\(\int (e+f x) \cot ^{-1}(\tanh (a+b x)) \, dx\) [185]

Optimal result

Integrand size = 13, antiderivative size = 159 \[ \int (e+f x) \cot ^{-1}(\tanh (a+b x)) \, dx=\frac {(e+f x)^2 \cot ^{-1}(\tanh (a+b x))}{2 f}+\frac {(e+f x)^2 \arctan \left (e^{2 a+2 b x}\right )}{2 f}-\frac {i (e+f x) \operatorname {PolyLog}\left (2,-i e^{2 a+2 b x}\right )}{4 b}+\frac {i (e+f x) \operatorname {PolyLog}\left (2,i e^{2 a+2 b x}\right )}{4 b}+\frac {i f \operatorname {PolyLog}\left (3,-i e^{2 a+2 b x}\right )}{8 b^2}-\frac {i f \operatorname {PolyLog}\left (3,i e^{2 a+2 b x}\right )}{8 b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {5292, 4265, 2611, 2320, 6724} \[ \int (e+f x) \cot ^{-1}(\tanh (a+b x)) \, dx=\frac {(e+f x)^2 \arctan \left (e^{2 a+2 b x}\right )}{2 f}+\frac {i f \operatorname {PolyLog}\left (3,-i e^{2 a+2 b x}\right )}{8 b^2}-\frac {i f \operatorname {PolyLog}\left (3,i e^{2 a+2 b x}\right )}{8 b^2}-\frac {i (e+f x) \operatorname {PolyLog}\left (2,-i e^{2 a+2 b x}\right )}{4 b}+\frac {i (e+f x) \operatorname {PolyLog}\left (2,i e^{2 a+2 b x}\right )}{4 b}+\frac {(e+f x)^2 \cot ^{-1}(\tanh (a+b x))}{2 f} \]

[In]

Int[(e + f*x)*ArcCot[Tanh[a + b*x]],x]

[Out]

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

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 5292

Int[ArcCot[Tanh[(a_.) + (b_.)*(x_)]]*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^(m + 1)*(ArcCot[T
anh[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]

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.49 \[ \int (e+f x) \cot ^{-1}(\tanh (a+b x)) \, dx=e x \cot ^{-1}(\tanh (a+b x))+\frac {1}{2} f x^2 \cot ^{-1}(\tanh (a+b x))+\frac {i e \left (2 b x \left (\log \left (1-i e^{2 (a+b x)}\right )-\log \left (1+i e^{2 (a+b x)}\right )\right )-\operatorname {PolyLog}\left (2,-i e^{2 (a+b x)}\right )+\operatorname {PolyLog}\left (2,i e^{2 (a+b x)}\right )\right )}{4 b}+\frac {i f \left (2 b^2 x^2 \log \left (1-i e^{2 (a+b x)}\right )-2 b^2 x^2 \log \left (1+i e^{2 (a+b x)}\right )-2 b x \operatorname {PolyLog}\left (2,-i e^{2 (a+b x)}\right )+2 b x \operatorname {PolyLog}\left (2,i e^{2 (a+b x)}\right )+\operatorname {PolyLog}\left (3,-i e^{2 (a+b x)}\right )-\operatorname {PolyLog}\left (3,i e^{2 (a+b x)}\right )\right )}{8 b^2} \]

[In]

Integrate[(e + f*x)*ArcCot[Tanh[a + b*x]],x]

[Out]

e*x*ArcCot[Tanh[a + b*x]] + (f*x^2*ArcCot[Tanh[a + b*x]])/2 + ((I/4)*e*(2*b*x*(Log[1 - I*E^(2*(a + b*x))] - Lo
g[1 + I*E^(2*(a + b*x))]) - PolyLog[2, (-I)*E^(2*(a + b*x))] + PolyLog[2, I*E^(2*(a + b*x))]))/b + ((I/8)*f*(2
*b^2*x^2*Log[1 - I*E^(2*(a + b*x))] - 2*b^2*x^2*Log[1 + I*E^(2*(a + b*x))] - 2*b*x*PolyLog[2, (-I)*E^(2*(a + b
*x))] + 2*b*x*PolyLog[2, I*E^(2*(a + b*x))] + PolyLog[3, (-I)*E^(2*(a + b*x))] - PolyLog[3, I*E^(2*(a + b*x))]
))/b^2

