, dx\) |
\(\Big \downarrow \) 5709 |
\(\displaystyle \frac {(e+f x)^4 \cot ^{-1}(\coth (a+b x))}{4 f}-\frac {b \int (e+f x)^4 \text {sech}(2 a+2 b x)dx}{4 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {(e+f x)^4 \cot ^{-1}(\coth (a+b x))}{4 f}-\frac {b \int (e+f x)^4 \csc \left (2 i a+2 i b x+\frac {\pi }{2}\right )dx}{4 f}\) |
\(\Big \downarrow \) 4668 |
\(\displaystyle \frac {(e+f x)^4 \cot ^{-1}(\coth (a+b x))}{4 f}-\frac {b \left (-\frac {2 i f \int (e+f x)^3 \log \left (1-i e^{2 a+2 b x}\right )dx}{b}+\frac {2 i f \int (e+f x)^3 \log \left (1+i e^{2 a+2 b x}\right )dx}{b}+\frac {(e+f x)^4 \arctan \left (e^{2 a+2 b x}\right )}{b}\right )}{4 f}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {(e+f x)^4 \cot ^{-1}(\coth (a+b x))}{4 f}-\frac {b \left (\frac {2 i f \left (\frac {3 f \int (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{2 a+2 b x}\right )dx}{2 b}-\frac {(e+f x)^3 \operatorname {PolyLog}\left (2,-i e^{2 a+2 b x}\right )}{2 b}\right )}{b}-\frac {2 i f \left (\frac {3 f \int (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{2 a+2 b x}\right )dx}{2 b}-\frac {(e+f x)^3 \operatorname {PolyLog}\left (2,i e^{2 a+2 b x}\right )}{2 b}\right )}{b}+\frac {(e+f x)^4 \arctan \left (e^{2 a+2 b x}\right )}{b}\right )}{4 f}\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle \frac {(e+f x)^4 \cot ^{-1}(\coth (a+b x))}{4 f}-\frac {b \left (\frac {2 i f \left (\frac {3 f \left (\frac {(e+f x)^2 \operatorname {PolyLog}\left (3,-i e^{2 a+2 b x}\right )}{2 b}-\frac {f \int (e+f x) \operatorname {PolyLog}\left (3,-i e^{2 a+2 b x}\right )dx}{b}\right )}{2 b}-\frac {(e+f x)^3 \operatorname {PolyLog}\left (2,-i e^{2 a+2 b x}\right )}{2 b}\right )}{b}-\frac {2 i f \left (\frac {3 f \left (\frac {(e+f x)^2 \operatorname {PolyLog}\left (3,i e^{2 a+2 b x}\right )}{2 b}-\frac {f \int (e+f x) \operatorname {PolyLog}\left (3,i e^{2 a+2 b x}\right )dx}{b}\right )}{2 b}-\frac {(e+f x)^3 \operatorname {PolyLog}\left (2,i e^{2 a+2 b x}\right )}{2 b}\right )}{b}+\frac {(e+f x)^4 \arctan \left (e^{2 a+2 b x}\right )}{b}\right )}{4 f}\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle \frac {(e+f x)^4 \cot ^{-1}(\coth (a+b x))}{4 f}-\frac {b \left (\frac {2 i f \left (\frac {3 f \left (\frac {(e+f x)^2 \operatorname {PolyLog}\left (3,-i e^{2 a+2 b x}\right )}{2 b}-\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (4,-i e^{2 a+2 b x}\right )}{2 b}-\frac {f \int \operatorname {PolyLog}\left (4,-i e^{2 a+2 b x}\right )dx}{2 b}\right )}{b}\right )}{2 b}-\frac {(e+f x)^3 \operatorname {PolyLog}\left (2,-i e^{2 a+2 b x}\right )}{2 b}\right )}{b}-\frac {2 i f \left (\frac {3 f \left (\frac {(e+f x)^2 \operatorname {PolyLog}\left (3,i e^{2 a+2 b x}\right )}{2 b}-\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (4,i e^{2 a+2 b x}\right )}{2 b}-\frac {f \int \operatorname {PolyLog}\left (4,i e^{2 a+2 b x}\right )dx}{2 b}\right )}{b}\right )}{2 b}-\frac {(e+f x)^3 \operatorname {PolyLog}\left (2,i e^{2 a+2 b x}\right )}{2 b}\right )}{b}+\frac {(e+f x)^4 \arctan \left (e^{2 a+2 b x}\right )}{b}\right )}{4 f}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {(e+f x)^4 \cot ^{-1}(\coth (a+b x))}{4 f}-\frac {b \left (\frac {2 i f \left (\frac {3 f \left (\frac {(e+f x)^2 \operatorname {PolyLog}\left (3,-i e^{2 a+2 b x}\right )}{2 b}-\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (4,-i e^{2 a+2 b x}\right )}{2 b}-\frac {f \int e^{-2 a-2 b x} \operatorname {PolyLog}\left (4,-i e^{2 a+2 b x}\right )de^{2 a+2 b x}}{4 b^2}\right )}{b}\right )}{2 b}-\frac {(e+f x)^3 \operatorname {PolyLog}\left (2,-i e^{2 a+2 b x}\right )}{2 b}\right )}{b}-\frac {2 i f \left (\frac {3 f \left (\frac {(e+f x)^2 \operatorname {PolyLog}\left (3,i e^{2 a+2 b x}\right )}{2 b}-\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (4,i e^{2 a+2 b x}\right )}{2 b}-\frac {f \int e^{-2 a-2 b x} \operatorname {PolyLog}\left (4,i e^{2 a+2 b x}\right )de^{2 a+2 b x}}{4 b^2}\right )}{b}\right )}{2 b}-\frac {(e+f x)^3 \operatorname {PolyLog}\left (2,i e^{2 a+2 b x}\right )}{2 b}\right )}{b}+\frac {(e+f x)^4 \arctan \left (e^{2 a+2 b x}\right )}{b}\right )}{4 f}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {(e+f x)^4 \cot ^{-1}(\coth (a+b x))}{4 f}-\frac {b \left (\frac {(e+f x)^4 \arctan \left (e^{2 a+2 b x}\right )}{b}+\frac {2 i f \left (\frac {3 f \left (\frac {(e+f x)^2 \operatorname {PolyLog}\left (3,-i e^{2 a+2 b x}\right )}{2 b}-\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (4,-i e^{2 a+2 b x}\right )}{2 b}-\frac {f \operatorname {PolyLog}\left (5,-i e^{2 a+2 b x}\right )}{4 b^2}\right )}{b}\right )}{2 b}-\frac {(e+f x)^3 \operatorname {PolyLog}\left (2,-i e^{2 a+2 b x}\right )}{2 b}\right )}{b}-\frac {2 i f \left (\frac {3 f \left (\frac {(e+f x)^2 \operatorname {PolyLog}\left (3,i e^{2 a+2 b x}\right )}{2 b}-\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (4,i e^{2 a+2 b x}\right )}{2 b}-\frac {f \operatorname {PolyLog}\left (5,i e^{2 a+2 b x}\right )}{4 b^2}\right )}{b}\right )}{2 b}-\frac {(e+f x)^3 \operatorname {PolyLog}\left (2,i e^{2 a+2 b x}\right )}{2 b}\right )}{b}\right )}{4 f}\) |
