Integrand size = 22, antiderivative size = 278 \[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x} \, dx=a b c d^2 x+b^2 c d^2 x \text {arctanh}(c x)+\frac {3}{2} d^2 (a+b \text {arctanh}(c x))^2+2 c d^2 x (a+b \text {arctanh}(c x))^2+\frac {1}{2} c^2 d^2 x^2 (a+b \text {arctanh}(c x))^2+2 d^2 (a+b \text {arctanh}(c x))^2 \text {arctanh}\left (1-\frac {2}{1-c x}\right )-4 b d^2 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )+\frac {1}{2} b^2 d^2 \log \left (1-c^2 x^2\right )-2 b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )-b d^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+b d^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c x}\right )+\frac {1}{2} b^2 d^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 d^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c x}\right ) \]
a*b*c*d^2*x+b^2*c*d^2*x*arctanh(c*x)+3/2*d^2*(a+b*arctanh(c*x))^2+2*c*d^2* x*(a+b*arctanh(c*x))^2+1/2*c^2*d^2*x^2*(a+b*arctanh(c*x))^2-2*d^2*(a+b*arc tanh(c*x))^2*arctanh(-1+2/(-c*x+1))-4*b*d^2*(a+b*arctanh(c*x))*ln(2/(-c*x+ 1))+1/2*b^2*d^2*ln(-c^2*x^2+1)-2*b^2*d^2*polylog(2,1-2/(-c*x+1))-b*d^2*(a+ b*arctanh(c*x))*polylog(2,1-2/(-c*x+1))+b*d^2*(a+b*arctanh(c*x))*polylog(2 ,-1+2/(-c*x+1))+1/2*b^2*d^2*polylog(3,1-2/(-c*x+1))-1/2*b^2*d^2*polylog(3, -1+2/(-c*x+1))
Result contains complex when optimal does not.
Time = 0.48 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.17 \[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x} \, dx=\frac {1}{2} d^2 \left (4 a^2 c x+a^2 c^2 x^2+2 a^2 \log (c x)+a b \left (2 c x+2 c^2 x^2 \text {arctanh}(c x)+\log (1-c x)-\log (1+c x)\right )+4 a b \left (2 c x \text {arctanh}(c x)+\log \left (1-c^2 x^2\right )\right )+b^2 \left (2 c x \text {arctanh}(c x)+\left (-1+c^2 x^2\right ) \text {arctanh}(c x)^2+\log \left (1-c^2 x^2\right )\right )+4 b^2 \left (\text {arctanh}(c x) \left ((-1+c x) \text {arctanh}(c x)-2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )\right )+2 a b (-\operatorname {PolyLog}(2,-c x)+\operatorname {PolyLog}(2,c x))+2 b^2 \left (\frac {i \pi ^3}{24}-\frac {2}{3} \text {arctanh}(c x)^3-\text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+\text {arctanh}(c x)^2 \log \left (1-e^{2 \text {arctanh}(c x)}\right )+\text {arctanh}(c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+\text {arctanh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c x)}\right )\right )\right ) \]
(d^2*(4*a^2*c*x + a^2*c^2*x^2 + 2*a^2*Log[c*x] + a*b*(2*c*x + 2*c^2*x^2*Ar cTanh[c*x] + Log[1 - c*x] - Log[1 + c*x]) + 4*a*b*(2*c*x*ArcTanh[c*x] + Lo g[1 - c^2*x^2]) + b^2*(2*c*x*ArcTanh[c*x] + (-1 + c^2*x^2)*ArcTanh[c*x]^2 + Log[1 - c^2*x^2]) + 4*b^2*(ArcTanh[c*x]*((-1 + c*x)*ArcTanh[c*x] - 2*Log [1 + E^(-2*ArcTanh[c*x])]) + PolyLog[2, -E^(-2*ArcTanh[c*x])]) + 2*a*b*(-P olyLog[2, -(c*x)] + PolyLog[2, c*x]) + 2*b^2*((I/24)*Pi^3 - (2*ArcTanh[c*x ]^3)/3 - ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + ArcTanh[c*x]^2*Log[ 1 - E^(2*ArcTanh[c*x])] + ArcTanh[c*x]*PolyLog[2, -E^(-2*ArcTanh[c*x])] + ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] + PolyLog[3, -E^(-2*ArcTanh[c* x])]/2 - PolyLog[3, E^(2*ArcTanh[c*x])]/2)))/2
Time = 0.85 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6502, 2009}
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.
