3.1.28 \(\int x (a+b \text {arctanh}(c x))^3 \, dx\) [28]

3.1.28.1 Optimal result

Integrand size = 12, antiderivative size = 123 \[ \int x (a+b \text {arctanh}(c x))^3 \, dx=\frac {3 b (a+b \text {arctanh}(c x))^2}{2 c^2}+\frac {3 b x (a+b \text {arctanh}(c x))^2}{2 c}-\frac {(a+b \text {arctanh}(c x))^3}{2 c^2}+\frac {1}{2} x^2 (a+b \text {arctanh}(c x))^3-\frac {3 b^2 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{c^2}-\frac {3 b^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{2 c^2} \]

output

3/2*b*(a+b*arctanh(c*x))^2/c^2+3/2*b*x*(a+b*arctanh(c*x))^2/c-1/2*(a+b*arc 
tanh(c*x))^3/c^2+1/2*x^2*(a+b*arctanh(c*x))^3-3*b^2*(a+b*arctanh(c*x))*ln( 
2/(-c*x+1))/c^2-3/2*b^3*polylog(2,1-2/(-c*x+1))/c^2
 

3.1.28.2 Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.31 \[ \int x (a+b \text {arctanh}(c x))^3 \, dx=\frac {6 b^2 (-1+c x) (a+b+a c x) \text {arctanh}(c x)^2+2 b^3 \left (-1+c^2 x^2\right ) \text {arctanh}(c x)^3+6 b \text {arctanh}(c x) \left (a c x (2 b+a c x)-2 b^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+a \left (6 a b c x+2 a^2 c^2 x^2+3 a b \log (1-c x)-3 a b \log (1+c x)+6 b^2 \log \left (1-c^2 x^2\right )\right )+6 b^3 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )}{4 c^2} \]

input

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

output

(6*b^2*(-1 + c*x)*(a + b + a*c*x)*ArcTanh[c*x]^2 + 2*b^3*(-1 + c^2*x^2)*Ar 
cTanh[c*x]^3 + 6*b*ArcTanh[c*x]*(a*c*x*(2*b + a*c*x) - 2*b^2*Log[1 + E^(-2 
*ArcTanh[c*x])]) + a*(6*a*b*c*x + 2*a^2*c^2*x^2 + 3*a*b*Log[1 - c*x] - 3*a 
*b*Log[1 + c*x] + 6*b^2*Log[1 - c^2*x^2]) + 6*b^3*PolyLog[2, -E^(-2*ArcTan 
h[c*x])])/(4*c^2)
 

3.1.28.3 Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6452, 6542, 6436, 6510, 6546, 6470, 2849, 2752}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x (a+b \text {arctanh}(c x))^3 \, dx\)

\(\Big \downarrow \) 6452

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arctanh}(c x))^3-\frac {3}{2} b c \int \frac {x^2 (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx\)

\(\Big \downarrow \) 6542

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arctanh}(c x))^3-\frac {3}{2} b c \left (\frac {\int \frac {(a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {\int (a+b \text {arctanh}(c x))^2dx}{c^2}\right )\)

\(\Big \downarrow \) 6436

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arctanh}(c x))^3-\frac {3}{2} b c \left (\frac {\int \frac {(a+b \text {arctanh}(c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {x (a+b \text {arctanh}(c x))^2-2 b c \int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}\right )\)

\(\Big \downarrow \) 6510

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arctanh}(c x))^3-\frac {3}{2} b c \left (\frac {(a+b \text {arctanh}(c x))^3}{3 b c^3}-\frac {x (a+b \text {arctanh}(c x))^2-2 b c \int \frac {x (a+b \text {arctanh}(c x))}{1-c^2 x^2}dx}{c^2}\right )\)

\(\Big \downarrow \) 6546

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arctanh}(c x))^3-\frac {3}{2} b c \left (\frac {(a+b \text {arctanh}(c x))^3}{3 b c^3}-\frac {x (a+b \text {arctanh}(c x))^2-2 b c \left (\frac {\int \frac {a+b \text {arctanh}(c x)}{1-c x}dx}{c}-\frac {(a+b \text {arctanh}(c x))^2}{2 b c^2}\right )}{c^2}\right )\)

\(\Big \downarrow \) 6470

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arctanh}(c x))^3-\frac {3}{2} b c \left (\frac {(a+b \text {arctanh}(c x))^3}{3 b c^3}-\frac {x (a+b \text {arctanh}(c x))^2-2 b c \left (\frac {\frac {\log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c}-b \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2}dx}{c}-\frac {(a+b \text {arctanh}(c x))^2}{2 b c^2}\right )}{c^2}\right )\)

