Integrand size = 16, antiderivative size = 160 \[ \int (d+e x) (a+b \text {arctanh}(c x))^2 \, dx=\frac {a b e x}{c}+\frac {b^2 e x \text {arctanh}(c x)}{c}+\frac {d (a+b \text {arctanh}(c x))^2}{c}-\frac {\left (d^2+\frac {e^2}{c^2}\right ) (a+b \text {arctanh}(c x))^2}{2 e}+\frac {(d+e x)^2 (a+b \text {arctanh}(c x))^2}{2 e}-\frac {2 b d (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{c}+\frac {b^2 e \log \left (1-c^2 x^2\right )}{2 c^2}-\frac {b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c} \] Output:
a*b*e*x/c+b^2*e*x*arctanh(c*x)/c+d*(a+b*arctanh(c*x))^2/c-1/2*(d^2+e^2/c^2 )*(a+b*arctanh(c*x))^2/e+1/2*(e*x+d)^2*(a+b*arctanh(c*x))^2/e-2*b*d*(a+b*a rctanh(c*x))*ln(2/(-c*x+1))/c+1/2*b^2*e*ln(-c^2*x^2+1)/c^2-b^2*d*polylog(2 ,1-2/(-c*x+1))/c
Time = 0.36 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.09 \[ \int (d+e x) (a+b \text {arctanh}(c x))^2 \, dx=\frac {2 a^2 c^2 d x+2 a b c e x+a^2 c^2 e x^2+b^2 (-1+c x) (2 c d+e+c e x) \text {arctanh}(c x)^2+2 b c \text {arctanh}(c x) \left (b e x+a c x (2 d+e x)-2 b d \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+a b e \log (1-c x)-a b e \log (1+c x)+2 a b c d \log \left (1-c^2 x^2\right )+b^2 e \log \left (1-c^2 x^2\right )+2 b^2 c d \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )}{2 c^2} \] Input:
Integrate[(d + e*x)*(a + b*ArcTanh[c*x])^2,x]
Output:
(2*a^2*c^2*d*x + 2*a*b*c*e*x + a^2*c^2*e*x^2 + b^2*(-1 + c*x)*(2*c*d + e + c*e*x)*ArcTanh[c*x]^2 + 2*b*c*ArcTanh[c*x]*(b*e*x + a*c*x*(2*d + e*x) - 2 *b*d*Log[1 + E^(-2*ArcTanh[c*x])]) + a*b*e*Log[1 - c*x] - a*b*e*Log[1 + c* x] + 2*a*b*c*d*Log[1 - c^2*x^2] + b^2*e*Log[1 - c^2*x^2] + 2*b^2*c*d*PolyL og[2, -E^(-2*ArcTanh[c*x])])/(2*c^2)
Time = 0.85 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.11, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, 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.
| \(\displaystyle \int (d+e x) (a+b \text {arctanh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6480 |
| \(\displaystyle \frac {(d+e x)^2 (a+b \text {arctanh}(c x))^2}{2 e}-\frac {b c \int \left (\frac {\left (d^2 c^2+2 d e x c^2+e^2\right ) (a+b \text {arctanh}(c x))}{c^2 \left (1-c^2 x^2\right )}-\frac {e^2 (a+b \text {arctanh}(c x))}{c^2}\right )dx}{e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^2 (a+b \text {arctanh}(c x))^2}{2 e}-\frac {b c \left (-\frac {d e (a+b \text {arctanh}(c x))^2}{b c^2}+\frac {2 d e \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c^2}+\frac {\left (c^2 d^2+e^2\right ) (a+b \text {arctanh}(c x))^2}{2 b c^3}-\frac {a e^2 x}{c^2}-\frac {b e^2 x \text {arctanh}(c x)}{c^2}+\frac {b d e \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c^2}-\frac {b e^2 \log \left (1-c^2 x^2\right )}{2 c^3}\right )}{e}\) |
Input:
Int[(d + e*x)*(a + b*ArcTanh[c*x])^2,x]
Output:
((d + e*x)^2*(a + b*ArcTanh[c*x])^2)/(2*e) - (b*c*(-((a*e^2*x)/c^2) - (b*e ^2*x*ArcTanh[c*x])/c^2 - (d*e*(a + b*ArcTanh[c*x])^2)/(b*c^2) + ((c^2*d^2 + e^2)*(a + b*ArcTanh[c*x])^2)/(2*b*c^3) + (2*d*e*(a + b*ArcTanh[c*x])*Log [2/(1 - c*x)])/c^2 - (b*e^2*Log[1 - c^2*x^2])/(2*c^3) + (b*d*e*PolyLog[2, 1 - 2/(1 - c*x)])/c^2))/e
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]