\(\displaystyle \int \frac {(c d x+d)^2 (a+b \text {arctanh}(c x))^2}{x} \, dx\) |
\(\Big \downarrow \) 6502 |
\(\displaystyle \int \left (c^2 d^2 x (a+b \text {arctanh}(c x))^2+2 c d^2 (a+b \text {arctanh}(c x))^2+\frac {d^2 (a+b \text {arctanh}(c x))^2}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} c^2 d^2 x^2 (a+b \text {arctanh}(c x))^2-b d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))+b d^2 \operatorname {PolyLog}\left (2,\frac {2}{1-c x}-1\right ) (a+b \text {arctanh}(c x))+2 c d^2 x (a+b \text {arctanh}(c x))^2+\frac {3}{2} d^2 (a+b \text {arctanh}(c x))^2+2 d^2 \text {arctanh}\left (1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^2-4 b d^2 \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))+a b c d^2 x+b^2 c d^2 x \text {arctanh}(c x)+\frac {1}{2} b^2 d^2 \log \left (1-c^2 x^2\right )-2 b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+\frac {1}{2} b^2 d^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 d^2 \operatorname {PolyLog}\left (3,\frac {2}{1-c x}-1\right )\) |
a*b*c*d^2*x + b^2*c*d^2*x*ArcTanh[c*x] + (3*d^2*(a + b*ArcTanh[c*x])^2)/2 + 2*c*d^2*x*(a + b*ArcTanh[c*x])^2 + (c^2*d^2*x^2*(a + b*ArcTanh[c*x])^2)/ 2 + 2*d^2*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)] - 4*b*d^2*(a + b *ArcTanh[c*x])*Log[2/(1 - c*x)] + (b^2*d^2*Log[1 - c^2*x^2])/2 - 2*b^2*d^2 *PolyLog[2, 1 - 2/(1 - c*x)] - b*d^2*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2 /(1 - c*x)] + b*d^2*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)] + (b ^2*d^2*PolyLog[3, 1 - 2/(1 - c*x)])/2 - (b^2*d^2*PolyLog[3, -1 + 2/(1 - c* x)])/2
3.1.80.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e _.)*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^p, ( f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 5.86 (sec) , antiderivative size = 895, normalized size of antiderivative = 3.22
method | result | size |
parts | \(\text {Expression too large to display}\) | \(895\) |
derivativedivides | \(\text {Expression too large to display}\) | \(897\) |
default | \(\text {Expression too large to display}\) | \(897\) |
d^2*a^2*(1/2*c^2*x^2+2*c*x+ln(x))+d^2*b^2*(1/2*c^2*x^2*arctanh(c*x)^2+2*c* x*arctanh(c*x)^2+ln(c*x)*arctanh(c*x)^2-arctanh(c*x)^2*ln((c*x+1)^2/(-c^2* x^2+1)-1)+arctanh(c*x)^2*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/2))+2*arctanh(c*x)*p olylog(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)*polylo g(2,(c*x+1)/(-c^2*x^2+1)^(1/2))-2*polylog(3,(c*x+1)/(-c^2*x^2+1)^(1/2))-ar ctanh(c*x)*polylog(2,-(c*x+1)^2/(-c^2*x^2+1))+1/2*polylog(3,-(c*x+1)^2/(-c ^2*x^2+1))-ln(1+(c*x+1)^2/(-c^2*x^2+1))+3/2*arctanh(c*x)^2-4*dilog(1+I*(c* x+1)/(-c^2*x^2+1)^(1/2))-4*dilog(1-I*(c*x+1)/(-c^2*x^2+1)^(1/2))-4*arctanh (c*x)*ln(1+I*(c*x+1)/(-c^2*x^2+1)^(1/2))-4*arctanh(c*x)*ln(1-I*(c*x+1)/(-c ^2*x^2+1)^(1/2))+(c*x+1)*arctanh(c*x)+1/2*I*Pi*csgn(I*(-(c*x+1)^2/(c^2*x^2 -1)-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^3*arctanh(c*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)-1))*csgn(I*(-(c*x+1)^2/ (c^2*x^2-1)-1)/(1-(c*x+1)^2/(c^2*x^2-1)))*arctanh(c*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)-1)/(1-(c*x+1)^2/( c^2*x^2-1)))^2*arctanh(c*x)^2-1/2*I*Pi*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1))* csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^2*arctanh(c*x )^2)+2*a*b*d^2*(1/2*c^2*x^2*arctanh(c*x)+2*c*x*arctanh(c*x)+ln(c*x)*arctan h(c*x)-1/2*dilog(c*x+1)-1/2*ln(c*x)*ln(c*x+1)-1/2*dilog(c*x)+1/2*c*x+5/4*l n(c*x-1)+3/4*ln(c*x+1))
\[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x} \, dx=\int { \frac {{\left (c d x + d\right )}^{2} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
integral((a^2*c^2*d^2*x^2 + 2*a^2*c*d^2*x + a^2*d^2 + (b^2*c^2*d^2*x^2 + 2 *b^2*c*d^2*x + b^2*d^2)*arctanh(c*x)^2 + 2*(a*b*c^2*d^2*x^2 + 2*a*b*c*d^2* x + a*b*d^2)*arctanh(c*x))/x, x)
\[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x} \, dx=d^{2} \left (\int 2 a^{2} c\, dx + \int \frac {a^{2}}{x}\, dx + \int a^{2} c^{2} x\, dx + \int 2 b^{2} c \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x}\, dx + \int 4 a b c \operatorname {atanh}{\left (c x \right )}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{x}\, dx + \int b^{2} c^{2} x \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b c^{2} x \operatorname {atanh}{\left (c x \right )}\, dx\right ) \]
d**2*(Integral(2*a**2*c, x) + Integral(a**2/x, x) + Integral(a**2*c**2*x, x) + Integral(2*b**2*c*atanh(c*x)**2, x) + Integral(b**2*atanh(c*x)**2/x, x) + Integral(4*a*b*c*atanh(c*x), x) + Integral(2*a*b*atanh(c*x)/x, x) + I ntegral(b**2*c**2*x*atanh(c*x)**2, x) + Integral(2*a*b*c**2*x*atanh(c*x), x))
\[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x} \, dx=\int { \frac {{\left (c d x + d\right )}^{2} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
1/2*a^2*c^2*d^2*x^2 + 2*a^2*c*d^2*x + 2*(2*c*x*arctanh(c*x) + log(-c^2*x^2 + 1))*a*b*d^2 + a^2*d^2*log(x) + 1/8*(b^2*c^2*d^2*x^2 + 4*b^2*c*d^2*x)*lo g(-c*x + 1)^2 - integrate(-1/4*((b^2*c^3*d^2*x^3 + b^2*c^2*d^2*x^2 - b^2*c *d^2*x - b^2*d^2)*log(c*x + 1)^2 + 4*(a*b*c^3*d^2*x^3 - a*b*c^2*d^2*x^2 + a*b*c*d^2*x - a*b*d^2)*log(c*x + 1) - (4*a*b*c*d^2*x - 4*a*b*d^2 + (4*a*b* c^3*d^2 + b^2*c^3*d^2)*x^3 - 4*(a*b*c^2*d^2 - b^2*c^2*d^2)*x^2 + 2*(b^2*c^ 3*d^2*x^3 + b^2*c^2*d^2*x^2 - b^2*c*d^2*x - b^2*d^2)*log(c*x + 1))*log(-c* x + 1))/(c*x^2 - x), x)
\[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x} \, dx=\int { \frac {{\left (c d x + d\right )}^{2} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
Timed out. \[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))^2}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^2}{x} \,d x \]