\(\int (e+f x)^2 \coth ^{-1}(\cot (a+b x)) \, dx\) [249]

Optimal result

Integrand size = 15, antiderivative size = 234 \[ \int (e+f x)^2 \coth ^{-1}(\cot (a+b x)) \, dx=\frac {(e+f x)^3 \coth ^{-1}(\cot (a+b x))}{3 f}+\frac {i (e+f x)^3 \arctan \left (e^{2 i (a+b x)}\right )}{3 f}-\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{2 i (a+b x)}\right )}{4 b}+\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{2 i (a+b x)}\right )}{4 b}+\frac {f (e+f x) \operatorname {PolyLog}\left (3,-i e^{2 i (a+b x)}\right )}{4 b^2}-\frac {f (e+f x) \operatorname {PolyLog}\left (3,i e^{2 i (a+b x)}\right )}{4 b^2}+\frac {i f^2 \operatorname {PolyLog}\left (4,-i e^{2 i (a+b x)}\right )}{8 b^3}-\frac {i f^2 \operatorname {PolyLog}\left (4,i e^{2 i (a+b x)}\right )}{8 b^3} \]

[Out]

1/3*(f*x+e)^3*arccoth(cot(b*x+a))/f+1/3*I*(f*x+e)^3*arctan(exp(2*I*(b*x+a)))/f-1/4*I*(f*x+e)^2*polylog(2,-I*ex
p(2*I*(b*x+a)))/b+1/4*I*(f*x+e)^2*polylog(2,I*exp(2*I*(b*x+a)))/b+1/4*f*(f*x+e)*polylog(3,-I*exp(2*I*(b*x+a)))
/b^2-1/4*f*(f*x+e)*polylog(3,I*exp(2*I*(b*x+a)))/b^2+1/8*I*f^2*polylog(4,-I*exp(2*I*(b*x+a)))/b^3-1/8*I*f^2*po
lylog(4,I*exp(2*I*(b*x+a)))/b^3

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6389, 4266, 2611, 6744, 2320, 6724} \[ \int (e+f x)^2 \coth ^{-1}(\cot (a+b x)) \, dx=\frac {i (e+f x)^3 \arctan \left (e^{2 i (a+b x)}\right )}{3 f}+\frac {i f^2 \operatorname {PolyLog}\left (4,-i e^{2 i (a+b x)}\right )}{8 b^3}-\frac {i f^2 \operatorname {PolyLog}\left (4,i e^{2 i (a+b x)}\right )}{8 b^3}+\frac {f (e+f x) \operatorname {PolyLog}\left (3,-i e^{2 i (a+b x)}\right )}{4 b^2}-\frac {f (e+f x) \operatorname {PolyLog}\left (3,i e^{2 i (a+b x)}\right )}{4 b^2}-\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{2 i (a+b x)}\right )}{4 b}+\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{2 i (a+b x)}\right )}{4 b}+\frac {(e+f x)^3 \coth ^{-1}(\cot (a+b x))}{3 f} \]

[In]

Int[(e + f*x)^2*ArcCoth[Cot[a + b*x]],x]

[Out]

((e + f*x)^3*ArcCoth[Cot[a + b*x]])/(3*f) + ((I/3)*(e + f*x)^3*ArcTan[E^((2*I)*(a + b*x))])/f - ((I/4)*(e + f*
x)^2*PolyLog[2, (-I)*E^((2*I)*(a + b*x))])/b + ((I/4)*(e + f*x)^2*PolyLog[2, I*E^((2*I)*(a + b*x))])/b + (f*(e
+ f*x)*PolyLog[3, (-I)*E^((2*I)*(a + b*x))])/(4*b^2) - (f*(e + f*x)*PolyLog[3, I*E^((2*I)*(a + b*x))])/(4*b^2
) + ((I/8)*f^2*PolyLog[4, (-I)*E^((2*I)*(a + b*x))])/b^3 - ((I/8)*f^2*PolyLog[4, I*E^((2*I)*(a + b*x))])/b^3

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 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 4266

