Integrand size = 16, antiderivative size = 210 \[ \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{x} \, dx=\frac {2}{3} \left (a+b \text {arctanh}\left (c x^3\right )\right )^3 \text {arctanh}\left (1-\frac {2}{1-c x^3}\right )-\frac {1}{2} b \left (a+b \text {arctanh}\left (c x^3\right )\right )^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^3}\right )+\frac {1}{2} b \left (a+b \text {arctanh}\left (c x^3\right )\right )^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c x^3}\right )+\frac {1}{2} b^2 \left (a+b \text {arctanh}\left (c x^3\right )\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x^3}\right )-\frac {1}{2} b^2 \left (a+b \text {arctanh}\left (c x^3\right )\right ) \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c x^3}\right )-\frac {1}{4} b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1-c x^3}\right )+\frac {1}{4} b^3 \operatorname {PolyLog}\left (4,-1+\frac {2}{1-c x^3}\right ) \]
-2/3*(a+b*arctanh(c*x^3))^3*arctanh(-1+2/(-c*x^3+1))-1/2*b*(a+b*arctanh(c* x^3))^2*polylog(2,1-2/(-c*x^3+1))+1/2*b*(a+b*arctanh(c*x^3))^2*polylog(2,- 1+2/(-c*x^3+1))+1/2*b^2*(a+b*arctanh(c*x^3))*polylog(3,1-2/(-c*x^3+1))-1/2 *b^2*(a+b*arctanh(c*x^3))*polylog(3,-1+2/(-c*x^3+1))-1/4*b^3*polylog(4,1-2 /(-c*x^3+1))+1/4*b^3*polylog(4,-1+2/(-c*x^3+1))
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.75 \[ \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{x} \, dx=a^3 \log (x)+\frac {1}{2} a^2 b \left (-\operatorname {PolyLog}\left (2,-c x^3\right )+\operatorname {PolyLog}\left (2,c x^3\right )\right )+a b^2 \left (\frac {i \pi ^3}{24}-\frac {2}{3} \text {arctanh}\left (c x^3\right )^3-\text {arctanh}\left (c x^3\right )^2 \log \left (1+e^{-2 \text {arctanh}\left (c x^3\right )}\right )+\text {arctanh}\left (c x^3\right )^2 \log \left (1-e^{2 \text {arctanh}\left (c x^3\right )}\right )+\text {arctanh}\left (c x^3\right ) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (c x^3\right )}\right )+\text {arctanh}\left (c x^3\right ) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}\left (c x^3\right )}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}\left (c x^3\right )}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}\left (c x^3\right )}\right )\right )+\frac {1}{192} b^3 \left (\pi ^4-32 \text {arctanh}\left (c x^3\right )^4-64 \text {arctanh}\left (c x^3\right )^3 \log \left (1+e^{-2 \text {arctanh}\left (c x^3\right )}\right )+64 \text {arctanh}\left (c x^3\right )^3 \log \left (1-e^{2 \text {arctanh}\left (c x^3\right )}\right )+96 \text {arctanh}\left (c x^3\right )^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (c x^3\right )}\right )+96 \text {arctanh}\left (c x^3\right )^2 \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}\left (c x^3\right )}\right )+96 \text {arctanh}\left (c x^3\right ) \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}\left (c x^3\right )}\right )-96 \text {arctanh}\left (c x^3\right ) \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}\left (c x^3\right )}\right )+48 \operatorname {PolyLog}\left (4,-e^{-2 \text {arctanh}\left (c x^3\right )}\right )+48 \operatorname {PolyLog}\left (4,e^{2 \text {arctanh}\left (c x^3\right )}\right )\right ) \]
a^3*Log[x] + (a^2*b*(-PolyLog[2, -(c*x^3)] + PolyLog[2, c*x^3]))/2 + a*b^2 *((I/24)*Pi^3 - (2*ArcTanh[c*x^3]^3)/3 - ArcTanh[c*x^3]^2*Log[1 + E^(-2*Ar cTanh[c*x^3])] + ArcTanh[c*x^3]^2*Log[1 - E^(2*ArcTanh[c*x^3])] + ArcTanh[ c*x^3]*PolyLog[2, -E^(-2*ArcTanh[c*x^3])] + ArcTanh[c*x^3]*PolyLog[2, E^(2 *ArcTanh[c*x^3])] + PolyLog[3, -E^(-2*ArcTanh[c*x^3])]/2 - PolyLog[3, E^(2 *ArcTanh[c*x^3])]/2) + (b^3*(Pi^4 - 32*ArcTanh[c*x^3]^4 - 64*ArcTanh[c*x^3 ]^3*Log[1 + E^(-2*ArcTanh[c*x^3])] + 64*ArcTanh[c*x^3]^3*Log[1 - E^(2*ArcT anh[c*x^3])] + 96*ArcTanh[c*x^3]^2*PolyLog[2, -E^(-2*ArcTanh[c*x^3])] + 96 *ArcTanh[c*x^3]^2*PolyLog[2, E^(2*ArcTanh[c*x^3])] + 96*ArcTanh[c*x^3]*Pol yLog[3, -E^(-2*ArcTanh[c*x^3])] - 96*ArcTanh[c*x^3]*PolyLog[3, E^(2*ArcTan h[c*x^3])] + 48*PolyLog[4, -E^(-2*ArcTanh[c*x^3])] + 48*PolyLog[4, E^(2*Ar cTanh[c*x^3])]))/192
Time = 1.19 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.14, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, 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.
