Integrand size = 19, antiderivative size = 279 \[ \int \frac {x (a+b \text {arctanh}(c x))^2}{d+e x} \, dx=\frac {(a+b \text {arctanh}(c x))^2}{c e}+\frac {x (a+b \text {arctanh}(c x))^2}{e}-\frac {2 b (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{c e}+\frac {d (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1+c x}\right )}{e^2}-\frac {d (a+b \text {arctanh}(c x))^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^2}-\frac {b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c e}-\frac {b d (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{e^2}+\frac {b d (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^2}-\frac {b^2 d \operatorname {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{2 e^2}+\frac {b^2 d \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e^2} \] Output:
(a+b*arctanh(c*x))^2/c/e+x*(a+b*arctanh(c*x))^2/e-2*b*(a+b*arctanh(c*x))*l n(2/(-c*x+1))/c/e+d*(a+b*arctanh(c*x))^2*ln(2/(c*x+1))/e^2-d*(a+b*arctanh( c*x))^2*ln(2*c*(e*x+d)/(c*d+e)/(c*x+1))/e^2-b^2*polylog(2,1-2/(-c*x+1))/c/ e-b*d*(a+b*arctanh(c*x))*polylog(2,1-2/(c*x+1))/e^2+b*d*(a+b*arctanh(c*x)) *polylog(2,1-2*c*(e*x+d)/(c*d+e)/(c*x+1))/e^2-1/2*b^2*d*polylog(3,1-2/(c*x +1))/e^2+1/2*b^2*d*polylog(3,1-2*c*(e*x+d)/(c*d+e)/(c*x+1))/e^2
Result contains complex when optimal does not.
Time = 11.53 (sec) , antiderivative size = 1153, normalized size of antiderivative = 4.13 \[ \int \frac {x (a+b \text {arctanh}(c x))^2}{d+e x} \, dx =\text {Too large to display} \] Input:
Integrate[(x*(a + b*ArcTanh[c*x])^2)/(d + e*x),x]
Output:
(6*a^2*e*x - 6*a^2*d*Log[d + e*x] + (6*a*b*((-I)*c*d*Pi*ArcTanh[c*x] + 2*c *e*x*ArcTanh[c*x] - 2*c*d*ArcTanh[(c*d)/e]*ArcTanh[c*x] + c*d*ArcTanh[c*x] ^2 - e*ArcTanh[c*x]^2 + (Sqrt[1 - (c^2*d^2)/e^2]*e*ArcTanh[c*x]^2)/E^ArcTa nh[(c*d)/e] + 2*c*d*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] + I*c*d*Pi*L og[1 + E^(2*ArcTanh[c*x])] - 2*c*d*ArcTanh[(c*d)/e]*Log[1 - E^(-2*(ArcTanh [(c*d)/e] + ArcTanh[c*x]))] - 2*c*d*ArcTanh[c*x]*Log[1 - E^(-2*(ArcTanh[(c *d)/e] + ArcTanh[c*x]))] + e*Log[1 - c^2*x^2] + (I/2)*c*d*Pi*Log[1 - c^2*x ^2] + 2*c*d*ArcTanh[(c*d)/e]*Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] - c*d*PolyLog[2, -E^(-2*ArcTanh[c*x])] + c*d*PolyLog[2, E^(-2*(ArcTanh[(c* d)/e] + ArcTanh[c*x]))]))/c + (b^2*(-6*e*ArcTanh[c*x]^2 + 6*c*e*x*ArcTanh[ c*x]^2 + 8*c*d*ArcTanh[c*x]^3 - 4*e*ArcTanh[c*x]^3 + (4*Sqrt[1 - (c^2*d^2) /e^2]*e*ArcTanh[c*x]^3)/E^ArcTanh[(c*d)/e] - 12*e*ArcTanh[c*x]*Log[1 + E^( -2*ArcTanh[c*x])] + 6*c*d*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] + (6 *I)*c*d*Pi*ArcTanh[c*x]*Log[(E^(-ArcTanh[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 + (Sqrt[c*d + e]*E^ArcTanh[c*x])/Sqrt[-(c*d) + e]] - 6*c*d*ArcTanh[c*x]^2*Log[1 - E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] - 6*c*d*ArcTanh[c*x]^2*Log[1 + E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] - 6*c* d*ArcTanh[c*x]^2*Log[1 - E^(2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] - 12*c*d *ArcTanh[(c*d)/e]*ArcTanh[c*x]*Log[(I/2)*E^(-ArcTanh[(c*d)/e] - ArcTanh...
Time = 0.57 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, 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.
