\(\int \coth ^{-1}(c+d \cot (a+b x)) \, dx\) [255]

Optimal result

Integrand size = 11, antiderivative size = 194 \[ \int \coth ^{-1}(c+d \cot (a+b x)) \, dx=x \coth ^{-1}(c+d \cot (a+b x))+\frac {1}{2} x \log \left (1-\frac {(1-c-i d) e^{2 i a+2 i b x}}{1-c+i d}\right )-\frac {1}{2} x \log \left (1-\frac {(1+c+i d) e^{2 i a+2 i b x}}{1+c-i d}\right )-\frac {i \operatorname {PolyLog}\left (2,\frac {(1-c-i d) e^{2 i a+2 i b x}}{1-c+i d}\right )}{4 b}+\frac {i \operatorname {PolyLog}\left (2,\frac {(1+c+i d) e^{2 i a+2 i b x}}{1+c-i d}\right )}{4 b} \]

[Out]

x*arccoth(c+d*cot(b*x+a))+1/2*x*ln(1-(1-c-I*d)*exp(2*I*a+2*I*b*x)/(1-c+I*d))-1/2*x*ln(1-(1+c+I*d)*exp(2*I*a+2*
I*b*x)/(1+c-I*d))-1/4*I*polylog(2,(1-c-I*d)*exp(2*I*a+2*I*b*x)/(1-c+I*d))/b+1/4*I*polylog(2,(1+c+I*d)*exp(2*I*
a+2*I*b*x)/(1+c-I*d))/b

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6397, 2221, 2317, 2438} \[ \int \coth ^{-1}(c+d \cot (a+b x)) \, dx=-\frac {i \operatorname {PolyLog}\left (2,\frac {(-c-i d+1) e^{2 i a+2 i b x}}{-c+i d+1}\right )}{4 b}+\frac {i \operatorname {PolyLog}\left (2,\frac {(c+i d+1) e^{2 i a+2 i b x}}{c-i d+1}\right )}{4 b}+\frac {1}{2} x \log \left (1-\frac {(-c-i d+1) e^{2 i a+2 i b x}}{-c+i d+1}\right )-\frac {1}{2} x \log \left (1-\frac {(c+i d+1) e^{2 i a+2 i b x}}{c-i d+1}\right )+x \coth ^{-1}(d \cot (a+b x)+c) \]

[In]

Int[ArcCoth[c + d*Cot[a + b*x]],x]

[Out]

x*ArcCoth[c + d*Cot[a + b*x]] + (x*Log[1 - ((1 - c - I*d)*E^((2*I)*a + (2*I)*b*x))/(1 - c + I*d)])/2 - (x*Log[
1 - ((1 + c + I*d)*E^((2*I)*a + (2*I)*b*x))/(1 + c - I*d)])/2 - ((I/4)*PolyLog[2, ((1 - c - I*d)*E^((2*I)*a +
(2*I)*b*x))/(1 - c + I*d)])/b + ((I/4)*PolyLog[2, ((1 + c + I*d)*E^((2*I)*a + (2*I)*b*x))/(1 + c - I*d)])/b

Rule 2221

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]
- Dist[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]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 6397

Int[ArcCoth[(c_.) + Cot[(a_.) + (b_.)*(x_)]*(d_.)], x_Symbol] :> Simp[x*ArcCoth[c + d*Cot[a + b*x]], x] + (-Di
st[I*b*(1 - c - I*d), Int[x*(E^(2*I*a + 2*I*b*x)/(1 - c + I*d - (1 - c - I*d)*E^(2*I*a + 2*I*b*x))), x], x] +
Dist[I*b*(1 + c + I*d), Int[x*(E^(2*I*a + 2*I*b*x)/(1 + c - I*d - (1 + c + I*d)*E^(2*I*a + 2*I*b*x))), x], x])
/; FreeQ[{a, b, c, d}, x] && NeQ[(c - I*d)^2, 1]

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

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(390\) vs. \(2(194)=388\).

