Integrand size = 16, antiderivative size = 125 \[ \int (d+e x)^3 (a+b \text {arctanh}(c x)) \, dx=\frac {b e \left (6 c^2 d^2+e^2\right ) x}{4 c^3}+\frac {b d e^2 x^2}{2 c}+\frac {b e^3 x^3}{12 c}+\frac {(d+e x)^4 (a+b \text {arctanh}(c x))}{4 e}+\frac {b (c d+e)^4 \log (1-c x)}{8 c^4 e}-\frac {b (c d-e)^4 \log (1+c x)}{8 c^4 e} \]
1/4*b*e*(6*c^2*d^2+e^2)*x/c^3+1/2*b*d*e^2*x^2/c+1/12*b*e^3*x^3/c+1/4*(e*x+ d)^4*(a+b*arctanh(c*x))/e+1/8*b*(c*d+e)^4*ln(-c*x+1)/c^4/e-1/8*b*(c*d-e)^4 *ln(c*x+1)/c^4/e
Time = 0.07 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.64 \[ \int (d+e x)^3 (a+b \text {arctanh}(c x)) \, dx=\frac {6 c \left (4 a c^3 d^3+b e \left (6 c^2 d^2+e^2\right )\right ) x+12 c^3 d e (3 a c d+b e) x^2+2 c^3 e^2 (12 a c d+b e) x^3+6 a c^4 e^3 x^4+6 b c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right ) \text {arctanh}(c x)+3 b \left (4 c^3 d^3+6 c^2 d^2 e+4 c d e^2+e^3\right ) \log (1-c x)+3 b \left (4 c^3 d^3-6 c^2 d^2 e+4 c d e^2-e^3\right ) \log (1+c x)}{24 c^4} \]
(6*c*(4*a*c^3*d^3 + b*e*(6*c^2*d^2 + e^2))*x + 12*c^3*d*e*(3*a*c*d + b*e)* x^2 + 2*c^3*e^2*(12*a*c*d + b*e)*x^3 + 6*a*c^4*e^3*x^4 + 6*b*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3)*ArcTanh[c*x] + 3*b*(4*c^3*d^3 + 6*c^2 *d^2*e + 4*c*d*e^2 + e^3)*Log[1 - c*x] + 3*b*(4*c^3*d^3 - 6*c^2*d^2*e + 4* c*d*e^2 - e^3)*Log[1 + c*x])/(24*c^4)
Time = 0.33 (sec) , antiderivative size = 122, 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.
| \(\displaystyle \int (d+e x)^3 (a+b \text {arctanh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6478 |
| \(\displaystyle \frac {(d+e x)^4 (a+b \text {arctanh}(c x))}{4 e}-\frac {b c \int \frac {(d+e x)^4}{1-c^2 x^2}dx}{4 e}\) |
| \(\Big \downarrow \) 477 |
| \(\displaystyle \frac {(d+e x)^4 (a+b \text {arctanh}(c x))}{4 e}-\frac {b c \int \left (\frac {(c d-e)^4}{2 c^4 (c x+1)}-\frac {e^4 x^2}{c^2}-\frac {e^2 \left (6 c^2 d^2+e^2\right )}{c^4}-\frac {4 d e^3 x}{c^2}+\frac {(c d+e)^4}{2 c^4 (1-c x)}\right )dx}{4 e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^4 (a+b \text {arctanh}(c x))}{4 e}-\frac {b c \left (\frac {(c d-e)^4 \log (c x+1)}{2 c^5}-\frac {(c d+e)^4 \log (1-c x)}{2 c^5}-\frac {2 d e^3 x^2}{c^2}-\frac {e^4 x^3}{3 c^2}-\frac {e^2 x \left (6 c^2 d^2+e^2\right )}{c^4}\right )}{4 e}\) |
((d + e*x)^4*(a + b*ArcTanh[c*x]))/(4*e) - (b*c*(-((e^2*(6*c^2*d^2 + e^2)* x)/c^4) - (2*d*e^3*x^2)/c^2 - (e^4*x^3)/(3*c^2) - ((c*d + e)^4*Log[1 - c*x ])/(2*c^5) + ((c*d - e)^4*Log[1 + c*x])/(2*c^5)))/(4*e)
