Integrand size = 18, antiderivative size = 480 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+e x)^3} \, dx=\frac {b c (a+b \text {arctanh}(c x))}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {(a+b \text {arctanh}(c x))^2}{2 e (d+e x)^2}+\frac {b c^2 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{2 e (c d+e)^2}+\frac {b^2 c^2 \log (1-c x)}{2 (c d-e) (c d+e)^2}-\frac {b c^2 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1+c x}\right )}{2 (c d-e)^2 e}+\frac {2 b c^3 d (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}-\frac {b^2 c^2 \log (1+c x)}{2 (c d-e)^2 (c d+e)}+\frac {b^2 c^2 e \log (d+e x)}{(c d-e)^2 (c d+e)^2}-\frac {2 b c^3 d (a+b \text {arctanh}(c x)) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}+\frac {b^2 c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{4 e (c d+e)^2}+\frac {b^2 c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{4 (c d-e)^2 e}-\frac {b^2 c^3 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}+\frac {b^2 c^3 d \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2} \] Output:
b*c*(a+b*arctanh(c*x))/(c^2*d^2-e^2)/(e*x+d)-1/2*(a+b*arctanh(c*x))^2/e/(e *x+d)^2+1/2*b*c^2*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/e/(c*d+e)^2+1/2*b^2*c^ 2*ln(-c*x+1)/(c*d-e)/(c*d+e)^2-1/2*b*c^2*(a+b*arctanh(c*x))*ln(2/(c*x+1))/ (c*d-e)^2/e+2*b*c^3*d*(a+b*arctanh(c*x))*ln(2/(c*x+1))/(c*d-e)^2/(c*d+e)^2 -1/2*b^2*c^2*ln(c*x+1)/(c*d-e)^2/(c*d+e)+b^2*c^2*e*ln(e*x+d)/(c*d-e)^2/(c* d+e)^2-2*b*c^3*d*(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+1/4*b^2*c^2*polylog(2,1-2/(-c*x+1))/e/(c*d+e)^2+1/4*b^2*c^2* polylog(2,1-2/(c*x+1))/(c*d-e)^2/e-b^2*c^3*d*polylog(2,1-2/(c*x+1))/(c*d-e )^2/(c*d+e)^2+b^2*c^3*d*polylog(2,1-2*c*(e*x+d)/(c*d+e)/(c*x+1))/(c*d-e)^2 /(c*d+e)^2
Result contains complex when optimal does not.
Time = 4.51 (sec) , antiderivative size = 470, normalized size of antiderivative = 0.98 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+e x)^3} \, dx=-\frac {a^2}{2 e (d+e x)^2}-\frac {a b c^2 \left (\frac {2 \text {arctanh}(c x)}{(c d+c e x)^2}+\frac {\log (1-c x)}{(c d+e)^2}+\frac {-\log (1+c x)+\frac {2 e \left (-c^2 d^2+e^2+2 c^2 d (d+e x) \log (c (d+e x))\right )}{c (c d+e)^2 (d+e x)}}{(-c d+e)^2}\right )}{2 e}+\frac {b^2 c^2 \left (-\frac {2 e^{-\text {arctanh}\left (\frac {c d}{e}\right )} \text {arctanh}(c x)^2}{\sqrt {1-\frac {c^2 d^2}{e^2}} e}-\frac {e \left (-1+c^2 x^2\right ) \text {arctanh}(c x)^2}{c^2 (d+e x)^2}+\frac {2 x \text {arctanh}(c x) (-e+c d \text {arctanh}(c x))}{c d (d+e x)}+\frac {2 e \left (-e \text {arctanh}(c x)+c d \log \left (\frac {c (d+e x)}{\sqrt {1-c^2 x^2}}\right )\right )}{c^3 d^3-c d e^2}+\frac {2 c d \left (i \pi \log \left (1+e^{2 \text {arctanh}(c x)}\right )-2 \text {arctanh}(c x) \log \left (1-e^{-2 \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )}\right )-i \pi \left (\text {arctanh}(c x)-\frac {1}{2} \log \left (1-c^2 x^2\right )\right )-2 \text {arctanh}\left (\frac {c d}{e}\right ) \left (\text {arctanh}(c x)+\log \left (1-e^{-2 \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )}\right )-\log \left (i \sinh \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )\right )\right )+\operatorname {PolyLog}\left (2,e^{-2 \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )}\right )\right )}{c^2 d^2-e^2}\right )}{2 (c d-e) (c d+e)} \] Input:
Integrate[(a + b*ArcTanh[c*x])^2/(d + e*x)^3,x]
Output:
-1/2*a^2/(e*(d + e*x)^2) - (a*b*c^2*((2*ArcTanh[c*x])/(c*d + c*e*x)^2 + Lo g[1 - c*x]/(c*d + e)^2 + (-Log[1 + c*x] + (2*e*(-(c^2*d^2) + e^2 + 2*c^2*d *(d + e*x)*Log[c*(d + e*x)]))/(c*(c*d + e)^2*(d + e*x)))/(-(c*d) + e)^2))/ (2*e) + (b^2*c^2*((-2*ArcTanh[c*x]^2)/(Sqrt[1 - (c^2*d^2)/e^2]*e*E^ArcTanh [(c*d)/e]) - (e*(-1 + c^2*x^2)*ArcTanh[c*x]^2)/(c^2*(d + e*x)^2) + (2*x*Ar cTanh[c*x]*(-e + c*d*ArcTanh[c*x]))/(c*d*(d + e*x)) + (2*e*(-(e*ArcTanh[c* x]) + c*d*Log[(c*(d + e*x))/Sqrt[1 - c^2*x^2]]))/(c^3*d^3 - c*d*e^2) + (2* c*d*(I*Pi*Log[1 + E^(2*ArcTanh[c*x])] - 2*ArcTanh[c*x]*Log[1 - E^(-2*(ArcT anh[(c*d)/e] + ArcTanh[c*x]))] - I*Pi*(ArcTanh[c*x] - Log[1 - c^2*x^2]/2) - 2*ArcTanh[(c*d)/e]*(ArcTanh[c*x] + Log[1 - E^(-2*(ArcTanh[(c*d)/e] + Arc Tanh[c*x]))] - Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]]) + PolyLog[2, E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]))/(c^2*d^2 - e^2)))/(2*(c*d - e) *(c*d + e))
Time = 0.87 (sec) , antiderivative size = 450, 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.
| \(\displaystyle \int \frac {(a+b \text {arctanh}(c x))^2}{(d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 6480 |
| \(\displaystyle \frac {b c \int \left (\frac {(a+b \text {arctanh}(c x)) c^2}{2 (c d+e)^2 (1-c x)}+\frac {(a+b \text {arctanh}(c x)) c^2}{2 (c d-e)^2 (c x+1)}-\frac {2 d e^2 (a+b \text {arctanh}(c x)) c^2}{(c d-e)^2 (c d+e)^2 (d+e x)}-\frac {e^2 (a+b \text {arctanh}(c x))}{(c d-e) (c d+e) (d+e x)^2}\right )dx}{e}-\frac {(a+b \text {arctanh}(c x))^2}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b c \left (\frac {e (a+b \text {arctanh}(c x))}{\left (c^2 d^2-e^2\right ) (d+e x)}+\frac {2 c^2 d e \log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{(c d-e)^2 (c d+e)^2}-\frac {2 c^2 d e (a+b \text {arctanh}(c x)) \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{(c d-e)^2 (c d+e)^2}+\frac {c \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{2 (c d+e)^2}-\frac {c \log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{2 (c d-e)^2}-\frac {b c^2 d e \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{(c d-e)^2 (c d+e)^2}+\frac {b c^2 d e \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{(c d-e)^2 (c d+e)^2}+\frac {b c e^2 \log (d+e x)}{(c d-e)^2 (c d+e)^2}+\frac {b c \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{4 (c d+e)^2}+\frac {b c \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{4 (c d-e)^2}+\frac {b c e \log (1-c x)}{2 (c d-e) (c d+e)^2}-\frac {b c e \log (c x+1)}{2 (c d-e)^2 (c d+e)}\right )}{e}-\frac {(a+b \text {arctanh}(c x))^2}{2 e (d+e x)^2}\) |
