Integrand size = 18, antiderivative size = 305 \[ \int \frac {\text {arctanh}(a x)^3}{x^3 (c+a c x)} \, dx=\frac {3 a^2 \text {arctanh}(a x)^2}{2 c}-\frac {3 a \text {arctanh}(a x)^2}{2 c x}-\frac {a^2 \text {arctanh}(a x)^3}{2 c}-\frac {\text {arctanh}(a x)^3}{2 c x^2}+\frac {a \text {arctanh}(a x)^3}{c x}+\frac {3 a^2 \text {arctanh}(a x) \log \left (2-\frac {2}{1+a x}\right )}{c}-\frac {3 a^2 \text {arctanh}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )}{c}+\frac {a^2 \text {arctanh}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}-\frac {3 a^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )}{2 c}+\frac {3 a^2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )}{c}-\frac {3 a^2 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )}{2 c}+\frac {3 a^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+a x}\right )}{2 c}-\frac {3 a^2 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+a x}\right )}{2 c}-\frac {3 a^2 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+a x}\right )}{4 c} \] Output:
3/2*a^2*arctanh(a*x)^2/c-3/2*a*arctanh(a*x)^2/c/x-1/2*a^2*arctanh(a*x)^3/c -1/2*arctanh(a*x)^3/c/x^2+a*arctanh(a*x)^3/c/x+3*a^2*arctanh(a*x)*ln(2-2/( a*x+1))/c-3*a^2*arctanh(a*x)^2*ln(2-2/(a*x+1))/c+a^2*arctanh(a*x)^3*ln(2-2 /(a*x+1))/c-3/2*a^2*polylog(2,-1+2/(a*x+1))/c+3*a^2*arctanh(a*x)*polylog(2 ,-1+2/(a*x+1))/c-3/2*a^2*arctanh(a*x)^2*polylog(2,-1+2/(a*x+1))/c+3/2*a^2* polylog(3,-1+2/(a*x+1))/c-3/2*a^2*arctanh(a*x)*polylog(3,-1+2/(a*x+1))/c-3 /4*a^2*polylog(4,-1+2/(a*x+1))/c
Result contains complex when optimal does not.
Time = 0.50 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.73 \[ \int \frac {\text {arctanh}(a x)^3}{x^3 (c+a c x)} \, dx=\frac {a^2 \left (-8 i \pi ^3+\pi ^4+96 \text {arctanh}(a x)^2-\frac {96 \text {arctanh}(a x)^2}{a x}+96 \text {arctanh}(a x)^3-\frac {32 \text {arctanh}(a x)^3}{a^2 x^2}+\frac {64 \text {arctanh}(a x)^3}{a x}-32 \text {arctanh}(a x)^4+192 \text {arctanh}(a x) \log \left (1-e^{-2 \text {arctanh}(a x)}\right )-192 \text {arctanh}(a x)^2 \log \left (1-e^{2 \text {arctanh}(a x)}\right )+64 \text {arctanh}(a x)^3 \log \left (1-e^{2 \text {arctanh}(a x)}\right )-96 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(a x)}\right )+96 (-2+\text {arctanh}(a x)) \text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(a x)}\right )+96 \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(a x)}\right )-96 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(a x)}\right )+48 \operatorname {PolyLog}\left (4,e^{2 \text {arctanh}(a x)}\right )\right )}{64 c} \] Input:
Integrate[ArcTanh[a*x]^3/(x^3*(c + a*c*x)),x]
Output:
(a^2*((-8*I)*Pi^3 + Pi^4 + 96*ArcTanh[a*x]^2 - (96*ArcTanh[a*x]^2)/(a*x) + 96*ArcTanh[a*x]^3 - (32*ArcTanh[a*x]^3)/(a^2*x^2) + (64*ArcTanh[a*x]^3)/( a*x) - 32*ArcTanh[a*x]^4 + 192*ArcTanh[a*x]*Log[1 - E^(-2*ArcTanh[a*x])] - 192*ArcTanh[a*x]^2*Log[1 - E^(2*ArcTanh[a*x])] + 64*ArcTanh[a*x]^3*Log[1 - E^(2*ArcTanh[a*x])] - 96*PolyLog[2, E^(-2*ArcTanh[a*x])] + 96*(-2 + ArcT anh[a*x])*ArcTanh[a*x]*PolyLog[2, E^(2*ArcTanh[a*x])] + 96*PolyLog[3, E^(2 *ArcTanh[a*x])] - 96*ArcTanh[a*x]*PolyLog[3, E^(2*ArcTanh[a*x])] + 48*Poly Log[4, E^(2*ArcTanh[a*x])]))/(64*c)
Time = 3.54 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.95, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6496, 27, 6452, 6496, 6452, 6494, 6544, 6452, 6510, 6550, 6494, 2897, 6618, 6622, 7164}
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.
