\(\int \frac {a+b \text {arctanh}(c x)}{(d+e x)^4} \, dx\) [8]

Optimal result

Integrand size = 16, antiderivative size = 174 \[ \int \frac {a+b \text {arctanh}(c x)}{(d+e x)^4} \, dx=\frac {b c}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {2 b c^3 d}{3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {a+b \text {arctanh}(c x)}{3 e (d+e x)^3}-\frac {b c^3 \log (1-c x)}{6 e (c d+e)^3}+\frac {b c^3 \log (1+c x)}{6 (c d-e)^3 e}-\frac {b c^3 \left (3 c^2 d^2+e^2\right ) \log (d+e x)}{3 \left (c^2 d^2-e^2\right )^3} \] Output:

1/6*b*c/(c^2*d^2-e^2)/(e*x+d)^2+2/3*b*c^3*d/(c^2*d^2-e^2)^2/(e*x+d)-1/3*(a 
+b*arctanh(c*x))/e/(e*x+d)^3-1/6*b*c^3*ln(-c*x+1)/e/(c*d+e)^3+1/6*b*c^3*ln 
(c*x+1)/(c*d-e)^3/e-1/3*b*c^3*(3*c^2*d^2+e^2)*ln(e*x+d)/(c^2*d^2-e^2)^3
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.99 \[ \int \frac {a+b \text {arctanh}(c x)}{(d+e x)^4} \, dx=\frac {1}{6} \left (-\frac {2 a}{e (d+e x)^3}+\frac {b c}{\left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {4 b c^3 d}{\left (-c^2 d^2+e^2\right )^2 (d+e x)}-\frac {2 b \text {arctanh}(c x)}{e (d+e x)^3}-\frac {b c^3 \log (1-c x)}{e (c d+e)^3}+\frac {b c^3 \log (1+c x)}{(c d-e)^3 e}-\frac {2 b c^3 \left (3 c^2 d^2+e^2\right ) \log (d+e x)}{\left (c^2 d^2-e^2\right )^3}\right ) \] Input:

Integrate[(a + b*ArcTanh[c*x])/(d + e*x)^4,x]
 

Output:

((-2*a)/(e*(d + e*x)^3) + (b*c)/((c^2*d^2 - e^2)*(d + e*x)^2) + (4*b*c^3*d 
)/((-(c^2*d^2) + e^2)^2*(d + e*x)) - (2*b*ArcTanh[c*x])/(e*(d + e*x)^3) - 
(b*c^3*Log[1 - c*x])/(e*(c*d + e)^3) + (b*c^3*Log[1 + c*x])/((c*d - e)^3*e 
) - (2*b*c^3*(3*c^2*d^2 + e^2)*Log[d + e*x])/(c^2*d^2 - e^2)^3)/6
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6478, 477, 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.

\(\displaystyle \int \frac {a+b \text {arctanh}(c x)}{(d+e x)^4} \, dx\)

\(\Big \downarrow \) 6478

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

\(\Big \downarrow \) 477

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

\(\Big \downarrow \) 2009

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

Input:

Int[(a + b*ArcTanh[c*x])/(d + e*x)^4,x]
 

Output:

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

Defintions of rubi rules used

rule 477

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
a^p   Int[ExpandIntegrand[(c + d*x)^n*(1 - Rt[-b/a, 2]*x)^p*(1 + Rt[-b/a, 2 
]*x)^p, x], x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[n] & 
& NiceSqrtQ[-b/a] &&  !FractionalPowerFactorQ[Rt[-b/a, 2]]
 

