\(\int (e+f x)^2 \cot ^{-1}(\coth (a+b x)) \, dx\) [201]

Optimal result

Integrand size = 15, antiderivative size = 229 \[ \int (e+f x)^2 \cot ^{-1}(\coth (a+b x)) \, dx=\frac {(e+f x)^3 \cot ^{-1}(\coth (a+b x))}{3 f}-\frac {(e+f x)^3 \arctan \left (e^{2 a+2 b x}\right )}{3 f}+\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{2 a+2 b x}\right )}{4 b}-\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{2 a+2 b x}\right )}{4 b}-\frac {i f (e+f x) \operatorname {PolyLog}\left (3,-i e^{2 a+2 b x}\right )}{4 b^2}+\frac {i f (e+f x) \operatorname {PolyLog}\left (3,i e^{2 a+2 b x}\right )}{4 b^2}+\frac {i f^2 \operatorname {PolyLog}\left (4,-i e^{2 a+2 b x}\right )}{8 b^3}-\frac {i f^2 \operatorname {PolyLog}\left (4,i e^{2 a+2 b x}\right )}{8 b^3} \]

[Out]

1/3*(f*x+e)^3*arccot(coth(b*x+a))/f-1/3*(f*x+e)^3*arctan(exp(2*b*x+2*a))/f+1/4*I*(f*x+e)^2*polylog(2,-I*exp(2*
b*x+2*a))/b-1/4*I*(f*x+e)^2*polylog(2,I*exp(2*b*x+2*a))/b-1/4*I*f*(f*x+e)*polylog(3,-I*exp(2*b*x+2*a))/b^2+1/4
*I*f*(f*x+e)*polylog(3,I*exp(2*b*x+2*a))/b^2+1/8*I*f^2*polylog(4,-I*exp(2*b*x+2*a))/b^3-1/8*I*f^2*polylog(4,I*
exp(2*b*x+2*a))/b^3

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 229, 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 = {5294, 4265, 2611, 6744, 2320, 6724} \[ \int (e+f x)^2 \cot ^{-1}(\coth (a+b x)) \, dx=-\frac {(e+f x)^3 \arctan \left (e^{2 a+2 b x}\right )}{3 f}+\frac {i f^2 \operatorname {PolyLog}\left (4,-i e^{2 a+2 b x}\right )}{8 b^3}-\frac {i f^2 \operatorname {PolyLog}\left (4,i e^{2 a+2 b x}\right )}{8 b^3}-\frac {i f (e+f x) \operatorname {PolyLog}\left (3,-i e^{2 a+2 b x}\right )}{4 b^2}+\frac {i f (e+f x) \operatorname {PolyLog}\left (3,i e^{2 a+2 b x}\right )}{4 b^2}+\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{2 a+2 b x}\right )}{4 b}-\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{2 a+2 b x}\right )}{4 b}+\frac {(e+f x)^3 \cot ^{-1}(\coth (a+b x))}{3 f} \]

[In]

Int[(e + f*x)^2*ArcCot[Coth[a + b*x]],x]

[Out]

((e + f*x)^3*ArcCot[Coth[a + b*x]])/(3*f) - ((e + f*x)^3*ArcTan[E^(2*a + 2*b*x)])/(3*f) + ((I/4)*(e + f*x)^2*P
olyLog[2, (-I)*E^(2*a + 2*b*x)])/b - ((I/4)*(e + f*x)^2*PolyLog[2, I*E^(2*a + 2*b*x)])/b - ((I/4)*f*(e + f*x)*
PolyLog[3, (-I)*E^(2*a + 2*b*x)])/b^2 + ((I/4)*f*(e + f*x)*PolyLog[3, I*E^(2*a + 2*b*x)])/b^2 + ((I/8)*f^2*Pol
yLog[4, (-I)*E^(2*a + 2*b*x)])/b^3 - ((I/8)*f^2*PolyLog[4, I*E^(2*a + 2*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 4265