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

Rule 6389

Int[ArcCoth[Cot[(a_.) + (b_.)*(x_)]]*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^(m + 1)*(ArcCoth[
Cot[a + b*x]]/(f*(m + 1))), x] - Dist[b/(f*(m + 1)), Int[(e + f*x)^(m + 1)*Sec[2*a + 2*b*x], x], x] /; FreeQ[{
a, b, e, f}, x] && IGtQ[m, 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*} \text {integral}& = \frac {(e+f x)^3 \coth ^{-1}(\cot (a+b x))}{3 f}-\frac {b \int (e+f x)^3 \sec (2 a+2 b x) \, dx}{3 f} \\ & = \frac {(e+f x)^3 \coth ^{-1}(\cot (a+b x))}{3 f}+\frac {i (e+f x)^3 \arctan \left (e^{2 i (a+b x)}\right )}{3 f}+\frac {1}{2} \int (e+f x)^2 \log \left (1-i e^{i (2 a+2 b x)}\right ) \, dx-\frac {1}{2} \int (e+f x)^2 \log \left (1+i e^{i (2 a+2 b x)}\right ) \, dx \\ & = \frac {(e+f x)^3 \coth ^{-1}(\cot (a+b x))}{3 f}+\frac {i (e+f x)^3 \arctan \left (e^{2 i (a+b x)}\right )}{3 f}-\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{2 i (a+b x)}\right )}{4 b}+\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{2 i (a+b x)}\right )}{4 b}+\frac {(i f) \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (2 a+2 b x)}\right ) \, dx}{2 b}-\frac {(i f) \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (2 a+2 b x)}\right ) \, dx}{2 b} \\ & = \frac {(e+f x)^3 \coth ^{-1}(\cot (a+b x))}{3 f}+\frac {i (e+f x)^3 \arctan \left (e^{2 i (a+b x)}\right )}{3 f}-\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{2 i (a+b x)}\right )}{4 b}+\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{2 i (a+b x)}\right )}{4 b}+\frac {f (e+f x) \operatorname {PolyLog}\left (3,-i e^{2 i (a+b x)}\right )}{4 b^2}-\frac {f (e+f x) \operatorname {PolyLog}\left (3,i e^{2 i (a+b x)}\right )}{4 b^2}-\frac {f^2 \int \operatorname {PolyLog}\left (3,-i e^{i (2 a+2 b x)}\right ) \, dx}{4 b^2}+\frac {f^2 \int \operatorname {PolyLog}\left (3,i e^{i (2 a+2 b x)}\right ) \, dx}{4 b^2} \\ & = \frac {(e+f x)^3 \coth ^{-1}(\cot (a+b x))}{3 f}+\frac {i (e+f x)^3 \arctan \left (e^{2 i (a+b x)}\right )}{3 f}-\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{2 i (a+b x)}\right )}{4 b}+\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{2 i (a+b x)}\right )}{4 b}+\frac {f (e+f x) \operatorname {PolyLog}\left (3,-i e^{2 i (a+b x)}\right )}{4 b^2}-\frac {f (e+f x) \operatorname {PolyLog}\left (3,i e^{2 i (a+b x)}\right )}{4 b^2}+\frac {\left (i f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{i (2 a+2 b x)}\right )}{8 b^3}-\frac {\left (i f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{i (2 a+2 b x)}\right )}{8 b^3} \\ & = \frac {(e+f x)^3 \coth ^{-1}(\cot (a+b x))}{3 f}+\frac {i (e+f x)^3 \arctan \left (e^{2 i (a+b x)}\right )}{3 f}-\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{2 i (a+b x)}\right )}{4 b}+\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{2 i (a+b x)}\right )}{4 b}+\frac {f (e+f x) \operatorname {PolyLog}\left (3,-i e^{2 i (a+b x)}\right )}{4 b^2}-\frac {f (e+f x) \operatorname {PolyLog}\left (3,i e^{2 i (a+b x)}\right )}{4 b^2}+\frac {i f^2 \operatorname {PolyLog}\left (4,-i e^{2 i (a+b x)}\right )}{8 b^3}-\frac {i f^2 \operatorname {PolyLog}\left (4,i e^{2 i (a+b x)}\right )}{8 b^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.75 \[ \int (e+f x)^2 \coth ^{-1}(\cot (a+b x)) \, dx=\frac {1}{3} x \left (3 e^2+3 e f x+f^2 x^2\right ) \coth ^{-1}(\cot (a+b x))+\frac {-12 b^3 e^2 x \log \left (1-i e^{2 i (a+b x)}\right )-12 b^3 e f x^2 \log \left (1-i e^{2 i (a+b x)}\right )-4 b^3 f^2 x^3 \log \left (1-i e^{2 i (a+b x)}\right )+12 b^3 e^2 x \log \left (1+i e^{2 i (a+b x)}\right )+12 b^3 e f x^2 \log \left (1+i e^{2 i (a+b x)}\right )+4 b^3 f^2 x^3 \log \left (1+i e^{2 i (a+b x)}\right )-6 i b^2 (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{2 i (a+b x)}\right )+6 i b^2 (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{2 i (a+b x)}\right )+6 b e f \operatorname {PolyLog}\left (3,-i e^{2 i (a+b x)}\right )+6 b f^2 x \operatorname {PolyLog}\left (3,-i e^{2 i (a+b x)}\right )-6 b e f \operatorname {PolyLog}\left (3,i e^{2 i (a+b x)}\right )-6 b f^2 x \operatorname {PolyLog}\left (3,i e^{2 i (a+b x)}\right )+3 i f^2 \operatorname {PolyLog}\left (4,-i e^{2 i (a+b x)}\right )-3 i f^2 \operatorname {PolyLog}\left (4,i e^{2 i (a+b x)}\right )}{24 b^3} \]