rule 2009

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

rule 6478

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

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.06

method	result	size
parts	\(-\frac {a}{3 \left (e x +d \right )^{3} e}+\frac {b \left (-\frac {c^{4} \operatorname {arctanh}\left (c x \right )}{3 \left (c e x +c d \right )^{3} e}+\frac {c^{4} \left (\frac {e}{2 \left (c d +e \right ) \left (c d -e \right ) \left (c e x +c d \right )^{2}}-\frac {e \left (3 c^{2} d^{2}+e^{2}\right ) \ln \left (c e x +c d \right )}{\left (c d +e \right )^{3} \left (c d -e \right )^{3}}+\frac {2 e d c}{\left (c d +e \right )^{2} \left (c d -e \right )^{2} \left (c e x +c d \right )}-\frac {\ln \left (c x -1\right )}{2 \left (c d +e \right )^{3}}+\frac {\ln \left (c x +1\right )}{2 \left (c d -e \right )^{3}}\right )}{3 e}\right )}{c}\)	\(184\)
derivativedivides	\(\frac {-\frac {a \,c^{4}}{3 \left (c e x +c d \right )^{3} e}+b \,c^{4} \left (-\frac {\operatorname {arctanh}\left (c x \right )}{3 \left (c e x +c d \right )^{3} e}+\frac {\frac {e}{2 \left (c d +e \right ) \left (c d -e \right ) \left (c e x +c d \right )^{2}}-\frac {e \left (3 c^{2} d^{2}+e^{2}\right ) \ln \left (c e x +c d \right )}{\left (c d +e \right )^{3} \left (c d -e \right )^{3}}+\frac {2 e d c}{\left (c d +e \right )^{2} \left (c d -e \right )^{2} \left (c e x +c d \right )}-\frac {\ln \left (c x -1\right )}{2 \left (c d +e \right )^{3}}+\frac {\ln \left (c x +1\right )}{2 \left (c d -e \right )^{3}}}{3 e}\right )}{c}\)	\(188\)
default	\(\frac {-\frac {a \,c^{4}}{3 \left (c e x +c d \right )^{3} e}+b \,c^{4} \left (-\frac {\operatorname {arctanh}\left (c x \right )}{3 \left (c e x +c d \right )^{3} e}+\frac {\frac {e}{2 \left (c d +e \right ) \left (c d -e \right ) \left (c e x +c d \right )^{2}}-\frac {e \left (3 c^{2} d^{2}+e^{2}\right ) \ln \left (c e x +c d \right )}{\left (c d +e \right )^{3} \left (c d -e \right )^{3}}+\frac {2 e d c}{\left (c d +e \right )^{2} \left (c d -e \right )^{2} \left (c e x +c d \right )}-\frac {\ln \left (c x -1\right )}{2 \left (c d +e \right )^{3}}+\frac {\ln \left (c x +1\right )}{2 \left (c d -e \right )^{3}}}{3 e}\right )}{c}\)	\(188\)
