Integrand size = 23, antiderivative size = 104 \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^2} \, dx=\frac {(a+b \text {arctanh}(c+d x))^2}{d e^2}-\frac {(a+b \text {arctanh}(c+d x))^2}{d e^2 (c+d x)}+\frac {2 b (a+b \text {arctanh}(c+d x)) \log \left (2-\frac {2}{1+c+d x}\right )}{d e^2}-\frac {b^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c+d x}\right )}{d e^2} \]
(a+b*arctanh(d*x+c))^2/d/e^2-(a+b*arctanh(d*x+c))^2/d/e^2/(d*x+c)+2*b*(a+b *arctanh(d*x+c))*ln(2-2/(d*x+c+1))/d/e^2-b^2*polylog(2,-1+2/(d*x+c+1))/d/e ^2
Time = 0.26 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.21 \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^2} \, dx=\frac {b^2 (-1+c+d x) \text {arctanh}(c+d x)^2+2 b \text {arctanh}(c+d x) \left (-a+b (c+d x) \log \left (1-e^{-2 \text {arctanh}(c+d x)}\right )\right )+a \left (-a+2 b (c+d x) \log \left (\frac {c+d x}{\sqrt {1-(c+d x)^2}}\right )\right )-b^2 (c+d x) \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c+d x)}\right )}{d e^2 (c+d x)} \]
(b^2*(-1 + c + d*x)*ArcTanh[c + d*x]^2 + 2*b*ArcTanh[c + d*x]*(-a + b*(c + d*x)*Log[1 - E^(-2*ArcTanh[c + d*x])]) + a*(-a + 2*b*(c + d*x)*Log[(c + d *x)/Sqrt[1 - (c + d*x)^2]]) - b^2*(c + d*x)*PolyLog[2, E^(-2*ArcTanh[c + d *x])])/(d*e^2*(c + d*x))
Time = 0.57 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.91, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6657, 27, 6452, 6550, 6494, 2897}
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.
| \(\displaystyle \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^2} \, dx\) |
| \(\Big \downarrow \) 6657 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arctanh}(c+d x))^2}{e^2 (c+d x)^2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arctanh}(c+d x))^2}{(c+d x)^2}d(c+d x)}{d e^2}\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {2 b \int \frac {a+b \text {arctanh}(c+d x)}{(c+d x) \left (1-(c+d x)^2\right )}d(c+d x)-\frac {(a+b \text {arctanh}(c+d x))^2}{c+d x}}{d e^2}\) |
| \(\Big \downarrow \) 6550 |
| \(\displaystyle \frac {2 b \left (\int \frac {a+b \text {arctanh}(c+d x)}{(c+d x) (c+d x+1)}d(c+d x)+\frac {(a+b \text {arctanh}(c+d x))^2}{2 b}\right )-\frac {(a+b \text {arctanh}(c+d x))^2}{c+d x}}{d e^2}\) |
| \(\Big \downarrow \) 6494 |
| \(\displaystyle \frac {2 b \left (-b \int \frac {\log \left (2-\frac {2}{c+d x+1}\right )}{1-(c+d x)^2}d(c+d x)+\frac {(a+b \text {arctanh}(c+d x))^2}{2 b}+\log \left (2-\frac {2}{c+d x+1}\right ) (a+b \text {arctanh}(c+d x))\right )-\frac {(a+b \text {arctanh}(c+d x))^2}{c+d x}}{d e^2}\) |
| \(\Big \downarrow \) 2897 |
| \(\displaystyle \frac {2 b \left (\frac {(a+b \text {arctanh}(c+d x))^2}{2 b}+\log \left (2-\frac {2}{c+d x+1}\right ) (a+b \text {arctanh}(c+d x))-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {2}{c+d x+1}-1\right )\right )-\frac {(a+b \text {arctanh}(c+d x))^2}{c+d x}}{d e^2}\) |
(-((a + b*ArcTanh[c + d*x])^2/(c + d*x)) + 2*b*((a + b*ArcTanh[c + d*x])^2 /(2*b) + (a + b*ArcTanh[c + d*x])*Log[2 - 2/(1 + c + d*x)] - (b*PolyLog[2, -1 + 2/(1 + c + d*x)])/2))/(d*e^2)
3.1.19.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, x][[2]], Expon[Pq, x]]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : > Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x ], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 ] && IntegerQ[m])) && NeQ[m, -1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x _Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Simp[b*c*(p/d) Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2 - 2/(1 + e*(x/d))] /(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c ^2*d^2 - e^2, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*d*(p + 1)), x] + Simp[1/ d Int[(a + b*ArcTanh[c*x])^p/(x*(1 + c*x)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
Int[((a_.) + ArcTanh[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[(f*(x/d))^m*(a + b*ArcTanh[x])^p, x ], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(251\) vs. \(2(104)=208\).