\(\Big \downarrow \) 2849

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arctanh}(c x))^3-\frac {3}{2} b c \left (\frac {(a+b \text {arctanh}(c x))^3}{3 b c^3}-\frac {x (a+b \text {arctanh}(c x))^2-2 b c \left (\frac {\frac {b \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-\frac {2}{1-c x}}d\frac {1}{1-c x}}{c}+\frac {\log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c}}{c}-\frac {(a+b \text {arctanh}(c x))^2}{2 b c^2}\right )}{c^2}\right )\)

\(\Big \downarrow \) 2752

\(\displaystyle \frac {1}{2} x^2 (a+b \text {arctanh}(c x))^3-\frac {3}{2} b c \left (\frac {(a+b \text {arctanh}(c x))^3}{3 b c^3}-\frac {x (a+b \text {arctanh}(c x))^2-2 b c \left (\frac {\frac {\log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{2 c}}{c}-\frac {(a+b \text {arctanh}(c x))^2}{2 b c^2}\right )}{c^2}\right )\)

input

Int[x*(a + b*ArcTanh[c*x])^3,x]
 

output

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

3.1.28.3.1 Defintions of rubi rules used

rule 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

rule 2849

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp 
[-e/g   Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ 
{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
 

rule 6436

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a 
 + b*ArcTanh[c*x^n])^p, x] - Simp[b*c*n*p   Int[x^n*((a + b*ArcTanh[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 6452

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

rule 6470

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol 
] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c 
*(p/e)   Int[(a + b*ArcTanh[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 6510

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symb 
ol] :> Simp[(a + b*ArcTanh[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 6542

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

rule 6546

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

3.1.28.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(487\) vs. \(2(113)=226\).

Time = 15.90 (sec) , antiderivative size = 488, normalized size of antiderivative = 3.97

method	result	size
risch	\(\frac {a^{3} x^{2}}{2}+\frac {b^{3} \left (c^{2} x^{2}-1\right ) \ln \left (c x +1\right )^{3}}{16 c^{2}}+\frac {3 b^{2} \left (-b \,x^{2} \ln \left (-c x +1\right ) c^{2}+2 a \,c^{2} x^{2}+2 b c x +b \ln \left (-c x +1\right )-2 a +2 b \right ) \ln \left (c x +1\right )^{2}}{16 c^{2}}-\frac {3 a^{2} b}{2 c^{2}}+\frac {b^{3} \ln \left (-c x +1\right )^{3}}{16 c^{2}}-\frac {3 b^{3} \ln \left (-c x +1\right )^{2}}{8 c^{2}}+\left (\frac {3 b^{3} \left (c^{2} x^{2}-1\right ) \ln \left (-c x +1\right )^{2}}{16 c^{2}}-\frac {3 b^{2} x \left (c x a +b \right ) \ln \left (-c x +1\right )}{4 c}-\frac {3 b \left (-c^{2} x^{2} a^{2}-2 a b c x -b \ln \left (-c x +1\right ) a -b^{2} \ln \left (-c x +1\right )\right )}{4 c^{2}}\right ) \ln \left (c x +1\right )-\frac {a^{3}}{2 c^{2}}-\frac {\ln \left (-c x +1\right )^{3} b^{3} x^{2}}{16}-\frac {3 b^{3} \ln \left (-c x +1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{2 c^{2}}+\frac {3 b^{3} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{2 c^{2}}-\frac {3 a^{2} b \ln \left (-c x -1\right )}{4 c^{2}}+\frac {3 a \,b^{2} \ln \left (-c x -1\right )}{2 c^{2}}+\frac {3 a^{2} b x}{2 c}-\frac {3 a \,b^{2} \ln \left (-c x +1\right )^{2}}{8 c^{2}}+\frac {3 a \,b^{2} \ln \left (-c x +1\right )}{2 c^{2}}+\frac {3 a^{2} b \ln \left (-c x +1\right )}{4 c^{2}}+\frac {3 b^{3} \ln \left (-c x +1\right )^{2} x}{8 c}+\frac {3 b^{3} \operatorname {dilog}\left (-\frac {c x}{2}+\frac {1}{2}\right )}{2 c^{2}}+\frac {3 \ln \left (-c x +1\right )^{2} a \,b^{2} x^{2}}{8}-\frac {3 \ln \left (-c x +1\right ) a^{2} b \,x^{2}}{4}-\frac {3 a \,b^{2} \ln \left (-c x +1\right ) x}{2 c}\)	\(488\)
derivativedivides	\(\text {Expression too large to display}\)	\(5534\)
default	\(\text {Expression too large to display}\)	\(5534\)
parts	\(\text {Expression too large to display}\)	\(5536\)