| \(\displaystyle \int \frac {\text {arctanh}(a x)^3}{x^3 (a c x+c)} \, dx\) |
| \(\Big \downarrow \) 6496 |
| \(\displaystyle \frac {\int \frac {\text {arctanh}(a x)^3}{x^3}dx}{c}-a \int \frac {\text {arctanh}(a x)^3}{c x^2 (a x+1)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\text {arctanh}(a x)^3}{x^3}dx}{c}-\frac {a \int \frac {\text {arctanh}(a x)^3}{x^2 (a x+1)}dx}{c}\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {\frac {3}{2} a \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)^3}{2 x^2}}{c}-\frac {a \int \frac {\text {arctanh}(a x)^3}{x^2 (a x+1)}dx}{c}\) |
| \(\Big \downarrow \) 6496 |
| \(\displaystyle \frac {\frac {3}{2} a \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)^3}{2 x^2}}{c}-\frac {a \left (\int \frac {\text {arctanh}(a x)^3}{x^2}dx-a \int \frac {\text {arctanh}(a x)^3}{x (a x+1)}dx\right )}{c}\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {\frac {3}{2} a \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)^3}{2 x^2}}{c}-\frac {a \left (3 a \int \frac {\text {arctanh}(a x)^2}{x \left (1-a^2 x^2\right )}dx-a \int \frac {\text {arctanh}(a x)^3}{x (a x+1)}dx-\frac {\text {arctanh}(a x)^3}{x}\right )}{c}\) |
| \(\Big \downarrow \) 6494 |
| \(\displaystyle \frac {\frac {3}{2} a \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)^3}{2 x^2}}{c}-\frac {a \left (3 a \int \frac {\text {arctanh}(a x)^2}{x \left (1-a^2 x^2\right )}dx-a \left (\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx\right )-\frac {\text {arctanh}(a x)^3}{x}\right )}{c}\) |
| \(\Big \downarrow \) 6544 |
| \(\displaystyle \frac {\frac {3}{2} a \left (a^2 \int \frac {\text {arctanh}(a x)^2}{1-a^2 x^2}dx+\int \frac {\text {arctanh}(a x)^2}{x^2}dx\right )-\frac {\text {arctanh}(a x)^3}{2 x^2}}{c}-\frac {a \left (3 a \int \frac {\text {arctanh}(a x)^2}{x \left (1-a^2 x^2\right )}dx-a \left (\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx\right )-\frac {\text {arctanh}(a x)^3}{x}\right )}{c}\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {\frac {3}{2} a \left (a^2 \int \frac {\text {arctanh}(a x)^2}{1-a^2 x^2}dx+2 a \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)^2}{x}\right )-\frac {\text {arctanh}(a x)^3}{2 x^2}}{c}-\frac {a \left (3 a \int \frac {\text {arctanh}(a x)^2}{x \left (1-a^2 x^2\right )}dx-a \left (\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx\right )-\frac {\text {arctanh}(a x)^3}{x}\right )}{c}\) |
| \(\Big \downarrow \) 6510 |
| \(\displaystyle \frac {\frac {3}{2} a \left (2 a \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )}dx+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\right )-\frac {\text {arctanh}(a x)^3}{2 x^2}}{c}-\frac {a \left (3 a \int \frac {\text {arctanh}(a x)^2}{x \left (1-a^2 x^2\right )}dx-a \left (\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx\right )-\frac {\text {arctanh}(a x)^3}{x}\right )}{c}\) |
| \(\Big \downarrow \) 6550 |
| \(\displaystyle \frac {\frac {3}{2} a \left (2 a \left (\int \frac {\text {arctanh}(a x)}{x (a x+1)}dx+\frac {1}{2} \text {arctanh}(a x)^2\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\right )-\frac {\text {arctanh}(a x)^3}{2 x^2}}{c}-\frac {a \left (-a \left (\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx\right )+3 a \left (\int \frac {\text {arctanh}(a x)^2}{x (a x+1)}dx+\frac {1}{3} \text {arctanh}(a x)^3\right )-\frac {\text {arctanh}(a x)^3}{x}\right )}{c}\) |
| \(\Big \downarrow \) 6494 |
| \(\displaystyle \frac {\frac {3}{2} a \left (2 a \left (-a \int \frac {\log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\right )-\frac {\text {arctanh}(a x)^3}{2 x^2}}{c}-\frac {a \left (3 a \left (-2 a \int \frac {\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{3} \text {arctanh}(a x)^3+\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )\right )-a \left (\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx\right )-\frac {\text {arctanh}(a x)^3}{x}\right )}{c}\) |
