Integrand size = 16, antiderivative size = 96 \[ \int (d+e x)^2 (a+b \text {arctanh}(c x)) \, dx=\frac {b d e x}{c}+\frac {b e^2 x^2}{6 c}+\frac {(d+e x)^3 (a+b \text {arctanh}(c x))}{3 e}+\frac {b (c d+e)^3 \log (1-c x)}{6 c^3 e}-\frac {b (c d-e)^3 \log (1+c x)}{6 c^3 e} \]
b*d*e*x/c+1/6*b*e^2*x^2/c+1/3*(e*x+d)^3*(a+b*arctanh(c*x))/e+1/6*b*(c*d+e) ^3*ln(-c*x+1)/c^3/e-1/6*b*(c*d-e)^3*ln(c*x+1)/c^3/e
Time = 0.06 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.34 \[ \int (d+e x)^2 (a+b \text {arctanh}(c x)) \, dx=\frac {1}{6} \left (\frac {6 d (a c d+b e) x}{c}+\frac {e (6 a c d+b e) x^2}{c}+2 a e^2 x^3+2 b x \left (3 d^2+3 d e x+e^2 x^2\right ) \text {arctanh}(c x)+\frac {b \left (3 c^2 d^2+3 c d e+e^2\right ) \log (1-c x)}{c^3}+\frac {b \left (3 c^2 d^2-3 c d e+e^2\right ) \log (1+c x)}{c^3}\right ) \]
((6*d*(a*c*d + b*e)*x)/c + (e*(6*a*c*d + b*e)*x^2)/c + 2*a*e^2*x^3 + 2*b*x *(3*d^2 + 3*d*e*x + e^2*x^2)*ArcTanh[c*x] + (b*(3*c^2*d^2 + 3*c*d*e + e^2) *Log[1 - c*x])/c^3 + (b*(3*c^2*d^2 - 3*c*d*e + e^2)*Log[1 + c*x])/c^3)/6
Time = 0.30 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.03, 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.
| \(\displaystyle \int (d+e x)^2 (a+b \text {arctanh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6478 |
| \(\displaystyle \frac {(d+e x)^3 (a+b \text {arctanh}(c x))}{3 e}-\frac {b c \int \frac {(d+e x)^3}{1-c^2 x^2}dx}{3 e}\) |
| \(\Big \downarrow \) 477 |
| \(\displaystyle \frac {(d+e x)^3 (a+b \text {arctanh}(c x))}{3 e}-\frac {b c \int \left (\frac {(c d-e)^3}{2 c^3 (c x+1)}-\frac {3 d e^2}{c^2}-\frac {e^3 x}{c^2}+\frac {(c d+e)^3}{2 c^3 (1-c x)}\right )dx}{3 e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^3 (a+b \text {arctanh}(c x))}{3 e}-\frac {b c \left (\frac {(c d-e)^3 \log (c x+1)}{2 c^4}-\frac {(c d+e)^3 \log (1-c x)}{2 c^4}-\frac {3 d e^2 x}{c^2}-\frac {e^3 x^2}{2 c^2}\right )}{3 e}\) |
((d + e*x)^3*(a + b*ArcTanh[c*x]))/(3*e) - (b*c*((-3*d*e^2*x)/c^2 - (e^3*x ^2)/(2*c^2) - ((c*d + e)^3*Log[1 - c*x])/(2*c^4) + ((c*d - e)^3*Log[1 + c* x])/(2*c^4)))/(3*e)
3.1.3.3.1 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 = 0.84 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.72
| method | result | size |
