Integrand size = 28, antiderivative size = 122 \[ \int \frac {(d+e x) (a+b \text {arctanh}(c x))^2}{1-c^2 x^2} \, dx=\frac {d (a+b \text {arctanh}(c x))^3}{3 b c}-\frac {e (a+b \text {arctanh}(c x))^3}{3 b c^2}+\frac {e (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1-c x}\right )}{c^2}+\frac {b e (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c^2}-\frac {b^2 e \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 c^2} \] Output:
1/3*d*(a+b*arctanh(c*x))^3/b/c-1/3*e*(a+b*arctanh(c*x))^3/b/c^2+e*(a+b*arc tanh(c*x))^2*ln(2/(-c*x+1))/c^2+b*e*(a+b*arctanh(c*x))*polylog(2,1-2/(-c*x +1))/c^2-1/2*b^2*e*polylog(3,1-2/(-c*x+1))/c^2
Time = 0.60 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.58 \[ \int \frac {(d+e x) (a+b \text {arctanh}(c x))^2}{1-c^2 x^2} \, dx=\frac {6 a b c d \text {arctanh}(c x)^2+6 a b e \text {arctanh}(c x)^2+2 b^2 c d \text {arctanh}(c x)^3+2 b^2 e \text {arctanh}(c x)^3+12 a b e \text {arctanh}(c x) \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+6 b^2 e \text {arctanh}(c x)^2 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )-3 a^2 c d \log (1-c x)-3 a^2 e \log (1-c x)+3 a^2 c d \log (1+c x)-3 a^2 e \log (1+c x)-6 b e (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )-3 b^2 e \operatorname {PolyLog}\left (3,-e^{-2 \text {arctanh}(c x)}\right )}{6 c^2} \] Input:
Integrate[((d + e*x)*(a + b*ArcTanh[c*x])^2)/(1 - c^2*x^2),x]
Output:
(6*a*b*c*d*ArcTanh[c*x]^2 + 6*a*b*e*ArcTanh[c*x]^2 + 2*b^2*c*d*ArcTanh[c*x ]^3 + 2*b^2*e*ArcTanh[c*x]^3 + 12*a*b*e*ArcTanh[c*x]*Log[1 + E^(-2*ArcTanh [c*x])] + 6*b^2*e*ArcTanh[c*x]^2*Log[1 + E^(-2*ArcTanh[c*x])] - 3*a^2*c*d* Log[1 - c*x] - 3*a^2*e*Log[1 - c*x] + 3*a^2*c*d*Log[1 + c*x] - 3*a^2*e*Log [1 + c*x] - 6*b*e*(a + b*ArcTanh[c*x])*PolyLog[2, -E^(-2*ArcTanh[c*x])] - 3*b^2*e*PolyLog[3, -E^(-2*ArcTanh[c*x])])/(6*c^2)
Time = 0.60 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {6610, 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.