[In]

Integrate[(e + f*x)^2*ArcCoth[Cot[a + b*x]],x]

[Out]

(x*(3*e^2 + 3*e*f*x + f^2*x^2)*ArcCoth[Cot[a + b*x]])/3 + (-12*b^3*e^2*x*Log[1 - I*E^((2*I)*(a + b*x))] - 12*b
^3*e*f*x^2*Log[1 - I*E^((2*I)*(a + b*x))] - 4*b^3*f^2*x^3*Log[1 - I*E^((2*I)*(a + b*x))] + 12*b^3*e^2*x*Log[1
+ I*E^((2*I)*(a + b*x))] + 12*b^3*e*f*x^2*Log[1 + I*E^((2*I)*(a + b*x))] + 4*b^3*f^2*x^3*Log[1 + I*E^((2*I)*(a
+ b*x))] - (6*I)*b^2*(e + f*x)^2*PolyLog[2, (-I)*E^((2*I)*(a + b*x))] + (6*I)*b^2*(e + f*x)^2*PolyLog[2, I*E^
((2*I)*(a + b*x))] + 6*b*e*f*PolyLog[3, (-I)*E^((2*I)*(a + b*x))] + 6*b*f^2*x*PolyLog[3, (-I)*E^((2*I)*(a + b*
x))] - 6*b*e*f*PolyLog[3, I*E^((2*I)*(a + b*x))] - 6*b*f^2*x*PolyLog[3, I*E^((2*I)*(a + b*x))] + (3*I)*f^2*Pol
yLog[4, (-I)*E^((2*I)*(a + b*x))] - (3*I)*f^2*PolyLog[4, I*E^((2*I)*(a + b*x))])/(24*b^3)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 22.22 (sec) , antiderivative size = 2719, normalized size of antiderivative = 11.62

method	result	size
risch	\(\text {Expression too large to display}\)	\(2719\)

[In]

int((f*x+e)^2*arccoth(cot(b*x+a)),x,method=_RETURNVERBOSE)

[Out]