Maple [C] (warning: unable to verify)

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

Time = 2.57 (sec) , antiderivative size = 1776, normalized size of antiderivative = 11.17

method	result	size
risch	\(\text {Expression too large to display}\)	\(1776\)

[In]

int((f*x+e)*arccot(tanh(b*x+a)),x,method=_RETURNVERBOSE)

[Out]

1/4*I*ln(exp(2*b*x+2*a)-I)*x^2*f+1/2*I*ln(exp(2*b*x+2*a)-I)*e*x-1/4*I*f*ln(1+I*exp(2*b*x+2*a))*x^2-1/2*I*e*ln(
1+exp(b*x+a)*(-1)^(3/4))*x-1/2*I*e*ln(1-exp(b*x+a)*(-1)^(3/4))*x-1/2*I*e/b*dilog(1+exp(b*x+a)*(-1)^(3/4))-1/2*
I*e/b*dilog(1-exp(b*x+a)*(-1)^(3/4))+1/4*Pi*(csgn(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))-csgn(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))-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/(ex
p(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))*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)+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)-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)-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-csg
n(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((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)*(1/2*f*x^2+e*x)-1/2*I*(1/2*f*x^2+e*x)*ln(exp(2*b*x+2*a)+I)+1/2*I*f/b*ln(1-I*exp(
2*b*x+2*a))*a*x-1/2*I*f/b*a*ln(((-I)^(1/2)-exp(b*x+a))/(-I)^(1/2))*x-1/2*I*f/b*a*ln(((-I)^(1/2)+exp(b*x+a))/(-
I)^(1/2))*x+1/8*I*f*polylog(3,-I*exp(2*b*x+2*a))/b^2+1/4*I*f*ln(1-I*exp(2*b*x+2*a))*x^2+1/2*I*e*ln(((-I)^(1/2)
-exp(b*x+a))/(-I)^(1/2))*x+1/2*I*e*ln(((-I)^(1/2)+exp(b*x+a))/(-I)^(1/2))*x+1/2*I*e/b*dilog(((-I)^(1/2)-exp(b*
x+a))/(-I)^(1/2))+1/2*I*e/b*dilog(((-I)^(1/2)+exp(b*x+a))/(-I)^(1/2))-1/4*I*f/b^2*ln(1+I*exp(2*b*x+2*a))*a^2-1
/4*I*f/b*polylog(2,-I*exp(2*b*x+2*a))*x-1/4*I*f/b^2*polylog(2,-I*exp(2*b*x+2*a))*a-1/4*I*f/b^2*a^2*ln(-exp(2*b
*x+2*a)+I)-1/2*I*e/b*ln(1+exp(b*x+a)*(-1)^(3/4))*a-1/2*I*e/b*ln(1-exp(b*x+a)*(-1)^(3/4))*a+1/2*I*e/b*a*ln(-exp
(2*b*x+2*a)+I)+1/2*I*f/b^2*a^2*ln(1+exp(b*x+a)*(-1)^(3/4))+1/2*I*f/b^2*a^2*ln(1-exp(b*x+a)*(-1)^(3/4))+1/2*I*f
/b^2*a*dilog(1+exp(b*x+a)*(-1)^(3/4))+1/2*I*f/b^2*a*dilog(1-exp(b*x+a)*(-1)^(3/4))+1/4*I*f/b^2*ln(1-I*exp(2*b*
x+2*a))*a^2+1/4*I*f/b*polylog(2,I*exp(2*b*x+2*a))*x+1/4*I*f/b^2*polylog(2,I*exp(2*b*x+2*a))*a+1/4*I*f/b^2*a^2*
ln(exp(2*b*x+2*a)+I)+1/2*I*e/b*ln(((-I)^(1/2)-exp(b*x+a))/(-I)^(1/2))*a+1/2*I*e/b*ln(((-I)^(1/2)+exp(b*x+a))/(
-I)^(1/2))*a-1/2*I*e/b*a*ln(exp(2*b*x+2*a)+I)-1/2*I*f/b^2*a^2*ln(((-I)^(1/2)-exp(b*x+a))/(-I)^(1/2))-1/2*I*f/b
^2*a^2*ln(((-I)^(1/2)+exp(b*x+a))/(-I)^(1/2))-1/2*I*f/b^2*a*dilog(((-I)^(1/2)-exp(b*x+a))/(-I)^(1/2))-1/2*I*f/
b^2*a*dilog(((-I)^(1/2)+exp(b*x+a))/(-I)^(1/2))-1/2*I*f/b*ln(1+I*exp(2*b*x+2*a))*a*x+1/2*I*f/b*a*ln(1+exp(b*x+
a)*(-1)^(3/4))*x+1/2*I*f/b*a*ln(1-exp(b*x+a)*(-1)^(3/4))*x-1/8*I*f*polylog(3,I*exp(2*b*x+2*a))/b^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. 600 vs. \(2 (130) = 260\).