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] + (-Dist[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] + Dist[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]

Rule 5294

Int[ArcCot[Coth[(a_.) + (b_.)*(x_)]]*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^(m + 1)*(ArcCot[C
oth[a + b*x]]/(f*(m + 1))), x] - Dist[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]

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 \cot ^{-1}(\coth (a+b x))}{3 f}-\frac {b \int (e+f x)^3 \text {sech}(2 a+2 b x) \, dx}{3 f} \\ & = \frac {(e+f x)^3 \cot ^{-1}(\coth (a+b x))}{3 f}-\frac {(e+f x)^3 \arctan \left (e^{2 a+2 b x}\right )}{3 f}+\frac {1}{2} i \int (e+f x)^2 \log \left (1-i e^{2 a+2 b x}\right ) \, dx-\frac {1}{2} i \int (e+f x)^2 \log \left (1+i e^{2 a+2 b x}\right ) \, dx \\ & = \frac {(e+f x)^3 \cot ^{-1}(\coth (a+b x))}{3 f}-\frac {(e+f x)^3 \arctan \left (e^{2 a+2 b x}\right )}{3 f}+\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{2 a+2 b x}\right )}{4 b}-\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{2 a+2 b x}\right )}{4 b}-\frac {(i f) \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{2 a+2 b x}\right ) \, dx}{2 b}+\frac {(i f) \int (e+f x) \operatorname {PolyLog}\left (2,i e^{2 a+2 b x}\right ) \, dx}{2 b} \\ & = \frac {(e+f x)^3 \cot ^{-1}(\coth (a+b x))}{3 f}-\frac {(e+f x)^3 \arctan \left (e^{2 a+2 b x}\right )}{3 f}+\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{2 a+2 b x}\right )}{4 b}-\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{2 a+2 b x}\right )}{4 b}-\frac {i f (e+f x) \operatorname {PolyLog}\left (3,-i e^{2 a+2 b x}\right )}{4 b^2}+\frac {i f (e+f x) \operatorname {PolyLog}\left (3,i e^{2 a+2 b x}\right )}{4 b^2}+\frac {\left (i f^2\right ) \int \operatorname {PolyLog}\left (3,-i e^{2 a+2 b x}\right ) \, dx}{4 b^2}-\frac {\left (i f^2\right ) \int \operatorname {PolyLog}\left (3,i e^{2 a+2 b x}\right ) \, dx}{4 b^2} \\ & = \frac {(e+f x)^3 \cot ^{-1}(\coth (a+b x))}{3 f}-\frac {(e+f x)^3 \arctan \left (e^{2 a+2 b x}\right )}{3 f}+\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{2 a+2 b x}\right )}{4 b}-\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{2 a+2 b x}\right )}{4 b}-\frac {i f (e+f x) \operatorname {PolyLog}\left (3,-i e^{2 a+2 b x}\right )}{4 b^2}+\frac {i f (e+f x) \operatorname {PolyLog}\left (3,i e^{2 a+2 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^{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^{2 a+2 b x}\right )}{8 b^3} \\ & = \frac {(e+f x)^3 \cot ^{-1}(\coth (a+b x))}{3 f}-\frac {(e+f x)^3 \arctan \left (e^{2 a+2 b x}\right )}{3 f}+\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{2 a+2 b x}\right )}{4 b}-\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{2 a+2 b x}\right )}{4 b}-\frac {i f (e+f x) \operatorname {PolyLog}\left (3,-i e^{2 a+2 b x}\right )}{4 b^2}+\frac {i f (e+f x) \operatorname {PolyLog}\left (3,i e^{2 a+2 b x}\right )}{4 b^2}+\frac {i f^2 \operatorname {PolyLog}\left (4,-i e^{2 a+2 b x}\right )}{8 b^3}-\frac {i f^2 \operatorname {PolyLog}\left (4,i e^{2 a+2 b x}\right )}{8 b^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.64 \[ \int (e+f x)^2 \cot ^{-1}(\coth (a+b x)) \, dx=\frac {1}{3} x \left (3 e^2+3 e f x+f^2 x^2\right ) \cot ^{-1}(\coth (a+b x))-\frac {i \left (12 b^3 e^2 x \log \left (1-i e^{2 (a+b x)}\right )+12 b^3 e f x^2 \log \left (1-i e^{2 (a+b x)}\right )+4 b^3 f^2 x^3 \log \left (1-i e^{2 (a+b x)}\right )-12 b^3 e^2 x \log \left (1+i e^{2 (a+b x)}\right )-12 b^3 e f x^2 \log \left (1+i e^{2 (a+b x)}\right )-4 b^3 f^2 x^3 \log \left (1+i e^{2 (a+b x)}\right )-6 b^2 (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{2 (a+b x)}\right )+6 b^2 (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{2 (a+b x)}\right )+6 b e f \operatorname {PolyLog}\left (3,-i e^{2 (a+b x)}\right )+6 b f^2 x \operatorname {PolyLog}\left (3,-i e^{2 (a+b x)}\right )-6 b e f \operatorname {PolyLog}\left (3,i e^{2 (a+b x)}\right )-6 b f^2 x \operatorname {PolyLog}\left (3,i e^{2 (a+b x)}\right )-3 f^2 \operatorname {PolyLog}\left (4,-i e^{2 (a+b x)}\right )+3 f^2 \operatorname {PolyLog}\left (4,i e^{2 (a+b x)}\right )\right )}{24 b^3} \]

