Integrand size = 23, antiderivative size = 119 \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^3} \, dx=-\frac {b (a+b \text {arctanh}(c+d x))}{d e^3 (c+d x)}+\frac {(a+b \text {arctanh}(c+d x))^2}{2 d e^3}-\frac {(a+b \text {arctanh}(c+d x))^2}{2 d e^3 (c+d x)^2}+\frac {b^2 \log (c+d x)}{d e^3}-\frac {b^2 \log \left (1-(c+d x)^2\right )}{2 d e^3} \] Output:
-b*(a+b*arctanh(d*x+c))/d/e^3/(d*x+c)+1/2*(a+b*arctanh(d*x+c))^2/d/e^3-1/2 *(a+b*arctanh(d*x+c))^2/d/e^3/(d*x+c)^2+b^2*ln(d*x+c)/d/e^3-1/2*b^2*ln(1-( d*x+c)^2)/d/e^3
Time = 0.18 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.14 \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^3} \, dx=\frac {-\frac {a^2}{(c+d x)^2}-\frac {2 a b}{c+d x}-\frac {2 b (a+b (c+d x)) \text {arctanh}(c+d x)}{(c+d x)^2}+\frac {b^2 \left (-1+c^2+2 c d x+d^2 x^2\right ) \text {arctanh}(c+d x)^2}{(c+d x)^2}-b (a+b) \log (1-c-d x)+2 b^2 \log (c+d x)+(a-b) b \log (1+c+d x)}{2 d e^3} \] Input:
Integrate[(a + b*ArcTanh[c + d*x])^2/(c*e + d*e*x)^3,x]
Output:
(-(a^2/(c + d*x)^2) - (2*a*b)/(c + d*x) - (2*b*(a + b*(c + d*x))*ArcTanh[c + d*x])/(c + d*x)^2 + (b^2*(-1 + c^2 + 2*c*d*x + d^2*x^2)*ArcTanh[c + d*x ]^2)/(c + d*x)^2 - b*(a + b)*Log[1 - c - d*x] + 2*b^2*Log[c + d*x] + (a - b)*b*Log[1 + c + d*x])/(2*d*e^3)
Time = 0.63 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.82, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {6657, 27, 6452, 6544, 6452, 243, 47, 14, 16, 6510}
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.
| \(\displaystyle \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^3} \, dx\) |
| \(\Big \downarrow \) 6657 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arctanh}(c+d x))^2}{e^3 (c+d x)^3}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arctanh}(c+d x))^2}{(c+d x)^3}d(c+d x)}{d e^3}\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {b \int \frac {a+b \text {arctanh}(c+d x)}{(c+d x)^2 \left (1-(c+d x)^2\right )}d(c+d x)-\frac {(a+b \text {arctanh}(c+d x))^2}{2 (c+d x)^2}}{d e^3}\) |
| \(\Big \downarrow \) 6544 |
| \(\displaystyle \frac {b \left (\int \frac {a+b \text {arctanh}(c+d x)}{(c+d x)^2}d(c+d x)+\int \frac {a+b \text {arctanh}(c+d x)}{1-(c+d x)^2}d(c+d x)\right )-\frac {(a+b \text {arctanh}(c+d x))^2}{2 (c+d x)^2}}{d e^3}\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {b \left (\int \frac {a+b \text {arctanh}(c+d x)}{1-(c+d x)^2}d(c+d x)+b \int \frac {1}{(c+d x) \left (1-(c+d x)^2\right )}d(c+d x)-\frac {a+b \text {arctanh}(c+d x)}{c+d x}\right )-\frac {(a+b \text {arctanh}(c+d x))^2}{2 (c+d x)^2}}{d e^3}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {b \left (\int \frac {a+b \text {arctanh}(c+d x)}{1-(c+d x)^2}d(c+d x)+\frac {1}{2} b \int \frac {1}{(-c-d x+1) (c+d x)^2}d(c+d x)^2-\frac {a+b \text {arctanh}(c+d x)}{c+d x}\right )-\frac {(a+b \text {arctanh}(c+d x))^2}{2 (c+d x)^2}}{d e^3}\) |
| \(\Big \downarrow \) 47 |
| \(\displaystyle \frac {b \left (\int \frac {a+b \text {arctanh}(c+d x)}{1-(c+d x)^2}d(c+d x)+\frac {1}{2} b \left (\int \frac {1}{-c-d x+1}d(c+d x)^2+\int \frac {1}{(c+d x)^2}d(c+d x)^2\right )-\frac {a+b \text {arctanh}(c+d x)}{c+d x}\right )-\frac {(a+b \text {arctanh}(c+d x))^2}{2 (c+d x)^2}}{d e^3}\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle \frac {b \left (\int \frac {a+b \text {arctanh}(c+d x)}{1-(c+d x)^2}d(c+d x)+\frac {1}{2} b \left (\int \frac {1}{-c-d x+1}d(c+d x)^2+\log \left ((c+d x)^2\right )\right )-\frac {a+b \text {arctanh}(c+d x)}{c+d x}\right )-\frac {(a+b \text {arctanh}(c+d x))^2}{2 (c+d x)^2}}{d e^3}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {b \left (\int \frac {a+b \text {arctanh}(c+d x)}{1-(c+d x)^2}d(c+d x)-\frac {a+b \text {arctanh}(c+d x)}{c+d x}+\frac {1}{2} b \left (\log \left ((c+d x)^2\right )-\log (-c-d x+1)\right )\right )-\frac {(a+b \text {arctanh}(c+d x))^2}{2 (c+d x)^2}}{d e^3}\) |