Time = 0.52 (sec) , antiderivative size = 390, normalized size of antiderivative = 2.01 \[ \int \coth ^{-1}(c+d \cot (a+b x)) \, dx=x \left (\coth ^{-1}(c+d \cot (a+b x))+\frac {2 a \log (d+(-1+c) \tan (a+b x))+i \log (1+i \tan (a+b x)) \log \left (-\frac {i (d+(-1+c) \tan (a+b x))}{-1+c-i d}\right )-i \log (1-i \tan (a+b x)) \log \left (\frac {i (d+(-1+c) \tan (a+b x))}{-1+c+i d}\right )-2 a \log (d+(1+c) \tan (a+b x))+i \log (1-i \tan (a+b x)) \log \left (\frac {i (d+(1+c) \tan (a+b x))}{1+c+i d}\right )-i \log (1+i \tan (a+b x)) \log \left (\frac {d+(1+c) \tan (a+b x)}{i (1+c)+d}\right )-i \operatorname {PolyLog}\left (2,\frac {(-1+c) (1-i \tan (a+b x))}{-1+c+i d}\right )+i \operatorname {PolyLog}\left (2,\frac {(1+c) (1-i \tan (a+b x))}{1+c+i d}\right )+i \operatorname {PolyLog}\left (2,\frac {(-1+c) (1+i \tan (a+b x))}{-1+c-i d}\right )-i \operatorname {PolyLog}\left (2,\frac {(1+c) (1+i \tan (a+b x))}{1+c-i d}\right )}{4 a-2 i \log (1-i \tan (a+b x))+2 i \log (1+i \tan (a+b x))}\right ) \]

[In]

Integrate[ArcCoth[c + d*Cot[a + b*x]],x]

[Out]

x*(ArcCoth[c + d*Cot[a + b*x]] + (2*a*Log[d + (-1 + c)*Tan[a + b*x]] + I*Log[1 + I*Tan[a + b*x]]*Log[((-I)*(d
+ (-1 + c)*Tan[a + b*x]))/(-1 + c - I*d)] - I*Log[1 - I*Tan[a + b*x]]*Log[(I*(d + (-1 + c)*Tan[a + b*x]))/(-1
+ c + I*d)] - 2*a*Log[d + (1 + c)*Tan[a + b*x]] + I*Log[1 - I*Tan[a + b*x]]*Log[(I*(d + (1 + c)*Tan[a + b*x]))
/(1 + c + I*d)] - I*Log[1 + I*Tan[a + b*x]]*Log[(d + (1 + c)*Tan[a + b*x])/(I*(1 + c) + d)] - I*PolyLog[2, ((-
1 + c)*(1 - I*Tan[a + b*x]))/(-1 + c + I*d)] + I*PolyLog[2, ((1 + c)*(1 - I*Tan[a + b*x]))/(1 + c + I*d)] + I*
PolyLog[2, ((-1 + c)*(1 + I*Tan[a + b*x]))/(-1 + c - I*d)] - I*PolyLog[2, ((1 + c)*(1 + I*Tan[a + b*x]))/(1 +
c - I*d)])/(4*a - (2*I)*Log[1 - I*Tan[a + b*x]] + (2*I)*Log[1 + I*Tan[a + b*x]]))

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 563 vs. \(2 (164 ) = 328\).

Time = 2.99 (sec) , antiderivative size = 564, normalized size of antiderivative = 2.91