Time = 0.31 (sec) , antiderivative size = 600, normalized size of antiderivative = 3.77 \[ \int (e+f x) \cot ^{-1}(\tanh (a+b x)) \, dx=\frac {2 \, {\left (b^{2} f x^{2} + 2 \, b^{2} e x\right )} \arctan \left (\frac {\cosh \left (b x + a\right )}{\sinh \left (b x + a\right )}\right ) - 2 \, {\left (-i \, b f x - i \, b e\right )} {\rm Li}_2\left (\frac {1}{2} \, \sqrt {4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - 2 \, {\left (-i \, b f x - i \, b e\right )} {\rm Li}_2\left (-\frac {1}{2} \, \sqrt {4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - 2 \, {\left (i \, b f x + i \, b e\right )} {\rm Li}_2\left (\frac {1}{2} \, \sqrt {-4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - 2 \, {\left (i \, b f x + i \, b e\right )} {\rm Li}_2\left (-\frac {1}{2} \, \sqrt {-4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) + {\left (i \, b^{2} f x^{2} + 2 i \, b^{2} e x + 2 i \, a b e - i \, a^{2} f\right )} \log \left (\frac {1}{2} \, \sqrt {4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) + {\left (i \, b^{2} f x^{2} + 2 i \, b^{2} e x + 2 i \, a b e - i \, a^{2} f\right )} \log \left (-\frac {1}{2} \, \sqrt {4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) + {\left (-i \, b^{2} f x^{2} - 2 i \, b^{2} e x - 2 i \, a b e + i \, a^{2} f\right )} \log \left (\frac {1}{2} \, \sqrt {-4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) + {\left (-i \, b^{2} f x^{2} - 2 i \, b^{2} e x - 2 i \, a b e + i \, a^{2} f\right )} \log \left (-\frac {1}{2} \, \sqrt {-4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) + {\left (-2 i \, a b e + i \, a^{2} f\right )} \log \left (i \, \sqrt {4 i} + 2 \, \cosh \left (b x + a\right ) + 2 \, \sinh \left (b x + a\right )\right ) + {\left (-2 i \, a b e + i \, a^{2} f\right )} \log \left (-i \, \sqrt {4 i} + 2 \, \cosh \left (b x + a\right ) + 2 \, \sinh \left (b x + a\right )\right ) + {\left (2 i \, a b e - i \, a^{2} f\right )} \log \left (i \, \sqrt {-4 i} + 2 \, \cosh \left (b x + a\right ) + 2 \, \sinh \left (b x + a\right )\right ) + {\left (2 i \, a b e - i \, a^{2} f\right )} \log \left (-i \, \sqrt {-4 i} + 2 \, \cosh \left (b x + a\right ) + 2 \, \sinh \left (b x + a\right )\right ) - 2 i \, f {\rm polylog}\left (3, \frac {1}{2} \, \sqrt {4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - 2 i \, f {\rm polylog}\left (3, -\frac {1}{2} \, \sqrt {4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) + 2 i \, f {\rm polylog}\left (3, \frac {1}{2} \, \sqrt {-4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) + 2 i \, f {\rm polylog}\left (3, -\frac {1}{2} \, \sqrt {-4 i} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right )}{4 \, b^{2}} \]

