\(\int \frac {x (a+b \text {arctanh}(c x))^2}{d+c d x} \, dx\) [97]

Optimal result

Integrand size = 20, antiderivative size = 172 \[ \int \frac {x (a+b \text {arctanh}(c x))^2}{d+c d x} \, dx=\frac {(a+b \text {arctanh}(c x))^2}{c^2 d}+\frac {x (a+b \text {arctanh}(c x))^2}{c d}-\frac {2 b (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{c^2 d}+\frac {(a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1+c x}\right )}{c^2 d}-\frac {b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c^2 d}-\frac {b (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{c^2 d}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{2 c^2 d} \] Output:

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

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.81 \[ \int \frac {x (a+b \text {arctanh}(c x))^2}{d+c d x} \, dx=\frac {2 b^2 \text {arctanh}(c x)^2 \left (-1+c x+\log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+4 b \text {arctanh}(c x) \left (a c x+(a-b) \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+2 a \left (a c x-a \log (1+c x)+b \log \left (1-c^2 x^2\right )\right )-2 b (a-b+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )-b^2 \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )}{2 c^2 d} \] Input:

Integrate[(x*(a + b*ArcTanh[c*x])^2)/(d + c*d*x),x]
 

Output:

(2*b^2*ArcTanh[c*x]^2*(-1 + c*x + Log[1 + E^(-2*ArcTanh[c*x])]) + 4*b*ArcT 
anh[c*x]*(a*c*x + (a - b)*Log[1 + E^(-2*ArcTanh[c*x])]) + 2*a*(a*c*x - a*L 
og[1 + c*x] + b*Log[1 - c^2*x^2]) - 2*b*(a - b + b*ArcTanh[c*x])*PolyLog[2 
, -E^(-2*ArcTanh[c*x])] - b^2*PolyLog[3, -E^(-2*ArcTanh[c*x])])/(2*c^2*d)
 

Rubi [A] (verified)

Time = 1.75 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6492, 27, 6436, 6470, 6546, 6470, 2849, 2752, 6618, 7164}

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.

\(\displaystyle \int \frac {x (a+b \text {arctanh}(c x))^2}{c d x+d} \, dx\)

\(\Big \downarrow \) 6492

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6436

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 6546

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

\(\Big \downarrow \) 6618

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

\(\Big \downarrow \) 7164

\(\displaystyle \frac {x (a+b \text {arctanh}(c x))^2-2 b c \left (\frac {\frac {\log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{2 c}}{c}-\frac {(a+b \text {arctanh}(c x))^2}{2 b c^2}\right )}{c d}-\frac {2 b \left (\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{2 c}+\frac {b \operatorname {PolyLog}\left (3,1-\frac {2}{c x+1}\right )}{4 c}\right )-\frac {\log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2}{c}}{c d}\)

Input:

Int[(x*(a + b*ArcTanh[c*x])^2)/(d + c*d*x),x]
 

Output:

(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*d) - (-(((a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/c) + 2*b 
*(((a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/(2*c) + (b*PolyLog[3, 
 1 - 2/(1 + c*x)])/(4*c)))/(c*d)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 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 6492

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

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]
 

rule 6618

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^ 
2), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x 
] - Simp[b*(p/2)   Int[(a + b*ArcTanh[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 7164

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]
 

Maple [C] (warning: unable to verify)

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

Time = 1.83 (sec) , antiderivative size = 2602, normalized size of antiderivative = 15.13

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2602\)
default	\(\text {Expression too large to display}\)	\(2602\)
parts	\(\text {Expression too large to display}\)	\(2609\)

Input:

int(x*(a+b*arctanh(c*x))^2/(c*d*x+d),x,method=_RETURNVERBOSE)
 

Output:

1/c^2*(a^2/d*(c*x-ln(c*x+1))+b^2/d*(-1/4*I*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1) 
)*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c^2*x^2-1)))*csgn(I/(1-(c*x+1 
)^2/(c^2*x^2-1)))*(2*arctanh(c*x)^2-2*arctanh(c*x)*ln(1+(c*x+1)^2/(-c^2*x^ 
2+1))-polylog(2,-(c*x+1)^2/(-c^2*x^2+1)))-arctanh(c*x)^2*ln(c*x+1)-arctanh 
(c*x)*ln(1+(c*x+1)^2/(-c^2*x^2+1))-arctanh(c*x)*ln(1+I*(c*x+1)/(-c^2*x^2+1 
)^(1/2))-arctanh(c*x)*ln(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))-1/2*ln(2)*polylog 
(2,-(c*x+1)^2/(-c^2*x^2+1))+ln(2)*dilog(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))+ln 
(2)*dilog(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))-dilog(1+I*(c*x+1)/(-c^2*x^2+1)^( 
1/2))-dilog(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))-1/2*I*Pi*csgn(I*(c*x+1)^2/(c^2 
*x^2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c^2*x^2-1)))*csgn(I/(1 
-(c*x+1)^2/(c^2*x^2-1)))*(arctanh(c*x)*ln(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))+ 
arctanh(c*x)*ln(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))+dilog(1+I*(c*x+1)/(-c^2*x^ 
2+1)^(1/2))+dilog(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2)))-2/3*arctanh(c*x)^3+arct 
anh(c*x)^2*c*x+arctanh(c*x)^2-1/2*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))-1/2*p 
olylog(3,-(c*x+1)^2/(-c^2*x^2+1))+arctanh(c*x)*polylog(2,-(c*x+1)^2/(-c^2* 
x^2+1))+ln(2)*arctanh(c*x)*ln(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))+ln(2)*arctan 
h(c*x)*ln(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))-ln(2)*arctanh(c*x)*ln(1+(c*x+1)^ 
2/(-c^2*x^2+1))+1/2*I*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c^2*x^ 
2-1)))^2*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))*(arctanh(c*x)*ln(1+I*(c*x+1)/(- 
c^2*x^2+1)^(1/2))+arctanh(c*x)*ln(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))+dilog...
 