method	result	size
derivativedivides	\(\frac {-d \left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (b x +a \right )\right )\right ) \operatorname {arccoth}\left (c +d \cot \left (b x +a \right )\right )-d^{2} \left (\frac {\arctan \left (-\frac {c +d \cot \left (b x +a \right )}{d}+\frac {c}{d}\right ) \ln \left (d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )+c +1\right )}{2 d}-\frac {\arctan \left (-\frac {c +d \cot \left (b x +a \right )}{d}+\frac {c}{d}\right ) \ln \left (d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )+c -1\right )}{2 d}+\frac {i \ln \left (d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )+c -1\right ) \left (\ln \left (\frac {i d -d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d +c -1}\right )-\ln \left (\frac {i d +d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d -c +1}\right )\right )}{4 d}+\frac {i \left (\operatorname {dilog}\left (\frac {i d -d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d +c -1}\right )-\operatorname {dilog}\left (\frac {i d +d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d -c +1}\right )\right )}{4 d}-\frac {i \ln \left (d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )+c +1\right ) \left (\ln \left (\frac {i d -d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d +c +1}\right )-\ln \left (\frac {i d +d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d -c -1}\right )\right )}{4 d}-\frac {i \left (\operatorname {dilog}\left (\frac {i d -d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d +c +1}\right )-\operatorname {dilog}\left (\frac {i d +d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d -c -1}\right )\right )}{4 d}\right )}{b d}\)	\(564\)
default	\(\frac {-d \left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (b x +a \right )\right )\right ) \operatorname {arccoth}\left (c +d \cot \left (b x +a \right )\right )-d^{2} \left (\frac {\arctan \left (-\frac {c +d \cot \left (b x +a \right )}{d}+\frac {c}{d}\right ) \ln \left (d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )+c +1\right )}{2 d}-\frac {\arctan \left (-\frac {c +d \cot \left (b x +a \right )}{d}+\frac {c}{d}\right ) \ln \left (d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )+c -1\right )}{2 d}+\frac {i \ln \left (d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )+c -1\right ) \left (\ln \left (\frac {i d -d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d +c -1}\right )-\ln \left (\frac {i d +d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d -c +1}\right )\right )}{4 d}+\frac {i \left (\operatorname {dilog}\left (\frac {i d -d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d +c -1}\right )-\operatorname {dilog}\left (\frac {i d +d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d -c +1}\right )\right )}{4 d}-\frac {i \ln \left (d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )+c +1\right ) \left (\ln \left (\frac {i d -d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d +c +1}\right )-\ln \left (\frac {i d +d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d -c -1}\right )\right )}{4 d}-\frac {i \left (\operatorname {dilog}\left (\frac {i d -d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d +c +1}\right )-\operatorname {dilog}\left (\frac {i d +d \left (\frac {c +d \cot \left (b x +a \right )}{d}-\frac {c}{d}\right )}{i d -c -1}\right )\right )}{4 d}\right )}{b d}\)	\(564\)
risch	\(\text {Expression too large to display}\)	\(3921\)

[In]

int(arccoth(c+d*cot(b*x+a)),x,method=_RETURNVERBOSE)

[Out]

1/b/d*(-d*(1/2*Pi-arccot(cot(b*x+a)))*arccoth(c+d*cot(b*x+a))-d^2*(1/2*arctan(-(c+d*cot(b*x+a))/d+c/d)/d*ln(d*
((c+d*cot(b*x+a))/d-c/d)+c+1)-1/2*arctan(-(c+d*cot(b*x+a))/d+c/d)/d*ln(d*((c+d*cot(b*x+a))/d-c/d)+c-1)+1/4*I*l
n(d*((c+d*cot(b*x+a))/d-c/d)+c-1)*(ln((I*d-d*((c+d*cot(b*x+a))/d-c/d))/(I*d+c-1))-ln((I*d+d*((c+d*cot(b*x+a))/
d-c/d))/(1-c+I*d)))/d+1/4*I*(dilog((I*d-d*((c+d*cot(b*x+a))/d-c/d))/(I*d+c-1))-dilog((I*d+d*((c+d*cot(b*x+a))/
d-c/d))/(1-c+I*d)))/d-1/4*I*ln(d*((c+d*cot(b*x+a))/d-c/d)+c+1)*(ln((I*d-d*((c+d*cot(b*x+a))/d-c/d))/(1+c+I*d))
-ln((I*d+d*((c+d*cot(b*x+a))/d-c/d))/(I*d-c-1)))/d-1/4*I*(dilog((I*d-d*((c+d*cot(b*x+a))/d-c/d))/(1+c+I*d))-di
log((I*d+d*((c+d*cot(b*x+a))/d-c/d))/(I*d-c-1)))/d))

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. 1098 vs. \(2 (136) = 272\).

Time = 0.38 (sec) , antiderivative size = 1098, normalized size of antiderivative = 5.66 \[ \int \coth ^{-1}(c+d \cot (a+b x)) \, dx=\text {Too large to display} \]

[In]

integrate(arccoth(c+d*cot(b*x+a)),x, algorithm="fricas")

[Out]