[In]

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

[Out]

(x*(3*e^2 + 3*e*f*x + f^2*x^2)*ArcCot[Coth[a + b*x]])/3 - ((I/24)*(12*b^3*e^2*x*Log[1 - I*E^(2*(a + b*x))] + 1
2*b^3*e*f*x^2*Log[1 - I*E^(2*(a + b*x))] + 4*b^3*f^2*x^3*Log[1 - I*E^(2*(a + b*x))] - 12*b^3*e^2*x*Log[1 + I*E
^(2*(a + b*x))] - 12*b^3*e*f*x^2*Log[1 + I*E^(2*(a + b*x))] - 4*b^3*f^2*x^3*Log[1 + I*E^(2*(a + b*x))] - 6*b^2
*(e + f*x)^2*PolyLog[2, (-I)*E^(2*(a + b*x))] + 6*b^2*(e + f*x)^2*PolyLog[2, I*E^(2*(a + b*x))] + 6*b*e*f*Poly
Log[3, (-I)*E^(2*(a + b*x))] + 6*b*f^2*x*PolyLog[3, (-I)*E^(2*(a + b*x))] - 6*b*e*f*PolyLog[3, I*E^(2*(a + b*x
))] - 6*b*f^2*x*PolyLog[3, I*E^(2*(a + b*x))] - 3*f^2*PolyLog[4, (-I)*E^(2*(a + b*x))] + 3*f^2*PolyLog[4, I*E^
(2*(a + b*x))]))/b^3

Maple [C] (warning: unable to verify)

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

Time = 21.01 (sec) , antiderivative size = 2668, normalized size of antiderivative = 11.65

method	result	size
risch	\(\text {Expression too large to display}\)	\(2668\)

[In]

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

[Out]

