\(\int x \coth ^{-1}(1-d-d \coth (a+b x)) \, dx\) [228]

Optimal result

Integrand size = 17, antiderivative size = 109 \[ \int x \coth ^{-1}(1-d-d \coth (a+b x)) \, dx=\frac {b x^3}{6}+\frac {1}{2} x^2 \coth ^{-1}(1-d-d \coth (a+b x))-\frac {1}{4} x^2 \log \left (1-(1-d) e^{2 a+2 b x}\right )-\frac {x \operatorname {PolyLog}\left (2,(1-d) e^{2 a+2 b x}\right )}{4 b}+\frac {\operatorname {PolyLog}\left (3,(1-d) e^{2 a+2 b x}\right )}{8 b^2} \]

[Out]

1/6*b*x^3+1/2*x^2*arccoth(1-d-d*coth(b*x+a))-1/4*x^2*ln(1-(1-d)*exp(2*b*x+2*a))-1/4*x*polylog(2,(1-d)*exp(2*b*
x+2*a))/b+1/8*polylog(3,(1-d)*exp(2*b*x+2*a))/b^2

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {6377, 2215, 2221, 2611, 2320, 6724} \[ \int x \coth ^{-1}(1-d-d \coth (a+b x)) \, dx=\frac {\operatorname {PolyLog}\left (3,(1-d) e^{2 a+2 b x}\right )}{8 b^2}-\frac {x \operatorname {PolyLog}\left (2,(1-d) e^{2 a+2 b x}\right )}{4 b}-\frac {1}{4} x^2 \log \left (1-(1-d) e^{2 a+2 b x}\right )+\frac {1}{2} x^2 \coth ^{-1}(d (-\coth (a+b x))-d+1)+\frac {b x^3}{6} \]

[In]

Int[x*ArcCoth[1 - d - d*Coth[a + b*x]],x]

[Out]

(b*x^3)/6 + (x^2*ArcCoth[1 - d - d*Coth[a + b*x]])/2 - (x^2*Log[1 - (1 - d)*E^(2*a + 2*b*x)])/4 - (x*PolyLog[2
, (1 - d)*E^(2*a + 2*b*x)])/(4*b) + PolyLog[3, (1 - d)*E^(2*a + 2*b*x)]/(8*b^2)

Rule 2215

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[(c
+ d*x)^(m + 1)/(a*d*(m + 1)), x] - Dist[b/a, Int[(c + d*x)^m*((F^(g*(e + f*x)))^n/(a + b*(F^(g*(e + f*x)))^n))
, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

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 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 6377

Int[ArcCoth[(c_.) + Coth[(a_.) + (b_.)*(x_)]*(d_.)]*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^(m
+ 1)*(ArcCoth[c + d*Coth[a + b*x]]/(f*(m + 1))), x] + Dist[b/(f*(m + 1)), Int[(e + f*x)^(m + 1)/(c - d - c*E^
(2*a + 2*b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, 1]

