\(\int \coth ^{-1}(c+d \coth (a+b x)) \, dx\) [219]

Optimal result

Integrand size = 11, antiderivative size = 150 \[ \int \coth ^{-1}(c+d \coth (a+b x)) \, dx=x \coth ^{-1}(c+d \coth (a+b x))+\frac {1}{2} x \log \left (1-\frac {(1-c-d) e^{2 a+2 b x}}{1-c+d}\right )-\frac {1}{2} x \log \left (1-\frac {(1+c+d) e^{2 a+2 b x}}{1+c-d}\right )+\frac {\operatorname {PolyLog}\left (2,\frac {(1-c-d) e^{2 a+2 b x}}{1-c+d}\right )}{4 b}-\frac {\operatorname {PolyLog}\left (2,\frac {(1+c+d) e^{2 a+2 b x}}{1+c-d}\right )}{4 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 150, 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 = {6373, 2221, 2317, 2438} \[ \int \coth ^{-1}(c+d \coth (a+b x)) \, dx=\frac {\operatorname {PolyLog}\left (2,\frac {(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right )}{4 b}-\frac {\operatorname {PolyLog}\left (2,\frac {(c+d+1) e^{2 a+2 b x}}{c-d+1}\right )}{4 b}+\frac {1}{2} x \log \left (1-\frac {(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right )-\frac {1}{2} x \log \left (1-\frac {(c+d+1) e^{2 a+2 b x}}{c-d+1}\right )+x \coth ^{-1}(d \coth (a+b x)+c) \]

[In]

Int[ArcCoth[c + d*Coth[a + b*x]],x]

[Out]

x*ArcCoth[c + d*Coth[a + b*x]] + (x*Log[1 - ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)])/2 - (x*Log[1 - ((1 + c
+ d)*E^(2*a + 2*b*x))/(1 + c - d)])/2 + PolyLog[2, ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)]/(4*b) - PolyLog
[2, ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)]/(4*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 6373

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

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.87 \[ \int \coth ^{-1}(c+d \coth (a+b x)) \, dx=x \coth ^{-1}(c+d \coth (a+b x))-\frac {-2 b x \left (\log \left (1-\frac {(-1+c+d) e^{2 (a+b x)}}{-1+c-d}\right )-\log \left (1-\frac {(1+c+d) e^{2 (a+b x)}}{1+c-d}\right )\right )-\operatorname {PolyLog}\left (2,\frac {(-1+c+d) e^{2 (a+b x)}}{-1+c-d}\right )+\operatorname {PolyLog}\left (2,\frac {(1+c+d) e^{2 (a+b x)}}{1+c-d}\right )}{4 b} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(347\) vs. \(2(138)=276\).

Time = 2.95 (sec) , antiderivative size = 348, normalized size of antiderivative = 2.32

