Integrand size = 22, antiderivative size = 77 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x (d+c d x)} \, dx=\frac {(a+b \text {arctanh}(c x))^2 \log \left (2-\frac {2}{1+c x}\right )}{d}-\frac {b (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )}{d}-\frac {b^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+c x}\right )}{2 d} \]
(a+b*arctanh(c*x))^2*ln(2-2/(c*x+1))/d-b*(a+b*arctanh(c*x))*polylog(2,-1+2 /(c*x+1))/d-1/2*b^2*polylog(3,-1+2/(c*x+1))/d
Result contains complex when optimal does not.
Time = 0.73 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.71 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x (d+c d x)} \, dx=\frac {a^2 \log (c x)-a^2 \log (1+c x)+a b \left (2 \text {arctanh}(c x) \log \left (1-e^{-2 \text {arctanh}(c x)}\right )-\operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )\right )+b^2 \left (\frac {i \pi ^3}{24}-\frac {2}{3} \text {arctanh}(c x)^3+\text {arctanh}(c x)^2 \log \left (1-e^{2 \text {arctanh}(c x)}\right )+\text {arctanh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c x)}\right )\right )}{d} \]
(a^2*Log[c*x] - a^2*Log[1 + c*x] + a*b*(2*ArcTanh[c*x]*Log[1 - E^(-2*ArcTa nh[c*x])] - PolyLog[2, E^(-2*ArcTanh[c*x])]) + b^2*((I/24)*Pi^3 - (2*ArcTa nh[c*x]^3)/3 + ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] + ArcTanh[c*x]*P olyLog[2, E^(2*ArcTanh[c*x])] - PolyLog[3, E^(2*ArcTanh[c*x])]/2))/d
Time = 0.51 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {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.
\(\displaystyle \int \frac {(a+b \text {arctanh}(c x))^2}{x (c d x+d)} \, dx\) |
\(\Big \downarrow \) 6494 |
\(\displaystyle \frac {\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2}{d}-\frac {2 b c \int \frac {(a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{c x+1}\right )}{1-c^2 x^2}dx}{d}\) |
\(\Big \downarrow \) 6618 |
\(\displaystyle \frac {\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2}{d}-\frac {2 b c \left (\frac {\operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right ) (a+b \text {arctanh}(c x))}{2 c}-\frac {1}{2} b \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right )}{1-c^2 x^2}dx\right )}{d}\) |
\(\Big \downarrow \) 7164 |
\(\displaystyle \frac {\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2}{d}-\frac {2 b c \left (\frac {\operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right ) (a+b \text {arctanh}(c x))}{2 c}+\frac {b \operatorname {PolyLog}\left (3,\frac {2}{c x+1}-1\right )}{4 c}\right )}{d}\) |
((a + b*ArcTanh[c*x])^2*Log[2 - 2/(1 + c*x)])/d - (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)))/d
3.1.99.3.1 Defintions of rubi rules used
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[(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 = 2.89 (sec) , antiderivative size = 1148, normalized size of antiderivative = 14.91
method | result | size |
parts | \(\text {Expression too large to display}\) | \(1148\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1150\) |
default | \(\text {Expression too large to display}\) | \(1150\) |
a^2/d*(-ln(c*x+1)+ln(x))+b^2/d*(-arctanh(c*x)^2*ln(c*x+1)+ln(c*x)*arctanh( c*x)^2+2*arctanh(c*x)^2*ln((c*x+1)/(-c^2*x^2+1)^(1/2))-arctanh(c*x)^2*ln(( c*x+1)^2/(-c^2*x^2+1)-1)-2/3*arctanh(c*x)^3+1/2*(I*Pi*csgn(I/(1-(c*x+1)^2/ (c^2*x^2-1)))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^2-I* Pi*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I* (c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c^2*x^2-1)))+I*csgn(I*(-(c*x+1)^2/(c^2 *x^2-1)-1)/(1-(c*x+1)^2/(c^2*x^2-1)))*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))*cs gn(I*(-(c*x+1)^2/(c^2*x^2-1)-1))*Pi-I*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1)/(1 -(c*x+1)^2/(c^2*x^2-1)))^2*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))*Pi+I*Pi*csgn( I*(c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^3-I*Pi*csgn(I*(c*x+1)^2 /(c^2*x^2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^2+I* Pi*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))+2*I* Pi*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))*csgn(I*(c*x+1)^2/(c^2*x^2-1))^2+I*Pi *csgn(I*(c*x+1)^2/(c^2*x^2-1))^3-I*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1)/(1-(c *x+1)^2/(c^2*x^2-1)))^2*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1))*Pi+I*csgn(I*(-( c*x+1)^2/(c^2*x^2-1)-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^3*Pi+2*ln(2))*arctanh(c *x)^2+arctanh(c*x)^2*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+2*arctanh(c*x)*polyl og(2,-(c*x+1)/(-c^2*x^2+1)^(1/2))-2*polylog(3,-(c*x+1)/(-c^2*x^2+1)^(1/2)) +arctanh(c*x)^2*ln(1-(c*x+1)/(-c^2*x^2+1)^(1/2))+2*arctanh(c*x)*polylog(2, (c*x+1)/(-c^2*x^2+1)^(1/2))-2*polylog(3,(c*x+1)/(-c^2*x^2+1)^(1/2)))+2*...
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x (d+c d x)} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )} x} \,d x } \]
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x (d+c d x)} \, dx=\frac {\int \frac {a^{2}}{c x^{2} + x}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{c x^{2} + x}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{c x^{2} + x}\, dx}{d} \]
(Integral(a**2/(c*x**2 + x), x) + Integral(b**2*atanh(c*x)**2/(c*x**2 + x) , x) + Integral(2*a*b*atanh(c*x)/(c*x**2 + x), x))/d
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x (d+c d x)} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )} x} \,d x } \]
-1/4*b^2*log(c*x + 1)*log(-c*x + 1)^2/d - a^2*(log(c*x + 1)/d - log(x)/d) + integrate(1/4*((b^2*c*x - b^2)*log(c*x + 1)^2 + 4*(a*b*c*x - a*b)*log(c* x + 1) - 2*(2*a*b*c*x - 2*a*b - (b^2*c^2*x^2 + b^2)*log(c*x + 1))*log(-c*x + 1))/(c^2*d*x^3 - d*x), x)
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x (d+c d x)} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )} x} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x (d+c d x)} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{x\,\left (d+c\,d\,x\right )} \,d x \]