parallelrisch	\(\frac {6 x a \,c^{6} d^{8}+6 \,\operatorname {arctanh}\left (c x \right ) b \,c^{5} d^{8}-6 x^{2} a d \,e^{7}-6 x a \,d^{2} e^{6}+2 \,\operatorname {arctanh}\left (c x \right ) b \,d^{3} e^{5}+6 \ln \left (c x -1\right ) b \,c^{5} d^{8}-6 \ln \left (e x +d \right ) b \,c^{5} d^{8}+2 x^{3} a \,c^{6} d^{6} e^{2}-5 x^{3} b \,c^{5} d^{5} e^{3}+6 x^{2} a \,c^{6} d^{7} e +6 x \,\operatorname {arctanh}\left (c x \right ) b \,c^{6} d^{8}-6 x^{3} a \,c^{4} d^{4} e^{4}-11 x^{2} b \,c^{5} d^{6} e^{2}+6 x^{3} b \,c^{3} d^{3} e^{5}-18 x^{2} a \,c^{4} d^{5} e^{3}-6 x b \,c^{5} d^{7} e +6 x^{3} a \,c^{2} d^{2} e^{6}+14 x^{2} b \,c^{3} d^{4} e^{4}-18 x a \,c^{4} d^{6} e^{2}+12 \,\operatorname {arctanh}\left (c x \right ) b \,c^{4} d^{7} e -x^{3} b c d \,e^{7}+18 x^{2} a \,c^{2} d^{3} e^{5}+8 x b \,c^{3} d^{5} e^{3}+2 \,\operatorname {arctanh}\left (c x \right ) b \,c^{3} d^{6} e^{2}-3 x^{2} b c \,d^{2} e^{6}+18 x a \,c^{2} d^{4} e^{4}-6 \,\operatorname {arctanh}\left (c x \right ) b \,c^{2} d^{5} e^{3}-2 x b c \,d^{3} e^{5}+2 \ln \left (c x -1\right ) b \,c^{3} d^{6} e^{2}-2 \ln \left (e x +d \right ) b \,c^{3} d^{6} e^{2}-2 x^{3} a \,e^{8}+2 x^{3} \operatorname {arctanh}\left (c x \right ) b \,c^{6} d^{6} e^{2}+6 x^{2} \operatorname {arctanh}\left (c x \right ) b \,c^{6} d^{7} e +6 x^{3} \operatorname {arctanh}\left (c x \right ) b \,c^{4} d^{4} e^{4}+18 x^{2} \operatorname {arctanh}\left (c x \right ) b \,c^{5} d^{6} e^{2}+2 x^{3} \operatorname {arctanh}\left (c x \right ) b \,c^{3} d^{3} e^{5}+18 x^{2} \operatorname {arctanh}\left (c x \right ) b \,c^{4} d^{5} e^{3}+18 x \,\operatorname {arctanh}\left (c x \right ) b \,c^{5} d^{7} e +6 x^{2} \operatorname {arctanh}\left (c x \right ) b \,c^{3} d^{4} e^{4}+18 x \,\operatorname {arctanh}\left (c x \right ) b \,c^{4} d^{6} e^{2}+6 x \,\operatorname {arctanh}\left (c x \right ) b \,c^{3} d^{5} e^{3}+6 x^{3} \operatorname {arctanh}\left (c x \right ) b \,c^{5} d^{5} e^{3}+6 \ln \left (c x -1\right ) x^{3} b \,c^{5} d^{5} e^{3}-6 \ln \left (e x +d \right ) x^{3} b \,c^{5} d^{5} e^{3}+18 \ln \left (c x -1\right ) x^{2} b \,c^{5} d^{6} e^{2}-18 \ln \left (e x +d \right ) x^{2} b \,c^{5} d^{6} e^{2}+2 \ln \left (c x -1\right ) x^{3} b \,c^{3} d^{3} e^{5}+18 \ln \left (c x -1\right ) x b \,c^{5} d^{7} e -2 \ln \left (e x +d \right ) x^{3} b \,c^{3} d^{3} e^{5}-18 \ln \left (e x +d \right ) x b \,c^{5} d^{7} e +6 \ln \left (c x -1\right ) x^{2} b \,c^{3} d^{4} e^{4}-6 \ln \left (e x +d \right ) x^{2} b \,c^{3} d^{4} e^{4}+6 \ln \left (c x -1\right ) x b \,c^{3} d^{5} e^{3}-6 \ln \left (e x +d \right ) x b \,c^{3} d^{5} e^{3}}{6 \left (c^{6} d^{6}-3 c^{4} d^{4} e^{2}+3 c^{2} d^{2} e^{4}-e^{6}\right ) \left (e x +d \right )^{3} d^{3}}\)	\(914\)
risch	\(\text {Expression too large to display}\)	\(1020\)

