Integrand size = 16, antiderivative size = 139 \[ \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{x^3} \, dx=-\frac {3 b \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2}{2 c^2}-\frac {3 b \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2}{2 c x}+\frac {\left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3}{2 c^2}-\frac {\left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3}{2 x^2}+\frac {3 b^2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right ) \log \left (\frac {2}{1-\frac {c}{x}}\right )}{c^2}+\frac {3 b^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {c}{x}}\right )}{2 c^2} \]
-3/2*b*(a+b*arccoth(x/c))^2/c^2-3/2*b*(a+b*arccoth(x/c))^2/c/x+1/2*(a+b*ar ccoth(x/c))^3/c^2-1/2*(a+b*arccoth(x/c))^3/x^2+3*b^2*(a+b*arccoth(x/c))*ln (2/(1-c/x))/c^2+3/2*b^3*polylog(2,1-2/(1-c/x))/c^2
Time = 0.23 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{x^3} \, dx=\frac {6 b^2 (-c+x) (b x+a (c+x)) \text {arctanh}\left (\frac {c}{x}\right )^2+2 b^3 \left (-c^2+x^2\right ) \text {arctanh}\left (\frac {c}{x}\right )^3+6 b \text {arctanh}\left (\frac {c}{x}\right ) \left (-a c (a c+2 b x)+2 b^2 x^2 \log \left (1+e^{-2 \text {arctanh}\left (\frac {c}{x}\right )}\right )\right )+a \left (12 b^2 x^2 \log \left (\frac {1}{\sqrt {1-\frac {c^2}{x^2}}}\right )-a \left (2 a c^2+6 b c x+3 b x^2 \log \left (1-\frac {c}{x}\right )-3 b x^2 \log \left (\frac {c+x}{x}\right )\right )\right )-6 b^3 x^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (\frac {c}{x}\right )}\right )}{4 c^2 x^2} \]
(6*b^2*(-c + x)*(b*x + a*(c + x))*ArcTanh[c/x]^2 + 2*b^3*(-c^2 + x^2)*ArcT anh[c/x]^3 + 6*b*ArcTanh[c/x]*(-(a*c*(a*c + 2*b*x)) + 2*b^2*x^2*Log[1 + E^ (-2*ArcTanh[c/x])]) + a*(12*b^2*x^2*Log[1/Sqrt[1 - c^2/x^2]] - a*(2*a*c^2 + 6*b*c*x + 3*b*x^2*Log[1 - c/x] - 3*b*x^2*Log[(c + x)/x])) - 6*b^3*x^2*Po lyLog[2, -E^(-2*ArcTanh[c/x])])/(4*c^2*x^2)
Time = 1.08 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {6454, 6452, 6542, 6436, 6510, 6546, 6470, 2849, 2752}
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.
| \(\displaystyle \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{x^3} \, dx\) |
| \(\Big \downarrow \) 6454 |
| \(\displaystyle -\int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{x}d\frac {1}{x}\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {3}{2} b c \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{\left (1-\frac {c^2}{x^2}\right ) x^2}d\frac {1}{x}-\frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{2 x^2}\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle \frac {3}{2} b c \left (\frac {\int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{1-\frac {c^2}{x^2}}d\frac {1}{x}}{c^2}-\frac {\int \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2d\frac {1}{x}}{c^2}\right )-\frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{2 x^2}\) |
| \(\Big \downarrow \) 6436 |
| \(\displaystyle \frac {3}{2} b c \left (\frac {\int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{1-\frac {c^2}{x^2}}d\frac {1}{x}}{c^2}-\frac {\frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{x}-2 b c \int \frac {a+b \text {arctanh}\left (\frac {c}{x}\right )}{\left (1-\frac {c^2}{x^2}\right ) x}d\frac {1}{x}}{c^2}\right )-\frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{2 x^2}\) |
| \(\Big \downarrow \) 6510 |
| \(\displaystyle \frac {3}{2} b c \left (\frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{3 b c^3}-\frac {\frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{x}-2 b c \int \frac {a+b \text {arctanh}\left (\frac {c}{x}\right )}{\left (1-\frac {c^2}{x^2}\right ) x}d\frac {1}{x}}{c^2}\right )-\frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{2 x^2}\) |
| \(\Big \downarrow \) 6546 |
| \(\displaystyle \frac {3}{2} b c \left (\frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{3 b c^3}-\frac {\frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{x}-2 b c \left (\frac {\int \frac {a+b \text {arctanh}\left (\frac {c}{x}\right )}{1-\frac {c}{x}}d\frac {1}{x}}{c}-\frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{2 b c^2}\right )}{c^2}\right )-\frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{2 x^2}\) |
| \(\Big \downarrow \) 6470 |
