\(\int \frac {(a+b \text {arctanh}(c x))^3}{(d+e x)^3} \, dx\) [20]

Optimal result

Integrand size = 18, antiderivative size = 953 \[ \int \frac {(a+b \text {arctanh}(c x))^3}{(d+e x)^3} \, dx =\text {Too large to display} \] Output:

3/2*b*c*(a+b*arctanh(c*x))^2/(c^2*d^2-e^2)/(e*x+d)-1/2*(a+b*arctanh(c*x))^ 
3/e/(e*x+d)^2-3/2*b^2*c^2*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/(c*d-e)/(c*d+e 
)^2+3/4*b*c^2*(a+b*arctanh(c*x))^2*ln(2/(-c*x+1))/e/(c*d+e)^2-3*b^2*c^2*e* 
(a+b*arctanh(c*x))*ln(2/(c*x+1))/(c*d-e)^2/(c*d+e)^2+3/2*b^2*c^2*(a+b*arct 
anh(c*x))*ln(2/(c*x+1))/(c*d-e)^2/(c*d+e)-3/4*b*c^2*(a+b*arctanh(c*x))^2*l 
n(2/(c*x+1))/(c*d-e)^2/e+3*b*c^3*d*(a+b*arctanh(c*x))^2*ln(2/(c*x+1))/(c*d 
-e)^2/(c*d+e)^2+3*b^2*c^2*e*(a+b*arctanh(c*x))*ln(2*c*(e*x+d)/(c*d+e)/(c*x 
+1))/(c*d-e)^2/(c*d+e)^2-3*b*c^3*d*(a+b*arctanh(c*x))^2*ln(2*c*(e*x+d)/(c* 
d+e)/(c*x+1))/(c*d-e)^2/(c*d+e)^2-3/4*b^3*c^2*polylog(2,1-2/(-c*x+1))/(c*d 
-e)/(c*d+e)^2+3/4*b^2*c^2*(a+b*arctanh(c*x))*polylog(2,1-2/(-c*x+1))/e/(c* 
d+e)^2+3/2*b^3*c^2*e*polylog(2,1-2/(c*x+1))/(c*d-e)^2/(c*d+e)^2-3/4*b^3*c^ 
2*polylog(2,1-2/(c*x+1))/(c*d-e)^2/(c*d+e)+3/4*b^2*c^2*(a+b*arctanh(c*x))* 
polylog(2,1-2/(c*x+1))/(c*d-e)^2/e-3*b^2*c^3*d*(a+b*arctanh(c*x))*polylog( 
2,1-2/(c*x+1))/(c*d-e)^2/(c*d+e)^2-3/2*b^3*c^2*e*polylog(2,1-2*c*(e*x+d)/( 
c*d+e)/(c*x+1))/(c*d-e)^2/(c*d+e)^2+3*b^2*c^3*d*(a+b*arctanh(c*x))*polylog 
(2,1-2*c*(e*x+d)/(c*d+e)/(c*x+1))/(c*d-e)^2/(c*d+e)^2-3/8*b^3*c^2*polylog( 
3,1-2/(-c*x+1))/e/(c*d+e)^2+3/8*b^3*c^2*polylog(3,1-2/(c*x+1))/(c*d-e)^2/e 
-3/2*b^3*c^3*d*polylog(3,1-2/(c*x+1))/(c*d-e)^2/(c*d+e)^2+3/2*b^3*c^3*d*po 
lylog(3,1-2*c*(e*x+d)/(c*d+e)/(c*x+1))/(c*d-e)^2/(c*d+e)^2
 

Mathematica [F]

\[ \int \frac {(a+b \text {arctanh}(c x))^3}{(d+e x)^3} \, dx=\int \frac {(a+b \text {arctanh}(c x))^3}{(d+e x)^3} \, dx \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 2.17 (sec) , antiderivative size = 896, normalized size of antiderivative = 0.94, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6480, 2009}

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.

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

\(\Big \downarrow \) 6480

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6480

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_S 
ymbol] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcTanh[c*x])^p/(e*(q + 1))), x] - 
 Simp[b*c*(p/(e*(q + 1)))   Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^(p - 1 
), (d + e*x)^(q + 1)/(1 - c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] 
 && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]
 

Maple [C] (warning: unable to verify)

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

Time = 10.63 (sec) , antiderivative size = 50477, normalized size of antiderivative = 52.97

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(50477\)
default	\(\text {Expression too large to display}\)	\(50477\)
parts	\(\text {Expression too large to display}\)	\(50485\)

Input:

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

Output:

result too large to display
 

Fricas [F]

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

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

Output:

integral((b^3*arctanh(c*x)^3 + 3*a*b^2*arctanh(c*x)^2 + 3*a^2*b*arctanh(c* 
x) + a^3)/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-3/4*((4*c^2*d*log(e*x + d)/(c^4*d^4 - 2*c^2*d^2*e^2 + e^4) - c*log(c*x + 
1)/(c^2*d^2*e - 2*c*d*e^2 + e^3) + c*log(c*x - 1)/(c^2*d^2*e + 2*c*d*e^2 + 
 e^3) - 2/(c^2*d^3 - d*e^2 + (c^2*d^2*e - e^3)*x))*c + 2*arctanh(c*x)/(e^3 
*x^2 + 2*d*e^2*x + d^2*e))*a^2*b - 1/2*a^3/(e^3*x^2 + 2*d*e^2*x + d^2*e) - 
 1/16*(((c^4*d^2*e^2 - 2*c^3*d*e^3 + c^2*e^4)*b^3*x^2 + 2*(c^4*d^3*e - 2*c 
^3*d^2*e^2 + c^2*d*e^3)*b^3*x - (2*c^3*d^3*e - 3*c^2*d^2*e^2 + e^4)*b^3)*l 
og(-c*x + 1)^3 - 3*(2*(c^3*d^2*e^2 - c*e^4)*b^3*x - 2*(c^4*d^4 - 2*c^2*d^2 
*e^2 + e^4)*a*b^2 + 2*(c^3*d^3*e - c*d*e^3)*b^3 + ((c^4*d^2*e^2 + 2*c^3*d* 
e^3 + c^2*e^4)*b^3*x^2 + 2*(c^4*d^3*e + 2*c^3*d^2*e^2 + c^2*d*e^3)*b^3*x + 
 (2*c^3*d^3*e + 3*c^2*d^2*e^2 - e^4)*b^3)*log(c*x + 1))*log(-c*x + 1)^2)/( 
c^4*d^6*e - 2*c^2*d^4*e^3 + d^2*e^5 + (c^4*d^4*e^3 - 2*c^2*d^2*e^5 + e^7)* 
x^2 + 2*(c^4*d^5*e^2 - 2*c^2*d^3*e^4 + d*e^6)*x) - integrate(1/8*(((c^4*d^ 
3*e - c^3*d^2*e^2 - c^2*d*e^3 + c*e^4)*b^3*x - (c^3*d^3*e - c^2*d^2*e^2 - 
c*d*e^3 + e^4)*b^3)*log(c*x + 1)^3 + 6*((c^4*d^3*e - c^3*d^2*e^2 - c^2*d*e 
^3 + c*e^4)*a*b^2*x - (c^3*d^3*e - c^2*d^2*e^2 - c*d*e^3 + e^4)*a*b^2)*log 
(c*x + 1)^2 - 3*(2*(c^3*d*e^3 - c^2*e^4)*b^3*x^2 - 2*(c^4*d^4 - c^3*d^3*e 
- c^2*d^2*e^2 + c*d*e^3)*a*b^2 + 2*(c^3*d^3*e - c^2*d^2*e^2)*b^3 + ((c^4*d 
^3*e - c^3*d^2*e^2 - c^2*d*e^3 + c*e^4)*b^3*x - (c^3*d^3*e - c^2*d^2*e^2 - 
 c*d*e^3 + e^4)*b^3)*log(c*x + 1)^2 - 2*((c^4*d^3*e - c^3*d^2*e^2 - c^2*d* 
e^3 + c*e^4)*a*b^2 - 2*(c^3*d^2*e^2 - c^2*d*e^3)*b^3)*x + ((c^4*d*e^3 +...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(a+b \text {arctanh}(c x))^3}{(d+e x)^3} \, dx=\text {too large to display} \] Input:

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

Output:

(36*atanh(c*x)**3*b**3*c**10*d**10*e*x + 18*atanh(c*x)**3*b**3*c**10*d**9* 
e**2*x**2 - 54*atanh(c*x)**3*b**3*c**8*d**9*e**2 - 48*atanh(c*x)**3*b**3*c 
**8*d**8*e**3*x - 24*atanh(c*x)**3*b**3*c**8*d**7*e**4*x**2 + 72*atanh(c*x 
)**3*b**3*c**6*d**7*e**4 - 8*atanh(c*x)**3*b**3*c**6*d**6*e**5*x - 4*atanh 
(c*x)**3*b**3*c**6*d**5*e**6*x**2 + 12*atanh(c*x)**3*b**3*c**4*d**5*e**6 + 
 16*atanh(c*x)**3*b**3*c**4*d**4*e**7*x + 8*atanh(c*x)**3*b**3*c**4*d**3*e 
**8*x**2 - 24*atanh(c*x)**3*b**3*c**2*d**3*e**8 + 4*atanh(c*x)**3*b**3*c** 
2*d**2*e**9*x + 2*atanh(c*x)**3*b**3*c**2*d*e**10*x**2 - 6*atanh(c*x)**3*b 
**3*d*e**10 + 108*atanh(c*x)**2*a*b**2*c**10*d**10*e*x + 54*atanh(c*x)**2* 
a*b**2*c**10*d**9*e**2*x**2 - 162*atanh(c*x)**2*a*b**2*c**8*d**9*e**2 - 14 
4*atanh(c*x)**2*a*b**2*c**8*d**8*e**3*x - 72*atanh(c*x)**2*a*b**2*c**8*d** 
7*e**4*x**2 + 216*atanh(c*x)**2*a*b**2*c**6*d**7*e**4 - 24*atanh(c*x)**2*a 
*b**2*c**6*d**6*e**5*x - 12*atanh(c*x)**2*a*b**2*c**6*d**5*e**6*x**2 + 36* 
atanh(c*x)**2*a*b**2*c**4*d**5*e**6 + 48*atanh(c*x)**2*a*b**2*c**4*d**4*e* 
*7*x + 24*atanh(c*x)**2*a*b**2*c**4*d**3*e**8*x**2 - 72*atanh(c*x)**2*a*b* 
*2*c**2*d**3*e**8 + 12*atanh(c*x)**2*a*b**2*c**2*d**2*e**9*x + 6*atanh(c*x 
)**2*a*b**2*c**2*d*e**10*x**2 - 18*atanh(c*x)**2*a*b**2*d*e**10 - 36*atanh 
(c*x)**2*b**3*c**9*d**10*e + 90*atanh(c*x)**2*b**3*c**9*d**9*e**2*x + 72*a 
tanh(c*x)**2*b**3*c**9*d**8*e**3*x**2 - 6*atanh(c*x)**2*b**3*c**7*d**8*e** 
3 - 228*atanh(c*x)**2*b**3*c**7*d**7*e**4*x - 150*atanh(c*x)**2*b**3*c*...