-1/2*I*f^2/b^3*a^2*dilog(((-I)^(1/2)-exp(b*x+a))/(-I)^(1/2))-1/2*I*f^2/b^3*a^2*dilog(((-I)^(1/2)+exp(b*x+a))/(
-I)^(1/2))+1/4*I*f/b^2*e*polylog(3,I*exp(2*b*x+2*a))+1/6*I*f^2/b^3*a^3*ln(exp(2*b*x+2*a)+I)+1/3*I*f^2/b^3*ln(1
-I*exp(2*b*x+2*a))*a^3-1/4*I*f^2/b*polylog(2,I*exp(2*b*x+2*a))*x^2+1/2*I/b*a*e^2*ln(exp(2*b*x+2*a)+I)-1/2*I/b*
e^2*ln(((-I)^(1/2)-exp(b*x+a))/(-I)^(1/2))*a-1/2*I/b*e^2*ln(((-I)^(1/2)+exp(b*x+a))/(-I)^(1/2))*a+1/8*I*f^2*po
lylog(4,-I*exp(2*b*x+2*a))/b^3-1/8*I*f^2*polylog(4,I*exp(2*b*x+2*a))/b^3-1/6*I*f^2/b^3*a^3*ln(-exp(2*b*x+2*a)+
I)+1/2*I*f*e*ln(1+I*exp(2*b*x+2*a))*x^2-1/3*I*f^2/b^3*ln(1+I*exp(2*b*x+2*a))*a^3+1/4*I*f^2/b*polylog(2,-I*exp(
2*b*x+2*a))*x^2-1/4*I*f^2/b^3*polylog(2,-I*exp(2*b*x+2*a))*a^2-1/4*I*f^2/b^2*polylog(3,-I*exp(2*b*x+2*a))*x+1/
2*I*f^2/b^3*a^3*ln(1+exp(b*x+a)*(-1)^(3/4))+1/2*I*f^2/b^3*a^3*ln(1-exp(b*x+a)*(-1)^(3/4))+1/2*I*f^2/b^3*a^2*di
log(1+exp(b*x+a)*(-1)^(3/4))+1/2*I*f^2/b^3*a^2*dilog(1-exp(b*x+a)*(-1)^(3/4))-1/4*I*f/b^2*e*polylog(3,-I*exp(2
*b*x+2*a))+1/2*I/b*e^2*ln(1+exp(b*x+a)*(-1)^(3/4))*a+1/2*I/b*e^2*ln(1-exp(b*x+a)*(-1)^(3/4))*a-1/2*I/b*a*e^2*l
n(-exp(2*b*x+2*a)+I)-1/2*I*f*ln(exp(2*b*x+2*a)-I)*x^2*e-1/2*I*f*e*ln(1-I*exp(2*b*x+2*a))*x^2+1/4*I*f^2/b^3*pol
ylog(2,I*exp(2*b*x+2*a))*a^2+1/4*I*f^2/b^2*polylog(3,I*exp(2*b*x+2*a))*x-1/2*I*f^2/b^3*a^3*ln(((-I)^(1/2)-exp(
b*x+a))/(-I)^(1/2))-1/2*I*f^2/b^3*a^3*ln(((-I)^(1/2)+exp(b*x+a))/(-I)^(1/2))+1/2*I*f/b^2*e*ln(1+I*exp(2*b*x+2*
a))*a^2+1/2*I*f/b*e*polylog(2,-I*exp(2*b*x+2*a))*x+1/2*I*f/b^2*e*polylog(2,-I*exp(2*b*x+2*a))*a-1/2*I*f^2/b^2*