-1/3*f^2/b^3*ln(1+I*exp(2*I*(b*x+a)))*a^3+1/4*f^2/b^2*polylog(3,-I*exp(2*I*(b*x+a)))*x+1/2*f*e*ln(1+I*exp(2*I*
(b*x+a)))*x^2+1/2*f^2/b^3*a^3*ln(1+exp(I*(b*x+a))*(-1)^(3/4))+1/2*f^2/b^3*a^3*ln(1-exp(I*(b*x+a))*(-1)^(3/4))-
1/6*f^2/b^3*a^3*ln(-exp(2*I*(b*x+a))+I)+1/4*f*e/b^2*polylog(3,-I*exp(2*I*(b*x+a)))-1/8*I*f^2*polylog(4,I*exp(2
*I*(b*x+a)))/b^3+1/8*I*f^2*polylog(4,-I*exp(2*I*(b*x+a)))/b^3-1/12*I*Pi*(csgn(I*(exp(2*I*(b*x+a))+I)/(exp(2*I*
(b*x+a))-1))*csgn((1-I)*(exp(2*I*(b*x+a))+I)/(exp(2*I*(b*x+a))-1))+csgn((1-I)*(exp(2*I*(b*x+a))+I)/(exp(2*I*(b
*x+a))-1))^2-csgn(I*(exp(2*I*(b*x+a))-I))*csgn(I/(exp(2*I*(b*x+a))-1))*csgn(I*(exp(2*I*(b*x+a))-I)/(exp(2*I*(b
*x+a))-1))+csgn(I*(exp(2*I*(b*x+a))-I))*csgn(I*(exp(2*I*(b*x+a))-I)/(exp(2*I*(b*x+a))-1))^2+csgn(I*(exp(2*I*(b
*x+a))+I))*csgn(I/(exp(2*I*(b*x+a))-1))*csgn(I*(exp(2*I*(b*x+a))+I)/(exp(2*I*(b*x+a))-1))-csgn(I*(exp(2*I*(b*x
+a))+I))*csgn(I*(exp(2*I*(b*x+a))+I)/(exp(2*I*(b*x+a))-1))^2+csgn(I/(exp(2*I*(b*x+a))-1))*csgn(I*(exp(2*I*(b*x
+a))-I)/(exp(2*I*(b*x+a))-1))^2-csgn(I/(exp(2*I*(b*x+a))-1))*csgn(I*(exp(2*I*(b*x+a))+I)/(exp(2*I*(b*x+a))-1))
^2-csgn(I*(exp(2*I*(b*x+a))-I)/(exp(2*I*(b*x+a))-1))^3-csgn(I*(exp(2*I*(b*x+a))-I)/(exp(2*I*(b*x+a))-1))*csgn(
(1+I)*(exp(2*I*(b*x+a))-I)/(exp(2*I*(b*x+a))-1))+csgn(I*(exp(2*I*(b*x+a))-I)/(exp(2*I*(b*x+a))-1))*csgn((1+I)*
(exp(2*I*(b*x+a))-I)/(exp(2*I*(b*x+a))-1))^2+csgn(I*(exp(2*I*(b*x+a))+I)/(exp(2*I*(b*x+a))-1))^3-csgn(I*(exp(2
*I*(b*x+a))+I)/(exp(2*I*(b*x+a))-1))*csgn((1-I)*(exp(2*I*(b*x+a))+I)/(exp(2*I*(b*x+a))-1))^2-csgn((1-I)*(exp(2
*I*(b*x+a))+I)/(exp(2*I*(b*x+a))-1))^3-csgn((1+I)*(exp(2*I*(b*x+a))-I)/(exp(2*I*(b*x+a))-1))^2+csgn((1+I)*(exp
(2*I*(b*x+a))-I)/(exp(2*I*(b*x+a))-1))^3+1)*(f*x+e)^3/f-1/2*f*e/b^2*ln(-I*exp(2*I*(b*x+a))+1)*a^2+1/2*f^2/b^2*
ln(-I*exp(2*I*(b*x+a))+1)*a^2*x+f*ln(((-I)^(1/2)-exp(I*(b*x+a)))/(-I)^(1/2))/b^2*a^2*e+f*ln(((-I)^(1/2)+exp(I*
(b*x+a)))/(-I)^(1/2))/b^2*a^2*e+1/2*f/b^2*a^2*e*ln(-exp(2*I*(b*x+a))+I)+1/2*f*e/b^2*ln(1+I*exp(2*I*(b*x+a)))*a
^2+1/2*f^2/b^2*a^2*ln(1+exp(I*(b*x+a))*(-1)^(3/4))*x+1/2*f^2/b^2*a^2*ln(1-exp(I*(b*x+a))*(-1)^(3/4))*x-f/b^2*a
^2*e*ln(1+exp(I*(b*x+a))*(-1)^(3/4))-f/b^2*a^2*e*ln(1-exp(I*(b*x+a))*(-1)^(3/4))-1/2*f^2/b^2*ln(1+I*exp(2*I*(b
*x+a)))*a^2*x+1/4*I*f^2*a^2/b^3*polylog(2,-I*exp(2*I*(b*x+a)))-1/2*I*f^2/b^3*a^2*dilog(1+exp(I*(b*x+a))*(-1)^(
3/4))-1/2*I*f^2/b^3*a^2*dilog(1-exp(I*(b*x+a))*(-1)^(3/4))-1/4*I*f^2/b*polylog(2,-I*exp(2*I*(b*x+a)))*x^2-1/2*
