Integrand size = 18, antiderivative size = 224 \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{x} \, dx=4 \text {arctanh}\left (1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-3 b \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c \sqrt {x}}\right )+3 b \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c \sqrt {x}}\right )+3 b^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{1-c \sqrt {x}}\right )-3 b^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c \sqrt {x}}\right )-\frac {3}{2} b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1-c \sqrt {x}}\right )+\frac {3}{2} b^3 \operatorname {PolyLog}\left (4,-1+\frac {2}{1-c \sqrt {x}}\right ) \]
-4*(a+b*arctanh(c*x^(1/2)))^3*arctanh(-1+2/(1-c*x^(1/2)))-3*b*(a+b*arctanh (c*x^(1/2)))^2*polylog(2,1-2/(1-c*x^(1/2)))+3*b*(a+b*arctanh(c*x^(1/2)))^2 *polylog(2,-1+2/(1-c*x^(1/2)))+3*b^2*(a+b*arctanh(c*x^(1/2)))*polylog(3,1- 2/(1-c*x^(1/2)))-3*b^2*(a+b*arctanh(c*x^(1/2)))*polylog(3,-1+2/(1-c*x^(1/2 )))-3/2*b^3*polylog(4,1-2/(1-c*x^(1/2)))+3/2*b^3*polylog(4,-1+2/(1-c*x^(1/ 2)))
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.89 \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{x} \, dx=a^3 \log (x)+3 a^2 b \left (-\operatorname {PolyLog}\left (2,-c \sqrt {x}\right )+\operatorname {PolyLog}\left (2,c \sqrt {x}\right )\right )+6 a b^2 \left (\frac {i \pi ^3}{24}-\frac {2}{3} \text {arctanh}\left (c \sqrt {x}\right )^3-\text {arctanh}\left (c \sqrt {x}\right )^2 \log \left (1+e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+\text {arctanh}\left (c \sqrt {x}\right )^2 \log \left (1-e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+\text {arctanh}\left (c \sqrt {x}\right ) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+\text {arctanh}\left (c \sqrt {x}\right ) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right )+\frac {1}{32} b^3 \left (\pi ^4-32 \text {arctanh}\left (c \sqrt {x}\right )^4-64 \text {arctanh}\left (c \sqrt {x}\right )^3 \log \left (1+e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+64 \text {arctanh}\left (c \sqrt {x}\right )^3 \log \left (1-e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+96 \text {arctanh}\left (c \sqrt {x}\right )^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+96 \text {arctanh}\left (c \sqrt {x}\right )^2 \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+96 \text {arctanh}\left (c \sqrt {x}\right ) \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )-96 \text {arctanh}\left (c \sqrt {x}\right ) \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+48 \operatorname {PolyLog}\left (4,-e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )+48 \operatorname {PolyLog}\left (4,e^{2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right ) \]
a^3*Log[x] + 3*a^2*b*(-PolyLog[2, -(c*Sqrt[x])] + PolyLog[2, c*Sqrt[x]]) + 6*a*b^2*((I/24)*Pi^3 - (2*ArcTanh[c*Sqrt[x]]^3)/3 - ArcTanh[c*Sqrt[x]]^2* Log[1 + E^(-2*ArcTanh[c*Sqrt[x]])] + ArcTanh[c*Sqrt[x]]^2*Log[1 - E^(2*Arc Tanh[c*Sqrt[x]])] + ArcTanh[c*Sqrt[x]]*PolyLog[2, -E^(-2*ArcTanh[c*Sqrt[x] ])] + ArcTanh[c*Sqrt[x]]*PolyLog[2, E^(2*ArcTanh[c*Sqrt[x]])] + PolyLog[3, -E^(-2*ArcTanh[c*Sqrt[x]])]/2 - PolyLog[3, E^(2*ArcTanh[c*Sqrt[x]])]/2) + (b^3*(Pi^4 - 32*ArcTanh[c*Sqrt[x]]^4 - 64*ArcTanh[c*Sqrt[x]]^3*Log[1 + E^ (-2*ArcTanh[c*Sqrt[x]])] + 64*ArcTanh[c*Sqrt[x]]^3*Log[1 - E^(2*ArcTanh[c* Sqrt[x]])] + 96*ArcTanh[c*Sqrt[x]]^2*PolyLog[2, -E^(-2*ArcTanh[c*Sqrt[x]]) ] + 96*ArcTanh[c*Sqrt[x]]^2*PolyLog[2, E^(2*ArcTanh[c*Sqrt[x]])] + 96*ArcT anh[c*Sqrt[x]]*PolyLog[3, -E^(-2*ArcTanh[c*Sqrt[x]])] - 96*ArcTanh[c*Sqrt[ x]]*PolyLog[3, E^(2*ArcTanh[c*Sqrt[x]])] + 48*PolyLog[4, -E^(-2*ArcTanh[c* Sqrt[x]])] + 48*PolyLog[4, E^(2*ArcTanh[c*Sqrt[x]])]))/32
Time = 1.20 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6450, 6448, 6614, 6620, 6624, 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.