| \(\Big \downarrow \) 6510 |
| \(\displaystyle \frac {b \left (\frac {(a+b \text {arctanh}(c+d x))^2}{2 b}-\frac {a+b \text {arctanh}(c+d x)}{c+d x}+\frac {1}{2} b \left (\log \left ((c+d x)^2\right )-\log (-c-d x+1)\right )\right )-\frac {(a+b \text {arctanh}(c+d x))^2}{2 (c+d x)^2}}{d e^3}\) |
Input:
Int[(a + b*ArcTanh[c + d*x])^2/(c*e + d*e*x)^3,x]
Output:
(-1/2*(a + b*ArcTanh[c + d*x])^2/(c + d*x)^2 + b*(-((a + b*ArcTanh[c + d*x ])/(c + d*x)) + (a + b*ArcTanh[c + d*x])^2/(2*b) + (b*(-Log[1 - c - d*x] + Log[(c + d*x)^2]))/2))/(d*e^3)
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c - a*d) Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[1/(c + d*x), x ], x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b , c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( e_.)*(x_)^2), x_Symbol] :> Simp[1/d Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x ], x] - Simp[e/(d*f^2) Int[(f*x)^(m + 2)*((a + b*ArcTanh[c*x])^p/(d + e*x ^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
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. \(237\) vs. \(2(113)=226\).
Time = 0.90 (sec) , antiderivative size = 238, normalized size of antiderivative = 2.00
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{2}}{2 e^{3} \left (d x +c \right )^{2}}+\frac {b^{2} \left (-\frac {\operatorname {arctanh}\left (d x +c \right )^{2}}{2 \left (d x +c \right )^{2}}-\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right )}{2}+\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right )}{2}-\frac {\operatorname {arctanh}\left (d x +c \right )}{d x +c}+\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (d x +c -1\right )^{2}}{8}-\frac {\ln \left (d x +c -1\right )}{2}+\ln \left (d x +c \right )-\frac {\ln \left (d x +c +1\right )}{2}-\frac {\ln \left (d x +c +1\right )^{2}}{8}+\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{4}\right )}{e^{3}}+\frac {2 b a \left (-\frac {\operatorname {arctanh}\left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {\ln \left (d x +c -1\right )}{4}+\frac {\ln \left (d x +c +1\right )}{4}-\frac {1}{2 \left (d x +c \right )}\right )}{e^{3}}}{d}\) | \(238\) |
| default | \(\frac {-\frac {a^{2}}{2 e^{3} \left (d x +c \right )^{2}}+\frac {b^{2} \left (-\frac {\operatorname {arctanh}\left (d x +c \right )^{2}}{2 \left (d x +c \right )^{2}}-\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right )}{2}+\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right )}{2}-\frac {\operatorname {arctanh}\left (d x +c \right )}{d x +c}+\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (d x +c -1\right )^{2}}{8}-\frac {\ln \left (d x +c -1\right )}{2}+\ln \left (d x +c \right )-\frac {\ln \left (d x +c +1\right )}{2}-\frac {\ln \left (d x +c +1\right )^{2}}{8}+\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{4}\right )}{e^{3}}+\frac {2 b a \left (-\frac {\operatorname {arctanh}\left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {\ln \left (d x +c -1\right )}{4}+\frac {\ln \left (d x +c +1\right )}{4}-\frac {1}{2 \left (d x +c \right )}\right )}{e^{3}}}{d}\) | \(238\) |
| parts | \(-\frac {a^{2}}{2 e^{3} \left (d x +c \right )^{2} d}+\frac {b^{2} \left (-\frac {\operatorname {arctanh}\left (d x +c \right )^{2}}{2 \left (d x +c \right )^{2}}-\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right )}{2}+\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right )}{2}-\frac {\operatorname {arctanh}\left (d x +c \right )}{d x +c}+\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (d x +c -1\right )^{2}}{8}-\frac {\ln \left (d x +c -1\right )}{2}+\ln \left (d x +c \right )-\frac {\ln \left (d x +c +1\right )}{2}-\frac {\ln \left (d x +c +1\right )^{2}}{8}+\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{4}\right )}{e^{3} d}+\frac {2 b a \left (-\frac {\operatorname {arctanh}\left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {\ln \left (d x +c -1\right )}{4}+\frac {\ln \left (d x +c +1\right )}{4}-\frac {1}{2 \left (d x +c \right )}\right )}{e^{3} d}\) | \(243\) |