ln(1+I*exp(2*b*x+2*a))*a^2*x-I*f/b^2*a^2*e*ln(1+exp(b*x+a)*(-1)^(3/4))-I*f/b^2*a^2*e*ln(1-exp(b*x+a)*(-1)^(3/4
))-I*f/b^2*a*e*dilog(1+exp(b*x+a)*(-1)^(3/4))-I*f/b^2*a*e*dilog(1-exp(b*x+a)*(-1)^(3/4))+1/2*I*f/b^2*a^2*e*ln(
-exp(2*b*x+2*a)+I)+1/2*I*f^2/b^2*a^2*ln(1+exp(b*x+a)*(-1)^(3/4))*x+1/2*I*f^2/b^2*a^2*ln(1-exp(b*x+a)*(-1)^(3/4
))*x+1/6*I/f*e^3*ln(-exp(2*b*x+2*a)+I)+1/6*I*f^2*ln(1+I*exp(2*b*x+2*a))*x^3+1/2*I/b*e^2*dilog(1+exp(b*x+a)*(-1
)^(3/4))+1/2*I/b*e^2*dilog(1-exp(b*x+a)*(-1)^(3/4))+1/2*I*e^2*ln(1+exp(b*x+a)*(-1)^(3/4))*x+1/2*I*e^2*ln(1-exp
(b*x+a)*(-1)^(3/4))*x-1/6*I*f^2*ln(exp(2*b*x+2*a)-I)*x^3-1/2*I*ln(exp(2*b*x+2*a)-I)*x*e^2-1/6*I/f*ln(exp(2*b*x
+2*a)-I)*e^3+1/6*I*(f*x+e)^3/f*ln(exp(2*b*x+2*a)+I)-1/12*Pi*(csgn(I*(exp(2*b*x+2*a)-I))*csgn(I/(exp(2*b*x+2*a)
-1))*csgn(I*(exp(2*b*x+2*a)-I)/(exp(2*b*x+2*a)-1))-csgn(I*(exp(2*b*x+2*a)-I))*csgn(I*(exp(2*b*x+2*a)-I)/(exp(2
*b*x+2*a)-1))^2-csgn(I*(exp(2*b*x+2*a)+I))*csgn(I/(exp(2*b*x+2*a)-1))*csgn(I*(exp(2*b*x+2*a)+I)/(exp(2*b*x+2*a
)-1))+csgn(I*(exp(2*b*x+2*a)+I))*csgn(I*(exp(2*b*x+2*a)+I)/(exp(2*b*x+2*a)-1))^2+csgn(I*(exp(2*b*x+2*a)-I)/(ex
p(2*b*x+2*a)-1))*csgn((1+I)*(exp(2*b*x+2*a)-I)/(exp(2*b*x+2*a)-1))-csgn((1+I)*(exp(2*b*x+2*a)-I)/(exp(2*b*x+2*
a)-1))^2-csgn(I*(exp(2*b*x+2*a)+I)/(exp(2*b*x+2*a)-1))*csgn((1-I)*(exp(2*b*x+2*a)+I)/(exp(2*b*x+2*a)-1))-csgn(
(1-I)*(exp(2*b*x+2*a)+I)/(exp(2*b*x+2*a)-1))^2-csgn(I/(exp(2*b*x+2*a)-1))*csgn(I*(exp(2*b*x+2*a)-I)/(exp(2*b*x
+2*a)-1))^2+csgn(I/(exp(2*b*x+2*a)-1))*csgn(I*(exp(2*b*x+2*a)+I)/(exp(2*b*x+2*a)-1))^2+csgn(I*(exp(2*b*x+2*a)-
