Integrand size = 21, antiderivative size = 412 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (d+e x)} \, dx=\frac {c (a+b \text {arctanh}(c x))^2}{d}-\frac {(a+b \text {arctanh}(c x))^2}{d x}-\frac {2 e (a+b \text {arctanh}(c x))^2 \text {arctanh}\left (1-\frac {2}{1-c x}\right )}{d^2}-\frac {e (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1+c x}\right )}{d^2}+\frac {e (a+b \text {arctanh}(c x))^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{d^2}+\frac {2 b c (a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{1+c x}\right )}{d}+\frac {b e (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{d^2}-\frac {b e (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c x}\right )}{d^2}+\frac {b e (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{d^2}-\frac {b^2 c \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )}{d}-\frac {b e (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{d^2}-\frac {b^2 e \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 d^2}+\frac {b^2 e \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c x}\right )}{2 d^2}+\frac {b^2 e \operatorname {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{2 d^2}-\frac {b^2 e \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 d^2} \]
c*(a+b*arctanh(c*x))^2/d-(a+b*arctanh(c*x))^2/d/x+2*e*(a+b*arctanh(c*x))^2 *arctanh(-1+2/(-c*x+1))/d^2-e*(a+b*arctanh(c*x))^2*ln(2/(c*x+1))/d^2+e*(a+ b*arctanh(c*x))^2*ln(2*c*(e*x+d)/(c*d+e)/(c*x+1))/d^2+2*b*c*(a+b*arctanh(c *x))*ln(2-2/(c*x+1))/d+b*e*(a+b*arctanh(c*x))*polylog(2,1-2/(-c*x+1))/d^2- b*e*(a+b*arctanh(c*x))*polylog(2,-1+2/(-c*x+1))/d^2+b*e*(a+b*arctanh(c*x)) *polylog(2,1-2/(c*x+1))/d^2-b^2*c*polylog(2,-1+2/(c*x+1))/d-b*e*(a+b*arcta nh(c*x))*polylog(2,1-2*c*(e*x+d)/(c*d+e)/(c*x+1))/d^2-1/2*b^2*e*polylog(3, 1-2/(-c*x+1))/d^2+1/2*b^2*e*polylog(3,-1+2/(-c*x+1))/d^2+1/2*b^2*e*polylog (3,1-2/(c*x+1))/d^2-1/2*b^2*e*polylog(3,1-2*c*(e*x+d)/(c*d+e)/(c*x+1))/d^2
Result contains complex when optimal does not.
Time = 15.53 (sec) , antiderivative size = 1270, normalized size of antiderivative = 3.08 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (d+e x)} \, dx =\text {Too large to display} \]
-(a^2/(d*x)) - (a^2*e*Log[x])/d^2 + (a^2*e*Log[d + e*x])/d^2 + (a*b*((-2*c *d^2*ArcTanh[c*x])/x + e^2*ArcTanh[c*x]^2 - (Sqrt[1 - (c^2*d^2)/e^2]*e^2*A rcTanh[c*x]^2)/E^ArcTanh[(c*d)/e] - c*d*e*ArcTanh[c*x]*(ArcTanh[c*x] + 2*L og[1 - E^(-2*ArcTanh[c*x])]) + c*d*e*ArcTanh[c*x]*(I*Pi + 2*ArcTanh[(c*d)/ e] + 2*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]) + c^2*d^2*(2*Log [c*x] - Log[1 - c^2*x^2]) - (I/2)*c*d*e*Pi*(2*Log[1 + E^(2*ArcTanh[c*x])] + Log[1 - c^2*x^2]) + 2*c*d*e*ArcTanh[(c*d)/e]*(Log[1 - E^(-2*(ArcTanh[(c* d)/e] + ArcTanh[c*x]))] - Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]]) + c*d*e*PolyLog[2, E^(-2*ArcTanh[c*x])] - c*d*e*PolyLog[2, E^(-2*(ArcTanh[(c *d)/e] + ArcTanh[c*x]))]))/(c*d^3) + (b^2*((-I)*c*d*e*Pi^3 + 24*c^2*d^2*Ar cTanh[c*x]^2 - (24*c*d^2*ArcTanh[c*x]^2)/x + 8*c*d*e*ArcTanh[c*x]^3 + 8*e^ 2*ArcTanh[c*x]^3 + 48*c^2*d^2*ArcTanh[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] - 24*c*d*e*ArcTanh[c*x]^2*Log[1 - E^(2*ArcTanh[c*x])] - 24*c^2*d^2*PolyLog[2 , E^(-2*ArcTanh[c*x])] - 24*c*d*e*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x ])] + 12*c*d*e*PolyLog[3, E^(2*ArcTanh[c*x])] + (24*(c*d - e)*e*(c*d + e)* (-6*c*d*ArcTanh[c*x]^3 + 2*e*ArcTanh[c*x]^3 - (4*Sqrt[1 - (c^2*d^2)/e^2]*e *ArcTanh[c*x]^3)/E^ArcTanh[(c*d)/e] - (6*I)*c*d*Pi*ArcTanh[c*x]*Log[(E^(-A rcTanh[c*x]) + E^ArcTanh[c*x])/2] - 6*c*d*ArcTanh[c*x]^2*Log[1 - (Sqrt[c*d + e]*E^ArcTanh[c*x])/Sqrt[-(c*d) + e]] - 6*c*d*ArcTanh[c*x]^2*Log[1 + (Sq rt[c*d + e]*E^ArcTanh[c*x])/Sqrt[-(c*d) + e]] + 6*c*d*ArcTanh[c*x]^2*Lo...
