Integrand size = 16, antiderivative size = 217 \[ \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3 \, dx=b^2 c^2 x \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )-\frac {1}{2} b c^3 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{2} b c x^2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2-\frac {1}{3} c^3 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3+\frac {1}{3} x^3 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3-b c^3 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2 \log \left (2-\frac {2}{1+\frac {c}{x}}\right )+\frac {1}{2} b^3 c^3 \log \left (1-\frac {c^2}{x^2}\right )+b^3 c^3 \log (x)+b^2 c^3 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right ) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+\frac {c}{x}}\right )+\frac {1}{2} b^3 c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+\frac {c}{x}}\right ) \]
b^2*c^2*x*(a+b*arccoth(x/c))-1/2*b*c^3*(a+b*arccoth(x/c))^2+1/2*b*c*x^2*(a +b*arccoth(x/c))^2-1/3*c^3*(a+b*arccoth(x/c))^3+1/3*x^3*(a+b*arccoth(x/c)) ^3-b*c^3*(a+b*arccoth(x/c))^2*ln(2-2/(1+c/x))+1/2*b^3*c^3*ln(1-c^2/x^2)+b^ 3*c^3*ln(x)+b^2*c^3*(a+b*arccoth(x/c))*polylog(2,-1+2/(1+c/x))+1/2*b^3*c^3 *polylog(3,-1+2/(1+c/x))
Result contains complex when optimal does not.
Time = 0.61 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.48 \[ \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3 \, dx=\frac {1}{6} \left (3 a^2 b c x^2+2 a^3 x^3+6 a^2 b x^3 \text {arctanh}\left (\frac {c}{x}\right )+3 a^2 b c^3 \log \left (-c^2+x^2\right )+6 a b^2 \left (c^2 x+\left (-c^3+x^3\right ) \text {arctanh}\left (\frac {c}{x}\right )^2+c \text {arctanh}\left (\frac {c}{x}\right ) \left (-c^2+x^2-2 c^2 \log \left (1-e^{-2 \text {arctanh}\left (\frac {c}{x}\right )}\right )\right )+c^3 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}\left (\frac {c}{x}\right )}\right )\right )+\frac {1}{4} b^3 \left (-i c^3 \pi ^3+24 c^2 x \text {arctanh}\left (\frac {c}{x}\right )-12 c^3 \text {arctanh}\left (\frac {c}{x}\right )^2+12 c x^2 \text {arctanh}\left (\frac {c}{x}\right )^2+8 c^3 \text {arctanh}\left (\frac {c}{x}\right )^3+8 x^3 \text {arctanh}\left (\frac {c}{x}\right )^3-24 c^3 \text {arctanh}\left (\frac {c}{x}\right )^2 \log \left (1-e^{2 \text {arctanh}\left (\frac {c}{x}\right )}\right )-24 c^3 \log \left (\frac {1}{\sqrt {1-\frac {c^2}{x^2}}}\right )-24 c^3 \log \left (\frac {c}{x}\right )-24 c^3 \text {arctanh}\left (\frac {c}{x}\right ) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}\left (\frac {c}{x}\right )}\right )+12 c^3 \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}\left (\frac {c}{x}\right )}\right )\right )\right ) \]
(3*a^2*b*c*x^2 + 2*a^3*x^3 + 6*a^2*b*x^3*ArcTanh[c/x] + 3*a^2*b*c^3*Log[-c ^2 + x^2] + 6*a*b^2*(c^2*x + (-c^3 + x^3)*ArcTanh[c/x]^2 + c*ArcTanh[c/x]* (-c^2 + x^2 - 2*c^2*Log[1 - E^(-2*ArcTanh[c/x])]) + c^3*PolyLog[2, E^(-2*A rcTanh[c/x])]) + (b^3*((-I)*c^3*Pi^3 + 24*c^2*x*ArcTanh[c/x] - 12*c^3*ArcT anh[c/x]^2 + 12*c*x^2*ArcTanh[c/x]^2 + 8*c^3*ArcTanh[c/x]^3 + 8*x^3*ArcTan h[c/x]^3 - 24*c^3*ArcTanh[c/x]^2*Log[1 - E^(2*ArcTanh[c/x])] - 24*c^3*Log[ 1/Sqrt[1 - c^2/x^2]] - 24*c^3*Log[c/x] - 24*c^3*ArcTanh[c/x]*PolyLog[2, E^ (2*ArcTanh[c/x])] + 12*c^3*PolyLog[3, E^(2*ArcTanh[c/x])]))/4)/6
Time = 1.97 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.938, Rules used = {6454, 6452, 6544, 6452, 6544, 6452, 243, 47, 14, 16, 6510, 6550, 6494, 6618, 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.