I)/(exp(2*b*x+2*a)-1))^3-csgn(I*(exp(2*b*x+2*a)-I)/(exp(2*b*x+2*a)-1))*csgn((1+I)*(exp(2*b*x+2*a)-I)/(exp(2*b*
x+2*a)-1))^2-csgn(I*(exp(2*b*x+2*a)+I)/(exp(2*b*x+2*a)-1))^3+csgn(I*(exp(2*b*x+2*a)+I)/(exp(2*b*x+2*a)-1))*csg
n((1-I)*(exp(2*b*x+2*a)+I)/(exp(2*b*x+2*a)-1))^2+csgn((1+I)*(exp(2*b*x+2*a)-I)/(exp(2*b*x+2*a)-1))^3+csgn((1-I
)*(exp(2*b*x+2*a)+I)/(exp(2*b*x+2*a)-1))^3-1)*(f*x+e)^3/f+I*f/b^2*a^2*e*ln(((-I)^(1/2)+exp(b*x+a))/(-I)^(1/2))
+I*f/b^2*a*e*dilog(((-I)^(1/2)-exp(b*x+a))/(-I)^(1/2))+I*f/b^2*a*e*dilog(((-I)^(1/2)+exp(b*x+a))/(-I)^(1/2))+I
*f/b^2*a^2*e*ln(((-I)^(1/2)-exp(b*x+a))/(-I)^(1/2))+1/2*I*f^2/b^2*ln(1-I*exp(2*b*x+2*a))*a^2*x-1/2*I*f^2/b^2*a
^2*ln(((-I)^(1/2)-exp(b*x+a))/(-I)^(1/2))*x-1/2*I*f^2/b^2*a^2*ln(((-I)^(1/2)+exp(b*x+a))/(-I)^(1/2))*x-1/2*I*f
/b^2*a^2*e*ln(exp(2*b*x+2*a)+I)-1/2*I*f/b^2*e*ln(1-I*exp(2*b*x+2*a))*a^2-1/2*I*f/b*e*polylog(2,I*exp(2*b*x+2*a
))*x-1/2*I*f/b^2*e*polylog(2,I*exp(2*b*x+2*a))*a-1/2*I*e^2*ln(((-I)^(1/2)-exp(b*x+a))/(-I)^(1/2))*x-1/2*I*e^2*
ln(((-I)^(1/2)+exp(b*x+a))/(-I)^(1/2))*x-1/2*I/b*e^2*dilog(((-I)^(1/2)-exp(b*x+a))/(-I)^(1/2))-1/2*I/b*e^2*dil
og(((-I)^(1/2)+exp(b*x+a))/(-I)^(1/2))-1/6*I*f^2*ln(1-I*exp(2*b*x+2*a))*x^3-1/6*I/f*e^3*ln(exp(2*b*x+2*a)+I)+I
*f/b*e*ln(1+I*exp(2*b*x+2*a))*a*x-I*f/b*a*e*ln(1+exp(b*x+a)*(-1)^(3/4))*x-I*f/b*a*e*ln(1-exp(b*x+a)*(-1)^(3/4)
)*x+I*f/b*a*e*ln(((-I)^(1/2)-exp(b*x+a))/(-I)^(1/2))*x+I*f/b*a*e*ln(((-I)^(1/2)+exp(b*x+a))/(-I)^(1/2))*x-I*f/
b*e*ln(1-I*exp(2*b*x+2*a))*a*x