Time = 0.95 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, 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.
\(\displaystyle \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (d+e x)} \, dx\) |
\(\Big \downarrow \) 6502 |
\(\displaystyle \int \left (\frac {e^2 (a+b \text {arctanh}(c x))^2}{d^2 (d+e x)}-\frac {e (a+b \text {arctanh}(c x))^2}{d^2 x}+\frac {(a+b \text {arctanh}(c x))^2}{d x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b e \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{d^2}-\frac {b e \operatorname {PolyLog}\left (2,\frac {2}{1-c x}-1\right ) (a+b \text {arctanh}(c x))}{d^2}+\frac {b e \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{d^2}-\frac {b e (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{d^2}-\frac {2 e \text {arctanh}\left (1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^2}{d^2}-\frac {e \log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2}{d^2}+\frac {e (a+b \text {arctanh}(c x))^2 \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{d^2}+\frac {c (a+b \text {arctanh}(c x))^2}{d}-\frac {(a+b \text {arctanh}(c x))^2}{d x}+\frac {2 b c \log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{d}-\frac {b^2 e \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 d^2}+\frac {b^2 e \operatorname {PolyLog}\left (3,\frac {2}{1-c x}-1\right )}{2 d^2}+\frac {b^2 e \operatorname {PolyLog}\left (3,1-\frac {2}{c x+1}\right )}{2 d^2}-\frac {b^2 e \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{2 d^2}-\frac {b^2 c \operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right )}{d}\) |
(c*(a + b*ArcTanh[c*x])^2)/d - (a + b*ArcTanh[c*x])^2/(d*x) - (2*e*(a + b* ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)])/d^2 - (e*(a + b*ArcTanh[c*x])^2* Log[2/(1 + c*x)])/d^2 + (e*(a + b*ArcTanh[c*x])^2*Log[(2*c*(d + e*x))/((c* d + e)*(1 + c*x))])/d^2 + (2*b*c*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)] )/d + (b*e*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/d^2 - (b*e*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)])/d^2 + (b*e*(a + b*ArcTanh [c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/d^2 - (b^2*c*PolyLog[2, -1 + 2/(1 + c* x)])/d - (b*e*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/d^2 - (b^2*e*PolyLog[3, 1 - 2/(1 - c*x)])/(2*d^2) + (b^2*e *PolyLog[3, -1 + 2/(1 - c*x)])/(2*d^2) + (b^2*e*PolyLog[3, 1 - 2/(1 + c*x) ])/(2*d^2) - (b^2*e*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))]) /(2*d^2)
3.2.58.3.1 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 = 15.33 (sec) , antiderivative size = 26440, normalized size of antiderivative = 64.17
method | result | size |
parts | \(\text {Expression too large to display}\) | \(26440\) |
derivativedivides | \(\text {Expression too large to display}\) | \(26451\) |
default | \(\text {Expression too large to display}\) | \(26451\) |
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (d+e x)} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )} x^{2}} \,d x } \]
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (d+e x)} \, dx=\int \frac {\left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{2}}{x^{2} \left (d + e x\right )}\, dx \]
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (d+e x)} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )} x^{2}} \,d x } \]
a^2*(e*log(e*x + d)/d^2 - e*log(x)/d^2 - 1/(d*x)) - 1/4*b^2*log(-c*x + 1)^ 2/(d*x) - integrate(-1/4*((b^2*c*d*x - b^2*d)*log(c*x + 1)^2 + 4*(a*b*c*d* x - a*b*d)*log(c*x + 1) + 2*(b^2*c*e*x^2 + 2*a*b*d - (2*a*b*c*d - b^2*c*d) *x - (b^2*c*d*x - b^2*d)*log(c*x + 1))*log(-c*x + 1))/(c*d*e*x^4 - d^2*x^2 + (c*d^2 - d*e)*x^3), x)
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (d+e x)} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^2 (d+e x)} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{x^2\,\left (d+e\,x\right )} \,d x \]