Time = 0.15 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.86
| method | result | size |
| parts | \(a^{2} \left (\frac {1}{2} e \,x^{2}+d x \right )+\frac {b^{2} \left (\frac {c \operatorname {arctanh}\left (c x \right )^{2} x^{2} e}{2}+\operatorname {arctanh}\left (c x \right )^{2} d c x -\frac {-\operatorname {arctanh}\left (c x \right ) e c x -\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) c d -\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) e}{2}-\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) c d +\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) e}{2}-\frac {\left (2 c d -e \right ) \left (-\frac {\ln \left (c x +1\right )^{2}}{4}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}\right )}{2}-\frac {\left (2 c d +e \right ) \left (\frac {\ln \left (c x -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{2}\right )}{2}-\frac {\ln \left (c x -1\right ) e}{2}-\frac {\ln \left (c x +1\right ) e}{2}}{c}\right )}{c}+\frac {2 a b \left (\frac {c \,\operatorname {arctanh}\left (c x \right ) x^{2} e}{2}+\operatorname {arctanh}\left (c x \right ) d c x -\frac {-c e x -\frac {\left (2 c d +e \right ) \ln \left (c x -1\right )}{2}+\frac {\left (-2 c d +e \right ) \ln \left (c x +1\right )}{2}}{2 c}\right )}{c}\) | \(298\) |
| derivativedivides | \(\frac {\frac {a^{2} \left (c^{2} d x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b^{2} \left (\operatorname {arctanh}\left (c x \right )^{2} d \,c^{2} x +\frac {\operatorname {arctanh}\left (c x \right )^{2} e \,c^{2} x^{2}}{2}+\operatorname {arctanh}\left (c x \right ) e c x +\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) c d +\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) e}{2}+\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) c d -\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) e}{2}+\frac {\left (2 c d -e \right ) \left (-\frac {\ln \left (c x +1\right )^{2}}{4}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}\right )}{2}+\frac {\left (2 c d +e \right ) \left (\frac {\ln \left (c x -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{2}\right )}{2}+\frac {\ln \left (c x -1\right ) e}{2}+\frac {\ln \left (c x +1\right ) e}{2}\right )}{c}+\frac {2 a b \left (\operatorname {arctanh}\left (c x \right ) d \,c^{2} x +\frac {\operatorname {arctanh}\left (c x \right ) e \,c^{2} x^{2}}{2}+\frac {c e x}{2}+\frac {\left (2 c d +e \right ) \ln \left (c x -1\right )}{4}-\frac {\left (-2 c d +e \right ) \ln \left (c x +1\right )}{4}\right )}{c}}{c}\) | \(304\) |
| default | \(\frac {\frac {a^{2} \left (c^{2} d x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b^{2} \left (\operatorname {arctanh}\left (c x \right )^{2} d \,c^{2} x +\frac {\operatorname {arctanh}\left (c x \right )^{2} e \,c^{2} x^{2}}{2}+\operatorname {arctanh}\left (c x \right ) e c x +\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) c d +\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right ) e}{2}+\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) c d -\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right ) e}{2}+\frac {\left (2 c d -e \right ) \left (-\frac {\ln \left (c x +1\right )^{2}}{4}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}\right )}{2}+\frac {\left (2 c d +e \right ) \left (\frac {\ln \left (c x -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{2}\right )}{2}+\frac {\ln \left (c x -1\right ) e}{2}+\frac {\ln \left (c x +1\right ) e}{2}\right )}{c}+\frac {2 a b \left (\operatorname {arctanh}\left (c x \right ) d \,c^{2} x +\frac {\operatorname {arctanh}\left (c x \right ) e \,c^{2} x^{2}}{2}+\frac {c e x}{2}+\frac {\left (2 c d +e \right ) \ln \left (c x -1\right )}{4}-\frac {\left (-2 c d +e \right ) \ln \left (c x +1\right )}{4}\right )}{c}}{c}\) | \(304\) |
| risch | \(\frac {a b e x}{c}+a^{2} d x +\frac {b^{2} \ln \left (-c x -1\right ) e}{2 c^{2}}+\frac {b^{2} \operatorname {dilog}\left (-\frac {c x}{2}+\frac {1}{2}\right ) d}{c}+\frac {e \,a^{2} x^{2}}{2}-\frac {b^{2} e \ln \left (-c x +1\right )^{2}}{8 c^{2}}+\frac {b^{2} e \ln \left (-c x +1\right )}{2 c^{2}}-\frac {\ln \left (-c x +1\right )^{2} b^{2} d}{4 c}+\frac {b^{2} e \ln \left (-c x +1\right )^{2} x^{2}}{8}+\frac {\ln \left (-c x +1\right )^{2} b^{2} d x}{4}+\frac {b^{2} \left (c^{2} e \,x^{2}+2 c^{2} d x +2 c d -e \right ) \ln \left (c x +1\right )^{2}}{8 c^{2}}+\frac {b \ln \left (-c x -1\right ) a d}{c}-\frac {b \ln \left (-c x -1\right ) a e}{2 c^{2}}-\frac {b a e \ln \left (-c x +1\right ) x^{2}}{2}-\ln \left (-c x +1\right ) a b d x +\frac {\ln \left (-c x +1\right ) a b d}{c}+\frac {b a e \ln \left (-c x +1\right )}{2 c^{2}}-\frac {b^{2} e \ln \left (-c x +1\right ) x}{2 c}+\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) d}{c}-\frac {b^{2} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right ) d}{c}-\frac {b a e}{c^{2}}+\left (-\frac {b^{2} x \left (e x +2 d \right ) \ln \left (-c x +1\right )}{4}+\frac {b \left (2 a \,c^{2} e \,x^{2}+4 a \,c^{2} d x +2 \ln \left (-c x +1\right ) b c d +2 b c e x +\ln \left (-c x +1\right ) b e \right )}{4 c^{2}}\right ) \ln \left (c x +1\right )-\frac {d \,a^{2}}{c}-\frac {e \,a^{2}}{2 c^{2}}\) | \(436\) |
Input:
int((e*x+d)*(a+b*arctanh(c*x))^2,x,method=_RETURNVERBOSE)
Output:
a^2*(1/2*e*x^2+d*x)+b^2/c*(1/2*c*arctanh(c*x)^2*x^2*e+arctanh(c*x)^2*d*c*x -1/c*(-arctanh(c*x)*e*c*x-arctanh(c*x)*ln(c*x-1)*c*d-1/2*arctanh(c*x)*ln(c *x-1)*e-arctanh(c*x)*ln(c*x+1)*c*d+1/2*arctanh(c*x)*ln(c*x+1)*e-1/2*(2*c*d -e)*(-1/4*ln(c*x+1)^2+1/2*(ln(c*x+1)-ln(1/2*c*x+1/2))*ln(-1/2*c*x+1/2)-1/2 *dilog(1/2*c*x+1/2))-1/2*(2*c*d+e)*(1/4*ln(c*x-1)^2-1/2*dilog(1/2*c*x+1/2) -1/2*ln(c*x-1)*ln(1/2*c*x+1/2))-1/2*ln(c*x-1)*e-1/2*ln(c*x+1)*e))+2*a*b/c* (1/2*c*arctanh(c*x)*x^2*e+arctanh(c*x)*d*c*x-1/2/c*(-c*e*x-1/2*(2*c*d+e)*l n(c*x-1)+1/2*(-2*c*d+e)*ln(c*x+1)))
\[ \int (d+e x) (a+b \text {arctanh}(c x))^2 \, dx=\int { {\left (e x + d\right )} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} \,d x } \] Input:
integrate((e*x+d)*(a+b*arctanh(c*x))^2,x, algorithm="fricas")
Output:
integral(a^2*e*x + a^2*d + (b^2*e*x + b^2*d)*arctanh(c*x)^2 + 2*(a*b*e*x + a*b*d)*arctanh(c*x), x)
\[ \int (d+e x) (a+b \text {arctanh}(c x))^2 \, dx=\int \left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{2} \left (d + e x\right )\, dx \] Input:
integrate((e*x+d)*(a+b*atanh(c*x))**2,x)
Output:
Integral((a + b*atanh(c*x))**2*(d + e*x), x)
Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (151) = 302\).