method	result	size
derivativedivides	\(\frac {-\frac {\operatorname {arccoth}\left (c +d \coth \left (b x +a \right )\right ) d \ln \left (-d \coth \left (b x +a \right )+d \right )}{2}+\frac {\operatorname {arccoth}\left (c +d \coth \left (b x +a \right )\right ) d \ln \left (-d \coth \left (b x +a \right )-d \right )}{2}-\frac {d^{2} \left (\frac {\frac {\operatorname {dilog}\left (\frac {-d \coth \left (b x +a \right )-c +1}{1-c -d}\right )}{2}+\frac {\ln \left (-d \coth \left (b x +a \right )+d \right ) \ln \left (\frac {-d \coth \left (b x +a \right )-c +1}{1-c -d}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {-d \coth \left (b x +a \right )-c -1}{-1-c -d}\right )}{2}-\frac {\ln \left (-d \coth \left (b x +a \right )+d \right ) \ln \left (\frac {-d \coth \left (b x +a \right )-c -1}{-1-c -d}\right )}{2}}{d}-\frac {-\frac {\operatorname {dilog}\left (\frac {-d \coth \left (b x +a \right )-c -1}{-1-c +d}\right )}{2}-\frac {\ln \left (-d \coth \left (b x +a \right )-d \right ) \ln \left (\frac {-d \coth \left (b x +a \right )-c -1}{-1-c +d}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {-d \coth \left (b x +a \right )-c +1}{1-c +d}\right )}{2}+\frac {\ln \left (-d \coth \left (b x +a \right )-d \right ) \ln \left (\frac {-d \coth \left (b x +a \right )-c +1}{1-c +d}\right )}{2}}{d}\right )}{2}}{b d}\)	\(348\)
default	\(\frac {-\frac {\operatorname {arccoth}\left (c +d \coth \left (b x +a \right )\right ) d \ln \left (-d \coth \left (b x +a \right )+d \right )}{2}+\frac {\operatorname {arccoth}\left (c +d \coth \left (b x +a \right )\right ) d \ln \left (-d \coth \left (b x +a \right )-d \right )}{2}-\frac {d^{2} \left (\frac {\frac {\operatorname {dilog}\left (\frac {-d \coth \left (b x +a \right )-c +1}{1-c -d}\right )}{2}+\frac {\ln \left (-d \coth \left (b x +a \right )+d \right ) \ln \left (\frac {-d \coth \left (b x +a \right )-c +1}{1-c -d}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {-d \coth \left (b x +a \right )-c -1}{-1-c -d}\right )}{2}-\frac {\ln \left (-d \coth \left (b x +a \right )+d \right ) \ln \left (\frac {-d \coth \left (b x +a \right )-c -1}{-1-c -d}\right )}{2}}{d}-\frac {-\frac {\operatorname {dilog}\left (\frac {-d \coth \left (b x +a \right )-c -1}{-1-c +d}\right )}{2}-\frac {\ln \left (-d \coth \left (b x +a \right )-d \right ) \ln \left (\frac {-d \coth \left (b x +a \right )-c -1}{-1-c +d}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {-d \coth \left (b x +a \right )-c +1}{1-c +d}\right )}{2}+\frac {\ln \left (-d \coth \left (b x +a \right )-d \right ) \ln \left (\frac {-d \coth \left (b x +a \right )-c +1}{1-c +d}\right )}{2}}{d}\right )}{2}}{b d}\)	\(348\)
risch	\(\text {Expression too large to display}\)	\(2951\)

[In]

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

[Out]

1/b/d*(-1/2*arccoth(c+d*coth(b*x+a))*d*ln(-d*coth(b*x+a)+d)+1/2*arccoth(c+d*coth(b*x+a))*d*ln(-d*coth(b*x+a)-d
)-1/2*d^2*(1/d*(1/2*dilog((-d*coth(b*x+a)-c+1)/(1-c-d))+1/2*ln(-d*coth(b*x+a)+d)*ln((-d*coth(b*x+a)-c+1)/(1-c-
d))-1/2*dilog((-d*coth(b*x+a)-c-1)/(-1-c-d))-1/2*ln(-d*coth(b*x+a)+d)*ln((-d*coth(b*x+a)-c-1)/(-1-c-d)))-1/d*(
-1/2*dilog((-d*coth(b*x+a)-c-1)/(-1-c+d))-1/2*ln(-d*coth(b*x+a)-d)*ln((-d*coth(b*x+a)-c-1)/(-1-c+d))+1/2*dilog
((-d*coth(b*x+a)-c+1)/(1-c+d))+1/2*ln(-d*coth(b*x+a)-d)*ln((-d*coth(b*x+a)-c+1)/(1-c+d)))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 539 vs. \(2 (128) = 256\).

Time = 0.28 (sec) , antiderivative size = 539, normalized size of antiderivative = 3.59 \[ \int \coth ^{-1}(c+d \coth (a+b x)) \, dx=\frac {b x \log \left (\frac {d \cosh \left (b x + a\right ) + {\left (c + 1\right )} \sinh \left (b x + a\right )}{d \cosh \left (b x + a\right ) + {\left (c - 1\right )} \sinh \left (b x + a\right )}\right ) + a \log \left (2 \, {\left (c + d + 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (c + d + 1\right )} \sinh \left (b x + a\right ) + 2 \, {\left (c - d + 1\right )} \sqrt {\frac {c + d + 1}{c - d + 1}}\right ) + a \log \left (2 \, {\left (c + d + 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (c + d + 1\right )} \sinh \left (b x + a\right ) - 2 \, {\left (c - d + 1\right )} \sqrt {\frac {c + d + 1}{c - d + 1}}\right ) - a \log \left (2 \, {\left (c + d - 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (c + d - 1\right )} \sinh \left (b x + a\right ) + 2 \, {\left (c - d - 1\right )} \sqrt {\frac {c + d - 1}{c - d - 1}}\right ) - a \log \left (2 \, {\left (c + d - 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (c + d - 1\right )} \sinh \left (b x + a\right ) - 2 \, {\left (c - d - 1\right )} \sqrt {\frac {c + d - 1}{c - d - 1}}\right ) - {\left (b x + a\right )} \log \left (\sqrt {\frac {c + d + 1}{c - d + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) - {\left (b x + a\right )} \log \left (-\sqrt {\frac {c + d + 1}{c - d + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) + {\left (b x + a\right )} \log \left (\sqrt {\frac {c + d - 1}{c - d - 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) + {\left (b x + a\right )} \log \left (-\sqrt {\frac {c + d - 1}{c - d - 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) - {\rm Li}_2\left (\sqrt {\frac {c + d + 1}{c - d + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - {\rm Li}_2\left (-\sqrt {\frac {c + d + 1}{c - d + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) + {\rm Li}_2\left (\sqrt {\frac {c + d - 1}{c - d - 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) + {\rm Li}_2\left (-\sqrt {\frac {c + d - 1}{c - d - 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right )}{2 \, b} \]