input

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

output

1/2*a^3*x^2+1/16*b^3*(c^2*x^2-1)/c^2*ln(c*x+1)^3+3/16*b^2*(-b*x^2*ln(-c*x+ 
1)*c^2+2*a*c^2*x^2+2*b*c*x+b*ln(-c*x+1)-2*a+2*b)/c^2*ln(c*x+1)^2-3/2/c^2*a 
^2*b+1/16/c^2*b^3*ln(-c*x+1)^3-3/8/c^2*b^3*ln(-c*x+1)^2+(3/16*b^3*(c^2*x^2 
-1)/c^2*ln(-c*x+1)^2-3/4*b^2*x*(a*c*x+b)/c*ln(-c*x+1)-3/4*b*(-c^2*x^2*a^2- 
2*a*b*c*x-b*ln(-c*x+1)*a-b^2*ln(-c*x+1))/c^2)*ln(c*x+1)-1/2/c^2*a^3-1/16*l 
n(-c*x+1)^3*b^3*x^2-3/2*b^3/c^2*ln(-c*x+1)*ln(1/2*c*x+1/2)+3/2*b^3/c^2*ln( 
-1/2*c*x+1/2)*ln(1/2*c*x+1/2)-3/4*a^2*b/c^2*ln(-c*x-1)+3/2*a*b^2/c^2*ln(-c 
*x-1)+3/2/c*a^2*b*x-3/8/c^2*a*b^2*ln(-c*x+1)^2+3/2/c^2*a*b^2*ln(-c*x+1)+3/ 
4/c^2*a^2*b*ln(-c*x+1)+3/8/c*b^3*ln(-c*x+1)^2*x+3/2*b^3/c^2*dilog(-1/2*c*x 
+1/2)+3/8*ln(-c*x+1)^2*a*b^2*x^2-3/4*ln(-c*x+1)*a^2*b*x^2-3/2/c*a*b^2*ln(- 
c*x+1)*x
 

3.1.28.5 Fricas [F]

\[ \int x (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} x \,d x } \]

input

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

output

integral(b^3*x*arctanh(c*x)^3 + 3*a*b^2*x*arctanh(c*x)^2 + 3*a^2*b*x*arcta 
nh(c*x) + a^3*x, x)
 

3.1.28.6 Sympy [F]

\[ \int x (a+b \text {arctanh}(c x))^3 \, dx=\int x \left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3}\, dx \]

input

integrate(x*(a+b*atanh(c*x))**3,x)
 

output

Integral(x*(a + b*atanh(c*x))**3, x)
 

3.1.28.7 Maxima [F]

\[ \int x (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} x \,d x } \]

input

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

output

3/2*a*b^2*x^2*arctanh(c*x)^2 + 1/2*a^3*x^2 + 3/4*(2*x^2*arctanh(c*x) + c*( 
2*x/c^2 - log(c*x + 1)/c^3 + log(c*x - 1)/c^3))*a^2*b + 3/8*(4*c*(2*x/c^2 
- log(c*x + 1)/c^3 + log(c*x - 1)/c^3)*arctanh(c*x) - (2*(log(c*x - 1) - 2 
)*log(c*x + 1) - log(c*x + 1)^2 - log(c*x - 1)^2 - 4*log(c*x - 1))/c^2)*a* 
b^2 - 1/64*(3*c^3*(x^2/c^3 + log(c^2*x^2 - 1)/c^5) + 21*c^2*(2*x/c^3 - log 
(c*x + 1)/c^4 + log(c*x - 1)/c^4) - 576*c*integrate(1/4*x*log(c*x + 1)/(c^ 
3*x^2 - c), x) - 2*(12*c*x*log(c*x + 1)^2 + 2*(c^2*x^2 - 1)*log(c*x + 1)^3 
 - 3*(c^2*x^2 - 2*c*x - 2*(c^2*x^2 - 1)*log(c*x + 1) + 1)*log(-c*x + 1)^2 
+ 3*(c^2*x^2 - 2*(c^2*x^2 - 1)*log(c*x + 1)^2 + 6*c*x - 8*(c*x + 1)*log(c* 
x + 1))*log(-c*x + 1))/c^2 + ((4*log(-c*x + 1)^3 - 6*log(-c*x + 1)^2 + 6*l 
og(-c*x + 1) - 3)*(c*x - 1)^2 + 8*(log(-c*x + 1)^3 - 3*log(-c*x + 1)^2 + 6 
*log(-c*x + 1) - 6)*(c*x - 1))/c^2 + 18*log(4*c^3*x^2 - 4*c)/c^2 - 192*int 
egrate(1/4*log(c*x + 1)/(c^3*x^2 - c), x))*b^3
 

3.1.28.8 Giac [F]

\[ \int x (a+b \text {arctanh}(c x))^3 \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3} x \,d x } \]

input

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

output

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

3.1.28.9 Mupad [F(-1)]

Timed out. \[ \int x (a+b \text {arctanh}(c x))^3 \, dx=\int x\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^3 \,d x \]

input

int(x*(a + b*atanh(c*x))^3,x)
 

output

int(x*(a + b*atanh(c*x))^3, x)