((e + f*x)^4*ArcCot[Coth[a + b*x]])/(4*f) - (b*(((e + f*x)^4*ArcTan[E^(2*a + 2*b*x)])/b + ((2*I)*f*(-1/2*((e + f*x)^3*PolyLog[2, (-I)*E^(2*a + 2*b*x )])/b + (3*f*(((e + f*x)^2*PolyLog[3, (-I)*E^(2*a + 2*b*x)])/(2*b) - (f*(( (e + f*x)*PolyLog[4, (-I)*E^(2*a + 2*b*x)])/(2*b) - (f*PolyLog[5, (-I)*E^( 2*a + 2*b*x)])/(4*b^2)))/b))/(2*b)))/b - ((2*I)*f*(-1/2*((e + f*x)^3*PolyL og[2, I*E^(2*a + 2*b*x)])/b + (3*f*(((e + f*x)^2*PolyLog[3, I*E^(2*a + 2*b *x)])/(2*b) - (f*(((e + f*x)*PolyLog[4, I*E^(2*a + 2*b*x)])/(2*b) - (f*Pol yLog[5, I*E^(2*a + 2*b*x)])/(4*b^2)))/b))/(2*b)))/b))/(4*f)
3.2.100.3.1 Defintions of rubi rules used
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[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ ))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[ 1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c , d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Int[ArcCot[Coth[(a_.) + (b_.)*(x_)]]*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^(m + 1)*(ArcCot[Coth[a + b*x]]/(f*(m + 1))), x] - Simp[b/ (f*(m + 1)) Int[(e + f*x)^(m + 1)*Sech[2*a + 2*b*x], x], x] /; FreeQ[{a, b, e, f}, x] && 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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 33.25 (sec) , antiderivative size = 3570, normalized size of antiderivative = 11.94
3/16*I*f^3*polylog(5,I*exp(2*b*x+2*a))/b^4-3/16*I*f^3*polylog(5,-I*exp(2*b *x+2*a))/b^4+3/2*I*f/b*e^2*ln(1+I*exp(2*b*x+2*a))*a*x+3/2*I*f^2/b^2*a^2*e* ln(1+exp(b*x+a)*(-1)^(3/4))*x+3/2*I*f^2/b^2*a^2*e*ln(1-exp(b*x+a)*(-1)^(3/ 4))*x-3/2*I*f/b*a*e^2*ln(1+exp(b*x+a)*(-1)^(3/4))*x-3/2*I*f/b*a*e^2*ln(1-e xp(b*x+a)*(-1)^(3/4))*x-3/8*I*f^3/b^3*polylog(4,I*exp(2*b*x+2*a))*x+1/2*I* f^3/b^4*a^4*ln(((-I)^(1/2)-exp(b*x+a))/(-I)^(1/2))+1/2*I*f^3/b^4*a^4*ln((( -I)^(1/2)+exp(b*x+a))/(-I)^(1/2))+1/2*I*f^3/b^4*a^3*dilog(((-I)^(1/2)-exp( b*x+a))/(-I)^(1/2))+1/2*I*f^3/b^4*a^3*dilog(((-I)^(1/2)+exp(b*x+a))/(-I)^( 1/2))-1/8*I*f^3/b^4*a^4*ln(exp(2*b*x+2*a)+I)-1/2*I*f^2*e*ln(1-I*exp(2*b*x+ 2*a))*x^3-3/4*I*f*e^2*ln(1-I*exp(2*b*x+2*a))*x^2-3/8*I*f^2/b^3*e*polylog(4 ,I*exp(2*b*x+2*a))+3/8*I*f/b^2*e^2*polylog(3,I*exp(2*b*x+2*a))-3/8*I*f^3/b ^4*ln(1-I*exp(2*b*x+2*a))*a^4-1/4*I*f^3/b*polylog(2,I*exp(2*b*x+2*a))*x^3- 1/4*I*f^3/b^4*polylog(2,I*exp(2*b*x+2*a))*a^3+3/8*I*f^3/b^2*polylog(3,I*ex p(2*b*x+2*a))*x^2+1/2*I/b*a*e^3*ln(exp(2*b*x+2*a)+I)-1/2*I/b*e^3*ln(((-I)^ (1/2)-exp(b*x+a))/(-I)^(1/2))*a-1/2*I/b*e^3*ln(((-I)^(1/2)+exp(b*x+a))/(-I )^(1/2))*a-3/2*I*f/b^2*a^2*e^2*ln(1+exp(b*x+a)*(-1)^(3/4))-3/2*I*f/b^2*a^2 *e^2*ln(1-exp(b*x+a)*(-1)^(3/4))-3/2*I*f/b^2*a*e^2*dilog(1+exp(b*x+a)*(-1) ^(3/4))-3/2*I*f/b^2*a*e^2*dilog(1-exp(b*x+a)*(-1)^(3/4))-1/2*I*f^3/b^4*a^4 *ln(1-exp(b*x+a)*(-1)^(3/4))-1/2*I*f^3/b^4*a^3*dilog(1+exp(b*x+a)*(-1)^(3/ 4))-1/2*I*f^3/b^4*a^3*dilog(1-exp(b*x+a)*(-1)^(3/4))+3/8*I*f^2/b^3*e*po...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1460 vs. \(2 (236) = 472\).