Input:

int((a+b*arctanh(c*x))/(e*x+d)^4,x,method=_RETURNVERBOSE)
 

Output:

-1/3*a/(e*x+d)^3/e+b/c*(-1/3*c^4/(c*e*x+c*d)^3/e*arctanh(c*x)+1/3*c^4/e*(1 
/2*e/(c*d+e)/(c*d-e)/(c*e*x+c*d)^2-e*(3*c^2*d^2+e^2)/(c*d+e)^3/(c*d-e)^3*l 
n(c*e*x+c*d)+2*e*d*c/(c*d+e)^2/(c*d-e)^2/(c*e*x+c*d)-1/2/(c*d+e)^3*ln(c*x- 
1)+1/2/(c*d-e)^3*ln(c*x+1)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 859 vs. \(2 (162) = 324\).

Time = 0.29 (sec) , antiderivative size = 859, normalized size of antiderivative = 4.94 \[ \int \frac {a+b \text {arctanh}(c x)}{(d+e x)^4} \, dx =\text {Too large to display} \] Input:

integrate((a+b*arctanh(c*x))/(e*x+d)^4,x, algorithm="fricas")
 

Output:

-1/6*(2*a*c^6*d^6 - 5*b*c^5*d^5*e - 6*a*c^4*d^4*e^2 + 6*b*c^3*d^3*e^3 + 6* 
a*c^2*d^2*e^4 - b*c*d*e^5 - 2*a*e^6 - 4*(b*c^5*d^3*e^3 - b*c^3*d*e^5)*x^2 
- (9*b*c^5*d^4*e^2 - 10*b*c^3*d^2*e^4 + b*c*e^6)*x - (b*c^6*d^6 + 3*b*c^5* 
d^5*e + 3*b*c^4*d^4*e^2 + b*c^3*d^3*e^3 + (b*c^6*d^3*e^3 + 3*b*c^5*d^2*e^4 
 + 3*b*c^4*d*e^5 + b*c^3*e^6)*x^3 + 3*(b*c^6*d^4*e^2 + 3*b*c^5*d^3*e^3 + 3 
*b*c^4*d^2*e^4 + b*c^3*d*e^5)*x^2 + 3*(b*c^6*d^5*e + 3*b*c^5*d^4*e^2 + 3*b 
*c^4*d^3*e^3 + b*c^3*d^2*e^4)*x)*log(c*x + 1) + (b*c^6*d^6 - 3*b*c^5*d^5*e 
 + 3*b*c^4*d^4*e^2 - b*c^3*d^3*e^3 + (b*c^6*d^3*e^3 - 3*b*c^5*d^2*e^4 + 3* 
b*c^4*d*e^5 - b*c^3*e^6)*x^3 + 3*(b*c^6*d^4*e^2 - 3*b*c^5*d^3*e^3 + 3*b*c^ 
4*d^2*e^4 - b*c^3*d*e^5)*x^2 + 3*(b*c^6*d^5*e - 3*b*c^5*d^4*e^2 + 3*b*c^4* 
d^3*e^3 - b*c^3*d^2*e^4)*x)*log(c*x - 1) + 2*(3*b*c^5*d^5*e + b*c^3*d^3*e^ 
3 + (3*b*c^5*d^2*e^4 + b*c^3*e^6)*x^3 + 3*(3*b*c^5*d^3*e^3 + b*c^3*d*e^5)* 
x^2 + 3*(3*b*c^5*d^4*e^2 + b*c^3*d^2*e^4)*x)*log(e*x + d) + (b*c^6*d^6 - 3 
*b*c^4*d^4*e^2 + 3*b*c^2*d^2*e^4 - b*e^6)*log(-(c*x + 1)/(c*x - 1)))/(c^6* 
d^9*e - 3*c^4*d^7*e^3 + 3*c^2*d^5*e^5 - d^3*e^7 + (c^6*d^6*e^4 - 3*c^4*d^4 
*e^6 + 3*c^2*d^2*e^8 - e^10)*x^3 + 3*(c^6*d^7*e^3 - 3*c^4*d^5*e^5 + 3*c^2* 
d^3*e^7 - d*e^9)*x^2 + 3*(c^6*d^8*e^2 - 3*c^4*d^6*e^4 + 3*c^2*d^4*e^6 - d^ 
2*e^8)*x)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 10946 vs. \(2 (150) = 300\).