| parallelrisch | \(-\frac {2 a b c \,d^{3} x +2 a^{2} d^{2} c +3 a b \,c^{2} d^{2}-a b \,d^{4} x^{2}-8 x \,\operatorname {arctanh}\left (d x +c \right ) a b \,c^{2} d^{3}-4 x^{2} \operatorname {arctanh}\left (d x +c \right ) a b c \,d^{4}+4 \,\operatorname {arctanh}\left (d x +c \right ) b^{2} c^{3} d^{2}+4 \,\operatorname {arctanh}\left (d x +c \right ) b^{2} c^{2} d^{2}+4 \ln \left (d x +c -1\right ) b^{2} c^{3} d^{2}-4 \ln \left (d x +c \right ) b^{2} c^{3} d^{2}-2 \operatorname {arctanh}\left (d x +c \right )^{2} b^{2} c^{3} d^{2}+2 b^{2} \operatorname {arctanh}\left (d x +c \right )^{2} c \,d^{2}-4 b^{2} c^{2} \operatorname {arctanh}\left (d x +c \right )^{2} x \,d^{3}-2 d^{4} b^{2} \operatorname {arctanh}\left (d x +c \right )^{2} x^{2} c -4 \ln \left (d x +c \right ) x^{2} b^{2} c \,d^{4}+8 \ln \left (d x +c -1\right ) x \,b^{2} c^{2} d^{3}-8 \ln \left (d x +c \right ) x \,b^{2} c^{2} d^{3}+4 \ln \left (d x +c -1\right ) x^{2} b^{2} c \,d^{4}+8 x \,\operatorname {arctanh}\left (d x +c \right ) b^{2} c^{2} d^{3}+4 x \,\operatorname {arctanh}\left (d x +c \right ) b^{2} c \,d^{3}+4 x^{2} \operatorname {arctanh}\left (d x +c \right ) b^{2} c \,d^{4}-4 \,\operatorname {arctanh}\left (d x +c \right ) a b \,c^{3} d^{2}+4 \,\operatorname {arctanh}\left (d x +c \right ) a b c \,d^{2}}{4 \left (d x +c \right )^{2} c \,d^{3} e^{3}}\) | \(393\) |
| risch | \(\frac {b^{2} \left (d^{2} x^{2}+2 c d x +c^{2}-1\right ) \ln \left (d x +c +1\right )^{2}}{8 e^{3} \left (d x +c \right )^{2} d}-\frac {b \left (b \,d^{2} x^{2} \ln \left (-d x -c +1\right )+2 b d x \ln \left (-d x -c +1\right ) c +\ln \left (-d x -c +1\right ) b \,c^{2}+2 b d x +2 b c -b \ln \left (-d x -c +1\right )+2 a \right ) \ln \left (d x +c +1\right )}{4 e^{3} \left (d x +c \right )^{2} d}+\frac {-4 a^{2}-8 a b d x -8 a b c -b^{2} \ln \left (-d x -c +1\right )^{2}+4 b^{2} d x \ln \left (-d x -c +1\right )-8 b a \ln \left (-d x -c +1\right ) c d x -4 b a \ln \left (-d x -c +1\right ) d^{2} x^{2}+2 b^{2} \ln \left (-d x -c +1\right )^{2} c d x -8 b^{2} \ln \left (-d x -c +1\right ) c d x +b^{2} \ln \left (-d x -c +1\right )^{2} c^{2}-4 b^{2} \ln \left (-d x -c +1\right ) c^{2}+4 b^{2} \ln \left (-d x -c +1\right ) c +4 b a \ln \left (-d x -c +1\right )-4 b a \ln \left (-d x -c +1\right ) c^{2}+b^{2} \ln \left (-d x -c +1\right )^{2} d^{2} x^{2}-4 b^{2} \ln \left (-d x -c +1\right ) d^{2} x^{2}+4 \ln \left (-d x -c -1\right ) a b \,c^{2}+8 \ln \left (-d x -c -1\right ) a b c d x -4 \ln \left (-d x -c -1\right ) b^{2} c^{2}+8 \ln \left (d x +c \right ) b^{2} c^{2}+4 \ln \left (-d x -c -1\right ) a b \,d^{2} x^{2}-8 \ln \left (-d x -c -1\right ) b^{2} c d x +16 \ln \left (d x +c \right ) b^{2} c d x -4 \ln \left (-d x -c -1\right ) b^{2} d^{2} x^{2}+8 \ln \left (d x +c \right ) b^{2} d^{2} x^{2}}{8 e^{3} \left (d x +c \right )^{2} d}\) | \(568\) |
Input:
int((a+b*arctanh(d*x+c))^2/(d*e*x+c*e)^3,x,method=_RETURNVERBOSE)
Output:
1/d*(-1/2*a^2/e^3/(d*x+c)^2+b^2/e^3*(-1/2/(d*x+c)^2*arctanh(d*x+c)^2-1/2*a rctanh(d*x+c)*ln(d*x+c-1)+1/2*arctanh(d*x+c)*ln(d*x+c+1)-1/(d*x+c)*arctanh (d*x+c)+1/4*ln(d*x+c-1)*ln(1/2*d*x+1/2*c+1/2)-1/8*ln(d*x+c-1)^2-1/2*ln(d*x +c-1)+ln(d*x+c)-1/2*ln(d*x+c+1)-1/8*ln(d*x+c+1)^2+1/4*(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))+2*b*a/e^3*(-1/2/(d*x+c)^2*arctanh( d*x+c)-1/4*ln(d*x+c-1)+1/4*ln(d*x+c+1)-1/2/(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 270 vs. \(2 (113) = 226\).