| \(\displaystyle \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3 \, dx\) |
| \(\Big \downarrow \) 6454 |
| \(\displaystyle -\int x^4 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3d\frac {1}{x}\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3-b c \int \frac {x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{1-\frac {c^2}{x^2}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 6544 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3-b c \left (c^2 \int \frac {x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{1-\frac {c^2}{x^2}}d\frac {1}{x}+\int x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2d\frac {1}{x}\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3-b c \left (c^2 \int \frac {x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{1-\frac {c^2}{x^2}}d\frac {1}{x}+b c \int \frac {x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}-\frac {1}{2} x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 6544 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3-b c \left (c^2 \int \frac {x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{1-\frac {c^2}{x^2}}d\frac {1}{x}+b c \left (c^2 \int \frac {a+b \text {arctanh}\left (\frac {c}{x}\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}+\int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )d\frac {1}{x}\right )-\frac {1}{2} x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3-b c \left (c^2 \int \frac {x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{1-\frac {c^2}{x^2}}d\frac {1}{x}+b c \left (c^2 \int \frac {a+b \text {arctanh}\left (\frac {c}{x}\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}+b c \int \frac {x}{1-\frac {c^2}{x^2}}d\frac {1}{x}-x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )\right )-\frac {1}{2} x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3-b c \left (c^2 \int \frac {x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{1-\frac {c^2}{x^2}}d\frac {1}{x}+b c \left (c^2 \int \frac {a+b \text {arctanh}\left (\frac {c}{x}\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}+\frac {1}{2} b c \int \frac {x}{1-\frac {c^2}{x^2}}d\frac {1}{x^2}-x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )\right )-\frac {1}{2} x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 47 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3-b c \left (c^2 \int \frac {x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{1-\frac {c^2}{x^2}}d\frac {1}{x}+b c \left (c^2 \int \frac {a+b \text {arctanh}\left (\frac {c}{x}\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}+\frac {1}{2} b c \left (c^2 \int \frac {1}{1-\frac {c^2}{x^2}}d\frac {1}{x^2}+\int xd\frac {1}{x^2}\right )-x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )\right )-\frac {1}{2} x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3-b c \left (c^2 \int \frac {x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{1-\frac {c^2}{x^2}}d\frac {1}{x}+b c \left (c^2 \int \frac {a+b \text {arctanh}\left (\frac {c}{x}\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}+\frac {1}{2} b c \left (c^2 \int \frac {1}{1-\frac {c^2}{x^2}}d\frac {1}{x^2}+\log \left (\frac {1}{x^2}\right )\right )-x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )\right )-\frac {1}{2} x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3-b c \left (c^2 \int \frac {x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{1-\frac {c^2}{x^2}}d\frac {1}{x}+b c \left (c^2 \int \frac {a+b \text {arctanh}\left (\frac {c}{x}\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}-x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )+\frac {1}{2} b c \left (\log \left (\frac {1}{x^2}\right )-\log \left (1-\frac {c^2}{x^2}\right )\right )\right )-\frac {1}{2} x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 