[In]

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

[Out]

1/2*(b*x*log((d*cosh(b*x + a) + (c + 1)*sinh(b*x + a))/(d*cosh(b*x + a) + (c - 1)*sinh(b*x + a))) + a*log(2*(c
+ d + 1)*cosh(b*x + a) + 2*(c + d + 1)*sinh(b*x + a) + 2*(c - d + 1)*sqrt((c + d + 1)/(c - d + 1))) + a*log(2
*(c + d + 1)*cosh(b*x + a) + 2*(c + d + 1)*sinh(b*x + a) - 2*(c - d + 1)*sqrt((c + d + 1)/(c - d + 1))) - a*lo
g(2*(c + d - 1)*cosh(b*x + a) + 2*(c + d - 1)*sinh(b*x + a) + 2*(c - d - 1)*sqrt((c + d - 1)/(c - d - 1))) - a
*log(2*(c + d - 1)*cosh(b*x + a) + 2*(c + d - 1)*sinh(b*x + a) - 2*(c - d - 1)*sqrt((c + d - 1)/(c - d - 1)))
- (b*x + a)*log(sqrt((c + d + 1)/(c - d + 1))*(cosh(b*x + a) + sinh(b*x + a)) + 1) - (b*x + a)*log(-sqrt((c +
d + 1)/(c - d + 1))*(cosh(b*x + a) + sinh(b*x + a)) + 1) + (b*x + a)*log(sqrt((c + d - 1)/(c - d - 1))*(cosh(b
*x + a) + sinh(b*x + a)) + 1) + (b*x + a)*log(-sqrt((c + d - 1)/(c - d - 1))*(cosh(b*x + a) + sinh(b*x + a)) +
1) - dilog(sqrt((c + d + 1)/(c - d + 1))*(cosh(b*x + a) + sinh(b*x + a))) - dilog(-sqrt((c + d + 1)/(c - d +
1))*(cosh(b*x + a) + sinh(b*x + a))) + dilog(sqrt((c + d - 1)/(c - d - 1))*(cosh(b*x + a) + sinh(b*x + a))) +
dilog(-sqrt((c + d - 1)/(c - d - 1))*(cosh(b*x + a) + sinh(b*x + a))))/b

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.46 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.95 \[ \int \coth ^{-1}(c+d \coth (a+b x)) \, dx=-\frac {1}{4} \, b d {\left (\frac {2 \, b x \log \left (-\frac {{\left (c + d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d + 1} + 1\right ) + {\rm Li}_2\left (\frac {{\left (c + d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d + 1}\right )}{b^{2} d} - \frac {2 \, b x \log \left (-\frac {{\left (c + d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d - 1} + 1\right ) + {\rm Li}_2\left (\frac {{\left (c + d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d - 1}\right )}{b^{2} d}\right )} + x \operatorname {arcoth}\left (d \coth \left (b x + a\right ) + c\right ) \]

[In]

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

[Out]

-1/4*b*d*((2*b*x*log(-(c + d + 1)*e^(2*b*x + 2*a)/(c - d + 1) + 1) + dilog((c + d + 1)*e^(2*b*x + 2*a)/(c - d
+ 1)))/(b^2*d) - (2*b*x*log(-(c + d - 1)*e^(2*b*x + 2*a)/(c - d - 1) + 1) + dilog((c + d - 1)*e^(2*b*x + 2*a)/
(c - d - 1)))/(b^2*d)) + x*arccoth(d*coth(b*x + a) + c)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(acoth(c + d*coth(a + b*x)),x)

[Out]

int(acoth(c + d*coth(a + b*x)), x)