| parallelrisch | \(\frac {2 x^{3} \operatorname {arctanh}\left (c x \right ) b \,c^{3} e^{2}+2 x^{3} a \,c^{3} e^{2}+6 x^{2} \operatorname {arctanh}\left (c x \right ) b \,c^{3} d e +6 x^{2} a \,c^{3} d e +6 d^{2} b \,\operatorname {arctanh}\left (c x \right ) x \,c^{3}+x^{2} b \,c^{2} e^{2}+6 x a \,c^{3} d^{2}+6 \ln \left (c x -1\right ) b \,c^{2} d^{2}+6 x b \,c^{2} d e +6 \,\operatorname {arctanh}\left (c x \right ) b \,c^{2} d^{2}-6 \,\operatorname {arctanh}\left (c x \right ) b c d e +2 \ln \left (c x -1\right ) b \,e^{2}+2 \,\operatorname {arctanh}\left (c x \right ) b \,e^{2}}{6 c^{3}}\) | \(165\) |
| parts | \(\frac {a \left (e x +d \right )^{3}}{3 e}+\frac {b \left (\frac {c \,e^{2} \operatorname {arctanh}\left (c x \right ) x^{3}}{3}+c e \,\operatorname {arctanh}\left (c x \right ) x^{2} d +\operatorname {arctanh}\left (c x \right ) c x \,d^{2}+\frac {c \,\operatorname {arctanh}\left (c x \right ) d^{3}}{3 e}-\frac {-3 c^{2} x d \,e^{2}-\frac {c^{2} x^{2} e^{3}}{2}-\frac {\left (c^{3} d^{3}+3 c^{2} d^{2} e +3 c d \,e^{2}+e^{3}\right ) \ln \left (c x -1\right )}{2}+\frac {\left (c^{3} d^{3}-3 c^{2} d^{2} e +3 c d \,e^{2}-e^{3}\right ) \ln \left (c x +1\right )}{2}}{3 c^{2} e}\right )}{c}\) | \(170\) |
| derivativedivides | \(\frac {\frac {a \left (e c x +c d \right )^{3}}{3 c^{2} e}+\frac {b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{3} d^{3}}{3 e}+\operatorname {arctanh}\left (c x \right ) c^{3} d^{2} x +e \,\operatorname {arctanh}\left (c x \right ) c^{3} d \,x^{2}+\frac {e^{2} \operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3}-\frac {-3 c^{2} x d \,e^{2}-\frac {c^{2} x^{2} e^{3}}{2}-\frac {\left (c^{3} d^{3}+3 c^{2} d^{2} e +3 c d \,e^{2}+e^{3}\right ) \ln \left (c x -1\right )}{2}+\frac {\left (c^{3} d^{3}-3 c^{2} d^{2} e +3 c d \,e^{2}-e^{3}\right ) \ln \left (c x +1\right )}{2}}{3 e}\right )}{c^{2}}}{c}\) | \(185\) |
| default | \(\frac {\frac {a \left (e c x +c d \right )^{3}}{3 c^{2} e}+\frac {b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{3} d^{3}}{3 e}+\operatorname {arctanh}\left (c x \right ) c^{3} d^{2} x +e \,\operatorname {arctanh}\left (c x \right ) c^{3} d \,x^{2}+\frac {e^{2} \operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3}-\frac {-3 c^{2} x d \,e^{2}-\frac {c^{2} x^{2} e^{3}}{2}-\frac {\left (c^{3} d^{3}+3 c^{2} d^{2} e +3 c d \,e^{2}+e^{3}\right ) \ln \left (c x -1\right )}{2}+\frac {\left (c^{3} d^{3}-3 c^{2} d^{2} e +3 c d \,e^{2}-e^{3}\right ) \ln \left (c x +1\right )}{2}}{3 e}\right )}{c^{2}}}{c}\) | \(185\) |