6510 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3-b c \left (c^2 \int \frac {x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{1-\frac {c^2}{x^2}}d\frac {1}{x}+b c \left (\frac {c \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{2 b}-x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )+\frac {1}{2} b c \left (\log \left (\frac {1}{x^2}\right )-\log \left (1-\frac {c^2}{x^2}\right )\right )\right )-\frac {1}{2} x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 6550 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3-b c \left (c^2 \left (\int \frac {x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{\frac {c}{x}+1}d\frac {1}{x}+\frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{3 b}\right )+b c \left (\frac {c \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{2 b}-x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )+\frac {1}{2} b c \left (\log \left (\frac {1}{x^2}\right )-\log \left (1-\frac {c^2}{x^2}\right )\right )\right )-\frac {1}{2} x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 6494 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3-b c \left (c^2 \left (-2 b c \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right ) \log \left (2-\frac {2}{\frac {c}{x}+1}\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}+\frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{3 b}+\log \left (2-\frac {2}{\frac {c}{x}+1}\right ) \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2\right )+b c \left (\frac {c \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{2 b}-x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )+\frac {1}{2} b c \left (\log \left (\frac {1}{x^2}\right )-\log \left (1-\frac {c^2}{x^2}\right )\right )\right )-\frac {1}{2} x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 6618 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3-b c \left (c^2 \left (-2 b c \left (\frac {\operatorname {PolyLog}\left (2,\frac {2}{\frac {c}{x}+1}-1\right ) \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )}{2 c}-\frac {1}{2} b \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{\frac {c}{x}+1}-1\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}\right )+\frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{3 b}+\log \left (2-\frac {2}{\frac {c}{x}+1}\right ) \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2\right )+b c \left (\frac {c \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{2 b}-x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )+\frac {1}{2} b c \left (\log \left (\frac {1}{x^2}\right )-\log \left (1-\frac {c^2}{x^2}\right )\right )\right )-\frac {1}{2} x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3-b c \left (c^2 \left (-2 b c \left (\frac {\operatorname {PolyLog}\left (2,\frac {2}{\frac {c}{x}+1}-1\right ) \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )}{2 c}+\frac {b \operatorname {PolyLog}\left (3,\frac {2}{\frac {c}{x}+1}-1\right )}{4 c}\right )+\frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{3 b}+\log \left (2-\frac {2}{\frac {c}{x}+1}\right ) \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2\right )+b c \left (\frac {c \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{2 b}-x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )+\frac {1}{2} b c \left (\log \left (\frac {1}{x^2}\right )-\log \left (1-\frac {c^2}{x^2}\right )\right )\right )-\frac {1}{2} x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2\right )\) |
(x^3*(a + b*ArcTanh[c/x])^3)/3 - b*c*(-1/2*(x^2*(a + b*ArcTanh[c/x])^2) + b*c*(-(x*(a + b*ArcTanh[c/x])) + (c*(a + b*ArcTanh[c/x])^2)/(2*b) + (b*c*( -Log[1 - c^2/x^2] + Log[x^(-2)]))/2) + c^2*((a + b*ArcTanh[c/x])^3/(3*b) + (a + b*ArcTanh[c/x])^2*Log[2 - 2/(1 + c/x)] - 2*b*c*(((a + b*ArcTanh[c/x] )*PolyLog[2, -1 + 2/(1 + c/x)])/(2*c) + (b*PolyLog[3, -1 + 2/(1 + c/x)])/( 4*c))))