\(\displaystyle \int \frac {x (a+b \text {arctanh}(c x))^2}{d+e x} \, dx\) |
\(\Big \downarrow \) 6502 |
\(\displaystyle \int \left (\frac {(a+b \text {arctanh}(c x))^2}{e}-\frac {d (a+b \text {arctanh}(c x))^2}{e (d+e x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {b d \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{e^2}+\frac {b d (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{e^2}+\frac {d \log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2}{e^2}-\frac {d (a+b \text {arctanh}(c x))^2 \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e^2}+\frac {x (a+b \text {arctanh}(c x))^2}{e}+\frac {(a+b \text {arctanh}(c x))^2}{c e}-\frac {2 b \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c e}-\frac {b^2 d \operatorname {PolyLog}\left (3,1-\frac {2}{c x+1}\right )}{2 e^2}+\frac {b^2 d \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{2 e^2}-\frac {b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c e}\) |
Input:
Int[(x*(a + b*ArcTanh[c*x])^2)/(d + e*x),x]
Output:
(a + b*ArcTanh[c*x])^2/(c*e) + (x*(a + b*ArcTanh[c*x])^2)/e - (2*b*(a + b* ArcTanh[c*x])*Log[2/(1 - c*x)])/(c*e) + (d*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/e^2 - (d*(a + b*ArcTanh[c*x])^2*Log[(2*c*(d + e*x))/((c*d + e)*( 1 + c*x))])/e^2 - (b^2*PolyLog[2, 1 - 2/(1 - c*x)])/(c*e) - (b*d*(a + b*Ar cTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/e^2 + (b*d*(a + b*ArcTanh[c*x])*P olyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e^2 - (b^2*d*PolyLog [3, 1 - 2/(1 + c*x)])/(2*e^2) + (b^2*d*PolyLog[3, 1 - (2*c*(d + e*x))/((c* d + e)*(1 + c*x))])/(2*e^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 = 5.59 (sec) , antiderivative size = 13674, normalized size of antiderivative = 49.01
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(13674\) |
default | \(\text {Expression too large to display}\) | \(13674\) |
parts | \(\text {Expression too large to display}\) | \(13680\) |
Input:
int(x*(a+b*arctanh(c*x))^2/(e*x+d),x,method=_RETURNVERBOSE)
Output:
result too large to display
\[ \int \frac {x (a+b \text {arctanh}(c x))^2}{d+e x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x}{e x + d} \,d x } \] Input:
integrate(x*(a+b*arctanh(c*x))^2/(e*x+d),x, algorithm="fricas")
Output:
integral((b^2*x*arctanh(c*x)^2 + 2*a*b*x*arctanh(c*x) + a^2*x)/(e*x + d), x)
\[ \int \frac {x (a+b \text {arctanh}(c x))^2}{d+e x} \, dx=\int \frac {x \left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{2}}{d + e x}\, dx \] Input:
integrate(x*(a+b*atanh(c*x))**2/(e*x+d),x)
Output:
Integral(x*(a + b*atanh(c*x))**2/(d + e*x), x)
\[ \int \frac {x (a+b \text {arctanh}(c x))^2}{d+e x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x}{e x + d} \,d x } \] Input:
integrate(x*(a+b*arctanh(c*x))^2/(e*x+d),x, algorithm="maxima")
Output:
1/4*b^2*x*log(-c*x + 1)^2/e + a^2*(x/e - d*log(e*x + d)/e^2) - integrate(- 1/4*((b^2*c*e*x^2 - b^2*e*x)*log(c*x + 1)^2 + 4*(a*b*c*e*x^2 - a*b*e*x)*lo g(c*x + 1) - 2*((2*a*b*c*e + b^2*c*e)*x^2 + (b^2*c*d - 2*a*b*e)*x + (b^2*c *e*x^2 - b^2*e*x)*log(c*x + 1))*log(-c*x + 1))/(c*e^2*x^2 - d*e + (c*d*e - e^2)*x), x)
\[ \int \frac {x (a+b \text {arctanh}(c x))^2}{d+e x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x}{e x + d} \,d x } \] Input:
integrate(x*(a+b*arctanh(c*x))^2/(e*x+d),x, algorithm="giac")
Output:
integrate((b*arctanh(c*x) + a)^2*x/(e*x + d), x)
Timed out. \[ \int \frac {x (a+b \text {arctanh}(c x))^2}{d+e x} \, dx=\int \frac {x\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{d+e\,x} \,d x \] Input:
int((x*(a + b*atanh(c*x))^2)/(d + e*x),x)
Output:
int((x*(a + b*atanh(c*x))^2)/(d + e*x), x)
\[ \int \frac {x (a+b \text {arctanh}(c x))^2}{d+e x} \, dx=\frac {2 \left (\int \frac {\mathit {atanh} \left (c x \right ) x}{e x +d}d x \right ) a b \,e^{2}+\left (\int \frac {\mathit {atanh} \left (c x \right )^{2} x}{e x +d}d x \right ) b^{2} e^{2}-\mathrm {log}\left (e x +d \right ) a^{2} d +a^{2} e x}{e^{2}} \] Input:
int(x*(a+b*atanh(c*x))^2/(e*x+d),x)
Output:
(2*int((atanh(c*x)*x)/(d + e*x),x)*a*b*e**2 + int((atanh(c*x)**2*x)/(d + e *x),x)*b**2*e**2 - log(d + e*x)*a**2*d + a**2*e*x)/e**2