3.1.2.3.1 Defintions of rubi rules used
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]]
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]
Time = 0.85 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.82
| method | result | size |
| parts | \(\frac {a \left (e x +d \right )^{4}}{4 e}+\frac {b \left (\frac {c \,e^{3} \operatorname {arctanh}\left (c x \right ) x^{4}}{4}+c \,e^{2} \operatorname {arctanh}\left (c x \right ) x^{3} d +\frac {3 c e \,\operatorname {arctanh}\left (c x \right ) x^{2} d^{2}}{2}+\operatorname {arctanh}\left (c x \right ) c x \,d^{3}+\frac {c \,\operatorname {arctanh}\left (c x \right ) d^{4}}{4 e}-\frac {-6 e^{2} c^{3} d^{2} x -2 c^{3} d \,e^{3} x^{2}-\frac {e^{4} c^{3} x^{3}}{3}-e^{4} c x -\frac {\left (c^{4} d^{4}+4 c^{3} d^{3} e +6 c^{2} d^{2} e^{2}+4 c d \,e^{3}+e^{4}\right ) \ln \left (c x -1\right )}{2}+\frac {\left (c^{4} d^{4}-4 c^{3} d^{3} e +6 c^{2} d^{2} e^{2}-4 c d \,e^{3}+e^{4}\right ) \ln \left (c x +1\right )}{2}}{4 c^{3} e}\right )}{c}\) | \(227\) |
| parallelrisch | \(\frac {3 x^{4} \operatorname {arctanh}\left (c x \right ) b \,c^{4} e^{3}+3 x^{4} a \,c^{4} e^{3}+12 x^{3} \operatorname {arctanh}\left (c x \right ) b \,c^{4} d \,e^{2}+12 x^{3} a \,c^{4} d \,e^{2}+18 x^{2} \operatorname {arctanh}\left (c x \right ) b \,c^{4} d^{2} e +x^{3} b \,c^{3} e^{3}+18 x^{2} a \,c^{4} d^{2} e +12 d^{3} b \,\operatorname {arctanh}\left (c x \right ) x \,c^{4}+6 x^{2} b \,c^{3} d \,e^{2}+12 x a \,c^{4} d^{3}+12 \ln \left (c x -1\right ) b \,c^{3} d^{3}+18 x b \,c^{3} d^{2} e +12 \,\operatorname {arctanh}\left (c x \right ) b \,c^{3} d^{3}-18 \,\operatorname {arctanh}\left (c x \right ) b \,c^{2} d^{2} e +12 \ln \left (c x -1\right ) b c d \,e^{2}+3 x b c \,e^{3}+12 \,\operatorname {arctanh}\left (c x \right ) b c d \,e^{2}-3 \,\operatorname {arctanh}\left (c x \right ) b \,e^{3}}{12 c^{4}}\) | \(240\) |
| derivativedivides | \(\frac {\frac {a \left (e c x +c d \right )^{4}}{4 c^{3} e}+\frac {b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{4} d^{4}}{4 e}+\operatorname {arctanh}\left (c x \right ) c^{4} d^{3} x +\frac {3 e \,\operatorname {arctanh}\left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \operatorname {arctanh}\left (c x \right ) c^{4} d \,x^{3}+\frac {e^{3} \operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {-6 e^{2} c^{3} d^{2} x -2 c^{3} d \,e^{3} x^{2}-\frac {e^{4} c^{3} x^{3}}{3}-e^{4} c x -\frac {\left (c^{4} d^{4}+4 c^{3} d^{3} e +6 c^{2} d^{2} e^{2}+4 c d \,e^{3}+e^{4}\right ) \ln \left (c x -1\right )}{2}+\frac {\left (c^{4} d^{4}-4 c^{3} d^{3} e +6 c^{2} d^{2} e^{2}-4 c d \,e^{3}+e^{4}\right ) \ln \left (c x +1\right )}{2}}{4 e}\right )}{c^{3}}}{c}\) | \(244\) |