| \(\displaystyle \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{x} \, dx\) |
| \(\Big \downarrow \) 6450 |
| \(\displaystyle 2 \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{\sqrt {x}}d\sqrt {x}\) |
| \(\Big \downarrow \) 6448 |
| \(\displaystyle 2 \left (2 \text {arctanh}\left (1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-6 b c \int \frac {\text {arctanh}\left (1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}\right )\) |
| \(\Big \downarrow \) 6614 |
| \(\displaystyle 2 \left (2 \text {arctanh}\left (1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-6 b c \left (\frac {1}{2} \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2 \log \left (2-\frac {2}{1-c \sqrt {x}}\right )}{1-c^2 x}d\sqrt {x}-\frac {1}{2} \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2 \log \left (\frac {2}{1-c \sqrt {x}}\right )}{1-c^2 x}d\sqrt {x}\right )\right )\) |
| \(\Big \downarrow \) 6620 |
| \(\displaystyle 2 \left (2 \text {arctanh}\left (1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-6 b c \left (\frac {1}{2} \left (\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 c}-b \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c \sqrt {x}}\right )}{1-c^2 x}d\sqrt {x}\right )+\frac {1}{2} \left (b \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \operatorname {PolyLog}\left (2,\frac {2}{1-c \sqrt {x}}-1\right )}{1-c^2 x}d\sqrt {x}-\frac {\operatorname {PolyLog}\left (2,\frac {2}{1-c \sqrt {x}}-1\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 c}\right )\right )\right )\) |
| \(\Big \downarrow \) 6624 |
| \(\displaystyle 2 \left (2 \text {arctanh}\left (1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-6 b c \left (\frac {1}{2} \left (\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 c}-b \left (\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{2 c}-\frac {1}{2} b \int \frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-c \sqrt {x}}\right )}{1-c^2 x}d\sqrt {x}\right )\right )+\frac {1}{2} \left (b \left (\frac {\operatorname {PolyLog}\left (3,\frac {2}{1-c \sqrt {x}}-1\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{2 c}-\frac {1}{2} b \int \frac {\operatorname {PolyLog}\left (3,\frac {2}{1-c \sqrt {x}}-1\right )}{1-c^2 x}d\sqrt {x}\right )-\frac {\operatorname {PolyLog}\left (2,\frac {2}{1-c \sqrt {x}}-1\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 c}\right )\right )\right )\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle 2 \left (2 \text {arctanh}\left (1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-6 b c \left (\frac {1}{2} \left (\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 c}-b \left (\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{2 c}-\frac {b \operatorname {PolyLog}\left (4,1-\frac {2}{1-c \sqrt {x}}\right )}{4 c}\right )\right )+\frac {1}{2} \left (b \left (\frac {\operatorname {PolyLog}\left (3,\frac {2}{1-c \sqrt {x}}-1\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{2 c}-\frac {b \operatorname {PolyLog}\left (4,\frac {2}{1-c \sqrt {x}}-1\right )}{4 c}\right )-\frac {\operatorname {PolyLog}\left (2,\frac {2}{1-c \sqrt {x}}-1\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 c}\right )\right )\right )\) |
2*(2*ArcTanh[1 - 2/(1 - c*Sqrt[x])]*(a + b*ArcTanh[c*Sqrt[x]])^3 - 6*b*c*( (((a + b*ArcTanh[c*Sqrt[x]])^2*PolyLog[2, 1 - 2/(1 - c*Sqrt[x])])/(2*c) - b*(((a + b*ArcTanh[c*Sqrt[x]])*PolyLog[3, 1 - 2/(1 - c*Sqrt[x])])/(2*c) - (b*PolyLog[4, 1 - 2/(1 - c*Sqrt[x])])/(4*c)))/2 + (-1/2*((a + b*ArcTanh[c* Sqrt[x]])^2*PolyLog[2, -1 + 2/(1 - c*Sqrt[x])])/c + b*(((a + b*ArcTanh[c*S qrt[x]])*PolyLog[3, -1 + 2/(1 - c*Sqrt[x])])/(2*c) - (b*PolyLog[4, -1 + 2/ (1 - c*Sqrt[x])])/(4*c)))/2))