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 {1}{2} x^2 \coth ^{-1}(1-d-d \coth (a+b x))+\frac {1}{2} b \int \frac {x^2}{1+(-1+d) e^{2 a+2 b x}} \, dx \\ & = \frac {b x^3}{6}+\frac {1}{2} x^2 \coth ^{-1}(1-d-d \coth (a+b x))+\frac {1}{2} (b (1-d)) \int \frac {e^{2 a+2 b x} x^2}{1+(-1+d) e^{2 a+2 b x}} \, dx \\ & = \frac {b x^3}{6}+\frac {1}{2} x^2 \coth ^{-1}(1-d-d \coth (a+b x))-\frac {1}{4} x^2 \log \left (1-(1-d) e^{2 a+2 b x}\right )+\frac {1}{2} \int x \log \left (1+(-1+d) e^{2 a+2 b x}\right ) \, dx \\ & = \frac {b x^3}{6}+\frac {1}{2} x^2 \coth ^{-1}(1-d-d \coth (a+b x))-\frac {1}{4} x^2 \log \left (1-(1-d) e^{2 a+2 b x}\right )-\frac {x \operatorname {PolyLog}\left (2,(1-d) e^{2 a+2 b x}\right )}{4 b}+\frac {\int \operatorname {PolyLog}\left (2,(1-d) e^{2 a+2 b x}\right ) \, dx}{4 b} \\ & = \frac {b x^3}{6}+\frac {1}{2} x^2 \coth ^{-1}(1-d-d \coth (a+b x))-\frac {1}{4} x^2 \log \left (1-(1-d) e^{2 a+2 b x}\right )-\frac {x \operatorname {PolyLog}\left (2,(1-d) e^{2 a+2 b x}\right )}{4 b}+\frac {\text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,(1-d) x)}{x} \, dx,x,e^{2 a+2 b x}\right )}{8 b^2} \\ & = \frac {b x^3}{6}+\frac {1}{2} x^2 \coth ^{-1}(1-d-d \coth (a+b x))-\frac {1}{4} x^2 \log \left (1-(1-d) e^{2 a+2 b x}\right )-\frac {x \operatorname {PolyLog}\left (2,(1-d) e^{2 a+2 b x}\right )}{4 b}+\frac {\operatorname {PolyLog}\left (3,(1-d) e^{2 a+2 b x}\right )}{8 b^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.86 \[ \int x \coth ^{-1}(1-d-d \coth (a+b x)) \, dx=\frac {2 b^2 x^2 \left (2 \coth ^{-1}(1-d-d \coth (a+b x))-\log \left (1+\frac {e^{-2 (a+b x)}}{-1+d}\right )\right )+2 b x \operatorname {PolyLog}\left (2,-\frac {e^{-2 (a+b x)}}{-1+d}\right )+\operatorname {PolyLog}\left (3,-\frac {e^{-2 (a+b x)}}{-1+d}\right )}{8 b^2} \]

[In]

Integrate[x*ArcCoth[1 - d - d*Coth[a + b*x]],x]

[Out]

(2*b^2*x^2*(2*ArcCoth[1 - d - d*Coth[a + b*x]] - Log[1 + 1/((-1 + d)*E^(2*(a + b*x)))]) + 2*b*x*PolyLog[2, -(1
/((-1 + d)*E^(2*(a + b*x))))] + PolyLog[3, -(1/((-1 + d)*E^(2*(a + b*x))))])/(8*b^2)

Maple [C] (warning: unable to verify)

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

Time = 1.57 (sec) , antiderivative size = 1640, normalized size of antiderivative = 15.05

method	result	size
risch	\(\text {Expression too large to display}\)	\(1640\)

[In]

int(x*arccoth(1-d-d*coth(b*x+a)),x,method=_RETURNVERBOSE)

[Out]