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. 1002 vs. \(2 (180) = 360\).

Time = 0.33 (sec) , antiderivative size = 1002, normalized size of antiderivative = 4.38 \[ \int (e+f x)^2 \cot ^{-1}(\coth (a+b x)) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/6*(-6*I*f^2*polylog(4, 1/2*sqrt(4*I)*(cosh(b*x + a) + sinh(b*x + a))) - 6*I*f^2*polylog(4, -1/2*sqrt(4*I)*(c
osh(b*x + a) + sinh(b*x + a))) + 6*I*f^2*polylog(4, 1/2*sqrt(-4*I)*(cosh(b*x + a) + sinh(b*x + a))) + 6*I*f^2*
polylog(4, -1/2*sqrt(-4*I)*(cosh(b*x + a) + sinh(b*x + a))) + 2*(b^3*f^2*x^3 + 3*b^3*e*f*x^2 + 3*b^3*e^2*x)*ar
ctan(sinh(b*x + a)/cosh(b*x + a)) - 3*(I*b^2*f^2*x^2 + 2*I*b^2*e*f*x + I*b^2*e^2)*dilog(1/2*sqrt(4*I)*(cosh(b*
x + a) + sinh(b*x + a))) - 3*(I*b^2*f^2*x^2 + 2*I*b^2*e*f*x + I*b^2*e^2)*dilog(-1/2*sqrt(4*I)*(cosh(b*x + a) +
sinh(b*x + a))) - 3*(-I*b^2*f^2*x^2 - 2*I*b^2*e*f*x - I*b^2*e^2)*dilog(1/2*sqrt(-4*I)*(cosh(b*x + a) + sinh(b
*x + a))) - 3*(-I*b^2*f^2*x^2 - 2*I*b^2*e*f*x - I*b^2*e^2)*dilog(-1/2*sqrt(-4*I)*(cosh(b*x + a) + sinh(b*x + a
))) + (-I*b^3*f^2*x^3 - 3*I*b^3*e*f*x^2 - 3*I*b^3*e^2*x - 3*I*a*b^2*e^2 + 3*I*a^2*b*e*f - I*a^3*f^2)*log(1/2*s
qrt(4*I)*(cosh(b*x + a) + sinh(b*x + a)) + 1) + (-I*b^3*f^2*x^3 - 3*I*b^3*e*f*x^2 - 3*I*b^3*e^2*x - 3*I*a*b^2*
e^2 + 3*I*a^2*b*e*f - I*a^3*f^2)*log(-1/2*sqrt(4*I)*(cosh(b*x + a) + sinh(b*x + a)) + 1) + (I*b^3*f^2*x^3 + 3*
I*b^3*e*f*x^2 + 3*I*b^3*e^2*x + 3*I*a*b^2*e^2 - 3*I*a^2*b*e*f + I*a^3*f^2)*log(1/2*sqrt(-4*I)*(cosh(b*x + a) +
sinh(b*x + a)) + 1) + (I*b^3*f^2*x^3 + 3*I*b^3*e*f*x^2 + 3*I*b^3*e^2*x + 3*I*a*b^2*e^2 - 3*I*a^2*b*e*f + I*a^
3*f^2)*log(-1/2*sqrt(-4*I)*(cosh(b*x + a) + sinh(b*x + a)) + 1) + (3*I*a*b^2*e^2 - 3*I*a^2*b*e*f + I*a^3*f^2)*
log(I*sqrt(4*I) + 2*cosh(b*x + a) + 2*sinh(b*x + a)) + (3*I*a*b^2*e^2 - 3*I*a^2*b*e*f + I*a^3*f^2)*log(-I*sqrt
(4*I) + 2*cosh(b*x + a) + 2*sinh(b*x + a)) + (-3*I*a*b^2*e^2 + 3*I*a^2*b*e*f - I*a^3*f^2)*log(I*sqrt(-4*I) + 2
*cosh(b*x + a) + 2*sinh(b*x + a)) + (-3*I*a*b^2*e^2 + 3*I*a^2*b*e*f - I*a^3*f^2)*log(-I*sqrt(-4*I) + 2*cosh(b*
x + a) + 2*sinh(b*x + a)) - 6*(-I*b*f^2*x - I*b*e*f)*polylog(3, 1/2*sqrt(4*I)*(cosh(b*x + a) + sinh(b*x + a)))
- 6*(-I*b*f^2*x - I*b*e*f)*polylog(3, -1/2*sqrt(4*I)*(cosh(b*x + a) + sinh(b*x + a))) - 6*(I*b*f^2*x + I*b*e*
f)*polylog(3, 1/2*sqrt(-4*I)*(cosh(b*x + a) + sinh(b*x + a))) - 6*(I*b*f^2*x + I*b*e*f)*polylog(3, -1/2*sqrt(-
4*I)*(cosh(b*x + a) + sinh(b*x + a))))/b^3

Sympy [F]

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

[In]

integrate((f*x+e)**2*acot(coth(b*x+a)),x)

[Out]

Integral((e + f*x)**2*acot(coth(a + b*x)), x)

Maxima [F]

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

[In]

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

[Out]

1/3*(f^2*x^3 + 3*e*f*x^2 + 3*e^2*x)*arctan((e^(2*b*x + 2*a) - 1)/(e^(2*b*x + 2*a) + 1)) - integrate(2/3*(b*f^2
*x^3*e^(2*a) + 3*b*e*f*x^2*e^(2*a) + 3*b*e^2*x*e^(2*a))*e^(2*b*x)/(e^(4*b*x + 4*a) + 1), x)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

int(acot(coth(a + b*x))*(e + f*x)^2,x)

[Out]

int(acot(coth(a + b*x))*(e + f*x)^2, x)