3.2.51.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c - a*d) Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[1/(c + d*x), x ], x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*ArcTanh[c*x])^p, x ], x, x^n], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 1] && IntegerQ[Simpl ify[(m + 1)/n]]
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_.)/((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[(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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 303.08 (sec) , antiderivative size = 1773, normalized size of antiderivative = 8.17
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1773\) |
| default | \(\text {Expression too large to display}\) | \(1773\) |
| parts | \(\text {Expression too large to display}\) | \(1775\) |
-c^3*(-1/3*a^3/c^3*x^3+b^3*(-1/3/c^3*x^3*arctanh(c/x)^3-1/2*arctanh(c/x)^2 *ln(1+c/x)-1/2*arctanh(c/x)^2*ln(c/x-1)-1/2/c^2*x^2*arctanh(c/x)^2+ln(c/x) *arctanh(c/x)^2+arctanh(c/x)^2*ln((1+c/x)/(1-c^2/x^2)^(1/2))-arctanh(c/x)^ 2*ln((1+c/x)^2/(1-c^2/x^2)-1)+arctanh(c/x)^2*ln(1-(1+c/x)/(1-c^2/x^2)^(1/2 ))+2*arctanh(c/x)*polylog(2,(1+c/x)/(1-c^2/x^2)^(1/2))-2*polylog(3,(1+c/x) /(1-c^2/x^2)^(1/2))+arctanh(c/x)^2*ln(1+(1+c/x)/(1-c^2/x^2)^(1/2))+2*arcta nh(c/x)*polylog(2,-(1+c/x)/(1-c^2/x^2)^(1/2))-2*polylog(3,-(1+c/x)/(1-c^2/ x^2)^(1/2))-1/12*arctanh(c/x)*(-3*I*arctanh(c/x)*csgn(I*(1+c/x)^2/(c^2/x^2 -1)/(1-(1+c/x)^2/(c^2/x^2-1)))^3*Pi*c/x-6*I*arctanh(c/x)*csgn(I*(1+c/x)^2/ (c^2/x^2-1))^2*csgn(I*(1+c/x)/(1-c^2/x^2)^(1/2))*Pi*c/x-6*I*arctanh(c/x)*P i*c/x+3*I*arctanh(c/x)*csgn(I*(1+c/x)^2/(c^2/x^2-1))*csgn(I*(1+c/x)^2/(c^2 /x^2-1)/(1-(1+c/x)^2/(c^2/x^2-1)))*csgn(I/(1-(1+c/x)^2/(c^2/x^2-1)))*Pi*c/ x+6*I*arctanh(c/x)*csgn(I*(-(1+c/x)^2/(c^2/x^2-1)-1)/(1-(1+c/x)^2/(c^2/x^2 -1)))^2*csgn(I/(1-(1+c/x)^2/(c^2/x^2-1)))*Pi*c/x-6*I*arctanh(c/x)*csgn(I*( -(1+c/x)^2/(c^2/x^2-1)-1)/(1-(1+c/x)^2/(c^2/x^2-1)))^3*Pi*c/x-3*I*arctanh( c/x)*csgn(I*(1+c/x)^2/(c^2/x^2-1))*csgn(I*(1+c/x)/(1-c^2/x^2)^(1/2))^2*Pi* c/x-6*I*arctanh(c/x)*csgn(I*(-(1+c/x)^2/(c^2/x^2-1)-1))*csgn(I*(-(1+c/x)^2 /(c^2/x^2-1)-1)/(1-(1+c/x)^2/(c^2/x^2-1)))*csgn(I/(1-(1+c/x)^2/(c^2/x^2-1) ))*Pi*c/x-3*I*arctanh(c/x)*csgn(I*(1+c/x)^2/(c^2/x^2-1)/(1-(1+c/x)^2/(c^2/ x^2-1)))^2*csgn(I/(1-(1+c/x)^2/(c^2/x^2-1)))*Pi*c/x-3*I*arctanh(c/x)*cs...
\[ \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3 \, dx=\int { {\left (b \operatorname {artanh}\left (\frac {c}{x}\right ) + a\right )}^{3} x^{2} \,d x } \]
integral(b^3*x^2*arctanh(c/x)^3 + 3*a*b^2*x^2*arctanh(c/x)^2 + 3*a^2*b*x^2 *arctanh(c/x) + a^3*x^2, x)
\[ \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3 \, dx=\int x^{2} \left (a + b \operatorname {atanh}{\left (\frac {c}{x} \right )}\right )^{3}\, dx \]
\[ \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3 \, dx=\int { {\left (b \operatorname {artanh}\left (\frac {c}{x}\right ) + a\right )}^{3} x^{2} \,d x } \]
1/3*a^3*x^3 + 1/2*(2*x^3*arctanh(c/x) + (c^2*log(-c^2 + x^2) + x^2)*c)*a^2 *b + 1/24*(b^3*c^3 - b^3*x^3)*log(-c + x)^3 + 1/8*(b^3*c*x^2 + 2*a*b^2*x^3 + (b^3*c^3 + b^3*x^3)*log(c + x))*log(-c + x)^2 - integrate(-1/8*((b^3*c* x^2 - b^3*x^3)*log(c + x)^3 + 6*(a*b^2*c*x^2 - a*b^2*x^3)*log(c + x)^2 + ( 2*b^3*c*x^2 + 4*a*b^2*x^3 - 3*(b^3*c*x^2 - b^3*x^3)*log(c + x)^2 + 2*(b^3* c^3 - 6*a*b^2*c*x^2 + (6*a*b^2 + b^3)*x^3)*log(c + x))*log(-c + x))/(c - x ), x)
\[ \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3 \, dx=\int { {\left (b \operatorname {artanh}\left (\frac {c}{x}\right ) + a\right )}^{3} x^{2} \,d x } \]
Timed out. \[ \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3 \, dx=\int x^2\,{\left (a+b\,\mathrm {atanh}\left (\frac {c}{x}\right )\right )}^3 \,d x \]