f/b^2*a^2*e*ln(exp(2*I*(b*x+a))+I)-1/2*f^2/b^2*a^2*ln(((-I)^(1/2)-exp(I*(b*x+a)))/(-I)^(1/2))*x-1/2*f^2/b^2*a^
2*ln(((-I)^(1/2)+exp(I*(b*x+a)))/(-I)^(1/2))*x+1/4*I*f^2/b*polylog(2,I*exp(2*I*(b*x+a)))*x^2+1/2*I*f^2/b^3*a^2
*dilog(((-I)^(1/2)-exp(I*(b*x+a)))/(-I)^(1/2))+1/2*I*f^2/b^3*a^2*dilog(((-I)^(1/2)+exp(I*(b*x+a)))/(-I)^(1/2))
-1/4*I*f^2*a^2/b^3*polylog(2,I*exp(2*I*(b*x+a)))+f*e/b*ln(1+I*exp(2*I*(b*x+a)))*a*x+I*f/b^2*a*e*dilog(1+exp(I*
(b*x+a))*(-1)^(3/4))+I*f/b^2*a*e*dilog(1-exp(I*(b*x+a))*(-1)^(3/4))-f/b*a*e*ln(1+exp(I*(b*x+a))*(-1)^(3/4))*x-
f/b*a*e*ln(1-exp(I*(b*x+a))*(-1)^(3/4))*x-1/2*I*f*e/b*polylog(2,-I*exp(2*I*(b*x+a)))*x-1/2*I*f*e/b^2*polylog(2
,-I*exp(2*I*(b*x+a)))*a+1/6*(f*x+e)^3/f*ln(exp(2*I*(b*x+a))+I)-1/6*f^2*ln(exp(2*I*(b*x+a))-I)*x^3-f*e/b*ln(-I*
exp(2*I*(b*x+a))+1)*a*x+f/b*a*e*ln(((-I)^(1/2)-exp(I*(b*x+a)))/(-I)^(1/2))*x+f/b*a*e*ln(((-I)^(1/2)+exp(I*(b*x
+a)))/(-I)^(1/2))*x-I*f/b^2*a*e*dilog(((-I)^(1/2)-exp(I*(b*x+a)))/(-I)^(1/2))-I*f/b^2*a*e*dilog(((-I)^(1/2)+ex
p(I*(b*x+a)))/(-I)^(1/2))+1/2*I*f*e/b*polylog(2,I*exp(2*I*(b*x+a)))*x+1/2*I*f*e/b^2*polylog(2,I*exp(2*I*(b*x+a
)))*a-1/2*f*ln(exp(2*I*(b*x+a))-I)*x^2*e-1/6/f*e^3*ln(exp(2*I*(b*x+a))+I)-1/6*f^2*ln(-I*exp(2*I*(b*x+a))+1)*x^
3-1/2*e^2*ln(((-I)^(1/2)-exp(I*(b*x+a)))/(-I)^(1/2))*x-1/2*e^2*ln(((-I)^(1/2)+exp(I*(b*x+a)))/(-I)^(1/2))*x+1/
2*e^2*ln(1+exp(I*(b*x+a))*(-1)^(3/4))*x+1/2*e^2*ln(1-exp(I*(b*x+a))*(-1)^(3/4))*x+1/6*f^2*ln(1+I*exp(2*I*(b*x+
a)))*x^3+1/6/f*e^3*ln(-exp(2*I*(b*x+a))+I)-1/2/b*a*e^2*ln(-exp(2*I*(b*x+a))+I)+1/2/b*e^2*ln(1+exp(I*(b*x+a))*(
-1)^(3/4))*a+1/2/b*e^2*ln(1-exp(I*(b*x+a))*(-1)^(3/4))*a-1/2*I/b*e^2*dilog(1+exp(I*(b*x+a))*(-1)^(3/4))-1/2*I/
b*e^2*dilog(1-exp(I*(b*x+a))*(-1)^(3/4))-1/2*ln(exp(2*I*(b*x+a))-I)*x*e^2-1/6/f*ln(exp(2*I*(b*x+a))-I)*e^3-1/2
*ln(((-I)^(1/2)-exp(I*(b*x+a)))/(-I)^(1/2))/b*a*e^2-1/2*ln(((-I)^(1/2)+exp(I*(b*x+a)))/(-I)^(1/2))/b*a*e^2+1/2
/b*a*e^2*ln(exp(2*I*(b*x+a))+I)+1/6*f^2/b^3*a^3*ln(exp(2*I*(b*x+a))+I)-1/4*f*e/b^2*polylog(3,I*exp(2*I*(b*x+a)
))+1/3*f^2/b^3*ln(-I*exp(2*I*(b*x+a))+1)*a^3-1/2*f^2*ln(((-I)^(1/2)-exp(I*(b*x+a)))/(-I)^(1/2))/b^3*a^3-1/2*f^
2*ln(((-I)^(1/2)+exp(I*(b*x+a)))/(-I)^(1/2))/b^3*a^3-1/2*f*e*ln(-I*exp(2*I*(b*x+a))+1)*x^2-1/4*f^2/b^2*polylog
(3,I*exp(2*I*(b*x+a)))*x+1/2*I/b*e^2*dilog(((-I)^(1/2)-exp(I*(b*x+a)))/(-I)^(1/2))+1/2*I/b*e^2*dilog(((-I)^(1/
2)+exp(I*(b*x+a)))/(-I)^(1/2))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1084 vs. \(2 (180) = 360\).