| \(\displaystyle \int \frac {(d+e x) (a+b \text {arctanh}(c x))^2}{1-c^2 x^2} \, dx\) |
| \(\Big \downarrow \) 6610 |
| \(\displaystyle \int \left (\frac {d (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}+\frac {e x (a+b \text {arctanh}(c x))^2}{1-c^2 x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b e \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{c^2}-\frac {e (a+b \text {arctanh}(c x))^3}{3 b c^2}+\frac {e \log \left (\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^2}{c^2}+\frac {d (a+b \text {arctanh}(c x))^3}{3 b c}-\frac {b^2 e \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 c^2}\) |
Input:
Int[((d + e*x)*(a + b*ArcTanh[c*x])^2)/(1 - c^2*x^2),x]
Output:
(d*(a + b*ArcTanh[c*x])^3)/(3*b*c) - (e*(a + b*ArcTanh[c*x])^3)/(3*b*c^2) + (e*(a + b*ArcTanh[c*x])^2*Log[2/(1 - c*x)])/c^2 + (b*e*(a + b*ArcTanh[c* x])*PolyLog[2, 1 - 2/(1 - c*x)])/c^2 - (b^2*e*PolyLog[3, 1 - 2/(1 - c*x)]) /(2*c^2)
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(m_.))/( (d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcTanh[c*x]) ^p/(d + e*x^2), (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && I GtQ[p, 0] && EqQ[c^2*d + e, 0] && IGtQ[m, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 18.46 (sec) , antiderivative size = 1435, normalized size of antiderivative = 11.76
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1435\) |
| default | \(\text {Expression too large to display}\) | \(1435\) |
| parts | \(\text {Expression too large to display}\) | \(1451\) |
Input:
int((e*x+d)*(a+b*arctanh(c*x))^2/(-c^2*x^2+1),x,method=_RETURNVERBOSE)
Output:
1/c*(-a^2/c*(1/2*(c*d+e)*ln(c*x-1)-1/2*(c*d-e)*ln(c*x+1))-b^2/c*(1/2*arcta nh(c*x)^2*ln(c*x-1)*c*d+1/2*arctanh(c*x)^2*ln(c*x-1)*e-1/2*arctanh(c*x)^2* ln(c*x+1)*c*d+1/2*arctanh(c*x)^2*ln(c*x+1)*e+(c*d-e)*arctanh(c*x)^2*ln((c* x+1)/(-c^2*x^2+1)^(1/2))-1/3*d*c*arctanh(c*x)^3+1/3*e*arctanh(c*x)^3-1/4*( 4*ln(2)*e-I*Pi*e*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+2*I*Pi*e*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*c*d*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1- (c*x+1)^2/(c^2*x^2-1)))^3+I*Pi*c*d*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*c*d*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+2*I*Pi*c*d-I*Pi*e*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)))+2*I*Pi*e *csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))^3-2*I*Pi*e*csgn(I/(1-(c*x+1)^2/(c^2*x^2 -1)))^2-2*I*Pi*c*d*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*c*d*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3+I*Pi*e*csgn(I*(c*x+1) /(-c^2*x^2+1)^(1/2))^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))+I*Pi*e*csgn(I*(c*x+1) ^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^3+I*Pi*e*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+2*I*P i*e+2*I*Pi*c*d*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))^3-2*I*Pi*c*d*csgn(I/(1-(c *x+1)^2/(c^2*x^2-1)))^2+I*Pi*e*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3-I*Pi*c*d...
\[ \int \frac {(d+e x) (a+b \text {arctanh}(c x))^2}{1-c^2 x^2} \, dx=\int { -\frac {{\left (e x + d\right )} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{c^{2} x^{2} - 1} \,d x } \] Input:
integrate((e*x+d)*(a+b*arctanh(c*x))^2/(-c^2*x^2+1),x, algorithm="fricas")
Output:
integral(-(a^2*e*x + a^2*d + (b^2*e*x + b^2*d)*arctanh(c*x)^2 + 2*(a*b*e*x + a*b*d)*arctanh(c*x))/(c^2*x^2 - 1), x)