Fricas [F]

\[ \int \frac {x (a+b \text {arctanh}(c x))^2}{d+c d x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x}{c d x + d} \,d x } \] Input:

integrate(x*(a+b*arctanh(c*x))^2/(c*d*x+d),x, algorithm="fricas")
 

Output:

integral((b^2*x*arctanh(c*x)^2 + 2*a*b*x*arctanh(c*x) + a^2*x)/(c*d*x + d) 
, x)
 

Sympy [F]

\[ \int \frac {x (a+b \text {arctanh}(c x))^2}{d+c d x} \, dx=\frac {\int \frac {a^{2} x}{c x + 1}\, dx + \int \frac {b^{2} x \operatorname {atanh}^{2}{\left (c x \right )}}{c x + 1}\, dx + \int \frac {2 a b x \operatorname {atanh}{\left (c x \right )}}{c x + 1}\, dx}{d} \] Input:

integrate(x*(a+b*atanh(c*x))**2/(c*d*x+d),x)
 

Output:

(Integral(a**2*x/(c*x + 1), x) + Integral(b**2*x*atanh(c*x)**2/(c*x + 1), 
x) + Integral(2*a*b*x*atanh(c*x)/(c*x + 1), x))/d
 

Maxima [F]

\[ \int \frac {x (a+b \text {arctanh}(c x))^2}{d+c d x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x}{c d x + d} \,d x } \] Input:

integrate(x*(a+b*arctanh(c*x))^2/(c*d*x+d),x, algorithm="maxima")
 

Output:

a^2*(x/(c*d) - log(c*x + 1)/(c^2*d)) + 1/4*(b^2*c*x - b^2*log(c*x + 1))*lo 
g(-c*x + 1)^2/(c^2*d) - integrate(-1/4*((b^2*c^2*x^2 - b^2*c*x)*log(c*x + 
1)^2 + 4*(a*b*c^2*x^2 - a*b*c*x)*log(c*x + 1) - 2*((2*a*b*c^2 + b^2*c^2)*x 
^2 - (2*a*b*c - b^2*c)*x + (b^2*c^2*x^2 - 2*b^2*c*x - b^2)*log(c*x + 1))*l 
og(-c*x + 1))/(c^3*d*x^2 - c*d), x)
 

Giac [F]

\[ \int \frac {x (a+b \text {arctanh}(c x))^2}{d+c d x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x}{c d x + d} \,d x } \] Input:

integrate(x*(a+b*arctanh(c*x))^2/(c*d*x+d),x, algorithm="giac")
 

Output:

integrate((b*arctanh(c*x) + a)^2*x/(c*d*x + d), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x (a+b \text {arctanh}(c x))^2}{d+c d x} \, dx=\int \frac {x\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{d+c\,d\,x} \,d x \] Input:

int((x*(a + b*atanh(c*x))^2)/(d + c*d*x),x)
 

Output:

int((x*(a + b*atanh(c*x))^2)/(d + c*d*x), x)
 

Reduce [F]

\[ \int \frac {x (a+b \text {arctanh}(c x))^2}{d+c d x} \, dx=\frac {2 \left (\int \frac {\mathit {atanh} \left (c x \right ) x}{c x +1}d x \right ) a b \,c^{2}+\left (\int \frac {\mathit {atanh} \left (c x \right )^{2} x}{c x +1}d x \right ) b^{2} c^{2}-\mathrm {log}\left (c x +1\right ) a^{2}+a^{2} c x}{c^{2} d} \] Input:

int(x*(a+b*atanh(c*x))^2/(c*d*x+d),x)
 

Output:

(2*int((atanh(c*x)*x)/(c*x + 1),x)*a*b*c**2 + int((atanh(c*x)**2*x)/(c*x + 
 1),x)*b**2*c**2 - log(c*x + 1)*a**2 + a**2*c*x)/(c**2*d)