\(\int (a+b \coth ^{-1}(c+d x))^3 \, dx\) [116]

Optimal result

Integrand size = 12, antiderivative size = 132 \[ \int \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d}-\frac {3 b \left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1-c-d x}\right )}{d}-\frac {3 b^2 \left (a+b \coth ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c-d x}\right )}{d}+\frac {3 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c-d x}\right )}{2 d} \]

[Out]

(a+b*arccoth(d*x+c))^3/d+(d*x+c)*(a+b*arccoth(d*x+c))^3/d-3*b*(a+b*arccoth(d*x+c))^2*ln(2/(-d*x-c+1))/d-3*b^2*
(a+b*arccoth(d*x+c))*polylog(2,1-2/(-d*x-c+1))/d+3/2*b^3*polylog(3,1-2/(-d*x-c+1))/d

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6239, 6022, 6132, 6056, 6096, 6206, 6745} \[ \int \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=-\frac {3 b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{d}+\frac {(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d}+\frac {\left (a+b \coth ^{-1}(c+d x)\right )^3}{d}-\frac {3 b \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d}+\frac {3 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{-c-d x+1}\right )}{2 d} \]

[In]

Int[(a + b*ArcCoth[c + d*x])^3,x]

[Out]

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

Rule 6022

Int[((a_.) + ArcCoth[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCoth[c*x^n])^p, x] - Dist[b
*c*n*p, Int[x^n*((a + b*ArcCoth[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p
, 0] && (EqQ[n, 1] || EqQ[p, 1])

Rule 6056

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

Rule 6096

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcCoth[c*x])^(p
+ 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]

Rule 6132

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

Rule 6206

Int[(Log[u_]*((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-(a + b*ArcC
oth[c*x])^p)*(PolyLog[2, 1 - u]/(2*c*d)), x] + Dist[b*(p/2), Int[(a + b*ArcCoth[c*x])^(p - 1)*(PolyLog[2, 1 -
u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1
- 2/(1 - c*x))^2, 0]

Rule 6239

Int[((a_.) + ArcCoth[(c_) + (d_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Dist[1/d, Subst[Int[(a + b*ArcCoth[x])^p, x
], x, c + d*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]

Rule 6745

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.24 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.58 \[ \int \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\frac {2 a^3 (c+d x)+6 a^2 b (c+d x) \coth ^{-1}(c+d x)+3 a^2 b \log \left (1-(c+d x)^2\right )+6 a b^2 \left (\coth ^{-1}(c+d x) \left ((-1+c+d x) \coth ^{-1}(c+d x)-2 \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )\right )+\operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(c+d x)}\right )\right )+2 b^3 \left (-\frac {i \pi ^3}{8}+\coth ^{-1}(c+d x)^3+(c+d x) \coth ^{-1}(c+d x)^3-3 \coth ^{-1}(c+d x)^2 \log \left (1-e^{2 \coth ^{-1}(c+d x)}\right )-3 \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,e^{2 \coth ^{-1}(c+d x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,e^{2 \coth ^{-1}(c+d x)}\right )\right )}{2 d} \]

[In]

Integrate[(a + b*ArcCoth[c + d*x])^3,x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(371\) vs. \(2(130)=260\).

Time = 0.85 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.82

method	result	size
derivativedivides	\(\frac {\left (d x +c \right ) a^{3}+b^{3} \left (\operatorname {arccoth}\left (d x +c \right )^{3} \left (d x +c -1\right )+2 \operatorname {arccoth}\left (d x +c \right )^{3}-3 \operatorname {arccoth}\left (d x +c \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-6 \,\operatorname {arccoth}\left (d x +c \right ) \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )+6 \operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-3 \operatorname {arccoth}\left (d x +c \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-6 \,\operatorname {arccoth}\left (d x +c \right ) \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )+6 \operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )\right )+3 a \,b^{2} \left (\operatorname {arccoth}\left (d x +c \right )^{2} \left (d x +c -1\right )+2 \operatorname {arccoth}\left (d x +c \right )^{2}-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )\right )+3 b \,a^{2} \left (\left (d x +c \right ) \operatorname {arccoth}\left (d x +c \right )+\frac {\ln \left (\left (d x +c \right )^{2}-1\right )}{2}\right )}{d}\)	\(372\)
default	\(\frac {\left (d x +c \right ) a^{3}+b^{3} \left (\operatorname {arccoth}\left (d x +c \right )^{3} \left (d x +c -1\right )+2 \operatorname {arccoth}\left (d x +c \right )^{3}-3 \operatorname {arccoth}\left (d x +c \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-6 \,\operatorname {arccoth}\left (d x +c \right ) \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )+6 \operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-3 \operatorname {arccoth}\left (d x +c \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-6 \,\operatorname {arccoth}\left (d x +c \right ) \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )+6 \operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )\right )+3 a \,b^{2} \left (\operatorname {arccoth}\left (d x +c \right )^{2} \left (d x +c -1\right )+2 \operatorname {arccoth}\left (d x +c \right )^{2}-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )\right )+3 b \,a^{2} \left (\left (d x +c \right ) \operatorname {arccoth}\left (d x +c \right )+\frac {\ln \left (\left (d x +c \right )^{2}-1\right )}{2}\right )}{d}\)	\(372\)
parts	\(a^{3} x +\frac {b^{3} \left (\operatorname {arccoth}\left (d x +c \right )^{3} \left (d x +c -1\right )+2 \operatorname {arccoth}\left (d x +c \right )^{3}-3 \operatorname {arccoth}\left (d x +c \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-6 \,\operatorname {arccoth}\left (d x +c \right ) \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )+6 \operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-3 \operatorname {arccoth}\left (d x +c \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-6 \,\operatorname {arccoth}\left (d x +c \right ) \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )+6 \operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )\right )}{d}+\frac {3 a \,b^{2} \left (\operatorname {arccoth}\left (d x +c \right )^{2} \left (d x +c -1\right )+2 \operatorname {arccoth}\left (d x +c \right )^{2}-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1-\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \,\operatorname {arccoth}\left (d x +c \right ) \ln \left (1+\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )-2 \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {d x +c -1}{d x +c +1}}}\right )\right )}{d}+\frac {3 b \,a^{2} \left (\left (d x +c \right ) \operatorname {arccoth}\left (d x +c \right )+\frac {\ln \left (\left (d x +c \right )^{2}-1\right )}{2}\right )}{d}\)	\(373\)