| default | \(\frac {\frac {a \left (e c x +c d \right )^{4}}{4 c^{3} e}+\frac {b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{4} d^{4}}{4 e}+\operatorname {arctanh}\left (c x \right ) c^{4} d^{3} x +\frac {3 e \,\operatorname {arctanh}\left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \operatorname {arctanh}\left (c x \right ) c^{4} d \,x^{3}+\frac {e^{3} \operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {-6 e^{2} c^{3} d^{2} x -2 c^{3} d \,e^{3} x^{2}-\frac {e^{4} c^{3} x^{3}}{3}-e^{4} c x -\frac {\left (c^{4} d^{4}+4 c^{3} d^{3} e +6 c^{2} d^{2} e^{2}+4 c d \,e^{3}+e^{4}\right ) \ln \left (c x -1\right )}{2}+\frac {\left (c^{4} d^{4}-4 c^{3} d^{3} e +6 c^{2} d^{2} e^{2}-4 c d \,e^{3}+e^{4}\right ) \ln \left (c x +1\right )}{2}}{4 e}\right )}{c^{3}}}{c}\) | \(244\) |
| risch | \(\frac {\left (e x +d \right )^{4} b \ln \left (c x +1\right )}{8 e}-\frac {e^{3} b \,x^{4} \ln \left (-c x +1\right )}{8}-\frac {e^{2} b d \,x^{3} \ln \left (-c x +1\right )}{2}+\frac {e^{3} a \,x^{4}}{4}-\frac {3 e b \,d^{2} x^{2} \ln \left (-c x +1\right )}{4}+e^{2} a d \,x^{3}-\frac {b \,d^{3} x \ln \left (-c x +1\right )}{2}+\frac {3 e a \,d^{2} x^{2}}{2}+\frac {b \,e^{3} x^{3}}{12 c}-\frac {\ln \left (c x +1\right ) b \,d^{4}}{8 e}+a \,d^{3} x +\frac {b d \,e^{2} x^{2}}{2 c}+\frac {\ln \left (c x +1\right ) b \,d^{3}}{2 c}+\frac {\ln \left (-c x +1\right ) b \,d^{3}}{2 c}+\frac {3 e b \,d^{2} x}{2 c}-\frac {3 e \ln \left (c x +1\right ) b \,d^{2}}{4 c^{2}}+\frac {3 e \ln \left (-c x +1\right ) b \,d^{2}}{4 c^{2}}+\frac {e^{2} \ln \left (c x +1\right ) b d}{2 c^{3}}+\frac {e^{2} \ln \left (-c x +1\right ) b d}{2 c^{3}}+\frac {e^{3} b x}{4 c^{3}}-\frac {e^{3} \ln \left (c x +1\right ) b}{8 c^{4}}+\frac {e^{3} \ln \left (-c x +1\right ) b}{8 c^{4}}\) | \(308\) |
1/4*a*(e*x+d)^4/e+b/c*(1/4*c*e^3*arctanh(c*x)*x^4+c*e^2*arctanh(c*x)*x^3*d +3/2*c*e*arctanh(c*x)*x^2*d^2+arctanh(c*x)*c*x*d^3+1/4*c/e*arctanh(c*x)*d^ 4-1/4/c^3/e*(-6*e^2*c^3*d^2*x-2*c^3*d*e^3*x^2-1/3*e^4*c^3*x^3-e^4*c*x-1/2* (c^4*d^4+4*c^3*d^3*e+6*c^2*d^2*e^2+4*c*d*e^3+e^4)*ln(c*x-1)+1/2*(c^4*d^4-4 *c^3*d^3*e+6*c^2*d^2*e^2-4*c*d*e^3+e^4)*ln(c*x+1)))
Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (113) = 226\).