1/8*(4*b*x*log((d*cos(2*b*x + 2*a) + (c + 1)*sin(2*b*x + 2*a) + d)/(d*cos(2*b*x + 2*a) + (c - 1)*sin(2*b*x + 2
*a) + d)) + 2*a*log(1/2*c^2 + I*(c + 1)*d - 1/2*d^2 - 1/2*(c^2 + d^2 + 2*c + 1)*cos(2*b*x + 2*a) + 1/2*(I*c^2
+ I*d^2 + 2*I*c + I)*sin(2*b*x + 2*a) + c + 1/2) - 2*a*log(1/2*c^2 + I*(c - 1)*d - 1/2*d^2 - 1/2*(c^2 + d^2 -
2*c + 1)*cos(2*b*x + 2*a) + 1/2*(I*c^2 + I*d^2 - 2*I*c + I)*sin(2*b*x + 2*a) - c + 1/2) + 2*a*log(-1/2*c^2 + I
*(c + 1)*d + 1/2*d^2 + 1/2*(c^2 + d^2 + 2*c + 1)*cos(2*b*x + 2*a) + 1/2*(I*c^2 + I*d^2 + 2*I*c + I)*sin(2*b*x
+ 2*a) - c - 1/2) - 2*a*log(-1/2*c^2 + I*(c - 1)*d + 1/2*d^2 + 1/2*(c^2 + d^2 - 2*c + 1)*cos(2*b*x + 2*a) + 1/
2*(I*c^2 + I*d^2 - 2*I*c + I)*sin(2*b*x + 2*a) + c - 1/2) - 2*(b*x + a)*log((c^2 + d^2 - (c^2 + 2*I*(c + 1)*d
- d^2 + 2*c + 1)*cos(2*b*x + 2*a) + (-I*c^2 + 2*(c + 1)*d + I*d^2 - 2*I*c - I)*sin(2*b*x + 2*a) + 2*c + 1)/(c^
2 + d^2 + 2*c + 1)) - 2*(b*x + a)*log((c^2 + d^2 - (c^2 - 2*I*(c + 1)*d - d^2 + 2*c + 1)*cos(2*b*x + 2*a) + (I
*c^2 + 2*(c + 1)*d - I*d^2 + 2*I*c + I)*sin(2*b*x + 2*a) + 2*c + 1)/(c^2 + d^2 + 2*c + 1)) + 2*(b*x + a)*log((
c^2 + d^2 - (c^2 + 2*I*(c - 1)*d - d^2 - 2*c + 1)*cos(2*b*x + 2*a) + (-I*c^2 + 2*(c - 1)*d + I*d^2 + 2*I*c - I
)*sin(2*b*x + 2*a) - 2*c + 1)/(c^2 + d^2 - 2*c + 1)) + 2*(b*x + a)*log((c^2 + d^2 - (c^2 - 2*I*(c - 1)*d - d^2
- 2*c + 1)*cos(2*b*x + 2*a) + (I*c^2 + 2*(c - 1)*d - I*d^2 - 2*I*c + I)*sin(2*b*x + 2*a) - 2*c + 1)/(c^2 + d^
2 - 2*c + 1)) + I*dilog(-(c^2 + d^2 - (c^2 + 2*I*(c + 1)*d - d^2 + 2*c + 1)*cos(2*b*x + 2*a) + (-I*c^2 + 2*(c
+ 1)*d + I*d^2 - 2*I*c - I)*sin(2*b*x + 2*a) + 2*c + 1)/(c^2 + d^2 + 2*c + 1) + 1) - I*dilog(-(c^2 + d^2 - (c^
2 - 2*I*(c + 1)*d - d^2 + 2*c + 1)*cos(2*b*x + 2*a) + (I*c^2 + 2*(c + 1)*d - I*d^2 + 2*I*c + I)*sin(2*b*x + 2*
a) + 2*c + 1)/(c^2 + d^2 + 2*c + 1) + 1) - I*dilog(-(c^2 + d^2 - (c^2 + 2*I*(c - 1)*d - d^2 - 2*c + 1)*cos(2*b
*x + 2*a) + (-I*c^2 + 2*(c - 1)*d + I*d^2 + 2*I*c - I)*sin(2*b*x + 2*a) - 2*c + 1)/(c^2 + d^2 - 2*c + 1) + 1)
+ I*dilog(-(c^2 + d^2 - (c^2 - 2*I*(c - 1)*d - d^2 - 2*c + 1)*cos(2*b*x + 2*a) + (I*c^2 + 2*(c - 1)*d - I*d^2
- 2*I*c + I)*sin(2*b*x + 2*a) - 2*c + 1)/(c^2 + d^2 - 2*c + 1) + 1))/b

Sympy [F]

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

[In]

integrate(acoth(c+d*cot(b*x+a)),x)

[Out]

Integral(acoth(c + d*cot(a + b*x)), x)

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 392 vs. \(2 (136) = 272\).