Time = 0.21 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.96 \[ \int (d+e x) (a+b \text {arctanh}(c x))^2 \, dx=\frac {1}{2} \, a^{2} e x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, x}{c^{2}} - \frac {\log \left (c x + 1\right )}{c^{3}} + \frac {\log \left (c x - 1\right )}{c^{3}}\right )}\right )} a b e + a^{2} d x + \frac {{\left (2 \, c x \operatorname {artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} a b d}{c} + \frac {{\left (\log \left (c x + 1\right ) \log \left (-\frac {1}{2} \, c x + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c x + \frac {1}{2}\right )\right )} b^{2} d}{c} + \frac {b^{2} e \log \left (c x + 1\right )}{2 \, c^{2}} + \frac {b^{2} e \log \left (c x - 1\right )}{2 \, c^{2}} + \frac {4 \, b^{2} c e x \log \left (c x + 1\right ) + {\left (b^{2} c^{2} e x^{2} + 2 \, b^{2} c^{2} d x + {\left (2 \, c d - e\right )} b^{2}\right )} \log \left (c x + 1\right )^{2} + {\left (b^{2} c^{2} e x^{2} + 2 \, b^{2} c^{2} d x - {\left (2 \, c d + e\right )} b^{2}\right )} \log \left (-c x + 1\right )^{2} - 2 \, {\left (2 \, b^{2} c e x + {\left (b^{2} c^{2} e x^{2} + 2 \, b^{2} c^{2} d x + {\left (2 \, c d - e\right )} b^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{8 \, c^{2}} \] Input:
integrate((e*x+d)*(a+b*arctanh(c*x))^2,x, algorithm="maxima")
Output:
1/2*a^2*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))*a*b*e + a^2*d*x + (2*c*x*arctanh(c*x) + log(-c^2*x^2 + 1))*a*b*d/c + (log(c*x + 1)*log(-1/2*c*x + 1/2) + dilog(1/2*c*x + 1/2))*b^ 2*d/c + 1/2*b^2*e*log(c*x + 1)/c^2 + 1/2*b^2*e*log(c*x - 1)/c^2 + 1/8*(4*b ^2*c*e*x*log(c*x + 1) + (b^2*c^2*e*x^2 + 2*b^2*c^2*d*x + (2*c*d - e)*b^2)* log(c*x + 1)^2 + (b^2*c^2*e*x^2 + 2*b^2*c^2*d*x - (2*c*d + e)*b^2)*log(-c* x + 1)^2 - 2*(2*b^2*c*e*x + (b^2*c^2*e*x^2 + 2*b^2*c^2*d*x + (2*c*d - e)*b ^2)*log(c*x + 1))*log(-c*x + 1))/c^2
\[ \int (d+e x) (a+b \text {arctanh}(c x))^2 \, dx=\int { {\left (e x + d\right )} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} \,d x } \] Input:
integrate((e*x+d)*(a+b*arctanh(c*x))^2,x, algorithm="giac")
Output:
integrate((e*x + d)*(b*arctanh(c*x) + a)^2, x)
Timed out. \[ \int (d+e x) (a+b \text {arctanh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,\left (d+e\,x\right ) \,d x \] Input:
int((a + b*atanh(c*x))^2*(d + e*x),x)
Output:
int((a + b*atanh(c*x))^2*(d + e*x), x)
\[ \int (d+e x) (a+b \text {arctanh}(c x))^2 \, dx=\frac {2 \mathit {atanh} \left (c x \right )^{2} b^{2} c^{2} d x +\mathit {atanh} \left (c x \right )^{2} b^{2} c^{2} e \,x^{2}-\mathit {atanh} \left (c x \right )^{2} b^{2} e +4 \mathit {atanh} \left (c x \right ) a b \,c^{2} d x +2 \mathit {atanh} \left (c x \right ) a b \,c^{2} e \,x^{2}+4 \mathit {atanh} \left (c x \right ) a b c d -2 \mathit {atanh} \left (c x \right ) a b e +2 \mathit {atanh} \left (c x \right ) b^{2} c e x +2 \mathit {atanh} \left (c x \right ) b^{2} e +4 \left (\int \frac {\mathit {atanh} \left (c x \right ) x}{c^{2} x^{2}-1}d x \right ) b^{2} c^{3} d +4 \,\mathrm {log}\left (c^{2} x -c \right ) a b c d +2 \,\mathrm {log}\left (c^{2} x -c \right ) b^{2} e +2 a^{2} c^{2} d x +a^{2} c^{2} e \,x^{2}+2 a b c e x}{2 c^{2}} \] Input:
int((e*x+d)*(a+b*atanh(c*x))^2,x)
Output:
(2*atanh(c*x)**2*b**2*c**2*d*x + atanh(c*x)**2*b**2*c**2*e*x**2 - atanh(c* x)**2*b**2*e + 4*atanh(c*x)*a*b*c**2*d*x + 2*atanh(c*x)*a*b*c**2*e*x**2 + 4*atanh(c*x)*a*b*c*d - 2*atanh(c*x)*a*b*e + 2*atanh(c*x)*b**2*c*e*x + 2*at anh(c*x)*b**2*e + 4*int((atanh(c*x)*x)/(c**2*x**2 - 1),x)*b**2*c**3*d + 4* log(c**2*x - c)*a*b*c*d + 2*log(c**2*x - c)*b**2*e + 2*a**2*c**2*d*x + a** 2*c**2*e*x**2 + 2*a*b*c*e*x)/(2*c**2)