Time = 0.26 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.95 \[ \int (d+e x)^3 (a+b \text {arctanh}(c x)) \, dx=\frac {6 \, a c^{4} e^{3} x^{4} + 2 \, {\left (12 \, a c^{4} d e^{2} + b c^{3} e^{3}\right )} x^{3} + 12 \, {\left (3 \, a c^{4} d^{2} e + b c^{3} d e^{2}\right )} x^{2} + 6 \, {\left (4 \, a c^{4} d^{3} + 6 \, b c^{3} d^{2} e + b c e^{3}\right )} x + 3 \, {\left (4 \, b c^{3} d^{3} - 6 \, b c^{2} d^{2} e + 4 \, b c d e^{2} - b e^{3}\right )} \log \left (c x + 1\right ) + 3 \, {\left (4 \, b c^{3} d^{3} + 6 \, b c^{2} d^{2} e + 4 \, b c d e^{2} + b e^{3}\right )} \log \left (c x - 1\right ) + 3 \, {\left (b c^{4} e^{3} x^{4} + 4 \, b c^{4} d e^{2} x^{3} + 6 \, b c^{4} d^{2} e x^{2} + 4 \, b c^{4} d^{3} x\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{24 \, c^{4}} \]
1/24*(6*a*c^4*e^3*x^4 + 2*(12*a*c^4*d*e^2 + b*c^3*e^3)*x^3 + 12*(3*a*c^4*d ^2*e + b*c^3*d*e^2)*x^2 + 6*(4*a*c^4*d^3 + 6*b*c^3*d^2*e + b*c*e^3)*x + 3* (4*b*c^3*d^3 - 6*b*c^2*d^2*e + 4*b*c*d*e^2 - b*e^3)*log(c*x + 1) + 3*(4*b* c^3*d^3 + 6*b*c^2*d^2*e + 4*b*c*d*e^2 + b*e^3)*log(c*x - 1) + 3*(b*c^4*e^3 *x^4 + 4*b*c^4*d*e^2*x^3 + 6*b*c^4*d^2*e*x^2 + 4*b*c^4*d^3*x)*log(-(c*x + 1)/(c*x - 1)))/c^4
Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (109) = 218\).
Time = 0.39 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.23 \[ \int (d+e x)^3 (a+b \text {arctanh}(c x)) \, dx=\begin {cases} a d^{3} x + \frac {3 a d^{2} e x^{2}}{2} + a d e^{2} x^{3} + \frac {a e^{3} x^{4}}{4} + b d^{3} x \operatorname {atanh}{\left (c x \right )} + \frac {3 b d^{2} e x^{2} \operatorname {atanh}{\left (c x \right )}}{2} + b d e^{2} x^{3} \operatorname {atanh}{\left (c x \right )} + \frac {b e^{3} x^{4} \operatorname {atanh}{\left (c x \right )}}{4} + \frac {b d^{3} \log {\left (x - \frac {1}{c} \right )}}{c} + \frac {b d^{3} \operatorname {atanh}{\left (c x \right )}}{c} + \frac {3 b d^{2} e x}{2 c} + \frac {b d e^{2} x^{2}}{2 c} + \frac {b e^{3} x^{3}}{12 c} - \frac {3 b d^{2} e \operatorname {atanh}{\left (c x \right )}}{2 c^{2}} + \frac {b d e^{2} \log {\left (x - \frac {1}{c} \right )}}{c^{3}} + \frac {b d e^{2} \operatorname {atanh}{\left (c x \right )}}{c^{3}} + \frac {b e^{3} x}{4 c^{3}} - \frac {b e^{3} \operatorname {atanh}{\left (c x \right )}}{4 c^{4}} & \text {for}\: c \neq 0 \\a \left (d^{3} x + \frac {3 d^{2} e x^{2}}{2} + d e^{2} x^{3} + \frac {e^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]