Time = 0.12 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.27 \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^3} \, dx=-\frac {8 \, a b d x + 8 \, a b c - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - b^{2}\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )^{2} + 4 \, a^{2} - 4 \, {\left ({\left (a b - b^{2}\right )} d^{2} x^{2} + 2 \, {\left (a b - b^{2}\right )} c d x + {\left (a b - b^{2}\right )} c^{2}\right )} \log \left (d x + c + 1\right ) - 8 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (d x + c\right ) + 4 \, {\left ({\left (a b + b^{2}\right )} d^{2} x^{2} + 2 \, {\left (a b + b^{2}\right )} c d x + {\left (a b + b^{2}\right )} c^{2}\right )} \log \left (d x + c - 1\right ) + 4 \, {\left (b^{2} d x + b^{2} c + a b\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{8 \, {\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} \] Input:
integrate((a+b*arctanh(d*x+c))^2/(d*e*x+c*e)^3,x, algorithm="fricas")
Output:
-1/8*(8*a*b*d*x + 8*a*b*c - (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - b^2)*lo g(-(d*x + c + 1)/(d*x + c - 1))^2 + 4*a^2 - 4*((a*b - b^2)*d^2*x^2 + 2*(a* b - b^2)*c*d*x + (a*b - b^2)*c^2)*log(d*x + c + 1) - 8*(b^2*d^2*x^2 + 2*b^ 2*c*d*x + b^2*c^2)*log(d*x + c) + 4*((a*b + b^2)*d^2*x^2 + 2*(a*b + b^2)*c *d*x + (a*b + b^2)*c^2)*log(d*x + c - 1) + 4*(b^2*d*x + b^2*c + a*b)*log(- (d*x + c + 1)/(d*x + c - 1)))/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3)
Leaf count of result is larger than twice the leaf count of optimal. 1102 vs. \(2 (100) = 200\).
Time = 1.54 (sec) , antiderivative size = 1102, normalized size of antiderivative = 9.26 \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^3} \, dx =\text {Too large to display} \] Input:
integrate((a+b*atanh(d*x+c))**2/(d*e*x+c*e)**3,x)
Output:
Piecewise((-a**2/(2*c**2*d*e**3 + 4*c*d**2*e**3*x + 2*d**3*e**3*x**2) + 2* a*b*c**2*atanh(c + d*x)/(2*c**2*d*e**3 + 4*c*d**2*e**3*x + 2*d**3*e**3*x** 2) + 4*a*b*c*d*x*atanh(c + d*x)/(2*c**2*d*e**3 + 4*c*d**2*e**3*x + 2*d**3* e**3*x**2) - 2*a*b*c/(2*c**2*d*e**3 + 4*c*d**2*e**3*x + 2*d**3*e**3*x**2) + 2*a*b*d**2*x**2*atanh(c + d*x)/(2*c**2*d*e**3 + 4*c*d**2*e**3*x + 2*d**3 *e**3*x**2) - 2*a*b*d*x/(2*c**2*d*e**3 + 4*c*d**2*e**3*x + 2*d**3*e**3*x** 2) - 2*a*b*atanh(c + d*x)/(2*c**2*d*e**3 + 4*c*d**2*e**3*x + 2*d**3*e**3*x **2) + 2*b**2*c**2*log(c/d + x)/(2*c**2*d*e**3 + 4*c*d**2*e**3*x + 2*d**3* e**3*x**2) - 2*b**2*c**2*log(c/d + x + 1/d)/(2*c**2*d*e**3 + 4*c*d**2*e**3 *x + 2*d**3*e**3*x**2) + b**2*c**2*atanh(c + d*x)**2/(2*c**2*d*e**3 + 4*c* d**2*e**3*x + 2*d**3*e**3*x**2) + 2*b**2*c**2*atanh(c + d*x)/(2*c**2*d*e** 3 + 4*c*d**2*e**3*x + 2*d**3*e**3*x**2) + 4*b**2*c*d*x*log(c/d + x)/(2*c** 2*d*e**3 + 4*c*d**2*e**3*x + 2*d**3*e**3*x**2) - 