\[ \int \frac {(d+e x) (a+b \text {arctanh}(c x))^2}{1-c^2 x^2} \, dx=- \int \frac {a^{2} d}{c^{2} x^{2} - 1}\, dx - \int \frac {a^{2} e x}{c^{2} x^{2} - 1}\, dx - \int \frac {b^{2} d \operatorname {atanh}^{2}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx - \int \frac {2 a b d \operatorname {atanh}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx - \int \frac {b^{2} e x \operatorname {atanh}^{2}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx - \int \frac {2 a b e x \operatorname {atanh}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx \] Input:
integrate((e*x+d)*(a+b*atanh(c*x))**2/(-c**2*x**2+1),x)
Output:
-Integral(a**2*d/(c**2*x**2 - 1), x) - Integral(a**2*e*x/(c**2*x**2 - 1), x) - Integral(b**2*d*atanh(c*x)**2/(c**2*x**2 - 1), x) - Integral(2*a*b*d* atanh(c*x)/(c**2*x**2 - 1), x) - Integral(b**2*e*x*atanh(c*x)**2/(c**2*x** 2 - 1), x) - Integral(2*a*b*e*x*atanh(c*x)/(c**2*x**2 - 1), x)
\[ \int \frac {(d+e x) (a+b \text {arctanh}(c x))^2}{1-c^2 x^2} \, dx=\int { -\frac {{\left (e x + d\right )} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{c^{2} x^{2} - 1} \,d x } \] Input:
integrate((e*x+d)*(a+b*arctanh(c*x))^2/(-c^2*x^2+1),x, algorithm="maxima")
Output:
a*b*d*(log(c*x + 1)/c - log(c*x - 1)/c)*arctanh(c*x) + 1/2*a^2*d*(log(c*x + 1)/c - log(c*x - 1)/c) - 1/4*(log(c*x + 1)^2 - 2*log(c*x + 1)*log(c*x - 1) + log(c*x - 1)^2)*a*b*d/c - 1/2*a^2*e*log(c^2*x^2 - 1)/c^2 + 1/24*(3*(c *d - e)*b^2*log(c*x + 1)*log(-c*x + 1)^2 - (c*d + e)*b^2*log(-c*x + 1)^3)/ c^2 - integrate(1/4*(4*a*b*c*e*x*log(c*x + 1) + (b^2*c*e*x + b^2*c*d)*log( c*x + 1)^2 - (4*a*b*c*e*x - ((c^2*d - 3*c*e)*b^2*x - (c*d + e)*b^2)*log(c* x + 1))*log(-c*x + 1))/(c^3*x^2 - c), x)
\[ \int \frac {(d+e x) (a+b \text {arctanh}(c x))^2}{1-c^2 x^2} \, dx=\int { -\frac {{\left (e x + d\right )} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{c^{2} x^{2} - 1} \,d x } \] Input:
integrate((e*x+d)*(a+b*arctanh(c*x))^2/(-c^2*x^2+1),x, algorithm="giac")
Output:
integrate(-(e*x + d)*(b*arctanh(c*x) + a)^2/(c^2*x^2 - 1), x)
Timed out. \[ \int \frac {(d+e x) (a+b \text {arctanh}(c x))^2}{1-c^2 x^2} \, dx=\int -\frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,\left (d+e\,x\right )}{c^2\,x^2-1} \,d x \] Input:
int(-((a + b*atanh(c*x))^2*(d + e*x))/(c^2*x^2 - 1),x)
Output:
int(-((a + b*atanh(c*x))^2*(d + e*x))/(c^2*x^2 - 1), x)
\[ \int \frac {(d+e x) (a+b \text {arctanh}(c x))^2}{1-c^2 x^2} \, dx=\frac {2 \mathit {atanh} \left (c x \right )^{3} b^{2} c d +6 \mathit {atanh} \left (c x \right )^{2} a b c d -12 \left (\int \frac {\mathit {atanh} \left (c x \right ) x}{c^{2} x^{2}-1}d x \right ) a b \,c^{2} e -6 \left (\int \frac {\mathit {atanh} \left (c x \right )^{2} x}{c^{2} x^{2}-1}d x \right ) b^{2} c^{2} e -3 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2} c d -3 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2} e +3 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2} c d -3 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2} e}{6 c^{2}} \] Input:
int((e*x+d)*(a+b*atanh(c*x))^2/(-c^2*x^2+1),x)
Output:
(2*atanh(c*x)**3*b**2*c*d + 6*atanh(c*x)**2*a*b*c*d - 12*int((atanh(c*x)*x )/(c**2*x**2 - 1),x)*a*b*c**2*e - 6*int((atanh(c*x)**2*x)/(c**2*x**2 - 1), x)*b**2*c**2*e - 3*log(c**2*x - c)*a**2*c*d - 3*log(c**2*x - c)*a**2*e + 3 *log(c**2*x + c)*a**2*c*d - 3*log(c**2*x + c)*a**2*e)/(6*c**2)