Piecewise((a*d**3*x + 3*a*d**2*e*x**2/2 + a*d*e**2*x**3 + a*e**3*x**4/4 + b*d**3*x*atanh(c*x) + 3*b*d**2*e*x**2*atanh(c*x)/2 + b*d*e**2*x**3*atanh(c *x) + b*e**3*x**4*atanh(c*x)/4 + b*d**3*log(x - 1/c)/c + b*d**3*atanh(c*x) /c + 3*b*d**2*e*x/(2*c) + b*d*e**2*x**2/(2*c) + b*e**3*x**3/(12*c) - 3*b*d **2*e*atanh(c*x)/(2*c**2) + b*d*e**2*log(x - 1/c)/c**3 + b*d*e**2*atanh(c* x)/c**3 + b*e**3*x/(4*c**3) - b*e**3*atanh(c*x)/(4*c**4), Ne(c, 0)), (a*(d **3*x + 3*d**2*e*x**2/2 + d*e**2*x**3 + e**3*x**4/4), True))
Time = 0.20 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.67 \[ \int (d+e x)^3 (a+b \text {arctanh}(c x)) \, dx=\frac {1}{4} \, a e^{3} x^{4} + a d e^{2} x^{3} + \frac {3}{2} \, a d^{2} e x^{2} + \frac {3}{4} \, {\left (2 \, x^{2} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, x}{c^{2}} - \frac {\log \left (c x + 1\right )}{c^{3}} + \frac {\log \left (c x - 1\right )}{c^{3}}\right )}\right )} b d^{2} e + \frac {1}{2} \, {\left (2 \, x^{3} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {x^{2}}{c^{2}} + \frac {\log \left (c^{2} x^{2} - 1\right )}{c^{4}}\right )}\right )} b d e^{2} + \frac {1}{24} \, {\left (6 \, x^{4} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4}} - \frac {3 \, \log \left (c x + 1\right )}{c^{5}} + \frac {3 \, \log \left (c x - 1\right )}{c^{5}}\right )}\right )} b e^{3} + a d^{3} x + \frac {{\left (2 \, c x \operatorname {artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} b d^{3}}{2 \, c} \]
1/4*a*e^3*x^4 + a*d*e^2*x^3 + 3/2*a*d^2*e*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))*b*d^2*e + 1/2*(2*x^3*ar ctanh(c*x) + c*(x^2/c^2 + log(c^2*x^2 - 1)/c^4))*b*d*e^2 + 1/24*(6*x^4*arc tanh(c*x) + c*(2*(c^2*x^3 + 3*x)/c^4 - 3*log(c*x + 1)/c^5 + 3*log(c*x - 1) /c^5))*b*e^3 + a*d^3*x + 1/2*(2*c*x*arctanh(c*x) + log(-c^2*x^2 + 1))*b*d^ 3/c
Leaf count of result is larger than twice the leaf count of optimal. 970 vs. \(2 (113) = 226\).
Time = 0.30 (sec) , antiderivative size = 970, normalized size of antiderivative = 7.76 \[ \int (d+e x)^3 (a+b \text {arctanh}(c x)) \, dx=\text {Too large to display} \]