Time = 0.32 (sec) , antiderivative size = 1084, normalized size of antiderivative = 4.63 \[ \int (e+f x)^2 \coth ^{-1}(\cot (a+b x)) \, dx=\text {Too large to display} \]

[In]

integrate((f*x+e)^2*arccoth(cot(b*x+a)),x, algorithm="fricas")

[Out]

1/48*(-3*I*f^2*polylog(4, I*cos(2*b*x + 2*a) + sin(2*b*x + 2*a)) - 3*I*f^2*polylog(4, I*cos(2*b*x + 2*a) - sin
(2*b*x + 2*a)) + 3*I*f^2*polylog(4, -I*cos(2*b*x + 2*a) + sin(2*b*x + 2*a)) + 3*I*f^2*polylog(4, -I*cos(2*b*x
+ 2*a) - sin(2*b*x + 2*a)) - 6*(-I*b^2*f^2*x^2 - 2*I*b^2*e*f*x - I*b^2*e^2)*dilog(I*cos(2*b*x + 2*a) + sin(2*b
*x + 2*a)) - 6*(-I*b^2*f^2*x^2 - 2*I*b^2*e*f*x - I*b^2*e^2)*dilog(I*cos(2*b*x + 2*a) - sin(2*b*x + 2*a)) - 6*(
I*b^2*f^2*x^2 + 2*I*b^2*e*f*x + I*b^2*e^2)*dilog(-I*cos(2*b*x + 2*a) + sin(2*b*x + 2*a)) - 6*(I*b^2*f^2*x^2 +
2*I*b^2*e*f*x + I*b^2*e^2)*dilog(-I*cos(2*b*x + 2*a) - sin(2*b*x + 2*a)) + 8*(b^3*f^2*x^3 + 3*b^3*e*f*x^2 + 3*
b^3*e^2*x)*log((cos(2*b*x + 2*a) + sin(2*b*x + 2*a) + 1)/(cos(2*b*x + 2*a) - sin(2*b*x + 2*a) + 1)) + 4*(3*a*b
^2*e^2 - 3*a^2*b*e*f + a^3*f^2)*log(cos(2*b*x + 2*a) + I*sin(2*b*x + 2*a) + I) - 4*(3*a*b^2*e^2 - 3*a^2*b*e*f
+ a^3*f^2)*log(cos(2*b*x + 2*a) - I*sin(2*b*x + 2*a) + I) - 4*(b^3*f^2*x^3 + 3*b^3*e*f*x^2 + 3*b^3*e^2*x + 3*a
*b^2*e^2 - 3*a^2*b*e*f + a^3*f^2)*log(I*cos(2*b*x + 2*a) + sin(2*b*x + 2*a) + 1) + 4*(b^3*f^2*x^3 + 3*b^3*e*f*
x^2 + 3*b^3*e^2*x + 3*a*b^2*e^2 - 3*a^2*b*e*f + a^3*f^2)*log(I*cos(2*b*x + 2*a) - sin(2*b*x + 2*a) + 1) - 4*(b
^3*f^2*x^3 + 3*b^3*e*f*x^2 + 3*b^3*e^2*x + 3*a*b^2*e^2 - 3*a^2*b*e*f + a^3*f^2)*log(-I*cos(2*b*x + 2*a) + sin(
2*b*x + 2*a) + 1) + 4*(b^3*f^2*x^3 + 3*b^3*e*f*x^2 + 3*b^3*e^2*x + 3*a*b^2*e^2 - 3*a^2*b*e*f + a^3*f^2)*log(-I
*cos(2*b*x + 2*a) - sin(2*b*x + 2*a) + 1) + 4*(3*a*b^2*e^2 - 3*a^2*b*e*f + a^3*f^2)*log(-cos(2*b*x + 2*a) + I*
sin(2*b*x + 2*a) + I) - 4*(3*a*b^2*e^2 - 3*a^2*b*e*f + a^3*f^2)*log(-cos(2*b*x + 2*a) - I*sin(2*b*x + 2*a) + I
) + 6*(b*f^2*x + b*e*f)*polylog(3, I*cos(2*b*x + 2*a) + sin(2*b*x + 2*a)) - 6*(b*f^2*x + b*e*f)*polylog(3, I*c
os(2*b*x + 2*a) - sin(2*b*x + 2*a)) + 6*(b*f^2*x + b*e*f)*polylog(3, -I*cos(2*b*x + 2*a) + sin(2*b*x + 2*a)) -
6*(b*f^2*x + b*e*f)*polylog(3, -I*cos(2*b*x + 2*a) - sin(2*b*x + 2*a)))/b^3