3.3.6.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)/(x_), x_Symbol] :> Simp[2*(a + b*ArcTanh[c*x])^p*ArcTanh[1 - 2/(1 - c*x)], x] - Simp[2*b*c*p Int[(a + b *ArcTanh[c*x])^(p - 1)*(ArcTanh[1 - 2/(1 - c*x)]/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_.)/(x_), x_Symbol] :> Simp[ 1/n Subst[Int[(a + b*ArcTanh[c*x])^p/x, x], x, x^n], x] /; FreeQ[{a, b, c , n}, x] && IGtQ[p, 0]
Int[(ArcTanh[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*( x_)^2), x_Symbol] :> Simp[1/2 Int[Log[1 + u]*((a + b*ArcTanh[c*x])^p/(d + e*x^2)), x], x] - Simp[1/2 Int[Log[1 - u]*((a + b*ArcTanh[c*x])^p/(d + e *x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 - c*x))^2, 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] && E qQ[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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 13.89 (sec) , antiderivative size = 1363, normalized size of antiderivative = 6.08
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1363\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1364\) |
| default | \(\text {Expression too large to display}\) | \(1364\) |
a^3*ln(x)+b^3*(2*ln(c*x^(1/2))*arctanh(c*x^(1/2))^3-2*arctanh(c*x^(1/2))^3 *ln((1+c*x^(1/2))^2/(-c^2*x+1)-1)+2*arctanh(c*x^(1/2))^3*ln(1+(1+c*x^(1/2) )/(-c^2*x+1)^(1/2))+6*arctanh(c*x^(1/2))^2*polylog(2,-(1+c*x^(1/2))/(-c^2* x+1)^(1/2))-12*arctanh(c*x^(1/2))*polylog(3,-(1+c*x^(1/2))/(-c^2*x+1)^(1/2 ))+12*polylog(4,-(1+c*x^(1/2))/(-c^2*x+1)^(1/2))+2*arctanh(c*x^(1/2))^3*ln (1-(1+c*x^(1/2))/(-c^2*x+1)^(1/2))+6*arctanh(c*x^(1/2))^2*polylog(2,(1+c*x ^(1/2))/(-c^2*x+1)^(1/2))-12*arctanh(c*x^(1/2))*polylog(3,(1+c*x^(1/2))/(- c^2*x+1)^(1/2))+12*polylog(4,(1+c*x^(1/2))/(-c^2*x+1)^(1/2))+I*Pi*csgn(I*( -(1+c*x^(1/2))^2/(c^2*x-1)-1)/(1-(1+c*x^(1/2))^2/(c^2*x-1)))*(csgn(I*(-(1+ c*x^(1/2))^2/(c^2*x-1)-1))*csgn(I/(1-(1+c*x^(1/2))^2/(c^2*x-1)))-csgn(I*(- (1+c*x^(1/2))^2/(c^2*x-1)-1))*csgn(I*(-(1+c*x^(1/2))^2/(c^2*x-1)-1)/(1-(1+ c*x^(1/2))^2/(c^2*x-1)))-csgn(I/(1-(1+c*x^(1/2))^2/(c^2*x-1)))*csgn(I*(-(1 +c*x^(1/2))^2/(c^2*x-1)-1)/(1-(1+c*x^(1/2))^2/(c^2*x-1)))+csgn(I*(-(1+c*x^ (1/2))^2/(c^2*x-1)-1)/(1-(1+c*x^(1/2))^2/(c^2*x-1)))^2)*arctanh(c*x^(1/2)) ^3-3*arctanh(c*x^(1/2))^2*polylog(2,-(1+c*x^(1/2))^2/(-c^2*x+1))+3*arctanh (c*x^(1/2))*polylog(3,-(1+c*x^(1/2))^2/(-c^2*x+1))-3/2*polylog(4,-(1+c*x^( 1/2))^2/(-c^2*x+1)))+3*a*b^2*(2*ln(c*x^(1/2))*arctanh(c*x^(1/2))^2-2*arcta nh(c*x^(1/2))*polylog(2,-(1+c*x^(1/2))^2/(-c^2*x+1))+polylog(3,-(1+c*x^(1/ 2))^2/(-c^2*x+1))-2*arctanh(c*x^(1/2))^2*ln((1+c*x^(1/2))^2/(-c^2*x+1)-1)+ 2*arctanh(c*x^(1/2))^2*ln(1+(1+c*x^(1/2))/(-c^2*x+1)^(1/2))+4*arctanh(c...
\[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )}^{3}}{x} \,d x } \]
integral((b^3*arctanh(c*sqrt(x))^3 + 3*a*b^2*arctanh(c*sqrt(x))^2 + 3*a^2* b*arctanh(c*sqrt(x)) + a^3)/x, x)
\[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{x} \, dx=\int \frac {\left (a + b \operatorname {atanh}{\left (c \sqrt {x} \right )}\right )^{3}}{x}\, dx \]
\[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )}^{3}}{x} \,d x } \]
1/8*b^3*integrate(log(c*sqrt(x) + 1)^3/x, x) - 3/8*b^3*integrate(log(c*sqr t(x) + 1)^2*log(-c*sqrt(x) + 1)/x, x) + 3/8*b^3*integrate(log(c*sqrt(x) + 1)*log(-c*sqrt(x) + 1)^2/x, x) - 1/8*b^3*integrate(log(-c*sqrt(x) + 1)^3/x , x) + 3/4*a*b^2*integrate(log(c*sqrt(x) + 1)^2/x, x) - 3/2*a*b^2*integrat e(log(c*sqrt(x) + 1)*log(-c*sqrt(x) + 1)/x, x) + 3/4*a*b^2*integrate(log(- c*sqrt(x) + 1)^2/x, x) + 3/2*a^2*b*integrate(log(c*sqrt(x) + 1)/x, x) - 3/ 2*a^2*b*integrate(log(-c*sqrt(x) + 1)/x, x) + a^3*log(x)
\[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )}^{3}}{x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )\right )}^3}{x} \,d x \]