1/2/b*a*d/(d-1)*x*ln(1+exp(b*x+a)*(1-d)^(1/2))+1/2/b*a*d/(d-1)*x*ln(1-exp(b*x+a)*(1-d)^(1/2))-1/2/b*d/(d-1)*ln
(1+(d-1)*exp(2*b*x+2*a))*a*x-1/2*x^2*ln(exp(b*x+a))+1/6*b*x^3+1/2/b^2*a^2*d/(d-1)*ln(1+exp(b*x+a)*(1-d)^(1/2))
+1/2/b^2*a^2*d/(d-1)*ln(1-exp(b*x+a)*(1-d)^(1/2))+1/2/b^2*a*d/(d-1)*dilog(1+exp(b*x+a)*(1-d)^(1/2))+1/2/b^2*a*
d/(d-1)*dilog(1-exp(b*x+a)*(1-d)^(1/2))-1/8/b^2/(d-1)*polylog(3,-(d-1)*exp(2*b*x+2*a))+1/4/(d-1)*ln(1+(d-1)*ex
p(2*b*x+2*a))*x^2+1/8/b^2*d/(d-1)*polylog(3,-(d-1)*exp(2*b*x+2*a))+1/4/b^2/(d-1)*ln(1+(d-1)*exp(2*b*x+2*a))*a^
2+1/4/b/(d-1)*polylog(2,-(d-1)*exp(2*b*x+2*a))*x+1/4/b^2/(d-1)*polylog(2,-(d-1)*exp(2*b*x+2*a))*a-1/2/b^2*a^2/
(d-1)*ln(1+exp(b*x+a)*(1-d)^(1/2))-1/2/b^2*a^2/(d-1)*ln(1-exp(b*x+a)*(1-d)^(1/2))-1/2/b^2*a/(d-1)*dilog(1+exp(
b*x+a)*(1-d)^(1/2))-1/2/b^2*a/(d-1)*dilog(1-exp(b*x+a)*(1-d)^(1/2))+1/4/b^2*a^2/(d-1)*ln(d*exp(2*b*x+2*a)-exp(
2*b*x+2*a)+1)-1/4*d/(d-1)*ln(1+(d-1)*exp(2*b*x+2*a))*x^2-1/4/b^2*d/(d-1)*ln(1+(d-1)*exp(2*b*x+2*a))*a^2-1/4/b^
2*a^2*d/(d-1)*ln(d*exp(2*b*x+2*a)-exp(2*b*x+2*a)+1)-1/4/b*d/(d-1)*polylog(2,-(d-1)*exp(2*b*x+2*a))*x-1/4/b^2*d
/(d-1)*polylog(2,-(d-1)*exp(2*b*x+2*a))*a+1/2/b/(d-1)*ln(1+(d-1)*exp(2*b*x+2*a))*a*x-1/2/b*a/(d-1)*x*ln(1+exp(
b*x+a)*(1-d)^(1/2))-1/2/b*a/(d-1)*x*ln(1-exp(b*x+a)*(1-d)^(1/2))+1/4*x^2*ln(d*exp(2*b*x+2*a)-exp(2*b*x+2*a)+1)
-1/8*(-I*Pi*csgn(I/(exp(2*b*x+2*a)-1)*(d*exp(2*b*x+2*a)-exp(2*b*x+2*a)+1))^3-I*Pi*csgn(I/(exp(2*b*x+2*a)-1))*c
sgn(I/(exp(2*b*x+2*a)-1)*(d*exp(2*b*x+2*a)-exp(2*b*x+2*a)+1))^2+2*I*Pi*csgn(I/(exp(2*b*x+2*a)-1)*(d*exp(2*b*x+
2*a)-exp(2*b*x+2*a)+1))^2+I*Pi*csgn(I/(exp(2*b*x+2*a)-1))*csgn(I*(d*exp(2*b*x+2*a)-exp(2*b*x+2*a)+1))*csgn(I/(
exp(2*b*x+2*a)-1)*(d*exp(2*b*x+2*a)-exp(2*b*x+2*a)+1))-I*Pi*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))-I*Pi*c
sgn(I*exp(2*b*x+2*a))^3+I*Pi*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)-1))^2+I*Pi*csgn(I*d/
(exp(2*b*x+2*a)-1)*exp(2*b*x+2*a))^3-I*Pi*csgn(I*(d*exp(2*b*x+2*a)-exp(2*b*x+2*a)+1))*csgn(I/(exp(2*b*x+2*a)-1
)*(d*exp(2*b*x+2*a)-exp(2*b*x+2*a)+1))^2+I*Pi*csgn(I/(exp(2*b*x+2*a)-1))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)
-1))^2-I*Pi*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)-1))*csgn(I*d)*csgn(I*d/(exp(2*b*x+2*a)-1)*exp(2*b*x+2*a))-I*
Pi*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)-1))^3+I*Pi*csgn(I*d)*csgn(I*d/(exp(2*b*x+2*a)-1)*exp(2*b*x+2*a))^2+2*
I*Pi*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2-2*I*Pi*csgn(I*d/(exp(2*b*x+2*a)-1)*exp(2*b*x+2*a))^2-I*Pi*csg
n(I*exp(2*b*x+2*a))*csgn(I/(exp(2*b*x+2*a)-1))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)-1))+I*Pi*csgn(I*exp(2*b*x
+2*a)/(exp(2*b*x+2*a)-1))*csgn(I*d/(exp(2*b*x+2*a)-1)*exp(2*b*x+2*a))^2+2*ln(d))*x^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 322 vs. \(2 (90) = 180\).