| \(\Big \downarrow \) 2897 |
| \(\displaystyle \frac {\frac {3}{2} a \left (2 a \left (\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\right )-\frac {\text {arctanh}(a x)^3}{2 x^2}}{c}-\frac {a \left (3 a \left (-2 a \int \frac {\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{3} \text {arctanh}(a x)^3+\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )\right )-a \left (\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )-3 a \int \frac {\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx\right )-\frac {\text {arctanh}(a x)^3}{x}\right )}{c}\) |
| \(\Big \downarrow \) 6618 |
| \(\displaystyle \frac {\frac {3}{2} a \left (2 a \left (\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\right )-\frac {\text {arctanh}(a x)^3}{2 x^2}}{c}-\frac {a \left (3 a \left (-2 a \left (\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}-\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{1-a^2 x^2}dx\right )+\frac {1}{3} \text {arctanh}(a x)^3+\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )\right )-a \left (\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )-3 a \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}-\int \frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{1-a^2 x^2}dx\right )\right )-\frac {\text {arctanh}(a x)^3}{x}\right )}{c}\) |
| \(\Big \downarrow \) 6622 |
| \(\displaystyle \frac {\frac {3}{2} a \left (2 a \left (\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\right )-\frac {\text {arctanh}(a x)^3}{2 x^2}}{c}-\frac {a \left (3 a \left (-2 a \left (\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}-\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{1-a^2 x^2}dx\right )+\frac {1}{3} \text {arctanh}(a x)^3+\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )\right )-a \left (\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )-3 a \left (-\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{1-a^2 x^2}dx+\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}+\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{2 a}\right )\right )-\frac {\text {arctanh}(a x)^3}{x}\right )}{c}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {\frac {3}{2} a \left (2 a \left (\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{a x+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )\right )+\frac {1}{3} a \text {arctanh}(a x)^3-\frac {\text {arctanh}(a x)^2}{x}\right )-\frac {\text {arctanh}(a x)^3}{2 x^2}}{c}-\frac {a \left (3 a \left (-2 a \left (\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}+\frac {\operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{4 a}\right )+\frac {1}{3} \text {arctanh}(a x)^3+\text {arctanh}(a x)^2 \log \left (2-\frac {2}{a x+1}\right )\right )-a \left (\text {arctanh}(a x)^3 \log \left (2-\frac {2}{a x+1}\right )-3 a \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{2 a}+\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{2 a}+\frac {\operatorname {PolyLog}\left (4,\frac {2}{a x+1}-1\right )}{4 a}\right )\right )-\frac {\text {arctanh}(a x)^3}{x}\right )}{c}\) |
Input:
Int[ArcTanh[a*x]^3/(x^3*(c + a*c*x)),x]
Output:
(-1/2*ArcTanh[a*x]^3/x^2 + (3*a*(-(ArcTanh[a*x]^2/x) + (a*ArcTanh[a*x]^3)/ 3 + 2*a*(ArcTanh[a*x]^2/2 + ArcTanh[a*x]*Log[2 - 2/(1 + a*x)] - PolyLog[2, -1 + 2/(1 + a*x)]/2)))/2)/c - (a*(-(ArcTanh[a*x]^3/x) + 3*a*(ArcTanh[a*x] ^3/3 + ArcTanh[a*x]^2*Log[2 - 2/(1 + a*x)] - 2*a*((ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 + a*x)])/(2*a) + PolyLog[3, -1 + 2/(1 + a*x)]/(4*a))) - a*(ArcT anh[a*x]^3*Log[2 - 2/(1 + a*x)] - 3*a*((ArcTanh[a*x]^2*PolyLog[2, -1 + 2/( 1 + a*x)])/(2*a) + (ArcTanh[a*x]*PolyLog[3, -1 + 2/(1 + a*x)])/(2*a) + Pol yLog[4, -1 + 2/(1 + a*x)]/(4*a)))))/c