1/3*c*(3*((c*x + 1)^3*b*c^3*d^3/(c*x - 1)^3 - 3*(c*x + 1)^2*b*c^3*d^3/(c*x - 1)^2 + 3*(c*x + 1)*b*c^3*d^3/(c*x - 1) - b*c^3*d^3 + 3*(c*x + 1)^3*b*c^ 2*d^2*e/(c*x - 1)^3 - 6*(c*x + 1)^2*b*c^2*d^2*e/(c*x - 1)^2 + 3*(c*x + 1)* b*c^2*d^2*e/(c*x - 1) + 3*(c*x + 1)^3*b*c*d*e^2/(c*x - 1)^3 - 3*(c*x + 1)^ 2*b*c*d*e^2/(c*x - 1)^2 + (c*x + 1)*b*c*d*e^2/(c*x - 1) - b*c*d*e^2 + (c*x + 1)^3*b*e^3/(c*x - 1)^3 + (c*x + 1)*b*e^3/(c*x - 1))*log(-(c*x + 1)/(c*x - 1))/((c*x + 1)^4*c^5/(c*x - 1)^4 - 4*(c*x + 1)^3*c^5/(c*x - 1)^3 + 6*(c *x + 1)^2*c^5/(c*x - 1)^2 - 4*(c*x + 1)*c^5/(c*x - 1) + c^5) + (6*(c*x + 1 )^3*a*c^3*d^3/(c*x - 1)^3 - 18*(c*x + 1)^2*a*c^3*d^3/(c*x - 1)^2 + 18*(c*x + 1)*a*c^3*d^3/(c*x - 1) - 6*a*c^3*d^3 + 18*(c*x + 1)^3*a*c^2*d^2*e/(c*x - 1)^3 - 36*(c*x + 1)^2*a*c^2*d^2*e/(c*x - 1)^2 + 18*(c*x + 1)*a*c^2*d^2*e /(c*x - 1) + 9*(c*x + 1)^3*b*c^2*d^2*e/(c*x - 1)^3 - 27*(c*x + 1)^2*b*c^2* d^2*e/(c*x - 1)^2 + 27*(c*x + 1)*b*c^2*d^2*e/(c*x - 1) - 9*b*c^2*d^2*e + 1 8*(c*x + 1)^3*a*c*d*e^2/(c*x - 1)^3 - 18*(c*x + 1)^2*a*c*d*e^2/(c*x - 1)^2 + 6*(c*x + 1)*a*c*d*e^2/(c*x - 1) - 6*a*c*d*e^2 + 6*(c*x + 1)^3*b*c*d*e^2 /(c*x - 1)^3 - 12*(c*x + 1)^2*b*c*d*e^2/(c*x - 1)^2 + 6*(c*x + 1)*b*c*d*e^ 2/(c*x - 1) + 6*(c*x + 1)^3*a*e^3/(c*x - 1)^3 + 6*(c*x + 1)*a*e^3/(c*x - 1 ) + 3*(c*x + 1)^3*b*e^3/(c*x - 1)^3 - 6*(c*x + 1)^2*b*e^3/(c*x - 1)^2 + 5* (c*x + 1)*b*e^3/(c*x - 1) - 2*b*e^3)/((c*x + 1)^4*c^5/(c*x - 1)^4 - 4*(c*x + 1)^3*c^5/(c*x - 1)^3 + 6*(c*x + 1)^2*c^5/(c*x - 1)^2 - 4*(c*x + 1)*c...
Time = 3.79 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.58 \[ \int (d+e x)^3 (a+b \text {arctanh}(c x)) \, dx=\frac {a\,e^3\,x^4}{4}+a\,d^3\,x+\frac {b\,d^3\,\ln \left (c^2\,x^2-1\right )}{2\,c}+\frac {b\,e^3\,x^3}{12\,c}+b\,d^3\,x\,\mathrm {atanh}\left (c\,x\right )+\frac {3\,a\,d^2\,e\,x^2}{2}+a\,d\,e^2\,x^3+\frac {b\,e^3\,x}{4\,c^3}-\frac {b\,e^3\,\mathrm {atanh}\left (c\,x\right )}{4\,c^4}+\frac {b\,e^3\,x^4\,\mathrm {atanh}\left (c\,x\right )}{4}+\frac {3\,b\,d^2\,e\,x}{2\,c}-\frac {3\,b\,d^2\,e\,\mathrm {atanh}\left (c\,x\right )}{2\,c^2}+\frac {3\,b\,d^2\,e\,x^2\,\mathrm {atanh}\left (c\,x\right )}{2}+b\,d\,e^2\,x^3\,\mathrm {atanh}\left (c\,x\right )+\frac {b\,d\,e^2\,\ln \left (c^2\,x^2-1\right )}{2\,c^3}+\frac {b\,d\,e^2\,x^2}{2\,c} \]
(a*e^3*x^4)/4 + a*d^3*x + (b*d^3*log(c^2*x^2 - 1))/(2*c) + (b*e^3*x^3)/(12 *c) + b*d^3*x*atanh(c*x) + (3*a*d^2*e*x^2)/2 + a*d*e^2*x^3 + (b*e^3*x)/(4* c^3) - (b*e^3*atanh(c*x))/(4*c^4) + (b*e^3*x^4*atanh(c*x))/4 + (3*b*d^2*e* x)/(2*c) - (3*b*d^2*e*atanh(c*x))/(2*c^2) + (3*b*d^2*e*x^2*atanh(c*x))/2 + b*d*e^2*x^3*atanh(c*x) + (b*d*e^2*log(c^2*x^2 - 1))/(2*c^3) + (b*d*e^2*x^ 2)/(2*c)