Input:
Int[(a + b*ArcTanh[c*x])^2/(d + e*x)^3,x]
Output:
-1/2*(a + b*ArcTanh[c*x])^2/(e*(d + e*x)^2) + (b*c*((e*(a + b*ArcTanh[c*x] ))/((c^2*d^2 - e^2)*(d + e*x)) + (c*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)]) /(2*(c*d + e)^2) + (b*c*e*Log[1 - c*x])/(2*(c*d - e)*(c*d + e)^2) - (c*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/(2*(c*d - e)^2) + (2*c^2*d*e*(a + b*Ar cTanh[c*x])*Log[2/(1 + c*x)])/((c*d - e)^2*(c*d + e)^2) - (b*c*e*Log[1 + c *x])/(2*(c*d - e)^2*(c*d + e)) + (b*c*e^2*Log[d + e*x])/((c*d - e)^2*(c*d + e)^2) - (2*c^2*d*e*(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) + (b*c*PolyLog[2, 1 - 2/(1 - c*x)])/ (4*(c*d + e)^2) + (b*c*PolyLog[2, 1 - 2/(1 + c*x)])/(4*(c*d - e)^2) - (b*c ^2*d*e*PolyLog[2, 1 - 2/(1 + c*x)])/((c*d - e)^2*(c*d + e)^2) + (b*c^2*d*e *PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/((c*d - e)^2*(c*d + e)^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 = 0.75 (sec) , antiderivative size = 592, normalized size of antiderivative = 1.23
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{2} c^{3}}{2 \left (c e x +c d \right )^{2} e}+b^{2} c^{3} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{2 \left (c e x +c d \right )^{2} e}+\frac {\frac {\operatorname {arctanh}\left (c x \right ) e}{\left (c d +e \right ) \left (c d -e \right ) \left (c e x +c d \right )}-\frac {2 \,\operatorname {arctanh}\left (c x \right ) e d c \ln \left (c e x +c d \right )}{\left (c d +e \right )^{2} \left (c d -e \right )^{2}}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{2 \left (c d +e \right )^{2}}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{2 \left (c d -e \right )^{2}}-\frac {\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}}{2 \left (c d +e \right )^{2}}+\frac {-\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}}{2 \left (c d -e \right )^{2}}+\frac {e \left (\frac {e \ln \left (c e x +c d \right )}{\left (c d +e \right ) \left (c d -e \right )}+\frac {\ln \left (c x -1\right )}{2 c d +2 e}-\frac {\ln \left (c x +1\right )}{2 c d -2 e}\right )}{\left (c d +e \right ) \left (c d -e \right )}-\frac {2 d c \left (\frac {e \left (\operatorname {dilog}\left (\frac {c e x -e}{-c d -e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x -e}{-c d -e}\right )\right )}{2}-\frac {e \left (\operatorname {dilog}\left (\frac {c e x +e}{-c d +e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x +e}{-c d +e}\right )\right )}{2}\right )}{\left (c d +e \right )^{2} \left (c d -e \right )^{2}}}{e}\right )+2 a b \,c^{3} \left (-\frac {\operatorname {arctanh}\left (c x \right )}{2 \left (c e x +c d \right )^{2} e}+\frac {\frac {e}{\left (c d +e \right ) \left (c d -e \right ) \left (c e x +c d \right )}-\frac {2 e d c \ln \left (c e x +c d \right )}{\left (c d +e \right )^{2} \left (c d -e \right )^{2}}-\frac {\ln \left (c x -1\right )}{2 \left (c d +e \right )^{2}}+\frac {\ln \left (c x +1\right )}{2 \left (c d -e \right )^{2}}}{2 e}\right )}{c}\) | \(592\) |