Time = 0.29 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.42
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{2}}{e^{2} \left (d x +c \right )}+\frac {b^{2} \left (-\frac {\operatorname {arctanh}\left (d x +c \right )^{2}}{d x +c}-\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right )+2 \ln \left (d x +c \right ) \operatorname {arctanh}\left (d x +c \right )-\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right )+\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )+\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (d x +c -1\right )^{2}}{4}+\frac {\ln \left (d x +c +1\right )^{2}}{4}-\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{2}-\operatorname {dilog}\left (d x +c \right )-\operatorname {dilog}\left (d x +c +1\right )-\ln \left (d x +c \right ) \ln \left (d x +c +1\right )\right )}{e^{2}}+\frac {2 a b \left (-\frac {\operatorname {arctanh}\left (d x +c \right )}{d x +c}-\frac {\ln \left (d x +c -1\right )}{2}+\ln \left (d x +c \right )-\frac {\ln \left (d x +c +1\right )}{2}\right )}{e^{2}}}{d}\) | \(252\) |
| default | \(\frac {-\frac {a^{2}}{e^{2} \left (d x +c \right )}+\frac {b^{2} \left (-\frac {\operatorname {arctanh}\left (d x +c \right )^{2}}{d x +c}-\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right )+2 \ln \left (d x +c \right ) \operatorname {arctanh}\left (d x +c \right )-\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right )+\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )+\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (d x +c -1\right )^{2}}{4}+\frac {\ln \left (d x +c +1\right )^{2}}{4}-\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{2}-\operatorname {dilog}\left (d x +c \right )-\operatorname {dilog}\left (d x +c +1\right )-\ln \left (d x +c \right ) \ln \left (d x +c +1\right )\right )}{e^{2}}+\frac {2 a b \left (-\frac {\operatorname {arctanh}\left (d x +c \right )}{d x +c}-\frac {\ln \left (d x +c -1\right )}{2}+\ln \left (d x +c \right )-\frac {\ln \left (d x +c +1\right )}{2}\right )}{e^{2}}}{d}\) | \(252\) |
| parts | \(-\frac {a^{2}}{e^{2} \left (d x +c \right ) d}+\frac {b^{2} \left (-\frac {\operatorname {arctanh}\left (d x +c \right )^{2}}{d x +c}-\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right )+2 \ln \left (d x +c \right ) \operatorname {arctanh}\left (d x +c \right )-\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right )+\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )+\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (d x +c -1\right )^{2}}{4}+\frac {\ln \left (d x +c +1\right )^{2}}{4}-\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{2}-\operatorname {dilog}\left (d x +c \right )-\operatorname {dilog}\left (d x +c +1\right )-\ln \left (d x +c \right ) \ln \left (d x +c +1\right )\right )}{e^{2} d}+\frac {2 a b \left (-\frac {\operatorname {arctanh}\left (d x +c \right )}{d x +c}-\frac {\ln \left (d x +c -1\right )}{2}+\ln \left (d x +c \right )-\frac {\ln \left (d x +c +1\right )}{2}\right )}{e^{2} d}\) | \(257\) |
1/d*(-a^2/e^2/(d*x+c)+b^2/e^2*(-1/(d*x+c)*arctanh(d*x+c)^2-arctanh(d*x+c)* ln(d*x+c-1)+2*ln(d*x+c)*arctanh(d*x+c)-arctanh(d*x+c)*ln(d*x+c+1)+dilog(1/ 2*d*x+1/2*c+1/2)+1/2*ln(d*x+c-1)*ln(1/2*d*x+1/2*c+1/2)-1/4*ln(d*x+c-1)^2+1 /4*ln(d*x+c+1)^2-1/2*(ln(d*x+c+1)-ln(1/2*d*x+1/2*c+1/2))*ln(-1/2*d*x-1/2*c +1/2)-dilog(d*x+c)-dilog(d*x+c+1)-ln(d*x+c)*ln(d*x+c+1))+2*a*b/e^2*(-1/(d* x+c)*arctanh(d*x+c)-1/2*ln(d*x+c-1)+ln(d*x+c)-1/2*ln(d*x+c+1)))
\[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^2} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )}^{2}}{{\left (d e x + c e\right )}^{2}} \,d x } \]
integral((b^2*arctanh(d*x + c)^2 + 2*a*b*arctanh(d*x + c) + a^2)/(d^2*e^2* x^2 + 2*c*d*e^2*x + c^2*e^2), x)
\[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^2} \, dx=\frac {\int \frac {a^{2}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx}{e^{2}} \]
(Integral(a**2/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral(b**2*atanh(c + d*x)**2/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral(2*a*b*atanh(c + d*x)/( c**2 + 2*c*d*x + d**2*x**2), x))/e**2
\[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^2} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )}^{2}}{{\left (d e x + c e\right )}^{2}} \,d x } \]
-(d*(log(d*x + c + 1)/(d^2*e^2) - 2*log(d*x + c)/(d^2*e^2) + log(d*x + c - 1)/(d^2*e^2)) + 2*arctanh(d*x + c)/(d^2*e^2*x + c*d*e^2))*a*b - 1/4*b^2*( log(-d*x - c + 1)^2/(d^2*e^2*x + c*d*e^2) + integrate(-((d*x + c - 1)*log( d*x + c + 1)^2 + 2*(d*x - (d*x + c - 1)*log(d*x + c + 1) + c)*log(-d*x - c + 1))/(d^3*e^2*x^3 + c^3*e^2 - c^2*e^2 + (3*c*d^2*e^2 - d^2*e^2)*x^2 + (3 *c^2*d*e^2 - 2*c*d*e^2)*x), x)) - a^2/(d^2*e^2*x + c*d*e^2)
\[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^2} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )}^{2}}{{\left (d e x + c e\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c+d\,x\right )\right )}^2}{{\left (c\,e+d\,e\,x\right )}^2} \,d x \]