[In]

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

[Out]

1/4*(2*(b^2*f*x^2 + 2*b^2*e*x)*arctan(cosh(b*x + a)/sinh(b*x + a)) - 2*(-I*b*f*x - I*b*e)*dilog(1/2*sqrt(4*I)*
(cosh(b*x + a) + sinh(b*x + a))) - 2*(-I*b*f*x - I*b*e)*dilog(-1/2*sqrt(4*I)*(cosh(b*x + a) + sinh(b*x + a)))
- 2*(I*b*f*x + I*b*e)*dilog(1/2*sqrt(-4*I)*(cosh(b*x + a) + sinh(b*x + a))) - 2*(I*b*f*x + I*b*e)*dilog(-1/2*s
qrt(-4*I)*(cosh(b*x + a) + sinh(b*x + a))) + (I*b^2*f*x^2 + 2*I*b^2*e*x + 2*I*a*b*e - I*a^2*f)*log(1/2*sqrt(4*
I)*(cosh(b*x + a) + sinh(b*x + a)) + 1) + (I*b^2*f*x^2 + 2*I*b^2*e*x + 2*I*a*b*e - I*a^2*f)*log(-1/2*sqrt(4*I)
*(cosh(b*x + a) + sinh(b*x + a)) + 1) + (-I*b^2*f*x^2 - 2*I*b^2*e*x - 2*I*a*b*e + I*a^2*f)*log(1/2*sqrt(-4*I)*
(cosh(b*x + a) + sinh(b*x + a)) + 1) + (-I*b^2*f*x^2 - 2*I*b^2*e*x - 2*I*a*b*e + I*a^2*f)*log(-1/2*sqrt(-4*I)*
(cosh(b*x + a) + sinh(b*x + a)) + 1) + (-2*I*a*b*e + I*a^2*f)*log(I*sqrt(4*I) + 2*cosh(b*x + a) + 2*sinh(b*x +
a)) + (-2*I*a*b*e + I*a^2*f)*log(-I*sqrt(4*I) + 2*cosh(b*x + a) + 2*sinh(b*x + a)) + (2*I*a*b*e - I*a^2*f)*lo
g(I*sqrt(-4*I) + 2*cosh(b*x + a) + 2*sinh(b*x + a)) + (2*I*a*b*e - I*a^2*f)*log(-I*sqrt(-4*I) + 2*cosh(b*x + a
) + 2*sinh(b*x + a)) - 2*I*f*polylog(3, 1/2*sqrt(4*I)*(cosh(b*x + a) + sinh(b*x + a))) - 2*I*f*polylog(3, -1/2
*sqrt(4*I)*(cosh(b*x + a) + sinh(b*x + a))) + 2*I*f*polylog(3, 1/2*sqrt(-4*I)*(cosh(b*x + a) + sinh(b*x + a)))
+ 2*I*f*polylog(3, -1/2*sqrt(-4*I)*(cosh(b*x + a) + sinh(b*x + a))))/b^2

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/2*(f*x^2 + 2*e*x)*arctan2(e^(2*b*x + 2*a) + 1, e^(2*b*x + 2*a) - 1) + integrate((b*f*x^2*e^(2*a) + 2*b*e*x*e
^(2*a))*e^(2*b*x)/(e^(4*b*x + 4*a) + 1), x)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

int(acot(tanh(a + b*x))*(e + f*x),x)

[Out]

int(acot(tanh(a + b*x))*(e + f*x), x)