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, x][[2]], Expon[Pq, x]]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : > Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x ], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 ] && IntegerQ[m])) && NeQ[m, -1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x _Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Simp[b*c*(p/d) Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2 - 2/(1 + e*(x/d))] /(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c ^2*d^2 - e^2, 0]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( e_.)*(x_)), x_Symbol] :> Simp[1/d Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x], x] - Simp[e/(d*f) Int[(f*x)^(m + 1)*((a + b*ArcTanh[c*x])^p/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0] && LtQ[m, -1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b , c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( e_.)*(x_)^2), x_Symbol] :> Simp[1/d Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x ], x] - Simp[e/(d*f^2) Int[(f*x)^(m + 2)*((a + b*ArcTanh[c*x])^p/(d + e*x ^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*d*(p + 1)), x] + Simp[1/ d Int[(a + b*ArcTanh[c*x])^p/(x*(1 + c*x)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^ 2), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x ] - Simp[b*(p/2) Int[(a + b*ArcTanh[c*x])^(p - 1)*(PolyLog[2, 1 - u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 + c*x))^2, 0]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_ .)*(x_)^2), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(PolyLog[k + 1, u]/ (2*c*d)), x] + Simp[b*(p/2) Int[(a + b*ArcTanh[c*x])^(p - 1)*(PolyLog[k + 1, u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] & & EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 + c*x))^2, 0]
Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /; !FalseQ[w]] /; FreeQ[n, x]
Leaf count of result is larger than twice the leaf count of optimal. \(601\) vs. \(2(287)=574\).
Time = 8.99 (sec) , antiderivative size = 602, normalized size of antiderivative = 1.97
| method | result | size |
| derivativedivides | \(a^{2} \left (-\frac {\operatorname {arctanh}\left (a x \right )^{2} \left (a x \,\operatorname {arctanh}\left (a x \right )-3 a x -\operatorname {arctanh}\left (a x \right )\right ) \left (a x -1\right )}{2 c \,a^{2} x^{2}}-\frac {\operatorname {arctanh}\left (a x \right )^{4}}{2 c}+\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {6 \operatorname {polylog}\left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {6 \operatorname {polylog}\left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {3 \operatorname {arctanh}\left (a x \right )^{2}}{c}+\frac {3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {3 \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {3 \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {2 \operatorname {arctanh}\left (a x \right )^{3}}{c}-\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {6 \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {6 \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}\right )\) | \(602\) |