Time = 5.15 (sec) , antiderivative size = 10946, normalized size of antiderivative = 62.91 \[ \int \frac {a+b \text {arctanh}(c x)}{(d+e x)^4} \, dx=\text {Too large to display} \] Input:

integrate((a+b*atanh(c*x))/(e*x+d)**4,x)
 

Output:

Piecewise((a*x/d**4, Eq(c, 0) & Eq(e, 0)), (-a/(3*d**3*e + 9*d**2*e**2*x + 
 9*d*e**3*x**2 + 3*e**4*x**3), Eq(c, 0)), ((a*x + b*x*atanh(c*x) + b*log(x 
 - 1/c)/c + b*atanh(c*x)/c)/d**4, Eq(e, 0)), (-24*a*d**3/(72*d**6*e + 216* 
d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) + 21*b*d**3*atanh(e* 
x/d)/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3 
) + 10*b*d**3/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3* 
e**4*x**3) - 9*b*d**2*e*x*atanh(e*x/d)/(72*d**6*e + 216*d**5*e**2*x + 216* 
d**4*e**3*x**2 + 72*d**3*e**4*x**3) + 9*b*d**2*e*x/(72*d**6*e + 216*d**5*e 
**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) - 9*b*d*e**2*x**2*atanh(e* 
x/d)/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3 
) + 3*b*d*e**2*x**2/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72 
*d**3*e**4*x**3) - 3*b*e**3*x**3*atanh(e*x/d)/(72*d**6*e + 216*d**5*e**2*x 
 + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3), Eq(c, -e/d)), (-24*a*d**3/(72* 
d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) - 21*b* 
d**3*atanh(e*x/d)/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d 
**3*e**4*x**3) - 10*b*d**3/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x* 
*2 + 72*d**3*e**4*x**3) + 9*b*d**2*e*x*atanh(e*x/d)/(72*d**6*e + 216*d**5* 
e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) - 9*b*d**2*e*x/(72*d**6*e 
 + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) + 9*b*d*e**2* 
x**2*atanh(e*x/d)/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 7...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 339 vs. \(2 (162) = 324\).

Time = 0.04 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.95 \[ \int \frac {a+b \text {arctanh}(c x)}{(d+e x)^4} \, dx=\frac {1}{6} \, {\left ({\left (\frac {c^{2} \log \left (c x + 1\right )}{c^{3} d^{3} e - 3 \, c^{2} d^{2} e^{2} + 3 \, c d e^{3} - e^{4}} - \frac {c^{2} \log \left (c x - 1\right )}{c^{3} d^{3} e + 3 \, c^{2} d^{2} e^{2} + 3 \, c d e^{3} + e^{4}} - \frac {2 \, {\left (3 \, c^{4} d^{2} + c^{2} e^{2}\right )} \log \left (e x + d\right )}{c^{6} d^{6} - 3 \, c^{4} d^{4} e^{2} + 3 \, c^{2} d^{2} e^{4} - e^{6}} + \frac {4 \, c^{2} d e x + 5 \, c^{2} d^{2} - e^{2}}{c^{4} d^{6} - 2 \, c^{2} d^{4} e^{2} + d^{2} e^{4} + {\left (c^{4} d^{4} e^{2} - 2 \, c^{2} d^{2} e^{4} + e^{6}\right )} x^{2} + 2 \, {\left (c^{4} d^{5} e - 2 \, c^{2} d^{3} e^{3} + d e^{5}\right )} x}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x\right )}{e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e}\right )} b - \frac {a}{3 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \] Input:

integrate((a+b*arctanh(c*x))/(e*x+d)^4,x, algorithm="maxima")
 

Output:

1/6*((c^2*log(c*x + 1)/(c^3*d^3*e - 3*c^2*d^2*e^2 + 3*c*d*e^3 - e^4) - c^2 
*log(c*x - 1)/(c^3*d^3*e + 3*c^2*d^2*e^2 + 3*c*d*e^3 + e^4) - 2*(3*c^4*d^2 
 + c^2*e^2)*log(e*x + d)/(c^6*d^6 - 3*c^4*d^4*e^2 + 3*c^2*d^2*e^4 - e^6) + 
 (4*c^2*d*e*x + 5*c^2*d^2 - e^2)/(c^4*d^6 - 2*c^2*d^4*e^2 + d^2*e^4 + (c^4 
*d^4*e^2 - 2*c^2*d^2*e^4 + e^6)*x^2 + 2*(c^4*d^5*e - 2*c^2*d^3*e^3 + d*e^5 
)*x))*c - 2*arctanh(c*x)/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e))*b 
- 1/3*a/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1792 vs. \(2 (162) = 324\).

Time = 0.17 (sec) , antiderivative size = 1792, normalized size of antiderivative = 10.30 \[ \int \frac {a+b \text {arctanh}(c x)}{(d+e x)^4} \, dx=\text {Too large to display} \] Input:

integrate((a+b*arctanh(c*x))/(e*x+d)^4,x, algorithm="giac")
 

Output:

-1/3*c*((3*b*c^4*d^2 + b*c^2*e^2)*log(-(c*x + 1)*c*d/(c*x - 1) + c*d - (c* 
x + 1)*e/(c*x - 1) - e)/(c^6*d^6 - 3*c^4*d^4*e^2 + 3*c^2*d^2*e^4 - e^6) - 
(3*(c*x + 1)^2*b*c^4*d^2/(c*x - 1)^2 - 6*(c*x + 1)*b*c^4*d^2/(c*x - 1) + 3 
*b*c^4*d^2 + 6*(c*x + 1)^2*b*c^3*d*e/(c*x - 1)^2 - 6*(c*x + 1)*b*c^3*d*e/( 
c*x - 1) + 3*(c*x + 1)^2*b*c^2*e^2/(c*x - 1)^2 + b*c^2*e^2)*log(-(c*x + 1) 
/(c*x - 1))/((c*x + 1)^3*c^6*d^6/(c*x - 1)^3 - 3*(c*x + 1)^2*c^6*d^6/(c*x 
- 1)^2 + 3*(c*x + 1)*c^6*d^6/(c*x - 1) - c^6*d^6 + 6*(c*x + 1)^3*c^5*d^5*e 
/(c*x - 1)^3 - 12*(c*x + 1)^2*c^5*d^5*e/(c*x - 1)^2 + 6*(c*x + 1)*c^5*d^5* 
e/(c*x - 1) + 15*(c*x + 1)^3*c^4*d^4*e^2/(c*x - 1)^3 - 15*(c*x + 1)^2*c^4* 
d^4*e^2/(c*x - 1)^2 - 3*(c*x + 1)*c^4*d^4*e^2/(c*x - 1) + 3*c^4*d^4*e^2 + 
20*(c*x + 1)^3*c^3*d^3*e^3/(c*x - 1)^3 - 12*(c*x + 1)*c^3*d^3*e^3/(c*x - 1 
) + 15*(c*x + 1)^3*c^2*d^2*e^4/(c*x - 1)^3 + 15*(c*x + 1)^2*c^2*d^2*e^4/(c 
*x - 1)^2 - 3*(c*x + 1)*c^2*d^2*e^4/(c*x - 1) - 3*c^2*d^2*e^4 + 6*(c*x + 1 
)^3*c*d*e^5/(c*x - 1)^3 + 12*(c*x + 1)^2*c*d*e^5/(c*x - 1)^2 + 6*(c*x + 1) 
*c*d*e^5/(c*x - 1) + (c*x + 1)^3*e^6/(c*x - 1)^3 + 3*(c*x + 1)^2*e^6/(c*x 
- 1)^2 + 3*(c*x + 1)*e^6/(c*x - 1) + e^6) - (3*b*c^4*d^2 + b*c^2*e^2)*log( 
-(c*x + 1)/(c*x - 1))/(c^6*d^6 - 3*c^4*d^4*e^2 + 3*c^2*d^2*e^4 - e^6) - 2* 
(3*(c*x + 1)^2*a*c^6*d^4/(c*x - 1)^2 - 6*(c*x + 1)*a*c^6*d^4/(c*x - 1) + 3 
*a*c^6*d^4 + 6*(c*x + 1)*a*c^5*d^3*e/(c*x - 1) - 6*a*c^5*d^3*e - 3*(c*x + 
1)^2*b*c^5*d^3*e/(c*x - 1)^2 + 6*(c*x + 1)*b*c^5*d^3*e/(c*x - 1) - 3*b*...
 