| \(\displaystyle \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{x} \, dx\) |
| \(\Big \downarrow \) 6450 |
| \(\displaystyle \frac {1}{3} \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{x^3}dx^3\) |
| \(\Big \downarrow \) 6448 |
| \(\displaystyle \frac {1}{3} \left (2 \text {arctanh}\left (1-\frac {2}{1-c x^3}\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-6 b c \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2 \text {arctanh}\left (1-\frac {2}{1-c x^3}\right )}{1-c^2 x^6}dx^3\right )\) |
| \(\Big \downarrow \) 6614 |
| \(\displaystyle \frac {1}{3} \left (2 \text {arctanh}\left (1-\frac {2}{1-c x^3}\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-6 b c \left (\frac {1}{2} \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2 \log \left (2-\frac {2}{1-c x^3}\right )}{1-c^2 x^6}dx^3-\frac {1}{2} \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2 \log \left (\frac {2}{1-c x^3}\right )}{1-c^2 x^6}dx^3\right )\right )\) |
| \(\Big \downarrow \) 6620 |
| \(\displaystyle \frac {1}{3} \left (2 \text {arctanh}\left (1-\frac {2}{1-c x^3}\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-6 b c \left (\frac {1}{2} \left (\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^3}\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 c}-b \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^3}\right )}{1-c^2 x^6}dx^3\right )+\frac {1}{2} \left (b \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right ) \operatorname {PolyLog}\left (2,\frac {2}{1-c x^3}-1\right )}{1-c^2 x^6}dx^3-\frac {\operatorname {PolyLog}\left (2,\frac {2}{1-c x^3}-1\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 c}\right )\right )\right )\) |
| \(\Big \downarrow \) 6624 |
| \(\displaystyle \frac {1}{3} \left (2 \text {arctanh}\left (1-\frac {2}{1-c x^3}\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-6 b c \left (\frac {1}{2} \left (\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^3}\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 c}-b \left (\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-c x^3}\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )}{2 c}-\frac {1}{2} b \int \frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-c x^3}\right )}{1-c^2 x^6}dx^3\right )\right )+\frac {1}{2} \left (b \left (\frac {\operatorname {PolyLog}\left (3,\frac {2}{1-c x^3}-1\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )}{2 c}-\frac {1}{2} b \int \frac {\operatorname {PolyLog}\left (3,\frac {2}{1-c x^3}-1\right )}{1-c^2 x^6}dx^3\right )-\frac {\operatorname {PolyLog}\left (2,\frac {2}{1-c x^3}-1\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 c}\right )\right )\right )\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {1}{3} \left (2 \text {arctanh}\left (1-\frac {2}{1-c x^3}\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )^3-6 b c \left (\frac {1}{2} \left (\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^3}\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 c}-b \left (\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-c x^3}\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )}{2 c}-\frac {b \operatorname {PolyLog}\left (4,1-\frac {2}{1-c x^3}\right )}{4 c}\right )\right )+\frac {1}{2} \left (b \left (\frac {\operatorname {PolyLog}\left (3,\frac {2}{1-c x^3}-1\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )}{2 c}-\frac {b \operatorname {PolyLog}\left (4,\frac {2}{1-c x^3}-1\right )}{4 c}\right )-\frac {\operatorname {PolyLog}\left (2,\frac {2}{1-c x^3}-1\right ) \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 c}\right )\right )\right )\) |
(2*(a + b*ArcTanh[c*x^3])^3*ArcTanh[1 - 2/(1 - c*x^3)] - 6*b*c*((((a + b*A rcTanh[c*x^3])^2*PolyLog[2, 1 - 2/(1 - c*x^3)])/(2*c) - b*(((a + b*ArcTanh [c*x^3])*PolyLog[3, 1 - 2/(1 - c*x^3)])/(2*c) - (b*PolyLog[4, 1 - 2/(1 - c *x^3)])/(4*c)))/2 + (-1/2*((a + b*ArcTanh[c*x^3])^2*PolyLog[2, -1 + 2/(1 - c*x^3)])/c + b*(((a + b*ArcTanh[c*x^3])*PolyLog[3, -1 + 2/(1 - c*x^3)])/( 2*c) - (b*PolyLog[4, -1 + 2/(1 - c*x^3)])/(4*c)))/2))/3
3.2.27.3.1 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]
\[\int \frac {{\left (a +b \,\operatorname {arctanh}\left (c \,x^{3}\right )\right )}^{3}}{x}d x\]
\[ \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x^{3}\right ) + a\right )}^{3}}{x} \,d x } \]
integral((b^3*arctanh(c*x^3)^3 + 3*a*b^2*arctanh(c*x^3)^2 + 3*a^2*b*arctan h(c*x^3) + a^3)/x, x)
\[ \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{x} \, dx=\int \frac {\left (a + b \operatorname {atanh}{\left (c x^{3} \right )}\right )^{3}}{x}\, dx \]
\[ \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x^{3}\right ) + a\right )}^{3}}{x} \,d x } \]
a^3*log(x) + integrate(1/8*b^3*(log(c*x^3 + 1) - log(-c*x^3 + 1))^3/x + 3/ 4*a*b^2*(log(c*x^3 + 1) - log(-c*x^3 + 1))^2/x + 3/2*a^2*b*(log(c*x^3 + 1) - log(-c*x^3 + 1))/x, x)
\[ \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x^{3}\right ) + a\right )}^{3}}{x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^3}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x^3\right )\right )}^3}{x} \,d x \]