| default | \(a^{2} \left (-\frac {\operatorname {arctanh}\left (a x \right )^{2} \left (a x \,\operatorname {arctanh}\left (a x \right )-3 a x -\operatorname {arctanh}\left (a x \right )\right ) \left (a x -1\right )}{2 c \,a^{2} x^{2}}-\frac {\operatorname {arctanh}\left (a x \right )^{4}}{2 c}+\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {6 \operatorname {polylog}\left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {\operatorname {arctanh}\left (a x \right )^{3} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {6 \operatorname {polylog}\left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {3 \operatorname {arctanh}\left (a x \right )^{2}}{c}+\frac {3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {3 \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {3 \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {2 \operatorname {arctanh}\left (a x \right )^{3}}{c}-\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {6 \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {6 \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}\right )\) | \(602\) |
Input:
int(arctanh(a*x)^3/x^3/(a*c*x+c),x,method=_RETURNVERBOSE)
Output:
a^2*(-1/2/c*arctanh(a*x)^2*(a*x*arctanh(a*x)-3*a*x-arctanh(a*x))*(a*x-1)/a ^2/x^2-1/2/c*arctanh(a*x)^4+1/c*arctanh(a*x)^3*ln(1-(a*x+1)/(-a^2*x^2+1)^( 1/2))+3/c*arctanh(a*x)^2*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-6/c*arctanh (a*x)*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))+6/c*polylog(4,(a*x+1)/(-a^2*x^ 2+1)^(1/2))+1/c*arctanh(a*x)^3*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+3/c*arctan h(a*x)^2*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))-6/c*arctanh(a*x)*polylog(3 ,-(a*x+1)/(-a^2*x^2+1)^(1/2))+6/c*polylog(4,-(a*x+1)/(-a^2*x^2+1)^(1/2))-3 /c*arctanh(a*x)^2+3/c*arctanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+3/c*po lylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+3/c*arctanh(a*x)*ln(1+(a*x+1)/(-a^2*x^ 2+1)^(1/2))+3/c*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+2/c*arctanh(a*x)^3- 3/c*arctanh(a*x)^2*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))-6/c*arctanh(a*x)*polyl og(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+6/c*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2)) -3/c*arctanh(a*x)^2*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-6/c*arctanh(a*x)*poly log(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+6/c*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/ 2)))
\[ \int \frac {\text {arctanh}(a x)^3}{x^3 (c+a c x)} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (a c x + c\right )} x^{3}} \,d x } \] Input:
integrate(arctanh(a*x)^3/x^3/(a*c*x+c),x, algorithm="fricas")
Output:
integral(arctanh(a*x)^3/(a*c*x^4 + c*x^3), x)
\[ \int \frac {\text {arctanh}(a x)^3}{x^3 (c+a c x)} \, dx=\frac {\int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{a x^{4} + x^{3}}\, dx}{c} \] Input:
integrate(atanh(a*x)**3/x**3/(a*c*x+c),x)
Output:
Integral(atanh(a*x)**3/(a*x**4 + x**3), x)/c
\[ \int \frac {\text {arctanh}(a x)^3}{x^3 (c+a c x)} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (a c x + c\right )} x^{3}} \,d x } \] Input:
integrate(arctanh(a*x)^3/x^3/(a*c*x+c),x, algorithm="maxima")
Output:
1/16*(2*a^2*x^2*log(a*x + 1) - 2*a*x + 1)*log(-a*x + 1)^3/(c*x^2) - 1/8*in tegrate(-1/2*(2*(a*x - 1)*log(a*x + 1)^3 - 6*(a*x - 1)*log(a*x + 1)^2*log( -a*x + 1) + 3*(2*a^3*x^3 + a^2*x^2 - a*x - 2*(a^4*x^4 + a^3*x^3 - a*x + 1) *log(a*x + 1))*log(-a*x + 1)^2)/(a^2*c*x^5 - c*x^3), x)
\[ \int \frac {\text {arctanh}(a x)^3}{x^3 (c+a c x)} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (a c x + c\right )} x^{3}} \,d x } \] Input:
integrate(arctanh(a*x)^3/x^3/(a*c*x+c),x, algorithm="giac")
Output:
integrate(arctanh(a*x)^3/((a*c*x + c)*x^3), x)
Timed out. \[ \int \frac {\text {arctanh}(a x)^3}{x^3 (c+a c x)} \, dx=\int \frac {{\mathrm {atanh}\left (a\,x\right )}^3}{x^3\,\left (c+a\,c\,x\right )} \,d x \] Input:
int(atanh(a*x)^3/(x^3*(c + a*c*x)),x)
Output:
int(atanh(a*x)^3/(x^3*(c + a*c*x)), x)
\[ \int \frac {\text {arctanh}(a x)^3}{x^3 (c+a c x)} \, dx=\frac {\int \frac {\mathit {atanh} \left (a x \right )^{3}}{a \,x^{4}+x^{3}}d x}{c} \] Input:
int(atanh(a*x)^3/x^3/(a*c*x+c),x)
Output:
int(atanh(a*x)**3/(a*x**4 + x**3),x)/c