[In]

int((a+b*arccoth(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*((d*x+c)*a^3+b^3*(arccoth(d*x+c)^3*(d*x+c-1)+2*arccoth(d*x+c)^3-3*arccoth(d*x+c)^2*ln(1+1/((d*x+c-1)/(d*x+
c+1))^(1/2))-6*arccoth(d*x+c)*polylog(2,-1/((d*x+c-1)/(d*x+c+1))^(1/2))+6*polylog(3,-1/((d*x+c-1)/(d*x+c+1))^(
1/2))-3*arccoth(d*x+c)^2*ln(1-1/((d*x+c-1)/(d*x+c+1))^(1/2))-6*arccoth(d*x+c)*polylog(2,1/((d*x+c-1)/(d*x+c+1)
)^(1/2))+6*polylog(3,1/((d*x+c-1)/(d*x+c+1))^(1/2)))+3*a*b^2*(arccoth(d*x+c)^2*(d*x+c-1)+2*arccoth(d*x+c)^2-2*
arccoth(d*x+c)*ln(1-1/((d*x+c-1)/(d*x+c+1))^(1/2))-2*polylog(2,1/((d*x+c-1)/(d*x+c+1))^(1/2))-2*arccoth(d*x+c)
*ln(1+1/((d*x+c-1)/(d*x+c+1))^(1/2))-2*polylog(2,-1/((d*x+c-1)/(d*x+c+1))^(1/2)))+3*b*a^2*((d*x+c)*arccoth(d*x
+c)+1/2*ln((d*x+c)^2-1)))

Fricas [F]

\[ \int \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{3} \,d x } \]

[In]

integrate((a+b*arccoth(d*x+c))^3,x, algorithm="fricas")

[Out]

integral(b^3*arccoth(d*x + c)^3 + 3*a*b^2*arccoth(d*x + c)^2 + 3*a^2*b*arccoth(d*x + c) + a^3, x)

Sympy [F]

\[ \int \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int \left (a + b \operatorname {acoth}{\left (c + d x \right )}\right )^{3}\, dx \]

[In]

integrate((a+b*acoth(d*x+c))**3,x)

[Out]

Integral((a + b*acoth(c + d*x))**3, x)

Maxima [F]

\[ \int \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{3} \,d x } \]

[In]

integrate((a+b*arccoth(d*x+c))^3,x, algorithm="maxima")

[Out]

a^3*x + 3/2*(2*(d*x + c)*arccoth(d*x + c) + log(-(d*x + c)^2 + 1))*a^2*b/d + 1/8*((b^3*d*x + b^3*(c + 1))*log(
d*x + c + 1)^3 + 3*(2*a*b^2*d*x - (b^3*d*x + b^3*(c - 1))*log(d*x + c - 1))*log(d*x + c + 1)^2)/d + integrate(
-1/8*((b^3*d*x + b^3*(c + 1))*log(d*x + c - 1)^3 - 6*(a*b^2*d*x + a*b^2*(c + 1))*log(d*x + c - 1)^2 + 3*(4*a*b
^2*d*x - (b^3*d*x + b^3*(c + 1))*log(d*x + c - 1)^2 + 2*(2*a*b^2*(c + 1) - b^3*(c - 1) + (2*a*b^2*d - b^3*d)*x
)*log(d*x + c - 1))*log(d*x + c + 1))/(d*x + c + 1), x)

Giac [F]

\[ \int \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{3} \,d x } \]

[In]

integrate((a+b*arccoth(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((b*arccoth(d*x + c) + a)^3, x)

Mupad [F(-1)]

Timed out. \[ \int \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int {\left (a+b\,\mathrm {acoth}\left (c+d\,x\right )\right )}^3 \,d x \]

[In]

int((a + b*acoth(c + d*x))^3,x)

[Out]

int((a + b*acoth(c + d*x))^3, x)