4*b**2*c*d*x*log(c/d + x + 1/d)/(2*c**2*d*e**3 + 4*c*d**2*e**3*x + 2*d**3*e**3*x**2) + 2*b**2*c*d*x *atanh(c + d*x)**2/(2*c**2*d*e**3 + 4*c*d**2*e**3*x + 2*d**3*e**3*x**2) + 4*b**2*c*d*x*atanh(c + d*x)/(2*c**2*d*e**3 + 4*c*d**2*e**3*x + 2*d**3*e**3 *x**2) - 2*b**2*c*atanh(c + d*x)/(2*c**2*d*e**3 + 4*c*d**2*e**3*x + 2*d**3 *e**3*x**2) + 2*b**2*d**2*x**2*log(c/d + x)/(2*c**2*d*e**3 + 4*c*d**2*e**3 *x + 2*d**3*e**3*x**2) - 2*b**2*d**2*x**2*log(c/d + x + 1/d)/(2*c**2*d*e** 3 + 4*c*d**2*e**3*x + 2*d**3*e**3*x**2) + b**2*d**2*x**2*atanh(c + d*x)...
Leaf count of result is larger than twice the leaf count of optimal. 329 vs. \(2 (113) = 226\).
Time = 0.04 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.76 \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^3} \, dx=-\frac {1}{2} \, {\left (d {\left (\frac {2}{d^{3} e^{3} x + c d^{2} e^{3}} - \frac {\log \left (d x + c + 1\right )}{d^{2} e^{3}} + \frac {\log \left (d x + c - 1\right )}{d^{2} e^{3}}\right )} + \frac {2 \, \operatorname {artanh}\left (d x + c\right )}{d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}}\right )} a b - \frac {1}{8} \, {\left (d^{2} {\left (\frac {\log \left (d x + c + 1\right )^{2} - 2 \, \log \left (d x + c + 1\right ) \log \left (d x + c - 1\right ) + \log \left (d x + c - 1\right )^{2} + 4 \, \log \left (d x + c - 1\right )}{d^{3} e^{3}} + \frac {4 \, \log \left (d x + c + 1\right )}{d^{3} e^{3}} - \frac {8 \, \log \left (d x + c\right )}{d^{3} e^{3}}\right )} + 4 \, d {\left (\frac {2}{d^{3} e^{3} x + c d^{2} e^{3}} - \frac {\log \left (d x + c + 1\right )}{d^{2} e^{3}} + \frac {\log \left (d x + c - 1\right )}{d^{2} e^{3}}\right )} \operatorname {artanh}\left (d x + c\right )\right )} b^{2} - \frac {b^{2} \operatorname {artanh}\left (d x + c\right )^{2}}{2 \, {\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} - \frac {a^{2}}{2 \, {\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} \] Input:
integrate((a+b*arctanh(d*x+c))^2/(d*e*x+c*e)^3,x, algorithm="maxima")
Output:
-1/2*(d*(2/(d^3*e^3*x + c*d^2*e^3) - log(d*x + c + 1)/(d^2*e^3) + log(d*x + c - 1)/(d^2*e^3)) + 2*arctanh(d*x + c)/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^ 2*d*e^3))*a*b - 1/8*(d^2*((log(d*x + c + 1)^2 - 2*log(d*x + c + 1)*log(d*x + c - 1) + log(d*x + c - 1)^2 + 4*log(d*x + c - 1))/(d^3*e^3) + 4*log(d*x + c + 1)/(d^3*e^3) - 8*log(d*x + c)/(d^3*e^3)) + 4*d*(2/(d^3*e^3*x + c*d^ 2*e^3) - log(d*x + c + 1)/(d^2*e^3) + log(d*x + c - 1)/(d^2*e^3))*arctanh( d*x + c))*b^2 - 1/2*b^2*arctanh(d*x + c)^2/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3) - 1/2*a^2/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3)
Leaf count of result is larger than twice the leaf count of optimal. 375 vs. \(2 (113) = 226\).
