Integrand size = 20, antiderivative size = 201 \[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))^2}{x^2} \, dx=c d (a+b \text {arctanh}(c x))^2-\frac {d (a+b \text {arctanh}(c x))^2}{x}+2 c d (a+b \text {arctanh}(c x))^2 \text {arctanh}\left (1-\frac {2}{1-c x}\right )+2 b c d (a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{1+c x}\right )-b c d (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+b c d (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c x}\right )-b^2 c d \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )+\frac {1}{2} b^2 c d \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 c d \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c x}\right ) \] Output:
c*d*(a+b*arctanh(c*x))^2-d*(a+b*arctanh(c*x))^2/x-2*c*d*(a+b*arctanh(c*x)) ^2*arctanh(-1+2/(-c*x+1))+2*b*c*d*(a+b*arctanh(c*x))*ln(2-2/(c*x+1))-b*c*d *(a+b*arctanh(c*x))*polylog(2,1-2/(-c*x+1))+b*c*d*(a+b*arctanh(c*x))*polyl og(2,-1+2/(-c*x+1))-b^2*c*d*polylog(2,-1+2/(c*x+1))+1/2*b^2*c*d*polylog(3, 1-2/(-c*x+1))-1/2*b^2*c*d*polylog(3,-1+2/(-c*x+1))
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.24 \[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))^2}{x^2} \, dx=-\frac {d \left (a^2-a^2 c x \log (x)+a b \left (2 \text {arctanh}(c x)+c x \left (-2 \log (c x)+\log \left (1-c^2 x^2\right )\right )\right )+b^2 \left (\text {arctanh}(c x) \left ((1-c x) \text {arctanh}(c x)-2 c x \log \left (1-e^{-2 \text {arctanh}(c x)}\right )\right )+c x \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )\right )+a b c x (\operatorname {PolyLog}(2,-c x)-\operatorname {PolyLog}(2,c x))-b^2 c x \left (\frac {i \pi ^3}{24}-\frac {2}{3} \text {arctanh}(c x)^3-\text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+\text {arctanh}(c x)^2 \log \left (1-e^{2 \text {arctanh}(c x)}\right )+\text {arctanh}(c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+\text {arctanh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c x)}\right )\right )\right )}{x} \] Input:
Integrate[((d + c*d*x)*(a + b*ArcTanh[c*x])^2)/x^2,x]
Output:
-((d*(a^2 - a^2*c*x*Log[x] + a*b*(2*ArcTanh[c*x] + c*x*(-2*Log[c*x] + Log[ 1 - c^2*x^2])) + b^2*(ArcTanh[c*x]*((1 - c*x)*ArcTanh[c*x] - 2*c*x*Log[1 - E^(-2*ArcTanh[c*x])]) + c*x*PolyLog[2, E^(-2*ArcTanh[c*x])]) + a*b*c*x*(P olyLog[2, -(c*x)] - PolyLog[2, c*x]) - b^2*c*x*((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*Lo g[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)))/x)
Time = 0.74 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.
| \(\displaystyle \int \frac {(c d x+d) (a+b \text {arctanh}(c x))^2}{x^2} \, dx\) |
| \(\Big \downarrow \) 6502 |
| \(\displaystyle \int \left (\frac {d (a+b \text {arctanh}(c x))^2}{x^2}+\frac {c d (a+b \text {arctanh}(c x))^2}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -b c d \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))+b c d \operatorname {PolyLog}\left (2,\frac {2}{1-c x}-1\right ) (a+b \text {arctanh}(c x))+c d (a+b \text {arctanh}(c x))^2-\frac {d (a+b \text {arctanh}(c x))^2}{x}+2 c d \text {arctanh}\left (1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^2+2 b c d \log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))+b^2 (-c) d \operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right )+\frac {1}{2} b^2 c d \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 c d \operatorname {PolyLog}\left (3,\frac {2}{1-c x}-1\right )\) |
Input:
Int[((d + c*d*x)*(a + b*ArcTanh[c*x])^2)/x^2,x]
Output:
c*d*(a + b*ArcTanh[c*x])^2 - (d*(a + b*ArcTanh[c*x])^2)/x + 2*c*d*(a + b*A rcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)] + 2*b*c*d*(a + b*ArcTanh[c*x])*Log [2 - 2/(1 + c*x)] - b*c*d*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)] + b*c*d*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)] - b^2*c*d*PolyL og[2, -1 + 2/(1 + c*x)] + (b^2*c*d*PolyLog[3, 1 - 2/(1 - c*x)])/2 - (b^2*c *d*PolyLog[3, -1 + 2/(1 - c*x)])/2
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 = 2.81 (sec) , antiderivative size = 1857, normalized size of antiderivative = 9.24
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1857\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1860\) |
| default | \(\text {Expression too large to display}\) | \(1860\) |
Input:
int((c*d*x+d)*(a+b*arctanh(c*x))^2/x^2,x,method=_RETURNVERBOSE)
Output:
d*a^2*(c*ln(x)-1/x)+d*b^2*c*(arctanh(c*x)^2*ln(c*x)-arctanh(c*x)^2/c/x-arc tanh(c*x)^2*ln((c*x+1)^2/(-c^2*x^2+1)-1)+arctanh(c*x)^2*ln(1-(c*x+1)^2/(-c ^2*x^2+1))+arctanh(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))-arctanh(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))-1/8*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*(4*arct anh(c*x)^2-2*arctanh(c*x)*ln(1-(c*x+1)^2/(-c^2*x^2+1))-2*arctanh(c*x)*ln(1 +(c*x+1)^2/(-c^2*x^2+1))-polylog(2,-(c*x+1)^2/(-c^2*x^2+1))-polylog(2,(c*x +1)^2/(-c^2*x^2+1)))-1/8*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*(2*arctanh(c*x)*ln(1+ (c*x+1)^2/(-c^2*x^2+1))-dilog((c*x+1)^2/(-c^2*x^2+1))+dilog(1+(c*x+1)^2/(- c^2*x^2+1)))+1/8*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*(4*arctanh(c*x)^2-2*arctanh(c*x)*ln(1-(c*x+1)^2/(-c^2*x^2+1))-2 *arctanh(c*x)*ln(1+(c*x+1)^2/(-c^2*x^2+1))-polylog(2,-(c*x+1)^2/(-c^2*x^2+ 1))-polylog(2,(c*x+1)^2/(-c^2*x^2+1)))+1/8*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)))*(4*arctanh(c*x)^2-2*arctanh(c*x)*ln(1-(c*x+1 )^2/(-c^2*x^2+1))-2*arctanh(c*x)*ln(1+(c*x+1)^2/(-c^2*x^2+1))-polylog(2,-( c*x+1)^2/(-c^2*x^2+1))-polylog(2,(c*x+1)^2/(-c^2*x^2+1)))-arctanh(c*x)^2+3 /2*arctanh(c*x)*ln(1-(c*x+1)^2/(-c^2*x^2+1))+3/4*polylog(2,(c*x+1)^2/(-...