| \(\displaystyle \frac {3}{2} b c \left (\frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{3 b c^3}-\frac {\frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{x}-2 b c \left (\frac {\frac {\log \left (\frac {2}{1-\frac {c}{x}}\right ) \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )}{c}-b \int \frac {\log \left (\frac {2}{1-\frac {c}{x}}\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}}{c}-\frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{2 b c^2}\right )}{c^2}\right )-\frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{2 x^2}\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {3}{2} b c \left (\frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{3 b c^3}-\frac {\frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{x}-2 b c \left (\frac {\frac {b \int \frac {\log \left (\frac {2}{1-\frac {c}{x}}\right )}{1-\frac {2}{1-\frac {c}{x}}}d\frac {1}{1-\frac {c}{x}}}{c}+\frac {\log \left (\frac {2}{1-\frac {c}{x}}\right ) \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )}{c}}{c}-\frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{2 b c^2}\right )}{c^2}\right )-\frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{2 x^2}\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {3}{2} b c \left (\frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{3 b c^3}-\frac {\frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{x}-2 b c \left (\frac {\frac {\log \left (\frac {2}{1-\frac {c}{x}}\right ) \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )}{c}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {c}{x}}\right )}{2 c}}{c}-\frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{2 b c^2}\right )}{c^2}\right )-\frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{2 x^2}\) |
-1/2*(a + b*ArcTanh[c/x])^3/x^2 + (3*b*c*((a + b*ArcTanh[c/x])^3/(3*b*c^3) - ((a + b*ArcTanh[c/x])^2/x - 2*b*c*(-1/2*(a + b*ArcTanh[c/x])^2/(b*c^2) + (((a + b*ArcTanh[c/x])*Log[2/(1 - c/x)])/c + (b*PolyLog[2, 1 - 2/(1 - c/ x)])/(2*c))/c))/c^2))/2
3.2.56.3.1 Defintions of rubi rules used
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp [-e/g Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ {c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTanh[c*x^n])^p, x] - Simp[b*c*n*p Int[x^n*((a + b*ArcTanh[c*x^n]) ^(p - 1)/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && (EqQ[n, 1] || EqQ[p, 1])
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_.)/((d_) + (e_.)*(x_)), x_Symbol ] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c *(p/e) Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[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[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcTanh[c* x])^p, x], x] - Simp[d*(f^2/e) 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] && GtQ[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*e*(p + 1)), x] + Simp[1/ (c*d) Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.70 (sec) , antiderivative size = 6081, normalized size of antiderivative = 43.75
\[\text {output too large to display}\]
\[ \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{x^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (\frac {c}{x}\right ) + a\right )}^{3}}{x^{3}} \,d x } \]
\[ \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{x^3} \, dx=\int \frac {\left (a + b \operatorname {atanh}{\left (\frac {c}{x} \right )}\right )^{3}}{x^{3}}\, dx \]
\[ \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{x^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (\frac {c}{x}\right ) + a\right )}^{3}}{x^{3}} \,d x } \]
3/4*(c*(log(c + x)/c^3 - log(-c + x)/c^3 - 2/(c^2*x)) - 2*arctanh(c/x)/x^2 )*a^2*b - 3/8*(c^2*((log(c + x)^2 - 2*(log(c + x) - 2)*log(-c + x) + log(- c + x)^2 + 4*log(c + x))/c^4 - 8*log(x)/c^4) - 4*c*(log(c + x)/c^3 - log(- c + x)/c^3 - 2/(c^2*x))*arctanh(c/x))*a*b^2 + 1/64*(32*c^4*integrate(-1/4* log(x)^3/(c^4*x^3 - c^2*x^5), x) - 3*c^3*(log(c + x)/c^5 - log(-c + x)/c^5 - 2/(c^4*x)) + 48*c^3*integrate(-1/4*x*log(x)^2/(c^4*x^3 - c^2*x^5), x) + 48*c^3*integrate(-1/4*x*log(x)/(c^4*x^3 - c^2*x^5), x) - 6*c*(2*log(-c + x)/c^3 - 2*log(x)/c^3 + (c + 2*x)/(c^2*x^2))*log(-c/x + 1)^2 + 21*c^2*(log (c + x)/c^4 + log(-c + x)/c^4 - 2*log(x)/c^4) - 32*c^2*integrate(-1/4*x^2* log(x)^3/(c^4*x^3 - c^2*x^5), x) + 48*c^2*integrate(-1/4*x^2*log(x)^2/(c^4 *x^3 - c^2*x^5), x) - 384*c^2*integrate(-1/4*x^2*log(c + x)/(c^4*x^3 - c^2 *x^5), x) + 144*c^2*integrate(-1/4*x^2*log(x)/(c^4*x^3 - c^2*x^5), x) - 18 *c*(log(c + x)/c^3 - log(-c + x)/c^3) + c*(6*(2*x^2*log(-c + x)^2 + 2*x^2* log(x)^2 - 6*x^2*log(x) + c^2 + 6*c*x - 2*(2*x^2*log(x) - 3*x^2)*log(-c + x))*log(-c/x + 1)/(c^3*x^2) - (4*x^2*log(-c + x)^3 - 4*x^2*log(x)^3 + 18*x ^2*log(x)^2 - 6*(2*x^2*log(x) - 3*x^2)*log(-c + x)^2 - 42*x^2*log(x) + 3*c ^2 + 42*c*x + 6*(2*x^2*log(x)^2 - 6*x^2*log(x) + 7*x^2)*log(-c + x))/(c^3* x^2)) - 48*c*integrate(-1/4*x^3*log(x)^2/(c^4*x^3 - c^2*x^5), x) - 192*c*i ntegrate(-1/4*x^3*log(c + x)/(c^4*x^3 - c^2*x^5), x) + 336*c*integrate(-1/ 4*x^3*log(x)/(c^4*x^3 - c^2*x^5), x) + 4*log(-c/x + 1)^3/x^2 - 2*(12*c*...
\[ \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{x^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (\frac {c}{x}\right ) + a\right )}^{3}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^3}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (\frac {c}{x}\right )\right )}^3}{x^3} \,d x \]