| risch | \(\frac {\left (e x +d \right )^{3} b \ln \left (c x +1\right )}{6 e}-\frac {e^{2} b \,x^{3} \ln \left (-c x +1\right )}{6}-\frac {e b d \,x^{2} \ln \left (-c x +1\right )}{2}+\frac {e^{2} a \,x^{3}}{3}-\frac {b \,d^{2} x \ln \left (-c x +1\right )}{2}+e a d \,x^{2}-\frac {\ln \left (c x +1\right ) b \,d^{3}}{6 e}+a \,d^{2} x +\frac {b \,e^{2} x^{2}}{6 c}+\frac {\ln \left (c x +1\right ) b \,d^{2}}{2 c}+\frac {\ln \left (-c x +1\right ) b \,d^{2}}{2 c}+\frac {b d e x}{c}-\frac {e \ln \left (c x +1\right ) b d}{2 c^{2}}+\frac {e \ln \left (-c x +1\right ) b d}{2 c^{2}}+\frac {e^{2} \ln \left (c x +1\right ) b}{6 c^{3}}+\frac {e^{2} \ln \left (-c x +1\right ) b}{6 c^{3}}\) | \(214\) |
1/6*(2*x^3*arctanh(c*x)*b*c^3*e^2+2*x^3*a*c^3*e^2+6*x^2*arctanh(c*x)*b*c^3 *d*e+6*x^2*a*c^3*d*e+6*d^2*b*arctanh(c*x)*x*c^3+x^2*b*c^2*e^2+6*x*a*c^3*d^ 2+6*ln(c*x-1)*b*c^2*d^2+6*x*b*c^2*d*e+6*arctanh(c*x)*b*c^2*d^2-6*arctanh(c *x)*b*c*d*e+2*ln(c*x-1)*b*e^2+2*arctanh(c*x)*b*e^2)/c^3
Time = 0.27 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.70 \[ \int (d+e x)^2 (a+b \text {arctanh}(c x)) \, dx=\frac {2 \, a c^{3} e^{2} x^{3} + {\left (6 \, a c^{3} d e + b c^{2} e^{2}\right )} x^{2} + 6 \, {\left (a c^{3} d^{2} + b c^{2} d e\right )} x + {\left (3 \, b c^{2} d^{2} - 3 \, b c d e + b e^{2}\right )} \log \left (c x + 1\right ) + {\left (3 \, b c^{2} d^{2} + 3 \, b c d e + b e^{2}\right )} \log \left (c x - 1\right ) + {\left (b c^{3} e^{2} x^{3} + 3 \, b c^{3} d e x^{2} + 3 \, b c^{3} d^{2} x\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{6 \, c^{3}} \]
1/6*(2*a*c^3*e^2*x^3 + (6*a*c^3*d*e + b*c^2*e^2)*x^2 + 6*(a*c^3*d^2 + b*c^ 2*d*e)*x + (3*b*c^2*d^2 - 3*b*c*d*e + b*e^2)*log(c*x + 1) + (3*b*c^2*d^2 + 3*b*c*d*e + b*e^2)*log(c*x - 1) + (b*c^3*e^2*x^3 + 3*b*c^3*d*e*x^2 + 3*b* c^3*d^2*x)*log(-(c*x + 1)/(c*x - 1)))/c^3
Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (82) = 164\).