Time = 0.38 (sec) , antiderivative size = 392, normalized size of antiderivative = 2.02 \[ \int \coth ^{-1}(c+d \cot (a+b x)) \, dx=\frac {4 \, {\left (b x + a\right )} \operatorname {arcoth}\left (c + \frac {d}{\tan \left (b x + a\right )}\right ) + {\left (\arctan \left (\frac {{\left (c + 1\right )} d + {\left (c^{2} + 2 \, c + 1\right )} \tan \left (b x + a\right )}{c^{2} + d^{2} + 2 \, c + 1}, \frac {{\left (c + 1\right )} d \tan \left (b x + a\right ) + d^{2}}{c^{2} + d^{2} + 2 \, c + 1}\right ) - \arctan \left (\frac {{\left (c - 1\right )} d + {\left (c^{2} - 2 \, c + 1\right )} \tan \left (b x + a\right )}{c^{2} + d^{2} - 2 \, c + 1}, \frac {{\left (c - 1\right )} d \tan \left (b x + a\right ) + d^{2}}{c^{2} + d^{2} - 2 \, c + 1}\right )\right )} \log \left (\tan \left (b x + a\right )^{2} + 1\right ) - {\left (b x + a\right )} \log \left (\frac {2 \, {\left (c + 1\right )} d \tan \left (b x + a\right ) + {\left (c^{2} + 2 \, c + 1\right )} \tan \left (b x + a\right )^{2} + d^{2}}{c^{2} + d^{2} + 2 \, c + 1}\right ) + {\left (b x + a\right )} \log \left (\frac {2 \, {\left (c - 1\right )} d \tan \left (b x + a\right ) + {\left (c^{2} - 2 \, c + 1\right )} \tan \left (b x + a\right )^{2} + d^{2}}{c^{2} + d^{2} - 2 \, c + 1}\right ) + i \, {\rm Li}_2\left (-\frac {{\left (c + 1\right )} \tan \left (b x + a\right ) - i \, c - i}{i \, c + d + i}\right ) - i \, {\rm Li}_2\left (-\frac {{\left (c - 1\right )} \tan \left (b x + a\right ) - i \, c + i}{i \, c + d - i}\right ) + i \, {\rm Li}_2\left (-\frac {{\left (c - 1\right )} \tan \left (b x + a\right ) + i \, c - i}{-i \, c + d + i}\right ) - i \, {\rm Li}_2\left (-\frac {{\left (c + 1\right )} \tan \left (b x + a\right ) + i \, c + i}{-i \, c + d - i}\right )}{4 \, b} \]

[In]

integrate(arccoth(c+d*cot(b*x+a)),x, algorithm="maxima")

[Out]

1/4*(4*(b*x + a)*arccoth(c + d/tan(b*x + a)) + (arctan2(((c + 1)*d + (c^2 + 2*c + 1)*tan(b*x + a))/(c^2 + d^2
+ 2*c + 1), ((c + 1)*d*tan(b*x + a) + d^2)/(c^2 + d^2 + 2*c + 1)) - arctan2(((c - 1)*d + (c^2 - 2*c + 1)*tan(b
*x + a))/(c^2 + d^2 - 2*c + 1), ((c - 1)*d*tan(b*x + a) + d^2)/(c^2 + d^2 - 2*c + 1)))*log(tan(b*x + a)^2 + 1)
- (b*x + a)*log((2*(c + 1)*d*tan(b*x + a) + (c^2 + 2*c + 1)*tan(b*x + a)^2 + d^2)/(c^2 + d^2 + 2*c + 1)) + (b
*x + a)*log((2*(c - 1)*d*tan(b*x + a) + (c^2 - 2*c + 1)*tan(b*x + a)^2 + d^2)/(c^2 + d^2 - 2*c + 1)) + I*dilog
(-((c + 1)*tan(b*x + a) - I*c - I)/(I*c + d + I)) - I*dilog(-((c - 1)*tan(b*x + a) - I*c + I)/(I*c + d - I)) +
I*dilog(-((c - 1)*tan(b*x + a) + I*c - I)/(-I*c + d + I)) - I*dilog(-((c + 1)*tan(b*x + a) + I*c + I)/(-I*c +
d - I)))/b

Giac [F]

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

[In]

integrate(arccoth(c+d*cot(b*x+a)),x, algorithm="giac")

[Out]

integrate(arccoth(d*cot(b*x + a) + c), x)

Mupad [F(-1)]

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

[In]

int(acoth(c + d*cot(a + b*x)),x)

[Out]

int(acoth(c + d*cot(a + b*x)), x)