Time = 0.26 (sec) , antiderivative size = 322, normalized size of antiderivative = 2.95 \[ \int x \coth ^{-1}(1-d-d \coth (a+b x)) \, dx=\frac {2 \, b^{3} x^{3} - 3 \, b^{2} x^{2} \log \left (\frac {d \cosh \left (b x + a\right ) + d \sinh \left (b x + a\right )}{d \cosh \left (b x + a\right ) + {\left (d - 2\right )} \sinh \left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_2\left (\frac {1}{2} \, \sqrt {-4 \, d + 4} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - 6 \, b x {\rm Li}_2\left (-\frac {1}{2} \, \sqrt {-4 \, d + 4} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - 3 \, a^{2} \log \left (2 \, {\left (d - 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (d - 1\right )} \sinh \left (b x + a\right ) + \sqrt {-4 \, d + 4}\right ) - 3 \, a^{2} \log \left (2 \, {\left (d - 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (d - 1\right )} \sinh \left (b x + a\right ) - \sqrt {-4 \, d + 4}\right ) - 3 \, {\left (b^{2} x^{2} - a^{2}\right )} \log \left (\frac {1}{2} \, \sqrt {-4 \, d + 4} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) - 3 \, {\left (b^{2} x^{2} - a^{2}\right )} \log \left (-\frac {1}{2} \, \sqrt {-4 \, d + 4} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) + 6 \, {\rm polylog}\left (3, \frac {1}{2} \, \sqrt {-4 \, d + 4} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) + 6 \, {\rm polylog}\left (3, -\frac {1}{2} \, \sqrt {-4 \, d + 4} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right )}{12 \, b^{2}} \]

[In]

integrate(x*arccoth(1-d-d*coth(b*x+a)),x, algorithm="fricas")

[Out]

1/12*(2*b^3*x^3 - 3*b^2*x^2*log((d*cosh(b*x + a) + d*sinh(b*x + a))/(d*cosh(b*x + a) + (d - 2)*sinh(b*x + a)))
- 6*b*x*dilog(1/2*sqrt(-4*d + 4)*(cosh(b*x + a) + sinh(b*x + a))) - 6*b*x*dilog(-1/2*sqrt(-4*d + 4)*(cosh(b*x
+ a) + sinh(b*x + a))) - 3*a^2*log(2*(d - 1)*cosh(b*x + a) + 2*(d - 1)*sinh(b*x + a) + sqrt(-4*d + 4)) - 3*a^
2*log(2*(d - 1)*cosh(b*x + a) + 2*(d - 1)*sinh(b*x + a) - sqrt(-4*d + 4)) - 3*(b^2*x^2 - a^2)*log(1/2*sqrt(-4*
d + 4)*(cosh(b*x + a) + sinh(b*x + a)) + 1) - 3*(b^2*x^2 - a^2)*log(-1/2*sqrt(-4*d + 4)*(cosh(b*x + a) + sinh(
b*x + a)) + 1) + 6*polylog(3, 1/2*sqrt(-4*d + 4)*(cosh(b*x + a) + sinh(b*x + a))) + 6*polylog(3, -1/2*sqrt(-4*
d + 4)*(cosh(b*x + a) + sinh(b*x + a))))/b^2

Sympy [F]

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

[In]

integrate(x*acoth(1-d-d*coth(b*x+a)),x)

[Out]

-Integral(x*acoth(d*coth(a + b*x) + d - 1), x)

Maxima [A] (verification not implemented)

none

Time = 0.83 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.93 \[ \int x \coth ^{-1}(1-d-d \coth (a+b x)) \, dx=\frac {1}{24} \, {\left (\frac {4 \, x^{3}}{d} - \frac {3 \, {\left (2 \, b^{2} x^{2} \log \left ({\left (d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-{\left (d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}\right ) - {\rm Li}_{3}(-{\left (d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )})\right )}}{b^{3} d}\right )} b d - \frac {1}{2} \, x^{2} \operatorname {arcoth}\left (d \coth \left (b x + a\right ) + d - 1\right ) \]

[In]

integrate(x*arccoth(1-d-d*coth(b*x+a)),x, algorithm="maxima")

[Out]

1/24*(4*x^3/d - 3*(2*b^2*x^2*log((d - 1)*e^(2*b*x + 2*a) + 1) + 2*b*x*dilog(-(d - 1)*e^(2*b*x + 2*a)) - polylo
g(3, -(d - 1)*e^(2*b*x + 2*a)))/(b^3*d))*b*d - 1/2*x^2*arccoth(d*coth(b*x + a) + d - 1)

Giac [F]

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

[In]

integrate(x*arccoth(1-d-d*coth(b*x+a)),x, algorithm="giac")

[Out]

integrate(x*arccoth(-d*coth(b*x + a) - d + 1), x)

Mupad [F(-1)]

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

[In]

int(-x*acoth(d + d*coth(a + b*x) - 1),x)

[Out]

int(-x*acoth(d + d*coth(a + b*x) - 1), x)