Time = 0.32 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.85 \[ \int (d+e x)^2 (a+b \text {arctanh}(c x)) \, dx=\begin {cases} a d^{2} x + a d e x^{2} + \frac {a e^{2} x^{3}}{3} + b d^{2} x \operatorname {atanh}{\left (c x \right )} + b d e x^{2} \operatorname {atanh}{\left (c x \right )} + \frac {b e^{2} x^{3} \operatorname {atanh}{\left (c x \right )}}{3} + \frac {b d^{2} \log {\left (x - \frac {1}{c} \right )}}{c} + \frac {b d^{2} \operatorname {atanh}{\left (c x \right )}}{c} + \frac {b d e x}{c} + \frac {b e^{2} x^{2}}{6 c} - \frac {b d e \operatorname {atanh}{\left (c x \right )}}{c^{2}} + \frac {b e^{2} \log {\left (x - \frac {1}{c} \right )}}{3 c^{3}} + \frac {b e^{2} \operatorname {atanh}{\left (c x \right )}}{3 c^{3}} & \text {for}\: c \neq 0 \\a \left (d^{2} x + d e x^{2} + \frac {e^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
Piecewise((a*d**2*x + a*d*e*x**2 + a*e**2*x**3/3 + b*d**2*x*atanh(c*x) + b *d*e*x**2*atanh(c*x) + b*e**2*x**3*atanh(c*x)/3 + b*d**2*log(x - 1/c)/c + b*d**2*atanh(c*x)/c + b*d*e*x/c + b*e**2*x**2/(6*c) - b*d*e*atanh(c*x)/c** 2 + b*e**2*log(x - 1/c)/(3*c**3) + b*e**2*atanh(c*x)/(3*c**3), Ne(c, 0)), (a*(d**2*x + d*e*x**2 + e**2*x**3/3), True))
Time = 0.19 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.43 \[ \int (d+e x)^2 (a+b \text {arctanh}(c x)) \, dx=\frac {1}{3} \, a e^{2} x^{3} + a d e x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, x}{c^{2}} - \frac {\log \left (c x + 1\right )}{c^{3}} + \frac {\log \left (c x - 1\right )}{c^{3}}\right )}\right )} b d e + \frac {1}{6} \, {\left (2 \, x^{3} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {x^{2}}{c^{2}} + \frac {\log \left (c^{2} x^{2} - 1\right )}{c^{4}}\right )}\right )} b e^{2} + a d^{2} x + \frac {{\left (2 \, c x \operatorname {artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} b d^{2}}{2 \, c} \]
1/3*a*e^2*x^3 + a*d*e*x^2 + 1/2*(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*e + 1/6*(2*x^3*arctanh(c*x) + c*(x^2/c^ 2 + log(c^2*x^2 - 1)/c^4))*b*e^2 + a*d^2*x + 1/2*(2*c*x*arctanh(c*x) + log (-c^2*x^2 + 1))*b*d^2/c
Leaf count of result is larger than twice the leaf count of optimal. 532 vs. \(2 (88) = 176\).
Time = 0.31 (sec) , antiderivative size = 532, normalized size of antiderivative = 5.54 \[ \int (d+e x)^2 (a+b \text {arctanh}(c x)) \, dx=\frac {1}{3} \, c {\left (\frac {{\left (\frac {3 \, {\left (c x + 1\right )}^{2} b c^{2} d^{2}}{{\left (c x - 1\right )}^{2}} - \frac {6 \, {\left (c x + 1\right )} b c^{2} d^{2}}{c x - 1} + 3 \, b c^{2} d^{2} + \frac {6 \, {\left (c x + 1\right )}^{2} b c d e}{{\left (c x - 1\right )}^{2}} - \frac {6 \, {\left (c x + 1\right )} b c d e}{c x - 1} + \frac {3 \, {\left (c x + 1\right )}^{2} b e^{2}}{{\left (c x - 1\right )}^{2}} + b e^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{\frac {{\left (c x + 1\right )}^{3} c^{4}}{{\left (c x - 1\right )}^{3}} - \frac {3 \, {\left (c x + 1\right )}^{2} c^{4}}{{\left (c x - 1\right )}^{2}} + \frac {3 \, {\left (c x + 1\right )} c^{4}}{c x - 1} - c^{4}} + \frac {2 \, {\left (\frac {3 \, {\left (c x + 1\right )}^{2} a c^{2} d^{2}}{{\left (c x - 1\right )}^{2}} - \frac {6 \, {\left (c x + 1\right )} a c^{2} d^{2}}{c x - 1} + 3 \, a c^{2} d^{2} + \frac {6 \, {\left (c x + 1\right )}^{2} a c d e}{{\left (c x - 1\right )}^{2}} - \frac {6 \, {\left (c x + 1\right )} a c d e}{c x - 1} + \frac {3 \, {\left (c x + 1\right )}^{2} b c d e}{{\left (c x - 1\right )}^{2}} - \frac {6 \, {\left (c x + 1\right )} b c d e}{c x - 1} + 3 \, b c d e + \frac {3 \, {\left (c x + 1\right )}^{2} a e^{2}}{{\left (c x - 1\right )}^{2}} + a e^{2} + \frac {{\left (c x + 1\right )}^{2} b e^{2}}{{\left (c x - 1\right )}^{2}} - \frac {{\left (c x + 1\right )} b e^{2}}{c x - 1}\right )}}{\frac {{\left (c x + 1\right )}^{3} c^{4}}{{\left (c x - 1\right )}^{3}} - \frac {3 \, {\left (c x + 1\right )}^{2} c^{4}}{{\left (c x - 1\right )}^{2}} + \frac {3 \, {\left (c x + 1\right )} c^{4}}{c x - 1} - c^{4}} - \frac {{\left (3 \, b c^{2} d^{2} + b e^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1} + 1\right )}{c^{4}} + \frac {{\left (3 \, b c^{2} d^{2} + b e^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{c^{4}}\right )} \]
1/3*c*((3*(c*x + 1)^2*b*c^2*d^2/(c*x - 1)^2 - 6*(c*x + 1)*b*c^2*d^2/(c*x - 1) + 3*b*c^2*d^2 + 6*(c*x + 1)^2*b*c*d*e/(c*x - 1)^2 - 6*(c*x + 1)*b*c*d* e/(c*x - 1) + 3*(c*x + 1)^2*b*e^2/(c*x - 1)^2 + b*e^2)*log(-(c*x + 1)/(c*x - 1))/((c*x + 1)^3*c^4/(c*x - 1)^3 - 3*(c*x + 1)^2*c^4/(c*x - 1)^2 + 3*(c *x + 1)*c^4/(c*x - 1) - c^4) + 2*(3*(c*x + 1)^2*a*c^2*d^2/(c*x - 1)^2 - 6* (c*x + 1)*a*c^2*d^2/(c*x - 1) + 3*a*c^2*d^2 + 6*(c*x + 1)^2*a*c*d*e/(c*x - 1)^2 - 6*(c*x + 1)*a*c*d*e/(c*x - 1) + 3*(c*x + 1)^2*b*c*d*e/(c*x - 1)^2 - 6*(c*x + 1)*b*c*d*e/(c*x - 1) + 3*b*c*d*e + 3*(c*x + 1)^2*a*e^2/(c*x - 1 )^2 + a*e^2 + (c*x + 1)^2*b*e^2/(c*x - 1)^2 - (c*x + 1)*b*e^2/(c*x - 1))/( (c*x + 1)^3*c^4/(c*x - 1)^3 - 3*(c*x + 1)^2*c^4/(c*x - 1)^2 + 3*(c*x + 1)* c^4/(c*x - 1) - c^4) - (3*b*c^2*d^2 + b*e^2)*log(-(c*x + 1)/(c*x - 1) + 1) /c^4 + (3*b*c^2*d^2 + b*e^2)*log(-(c*x + 1)/(c*x - 1))/c^4)
Time = 3.56 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.32 \[ \int (d+e x)^2 (a+b \text {arctanh}(c x)) \, dx=\frac {a\,e^2\,x^3}{3}+a\,d^2\,x+\frac {b\,d^2\,\ln \left (c^2\,x^2-1\right )}{2\,c}+\frac {b\,e^2\,\ln \left (c^2\,x^2-1\right )}{6\,c^3}+\frac {b\,e^2\,x^2}{6\,c}+a\,d\,e\,x^2+b\,d^2\,x\,\mathrm {atanh}\left (c\,x\right )+\frac {b\,e^2\,x^3\,\mathrm {atanh}\left (c\,x\right )}{3}+\frac {b\,d\,e\,x}{c}-\frac {b\,d\,e\,\mathrm {atanh}\left (c\,x\right )}{c^2}+b\,d\,e\,x^2\,\mathrm {atanh}\left (c\,x\right ) \]