Time = 0.14 (sec) , antiderivative size = 375, normalized size of antiderivative = 3.15 \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^3} \, dx=\frac {1}{4} \, {\left (\frac {{\left (d x + c + 1\right )} b^{2} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )^{2}}{{\left (\frac {{\left (d x + c + 1\right )}^{2} d^{2} e^{3}}{{\left (d x + c - 1\right )}^{2}} + \frac {2 \, {\left (d x + c + 1\right )} d^{2} e^{3}}{d x + c - 1} + d^{2} e^{3}\right )} {\left (d x + c - 1\right )}} + \frac {2 \, {\left (\frac {2 \, {\left (d x + c + 1\right )} a b}{d x + c - 1} + \frac {{\left (d x + c + 1\right )} b^{2}}{d x + c - 1} + b^{2}\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{\frac {{\left (d x + c + 1\right )}^{2} d^{2} e^{3}}{{\left (d x + c - 1\right )}^{2}} + \frac {2 \, {\left (d x + c + 1\right )} d^{2} e^{3}}{d x + c - 1} + d^{2} e^{3}} + \frac {4 \, {\left (\frac {{\left (d x + c + 1\right )} a^{2}}{d x + c - 1} + \frac {{\left (d x + c + 1\right )} a b}{d x + c - 1} + a b\right )}}{\frac {{\left (d x + c + 1\right )}^{2} d^{2} e^{3}}{{\left (d x + c - 1\right )}^{2}} + \frac {2 \, {\left (d x + c + 1\right )} d^{2} e^{3}}{d x + c - 1} + d^{2} e^{3}} + \frac {2 \, b^{2} \log \left (-\frac {d x + c + 1}{d x + c - 1} - 1\right )}{d^{2} e^{3}} - \frac {2 \, b^{2} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{d^{2} e^{3}}\right )} {\left ({\left (c + 1\right )} d - {\left (c - 1\right )} d\right )} \] Input:
integrate((a+b*arctanh(d*x+c))^2/(d*e*x+c*e)^3,x, algorithm="giac")
Output:
1/4*((d*x + c + 1)*b^2*log(-(d*x + c + 1)/(d*x + c - 1))^2/(((d*x + c + 1) ^2*d^2*e^3/(d*x + c - 1)^2 + 2*(d*x + c + 1)*d^2*e^3/(d*x + c - 1) + d^2*e ^3)*(d*x + c - 1)) + 2*(2*(d*x + c + 1)*a*b/(d*x + c - 1) + (d*x + c + 1)* b^2/(d*x + c - 1) + b^2)*log(-(d*x + c + 1)/(d*x + c - 1))/((d*x + c + 1)^ 2*d^2*e^3/(d*x + c - 1)^2 + 2*(d*x + c + 1)*d^2*e^3/(d*x + c - 1) + d^2*e^ 3) + 4*((d*x + c + 1)*a^2/(d*x + c - 1) + (d*x + c + 1)*a*b/(d*x + c - 1) + a*b)/((d*x + c + 1)^2*d^2*e^3/(d*x + c - 1)^2 + 2*(d*x + c + 1)*d^2*e^3/ (d*x + c - 1) + d^2*e^3) + 2*b^2*log(-(d*x + c + 1)/(d*x + c - 1) - 1)/(d^ 2*e^3) - 2*b^2*log(-(d*x + c + 1)/(d*x + c - 1))/(d^2*e^3))*((c + 1)*d - ( c - 1)*d)
Time = 5.74 (sec) , antiderivative size = 776, normalized size of antiderivative = 6.52 \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^3} \, dx={\ln \left (1-d\,x-c\right )}^2\,\left (\frac {b^2}{8\,d\,e^3}-\frac {b^2}{2\,d\,\left (4\,c^2\,e^3+8\,c\,d\,e^3\,x+4\,d^2\,e^3\,x^2\right )}\right )+{\ln \left (c+d\,x+1\right )}^2\,\left (\frac {b^2}{8\,d\,e^3}-\frac {b^2}{8\,d^2\,e^3\,\left (2\,c\,x+d\,x^2+\frac {c^2}{d}\right )}\right )+\ln \left (1-d\,x-c\right )\,\left (\ln \left (c+d\,x+1\right )\,\left (\frac {b^2}{2\,d\,\left (2\,c^2\,e^3+4\,c\,d\,e^3\,x+2\,d^2\,e^3\,x^2\right )}-\frac {b^2\,\left (c^2+2\,c\,d\,x+d^2\,x^2\right )}{2\,d\,\left (2\,c^2\,e^3+4\,c\,d\,e^3\,x+2\,d^2\,e^3\,x^2\right )}\right )+\frac {b^2}{2\,d\,\left (4\,c^2\,e^3+8\,c\,d\,e^3\,x+4\,d^2\,e^3\,x^2\right )}+\frac {b\,\left (4\,a-b\right )}{2\,d\,\left (4\,c^2\,e^3+8\,c\,d\,e^3\,x+4\,d^2\,e^3\,x^2\right )}-\frac {b^2\,\left (x\,\left (4\,c\,d-d+d\,\left (2\,c-1\right )\right )-c+c^2+c\,\left (2\,c-1\right )+3\,d^2\,x^2+1\right )}{2\,d\,\left (4\,c^2\,e^3+8\,c\,d\,e^3\,x+4\,d^2\,e^3\,x^2\right )}+\frac {b^2\,\left (x\,\left (2\,d\,e^3+d\,\left (4\,c\,e^3+2\,e^3\right )+8\,c\,d\,e^3\right )+2\,c\,e^3+2\,e^3+c\,\left (4\,c\,e^3+2\,e^3\right )+2\,c^2\,e^3+6\,d^2\,e^3\,x^2\right )}{4\,d\,e^3\,\left (4\,c^2\,e^3+8\,c\,d\,e^3\,x+4\,d^2\,e^3\,x^2\right )}\right )-\frac {\frac {a^2+2\,b\,c\,a}{2\,d}+a\,b\,x}{c^2\,e^3+2\,c\,d\,e^3\,x+d^2\,e^3\,x^2}-\frac {\ln \left (c+d\,x+1\right )\,\left (x\,\left (\frac {2\,b^2\,c+b^2}{4\,d\,e^3}+\frac {b^2\,c}{4\,d\,e^3}-\frac {b^2\,\left (3\,c-1\right )}{4\,d\,e^3}\right )+\frac {b^2\,c^2+b^2\,c+b^2+4\,a\,b}{8\,d^2\,e^3}-\frac {b^2\,\left (\frac {c^2-c+1}{2\,d}+\frac {c\,\left (2\,c-1\right )}{2\,d}\right )}{4\,d\,e^3}+\frac {c\,\left (2\,b^2\,c+b^2\right )}{8\,d^2\,e^3}\right )}{2\,c\,x+d\,x^2+\frac {c^2}{d}}+\frac {b^2\,\ln \left (c+d\,x\right )}{d\,e^3}-\frac {\ln \left (c+d\,x-1\right )\,\left (b^2+a\,b\right )}{2\,d\,e^3}+\frac {\ln \left (c+d\,x+1\right )\,\left (a\,b-b^2\right )}{2\,d\,e^3} \] Input:
int((a + b*atanh(c + d*x))^2/(c*e + d*e*x)^3,x)
Output:
log(1 - d*x - c)^2*(b^2/(8*d*e^3) - b^2/(2*d*(4*c^2*e^3 + 4*d^2*e^3*x^2 + 8*c*d*e^3*x))) + log(c + d*x + 1)^2*(b^2/(8*d*e^3) - b^2/(8*d^2*e^3*(2*c*x + d*x^2 + c^2/d))) + log(1 - d*x - c)*(log(c + d*x + 1)*(b^2/(2*d*(2*c^2* e^3 + 2*d^2*e^3*x^2 + 4*c*d*e^3*x)) - (b^2*(c^2 + d^2*x^2 + 2*c*d*x))/(2*d *(2*c^2*e^3 + 2*d^2*e^3*x^2 + 4*c*d*e^3*x))) + b^2/(2*d*(4*c^2*e^3 + 4*d^2 *e^3*x^2 + 8*c*d*e^3*x)) + (b*(4*a - b))/(2*d*(4*c^2*e^3 + 4*d^2*e^3*x^2 + 8*c*d*e^3*x)) - (b^2*(x*(4*c*d - d + d*(2*c - 1)) - c + c^2 + c*(2*c - 1) + 3*d^2*x^2 + 1))/(2*d*(4*c^2*e^3 + 4*d^2*e^3*x^2 + 8*c*d*e^3*x)) + (b^2* (x*(2*d*e^3 + d*(4*c*e^3 + 2*e^3) + 8*c*d*e^3) + 2*c*e^3 + 2*e^3 + c*(4*c* e^3 + 2*e^3) + 2*c^2*e^3 + 6*d^2*e^3*x^2))/(4*d*e^3*(4*c^2*e^3 + 4*d^2*e^3 *x^2 + 8*c*d*e^3*x))) - ((a^2 + 2*a*b*c)/(2*d) + a*b*x)/(c^2*e^3 + d^2*e^3 *x^2 + 2*c*d*e^3*x) - (log(c + d*x + 1)*(x*((2*b^2*c + b^2)/(4*d*e^3) + (b ^2*c)/(4*d*e^3) - (b^2*(3*c - 1))/(4*d*e^3)) + (4*a*b + b^2*c + b^2 + b^2* c^2)/(8*d^2*e^3) - (b^2*((c^2 - c + 1)/(2*d) + (c*(2*c - 1))/(2*d)))/(4*d* e^3) + (c*(2*b^2*c + b^2))/(8*d^2*e^3)))/(2*c*x + d*x^2 + c^2/d) + (b^2*lo g(c + d*x))/(d*e^3) - (log(c + d*x - 1)*(a*b + b^2))/(2*d*e^3) + (log(c + d*x + 1)*(a*b - b^2))/(2*d*e^3)