Sympy [F]

\[ \int (e+f x)^2 \coth ^{-1}(\cot (a+b x)) \, dx=\int \left (e + f x\right )^{2} \operatorname {acoth}{\left (\cot {\left (a + b x \right )} \right )}\, dx \]

[In]

integrate((f*x+e)**2*acoth(cot(b*x+a)),x)

[Out]

Integral((e + f*x)**2*acoth(cot(a + b*x)), x)

Maxima [F]

\[ \int (e+f x)^2 \coth ^{-1}(\cot (a+b x)) \, dx=\int { {\left (f x + e\right )}^{2} \operatorname {arcoth}\left (\cot \left (b x + a\right )\right ) \,d x } \]

[In]

integrate((f*x+e)^2*arccoth(cot(b*x+a)),x, algorithm="maxima")

[Out]

1/12*(f^2*x^3 + 3*e*f*x^2 + 3*e^2*x)*log(2*cos(2*b*x + 2*a)^2 + 2*sin(2*b*x + 2*a)^2 + 4*sin(2*b*x + 2*a) + 2)
- 1/12*(f^2*x^3 + 3*e*f*x^2 + 3*e^2*x)*log(2*cos(2*b*x + 2*a)^2 + 2*sin(2*b*x + 2*a)^2 - 4*sin(2*b*x + 2*a) +
2) - integrate(2/3*((b*f^2*x^3 + 3*b*e*f*x^2 + 3*b*e^2*x)*cos(4*b*x + 4*a)*cos(2*b*x + 2*a) + (b*f^2*x^3 + 3*
b*e*f*x^2 + 3*b*e^2*x)*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + (b*f^2*x^3 + 3*b*e*f*x^2 + 3*b*e^2*x)*cos(2*b*x + 2
*a))/(cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1), x)

Giac [F]

\[ \int (e+f x)^2 \coth ^{-1}(\cot (a+b x)) \, dx=\int { {\left (f x + e\right )}^{2} \operatorname {arcoth}\left (\cot \left (b x + a\right )\right ) \,d x } \]

[In]

integrate((f*x+e)^2*arccoth(cot(b*x+a)),x, algorithm="giac")

[Out]

integrate((f*x + e)^2*arccoth(cot(b*x + a)), x)

Mupad [F(-1)]

Timed out. \[ \int (e+f x)^2 \coth ^{-1}(\cot (a+b x)) \, dx=\int \mathrm {acoth}\left (\mathrm {cot}\left (a+b\,x\right )\right )\,{\left (e+f\,x\right )}^2 \,d x \]

[In]

int(acoth(cot(a + b*x))*(e + f*x)^2,x)

[Out]

int(acoth(cot(a + b*x))*(e + f*x)^2, x)