Mupad [B] (verification not implemented)

Time = 5.04 (sec) , antiderivative size = 418, normalized size of antiderivative = 2.40 \[ \int \frac {a+b \text {arctanh}(c x)}{(d+e x)^4} \, dx=\ln \left (d+e\,x\right )\,\left (\frac {b\,c^3}{6\,e\,{\left (e+c\,d\right )}^3}+\frac {b\,c^3}{6\,e\,{\left (e-c\,d\right )}^3}\right )-\frac {\frac {2\,a\,c^4\,d^4-5\,b\,c^3\,d^3\,e-4\,a\,c^2\,d^2\,e^2+b\,c\,d\,e^3+2\,a\,e^4}{2\,\left (c^4\,d^4-2\,c^2\,d^2\,e^2+e^4\right )}+\frac {x\,\left (b\,c\,e^4-9\,b\,c^3\,d^2\,e^2\right )}{2\,\left (c^4\,d^4-2\,c^2\,d^2\,e^2+e^4\right )}-\frac {2\,b\,c^3\,d\,e^3\,x^2}{c^4\,d^4-2\,c^2\,d^2\,e^2+e^4}}{3\,d^3\,e+9\,d^2\,e^2\,x+9\,d\,e^3\,x^2+3\,e^4\,x^3}-\frac {b\,c^3\,\ln \left (c\,x-1\right )}{6\,c^3\,d^3\,e+18\,c^2\,d^2\,e^2+18\,c\,d\,e^3+6\,e^4}-\frac {b\,c^3\,\ln \left (c\,x+1\right )}{-6\,c^3\,d^3\,e+18\,c^2\,d^2\,e^2-18\,c\,d\,e^3+6\,e^4}-\frac {b\,\ln \left (c\,x+1\right )}{6\,e\,\left (d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3\right )}+\frac {b\,\ln \left (1-c\,x\right )}{3\,e\,\left (2\,d^3+6\,d^2\,e\,x+6\,d\,e^2\,x^2+2\,e^3\,x^3\right )} \] Input:

int((a + b*atanh(c*x))/(d + e*x)^4,x)
 

Output:

log(d + e*x)*((b*c^3)/(6*e*(e + c*d)^3) + (b*c^3)/(6*e*(e - c*d)^3)) - ((2 
*a*e^4 + 2*a*c^4*d^4 - 4*a*c^2*d^2*e^2 + b*c*d*e^3 - 5*b*c^3*d^3*e)/(2*(e^ 
4 + c^4*d^4 - 2*c^2*d^2*e^2)) + (x*(b*c*e^4 - 9*b*c^3*d^2*e^2))/(2*(e^4 + 
c^4*d^4 - 2*c^2*d^2*e^2)) - (2*b*c^3*d*e^3*x^2)/(e^4 + c^4*d^4 - 2*c^2*d^2 
*e^2))/(3*d^3*e + 3*e^4*x^3 + 9*d^2*e^2*x + 9*d*e^3*x^2) - (b*c^3*log(c*x 
- 1))/(6*e^4 + 6*c^3*d^3*e + 18*c^2*d^2*e^2 + 18*c*d*e^3) - (b*c^3*log(c*x 
 + 1))/(6*e^4 - 6*c^3*d^3*e + 18*c^2*d^2*e^2 - 18*c*d*e^3) - (b*log(c*x + 
1))/(6*e*(d^3 + e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x)) + (b*log(1 - c*x))/(3* 
e*(2*d^3 + 2*e^3*x^3 + 6*d*e^2*x^2 + 6*d^2*e*x))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 1565, normalized size of antiderivative = 8.99 \[ \int \frac {a+b \text {arctanh}(c x)}{(d+e x)^4} \, dx =\text {Too large to display} \] Input:

int((a+b*atanh(c*x))/(e*x+d)^4,x)
 

Output:

(18*atanh(c*x)*b*c**6*d**8*e*x + 18*atanh(c*x)*b*c**6*d**7*e**2*x**2 + 6*a 
tanh(c*x)*b*c**6*d**6*e**3*x**3 - 54*atanh(c*x)*b*c**4*d**6*e**3*x - 54*at 
anh(c*x)*b*c**4*d**5*e**4*x**2 - 18*atanh(c*x)*b*c**4*d**4*e**5*x**3 + 54* 
atanh(c*x)*b*c**2*d**4*e**5*x + 54*atanh(c*x)*b*c**2*d**3*e**6*x**2 + 18*a 
tanh(c*x)*b*c**2*d**2*e**7*x**3 - 18*atanh(c*x)*b*d**2*e**7*x - 18*atanh(c 
*x)*b*d*e**8*x**2 - 6*atanh(c*x)*b*e**9*x**3 + 9*log(c**2*x - c)*b*c**5*d* 
*8*e + 27*log(c**2*x - c)*b*c**5*d**7*e**2*x + 27*log(c**2*x - c)*b*c**5*d 
**6*e**3*x**2 + 9*log(c**2*x - c)*b*c**5*d**5*e**4*x**3 - 18*log(c**2*x - 
c)*b*c**4*d**7*e**2 - 54*log(c**2*x - c)*b*c**4*d**6*e**3*x - 54*log(c**2* 
x - c)*b*c**4*d**5*e**4*x**2 - 18*log(c**2*x - c)*b*c**4*d**4*e**5*x**3 + 
3*log(c**2*x - c)*b*c**3*d**6*e**3 + 9*log(c**2*x - c)*b*c**3*d**5*e**4*x 
+ 9*log(c**2*x - c)*b*c**3*d**4*e**5*x**2 + 3*log(c**2*x - c)*b*c**3*d**3* 
e**6*x**3 + 9*log(c**2*x - c)*b*c**2*d**5*e**4 + 27*log(c**2*x - c)*b*c**2 
*d**4*e**5*x + 27*log(c**2*x - c)*b*c**2*d**3*e**6*x**2 + 9*log(c**2*x - c 
)*b*c**2*d**2*e**7*x**3 - 3*log(c**2*x - c)*b*d**3*e**6 - 9*log(c**2*x - c 
)*b*d**2*e**7*x - 9*log(c**2*x - c)*b*d*e**8*x**2 - 3*log(c**2*x - c)*b*e* 
*9*x**3 + 9*log(c**2*x + c)*b*c**5*d**8*e + 27*log(c**2*x + c)*b*c**5*d**7 
*e**2*x + 27*log(c**2*x + c)*b*c**5*d**6*e**3*x**2 + 9*log(c**2*x + c)*b*c 
**5*d**5*e**4*x**3 + 18*log(c**2*x + c)*b*c**4*d**7*e**2 + 54*log(c**2*x + 
 c)*b*c**4*d**6*e**3*x + 54*log(c**2*x + c)*b*c**4*d**5*e**4*x**2 + 18*...