Time = 0.33 (sec) , antiderivative size = 1460, normalized size of antiderivative = 4.88 \[ \int (e+f x)^3 \cot ^{-1}(\coth (a+b x)) \, dx=\text {Too large to display} \]
1/8*(24*I*f^3*polylog(5, 1/2*sqrt(4*I)*(cosh(b*x + a) + sinh(b*x + a))) + 24*I*f^3*polylog(5, -1/2*sqrt(4*I)*(cosh(b*x + a) + sinh(b*x + a))) - 24*I *f^3*polylog(5, 1/2*sqrt(-4*I)*(cosh(b*x + a) + sinh(b*x + a))) - 24*I*f^3 *polylog(5, -1/2*sqrt(-4*I)*(cosh(b*x + a) + sinh(b*x + a))) + 2*(b^4*f^3* x^4 + 4*b^4*e*f^2*x^3 + 6*b^4*e^2*f*x^2 + 4*b^4*e^3*x)*arctan(sinh(b*x + a )/cosh(b*x + a)) - 4*(I*b^3*f^3*x^3 + 3*I*b^3*e*f^2*x^2 + 3*I*b^3*e^2*f*x + I*b^3*e^3)*dilog(1/2*sqrt(4*I)*(cosh(b*x + a) + sinh(b*x + a))) - 4*(I*b ^3*f^3*x^3 + 3*I*b^3*e*f^2*x^2 + 3*I*b^3*e^2*f*x + I*b^3*e^3)*dilog(-1/2*s qrt(4*I)*(cosh(b*x + a) + sinh(b*x + a))) - 4*(-I*b^3*f^3*x^3 - 3*I*b^3*e* f^2*x^2 - 3*I*b^3*e^2*f*x - I*b^3*e^3)*dilog(1/2*sqrt(-4*I)*(cosh(b*x + a) + sinh(b*x + a))) - 4*(-I*b^3*f^3*x^3 - 3*I*b^3*e*f^2*x^2 - 3*I*b^3*e^2*f *x - I*b^3*e^3)*dilog(-1/2*sqrt(-4*I)*(cosh(b*x + a) + sinh(b*x + a))) + ( -I*b^4*f^3*x^4 - 4*I*b^4*e*f^2*x^3 - 6*I*b^4*e^2*f*x^2 - 4*I*b^4*e^3*x - 4 *I*a*b^3*e^3 + 6*I*a^2*b^2*e^2*f - 4*I*a^3*b*e*f^2 + I*a^4*f^3)*log(1/2*sq rt(4*I)*(cosh(b*x + a) + sinh(b*x + a)) + 1) + (-I*b^4*f^3*x^4 - 4*I*b^4*e *f^2*x^3 - 6*I*b^4*e^2*f*x^2 - 4*I*b^4*e^3*x - 4*I*a*b^3*e^3 + 6*I*a^2*b^2 *e^2*f - 4*I*a^3*b*e*f^2 + I*a^4*f^3)*log(-1/2*sqrt(4*I)*(cosh(b*x + a) + sinh(b*x + a)) + 1) + (I*b^4*f^3*x^4 + 4*I*b^4*e*f^2*x^3 + 6*I*b^4*e^2*f*x ^2 + 4*I*b^4*e^3*x + 4*I*a*b^3*e^3 - 6*I*a^2*b^2*e^2*f + 4*I*a^3*b*e*f^2 - I*a^4*f^3)*log(1/2*sqrt(-4*I)*(cosh(b*x + a) + sinh(b*x + a)) + 1) + (...
\[ \int (e+f x)^3 \cot ^{-1}(\coth (a+b x)) \, dx=\int \left (e + f x\right )^{3} \operatorname {acot}{\left (\coth {\left (a + b x \right )} \right )}\, dx \]
\[ \int (e+f x)^3 \cot ^{-1}(\coth (a+b x)) \, dx=\int { {\left (f x + e\right )}^{3} \operatorname {arccot}\left (\coth \left (b x + a\right )\right ) \,d x } \]
1/4*(f^3*x^4 + 4*e*f^2*x^3 + 6*e^2*f*x^2 + 4*e^3*x)*arctan((e^(2*b*x + 2*a ) - 1)/(e^(2*b*x + 2*a) + 1)) - integrate(1/2*(b*f^3*x^4*e^(2*a) + 4*b*e*f ^2*x^3*e^(2*a) + 6*b*e^2*f*x^2*e^(2*a) + 4*b*e^3*x*e^(2*a))*e^(2*b*x)/(e^( 4*b*x + 4*a) + 1), x)
Timed out. \[ \int (e+f x)^3 \cot ^{-1}(\coth (a+b x)) \, dx=\text {Timed out} \]
Timed out. \[ \int (e+f x)^3 \cot ^{-1}(\coth (a+b x)) \, dx=\int \mathrm {acot}\left (\mathrm {coth}\left (a+b\,x\right )\right )\,{\left (e+f\,x\right )}^3 \,d x \]