Time = 0.17 (sec) , antiderivative size = 503, normalized size of antiderivative = 4.23 \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^3} \, dx=\frac {-2 a b \,c^{2}+2 a b \,d^{2} x^{2}-2 a^{2} c -2 \,\mathrm {log}\left (d x +c +1\right ) b^{2} c \,d^{2} x^{2}+4 \,\mathrm {log}\left (d x +c \right ) b^{2} c \,d^{2} x^{2}+2 \mathit {atanh} \left (d x +c \right )^{2} b^{2} c^{3}-2 \mathit {atanh} \left (d x +c \right )^{2} b^{2} c -2 \mathit {atanh} \left (d x +c \right ) b^{2} c^{2}+2 \mathit {atanh} \left (d x +c \right ) b^{2} d^{2} x^{2}-2 \,\mathrm {log}\left (d x +c -1\right ) b^{2} c \,d^{2} x^{2}-2 \,\mathrm {log}\left (d x +c -1\right ) b^{2} c^{3}+\mathrm {log}\left (d x +c -1\right ) b^{2} c^{2}-2 \,\mathrm {log}\left (d x +c +1\right ) b^{2} c^{3}-\mathrm {log}\left (d x +c +1\right ) b^{2} c^{2}-4 \,\mathrm {log}\left (d x +c -1\right ) a b \,c^{2} d x -2 \,\mathrm {log}\left (d x +c -1\right ) a b c \,d^{2} x^{2}+4 \,\mathrm {log}\left (d x +c +1\right ) a b \,c^{2} d x +8 \,\mathrm {log}\left (d x +c \right ) b^{2} c^{2} d x -4 \,\mathrm {log}\left (d x +c -1\right ) b^{2} c^{2} d x +2 \,\mathrm {log}\left (d x +c -1\right ) b^{2} c d x -4 \,\mathrm {log}\left (d x +c +1\right ) b^{2} c^{2} d x -2 \,\mathrm {log}\left (d x +c +1\right ) b^{2} c d x +\mathrm {log}\left (d x +c -1\right ) b^{2} d^{2} x^{2}-\mathrm {log}\left (d x +c +1\right ) b^{2} d^{2} x^{2}-4 \mathit {atanh} \left (d x +c \right ) a b c -2 \,\mathrm {log}\left (d x +c -1\right ) a b \,c^{3}+2 \,\mathrm {log}\left (d x +c +1\right ) a b \,c^{3}+4 \,\mathrm {log}\left (d x +c \right ) b^{2} c^{3}+2 \,\mathrm {log}\left (d x +c +1\right ) a b c \,d^{2} x^{2}+4 \mathit {atanh} \left (d x +c \right )^{2} b^{2} c^{2} d x +2 \mathit {atanh} \left (d x +c \right )^{2} b^{2} c \,d^{2} x^{2}}{4 c d \,e^{3} \left (d^{2} x^{2}+2 c d x +c^{2}\right )} \] Input:
int((a+b*atanh(d*x+c))^2/(d*e*x+c*e)^3,x)
Output:
(2*atanh(c + d*x)**2*b**2*c**3 + 4*atanh(c + d*x)**2*b**2*c**2*d*x + 2*ata nh(c + d*x)**2*b**2*c*d**2*x**2 - 2*atanh(c + d*x)**2*b**2*c - 4*atanh(c + d*x)*a*b*c - 2*atanh(c + d*x)*b**2*c**2 + 2*atanh(c + d*x)*b**2*d**2*x**2 - 2*log(c + d*x - 1)*a*b*c**3 - 4*log(c + d*x - 1)*a*b*c**2*d*x - 2*log(c + d*x - 1)*a*b*c*d**2*x**2 - 2*log(c + d*x - 1)*b**2*c**3 - 4*log(c + d*x - 1)*b**2*c**2*d*x + log(c + d*x - 1)*b**2*c**2 - 2*log(c + d*x - 1)*b**2 *c*d**2*x**2 + 2*log(c + d*x - 1)*b**2*c*d*x + log(c + d*x - 1)*b**2*d**2* x**2 + 2*log(c + d*x + 1)*a*b*c**3 + 4*log(c + d*x + 1)*a*b*c**2*d*x + 2*l og(c + d*x + 1)*a*b*c*d**2*x**2 - 2*log(c + d*x + 1)*b**2*c**3 - 4*log(c + d*x + 1)*b**2*c**2*d*x - log(c + d*x + 1)*b**2*c**2 - 2*log(c + d*x + 1)* b**2*c*d**2*x**2 - 2*log(c + d*x + 1)*b**2*c*d*x - log(c + d*x + 1)*b**2*d **2*x**2 + 4*log(c + d*x)*b**2*c**3 + 8*log(c + d*x)*b**2*c**2*d*x + 4*log (c + d*x)*b**2*c*d**2*x**2 - 2*a**2*c - 2*a*b*c**2 + 2*a*b*d**2*x**2)/(4*c *d*e**3*(c**2 + 2*c*d*x + d**2*x**2))