\[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))^2}{x^2} \, dx=\int { \frac {{\left (c d x + d\right )} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \] Input:
integrate((c*d*x+d)*(a+b*arctanh(c*x))^2/x^2,x, algorithm="fricas")
Output:
integral((a^2*c*d*x + a^2*d + (b^2*c*d*x + b^2*d)*arctanh(c*x)^2 + 2*(a*b* c*d*x + a*b*d)*arctanh(c*x))/x^2, x)
\[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))^2}{x^2} \, dx=d \left (\int \frac {a^{2}}{x^{2}}\, dx + \int \frac {a^{2} c}{x}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {b^{2} c \operatorname {atanh}^{2}{\left (c x \right )}}{x}\, dx + \int \frac {2 a b c \operatorname {atanh}{\left (c x \right )}}{x}\, dx\right ) \] Input:
integrate((c*d*x+d)*(a+b*atanh(c*x))**2/x**2,x)
Output:
d*(Integral(a**2/x**2, x) + Integral(a**2*c/x, x) + Integral(b**2*atanh(c* x)**2/x**2, x) + Integral(2*a*b*atanh(c*x)/x**2, x) + Integral(b**2*c*atan h(c*x)**2/x, x) + Integral(2*a*b*c*atanh(c*x)/x, x))
\[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))^2}{x^2} \, dx=\int { \frac {{\left (c d x + d\right )} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \] Input:
integrate((c*d*x+d)*(a+b*arctanh(c*x))^2/x^2,x, algorithm="maxima")
Output:
a^2*c*d*log(x) - (c*(log(c^2*x^2 - 1) - log(x^2)) + 2*arctanh(c*x)/x)*a*b* d - 1/4*b^2*d*log(-c*x + 1)^2/x - a^2*d/x - integrate(-1/4*((b^2*c^2*d*x^2 - b^2*d)*log(c*x + 1)^2 + 4*(a*b*c^2*d*x^2 - a*b*c*d*x)*log(c*x + 1) - 2* (2*a*b*c^2*d*x^2 - (2*a*b*c*d + b^2*c*d)*x + (b^2*c^2*d*x^2 - b^2*d)*log(c *x + 1))*log(-c*x + 1))/(c*x^3 - x^2), x)
\[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))^2}{x^2} \, dx=\int { \frac {{\left (c d x + d\right )} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \] Input:
integrate((c*d*x+d)*(a+b*arctanh(c*x))^2/x^2,x, algorithm="giac")
Output:
integrate((c*d*x + d)*(b*arctanh(c*x) + a)^2/x^2, x)
Timed out. \[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))^2}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,\left (d+c\,d\,x\right )}{x^2} \,d x \] Input:
int(((a + b*atanh(c*x))^2*(d + c*d*x))/x^2,x)
Output:
int(((a + b*atanh(c*x))^2*(d + c*d*x))/x^2, x)
\[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))^2}{x^2} \, dx=\frac {d \left (-\mathit {atanh} \left (c x \right )^{2} b^{2}-2 \mathit {atanh} \left (c x \right ) a b c x -2 \mathit {atanh} \left (c x \right ) a b -2 \left (\int \frac {\mathit {atanh} \left (c x \right )}{c^{2} x^{3}-x}d x \right ) b^{2} c x +2 \left (\int \frac {\mathit {atanh} \left (c x \right )}{x}d x \right ) a b c x +\left (\int \frac {\mathit {atanh} \left (c x \right )^{2}}{x}d x \right ) b^{2} c x -2 \,\mathrm {log}\left (c^{2} x -c \right ) a b c x +\mathrm {log}\left (x \right ) a^{2} c x +2 \,\mathrm {log}\left (x \right ) a b c x -a^{2}\right )}{x} \] Input:
int((c*d*x+d)*(a+b*atanh(c*x))^2/x^2,x)
Output:
(d*( - atanh(c*x)**2*b**2 - 2*atanh(c*x)*a*b*c*x - 2*atanh(c*x)*a*b - 2*in t(atanh(c*x)/(c**2*x**3 - x),x)*b**2*c*x + 2*int(atanh(c*x)/x,x)*a*b*c*x + int(atanh(c*x)**2/x,x)*b**2*c*x - 2*log(c**2*x - c)*a*b*c*x + log(x)*a**2 *c*x + 2*log(x)*a*b*c*x - a**2))/x