| default | \(\frac {-\frac {a^{2} c^{3}}{2 \left (c e x +c d \right )^{2} e}+b^{2} c^{3} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{2 \left (c e x +c d \right )^{2} e}+\frac {\frac {\operatorname {arctanh}\left (c x \right ) e}{\left (c d +e \right ) \left (c d -e \right ) \left (c e x +c d \right )}-\frac {2 \,\operatorname {arctanh}\left (c x \right ) e d c \ln \left (c e x +c d \right )}{\left (c d +e \right )^{2} \left (c d -e \right )^{2}}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{2 \left (c d +e \right )^{2}}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{2 \left (c d -e \right )^{2}}-\frac {\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}}{2 \left (c d +e \right )^{2}}+\frac {-\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}}{2 \left (c d -e \right )^{2}}+\frac {e \left (\frac {e \ln \left (c e x +c d \right )}{\left (c d +e \right ) \left (c d -e \right )}+\frac {\ln \left (c x -1\right )}{2 c d +2 e}-\frac {\ln \left (c x +1\right )}{2 c d -2 e}\right )}{\left (c d +e \right ) \left (c d -e \right )}-\frac {2 d c \left (\frac {e \left (\operatorname {dilog}\left (\frac {c e x -e}{-c d -e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x -e}{-c d -e}\right )\right )}{2}-\frac {e \left (\operatorname {dilog}\left (\frac {c e x +e}{-c d +e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x +e}{-c d +e}\right )\right )}{2}\right )}{\left (c d +e \right )^{2} \left (c d -e \right )^{2}}}{e}\right )+2 a b \,c^{3} \left (-\frac {\operatorname {arctanh}\left (c x \right )}{2 \left (c e x +c d \right )^{2} e}+\frac {\frac {e}{\left (c d +e \right ) \left (c d -e \right ) \left (c e x +c d \right )}-\frac {2 e d c \ln \left (c e x +c d \right )}{\left (c d +e \right )^{2} \left (c d -e \right )^{2}}-\frac {\ln \left (c x -1\right )}{2 \left (c d +e \right )^{2}}+\frac {\ln \left (c x +1\right )}{2 \left (c d -e \right )^{2}}}{2 e}\right )}{c}\) | \(592\) |
| parts | \(-\frac {a^{2}}{2 \left (e x +d \right )^{2} e}+\frac {b^{2} \left (-\frac {c^{3} \operatorname {arctanh}\left (c x \right )^{2}}{2 \left (c e x +c d \right )^{2} e}+\frac {c^{3} \left (\frac {\operatorname {arctanh}\left (c x \right ) e}{\left (c d +e \right ) \left (c d -e \right ) \left (c e x +c d \right )}-\frac {2 \,\operatorname {arctanh}\left (c x \right ) e d c \ln \left (c e x +c d \right )}{\left (c d +e \right )^{2} \left (c d -e \right )^{2}}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{2 \left (c d +e \right )^{2}}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{2 \left (c d -e \right )^{2}}-\frac {\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}}{2 \left (c d +e \right )^{2}}+\frac {-\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}}{2 \left (c d -e \right )^{2}}+\frac {e \left (\frac {e \ln \left (c e x +c d \right )}{\left (c d +e \right ) \left (c d -e \right )}+\frac {\ln \left (c x -1\right )}{2 c d +2 e}-\frac {\ln \left (c x +1\right )}{2 c d -2 e}\right )}{\left (c d +e \right ) \left (c d -e \right )}-\frac {2 d c \left (\frac {e \left (\operatorname {dilog}\left (\frac {c e x -e}{-c d -e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x -e}{-c d -e}\right )\right )}{2}-\frac {e \left (\operatorname {dilog}\left (\frac {c e x +e}{-c d +e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x +e}{-c d +e}\right )\right )}{2}\right )}{\left (c d +e \right )^{2} \left (c d -e \right )^{2}}\right )}{e}\right )}{c}+\frac {2 a b \left (-\frac {c^{3} \operatorname {arctanh}\left (c x \right )}{2 \left (c e x +c d \right )^{2} e}+\frac {c^{3} \left (\frac {e}{\left (c d +e \right ) \left (c d -e \right ) \left (c e x +c d \right )}-\frac {2 e d c \ln \left (c e x +c d \right )}{\left (c d +e \right )^{2} \left (c d -e \right )^{2}}-\frac {\ln \left (c x -1\right )}{2 \left (c d +e \right )^{2}}+\frac {\ln \left (c x +1\right )}{2 \left (c d -e \right )^{2}}\right )}{2 e}\right )}{c}\) | \(594\) |
Input:
int((a+b*arctanh(c*x))^2/(e*x+d)^3,x,method=_RETURNVERBOSE)
Output:
1/c*(-1/2*a^2*c^3/(c*e*x+c*d)^2/e+b^2*c^3*(-1/2/(c*e*x+c*d)^2/e*arctanh(c* x)^2+1/e*(arctanh(c*x)*e/(c*d+e)/(c*d-e)/(c*e*x+c*d)-2*arctanh(c*x)*e*d*c/ (c*d+e)^2/(c*d-e)^2*ln(c*e*x+c*d)-1/2*arctanh(c*x)/(c*d+e)^2*ln(c*x-1)+1/2 *arctanh(c*x)/(c*d-e)^2*ln(c*x+1)-1/2/(c*d+e)^2*(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/(c*d-e)^2*(-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) )+e/(c*d+e)/(c*d-e)*(e/(c*d+e)/(c*d-e)*ln(c*e*x+c*d)+1/(2*c*d+2*e)*ln(c*x- 1)-1/(2*c*d-2*e)*ln(c*x+1))-2*d*c/(c*d+e)^2/(c*d-e)^2*(1/2*e*(dilog((c*e*x -e)/(-c*d-e))+ln(c*e*x+c*d)*ln((c*e*x-e)/(-c*d-e)))-1/2*e*(dilog((c*e*x+e) /(-c*d+e))+ln(c*e*x+c*d)*ln((c*e*x+e)/(-c*d+e))))))+2*a*b*c^3*(-1/2/(c*e*x +c*d)^2/e*arctanh(c*x)+1/2/e*(e/(c*d+e)/(c*d-e)/(c*e*x+c*d)-2*e*d*c/(c*d+e )^2/(c*d-e)^2*ln(c*e*x+c*d)-1/2/(c*d+e)^2*ln(c*x-1)+1/2/(c*d-e)^2*ln(c*x+1 ))))
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+e x)^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3}} \,d x } \] Input:
integrate((a+b*arctanh(c*x))^2/(e*x+d)^3,x, algorithm="fricas")
Output:
integral((b^2*arctanh(c*x)^2 + 2*a*b*arctanh(c*x) + a^2)/(e^3*x^3 + 3*d*e^ 2*x^2 + 3*d^2*e*x + d^3), x)
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+e x)^3} \, dx=\int \frac {\left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{2}}{\left (d + e x\right )^{3}}\, dx \] Input:
integrate((a+b*atanh(c*x))**2/(e*x+d)**3,x)
Output:
Integral((a + b*atanh(c*x))**2/(d + e*x)**3, x)
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+e x)^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3}} \,d x } \] Input:
integrate((a+b*arctanh(c*x))^2/(e*x+d)^3,x, algorithm="maxima")
Output:
-1/2*((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*b - 1/8*b^2*(log(-c*x + 1)^2/(e^3*x^2 + 2*d*e ^2*x + d^2*e) + 2*integrate(-((c*e*x - e)*log(c*x + 1)^2 + (c*e*x + c*d - 2*(c*e*x - e)*log(c*x + 1))*log(-c*x + 1))/(c*e^4*x^4 - d^3*e + (3*c*d*e^3 - e^4)*x^3 + 3*(c*d^2*e^2 - d*e^3)*x^2 + (c*d^3*e - 3*d^2*e^2)*x), x)) - 1/2*a^2/(e^3*x^2 + 2*d*e^2*x + d^2*e)
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+e x)^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3}} \,d x } \] Input:
integrate((a+b*arctanh(c*x))^2/(e*x+d)^3,x, algorithm="giac")
Output:
integrate((b*arctanh(c*x) + a)^2/(e*x + d)^3, x)
Timed out. \[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+e x)^3} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{{\left (d+e\,x\right )}^3} \,d x \] Input:
int((a + b*atanh(c*x))^2/(d + e*x)^3,x)
Output:
int((a + b*atanh(c*x))^2/(d + e*x)^3, x)
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+e x)^3} \, dx=\text {too large to display} \] Input:
int((a+b*atanh(c*x))^2/(e*x+d)^3,x)
Output:
(12*atanh(c*x)**2*b**2*c**8*d**8*e*x + 6*atanh(c*x)**2*b**2*c**8*d**7*e**2 *x**2 - 18*atanh(c*x)**2*b**2*c**6*d**7*e**2 - 20*atanh(c*x)**2*b**2*c**6* d**6*e**3*x - 10*atanh(c*x)**2*b**2*c**6*d**5*e**4*x**2 + 30*atanh(c*x)**2 *b**2*c**4*d**5*e**4 + 4*atanh(c*x)**2*b**2*c**4*d**4*e**5*x + 2*atanh(c*x )**2*b**2*c**4*d**3*e**6*x**2 - 6*atanh(c*x)**2*b**2*c**2*d**3*e**6 + 4*at anh(c*x)**2*b**2*c**2*d**2*e**7*x + 2*atanh(c*x)**2*b**2*c**2*d*e**8*x**2 - 6*atanh(c*x)**2*b**2*d*e**8 - 12*atanh(c*x)*a*b*c**8*d**9 - 16*atanh(c*x )*a*b*c**6*d**7*e**2 + 56*atanh(c*x)*a*b*c**4*d**5*e**4 - 16*atanh(c*x)*a* b*c**2*d**3*e**6 - 12*atanh(c*x)*a*b*d*e**8 - 22*atanh(c*x)*b**2*c**7*d**8 *e - 8*atanh(c*x)*b**2*c**7*d**7*e**2*x + 2*atanh(c*x)*b**2*c**7*d**6*e**3 *x**2 + 42*atanh(c*x)*b**2*c**5*d**6*e**3 + 24*atanh(c*x)*b**2*c**5*d**5*e **4*x + 2*atanh(c*x)*b**2*c**5*d**4*e**5*x**2 - 18*atanh(c*x)*b**2*c**3*d* *4*e**5 - 24*atanh(c*x)*b**2*c**3*d**3*e**6*x - 10*atanh(c*x)*b**2*c**3*d* *2*e**7*x**2 - 2*atanh(c*x)*b**2*c*d**2*e**7 + 8*atanh(c*x)*b**2*c*d*e**8* x + 6*atanh(c*x)*b**2*c*e**9*x**2 + 24*int((atanh(c*x)*x)/(c**4*d**5*x**2 + 3*c**4*d**4*e*x**3 + 3*c**4*d**3*e**2*x**4 + c**4*d**2*e**3*x**5 - c**2* d**5 - 3*c**2*d**4*e*x + 8*c**2*d**2*e**3*x**3 + 9*c**2*d*e**4*x**4 + 3*c* *2*e**5*x**5 - 3*d**3*e**2 - 9*d**2*e**3*x - 9*d*e**4*x**2 - 3*e**5*x**3), x)*b**2*c**11*d**13*e + 48*int((atanh(c*x)*x)/(c**4*d**5*x**2 + 3*c**4*d** 4*e*x**3 + 3*c**4*d**3*e**2*x**4 + c**4*d**2*e**3*x**5 - c**2*d**5 - 3*...