Integrand size = 23, antiderivative size = 172 \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^5} \, dx=-\frac {b^2}{12 d e^5 (c+d x)^2}-\frac {b (a+b \text {arctanh}(c+d x))}{6 d e^5 (c+d x)^3}-\frac {b (a+b \text {arctanh}(c+d x))}{2 d e^5 (c+d x)}+\frac {(a+b \text {arctanh}(c+d x))^2}{4 d e^5}-\frac {(a+b \text {arctanh}(c+d x))^2}{4 d e^5 (c+d x)^4}+\frac {2 b^2 \log (c+d x)}{3 d e^5}-\frac {b^2 \log \left (1-(c+d x)^2\right )}{3 d e^5} \]
-1/12*b^2/d/e^5/(d*x+c)^2-1/6*b*(a+b*arctanh(d*x+c))/d/e^5/(d*x+c)^3-1/2*b *(a+b*arctanh(d*x+c))/d/e^5/(d*x+c)+1/4*(a+b*arctanh(d*x+c))^2/d/e^5-1/4*( a+b*arctanh(d*x+c))^2/d/e^5/(d*x+c)^4+2/3*b^2*ln(d*x+c)/d/e^5-1/3*b^2*ln(1 -(d*x+c)^2)/d/e^5
Time = 0.26 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.27 \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^5} \, dx=-\frac {\frac {3 a^2}{(c+d x)^4}+\frac {2 a b}{(c+d x)^3}+\frac {b^2}{(c+d x)^2}+\frac {6 a b}{c+d x}+\frac {2 b \left (3 a+b \left (c+3 c^3+d x+9 c^2 d x+9 c d^2 x^2+3 d^3 x^3\right )\right ) \text {arctanh}(c+d x)}{(c+d x)^4}-\frac {3 b^2 \left (-1+c^4+4 c^3 d x+6 c^2 d^2 x^2+4 c d^3 x^3+d^4 x^4\right ) \text {arctanh}(c+d x)^2}{(c+d x)^4}+b (3 a+4 b) \log (1-c-d x)-8 b^2 \log (c+d x)-(3 a-4 b) b \log (1+c+d x)}{12 d e^5} \]
-1/12*((3*a^2)/(c + d*x)^4 + (2*a*b)/(c + d*x)^3 + b^2/(c + d*x)^2 + (6*a* b)/(c + d*x) + (2*b*(3*a + b*(c + 3*c^3 + d*x + 9*c^2*d*x + 9*c*d^2*x^2 + 3*d^3*x^3))*ArcTanh[c + d*x])/(c + d*x)^4 - (3*b^2*(-1 + c^4 + 4*c^3*d*x + 6*c^2*d^2*x^2 + 4*c*d^3*x^3 + d^4*x^4)*ArcTanh[c + d*x]^2)/(c + d*x)^4 + b*(3*a + 4*b)*Log[1 - c - d*x] - 8*b^2*Log[c + d*x] - (3*a - 4*b)*b*Log[1 + c + d*x])/(d*e^5)
Time = 0.99 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.91, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {6657, 27, 6452, 6544, 6452, 243, 54, 2009, 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.
\(\displaystyle \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^5} \, dx\) |
\(\Big \downarrow \) 6657 |
\(\displaystyle \frac {\int \frac {(a+b \text {arctanh}(c+d x))^2}{e^5 (c+d x)^5}d(c+d x)}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {(a+b \text {arctanh}(c+d x))^2}{(c+d x)^5}d(c+d x)}{d e^5}\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle \frac {\frac {1}{2} b \int \frac {a+b \text {arctanh}(c+d x)}{(c+d x)^4 \left (1-(c+d x)^2\right )}d(c+d x)-\frac {(a+b \text {arctanh}(c+d x))^2}{4 (c+d x)^4}}{d e^5}\) |
\(\Big \downarrow \) 6544 |
\(\displaystyle \frac {\frac {1}{2} b \left (\int \frac {a+b \text {arctanh}(c+d x)}{(c+d x)^4}d(c+d x)+\int \frac {a+b \text {arctanh}(c+d x)}{(c+d x)^2 \left (1-(c+d x)^2\right )}d(c+d x)\right )-\frac {(a+b \text {arctanh}(c+d x))^2}{4 (c+d x)^4}}{d e^5}\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle \frac {\frac {1}{2} b \left (\int \frac {a+b \text {arctanh}(c+d x)}{(c+d x)^2 \left (1-(c+d x)^2\right )}d(c+d x)+\frac {1}{3} b \int \frac {1}{(c+d x)^3 \left (1-(c+d x)^2\right )}d(c+d x)-\frac {a+b \text {arctanh}(c+d x)}{3 (c+d x)^3}\right )-\frac {(a+b \text {arctanh}(c+d x))^2}{4 (c+d x)^4}}{d e^5}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {\frac {1}{2} b \left (\int \frac {a+b \text {arctanh}(c+d x)}{(c+d x)^2 \left (1-(c+d x)^2\right )}d(c+d x)+\frac {1}{6} b \int \frac {1}{(-c-d x+1) (c+d x)^4}d(c+d x)^2-\frac {a+b \text {arctanh}(c+d x)}{3 (c+d x)^3}\right )-\frac {(a+b \text {arctanh}(c+d x))^2}{4 (c+d x)^4}}{d e^5}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {\frac {1}{2} b \left (\int \frac {a+b \text {arctanh}(c+d x)}{(c+d x)^2 \left (1-(c+d x)^2\right )}d(c+d x)+\frac {1}{6} b \int \left (\frac {1}{(c+d x)^2}+\frac {1}{(c+d x)^4}+\frac {1}{-c-d x+1}\right )d(c+d x)^2-\frac {a+b \text {arctanh}(c+d x)}{3 (c+d x)^3}\right )-\frac {(a+b \text {arctanh}(c+d x))^2}{4 (c+d x)^4}}{d e^5}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {1}{2} b \left (\int \frac {a+b \text {arctanh}(c+d x)}{(c+d x)^2 \left (1-(c+d x)^2\right )}d(c+d x)-\frac {a+b \text {arctanh}(c+d x)}{3 (c+d x)^3}+\frac {1}{6} b \left (-\frac {1}{(c+d x)^2}-\log (-c-d x+1)+\log \left ((c+d x)^2\right )\right )\right )-\frac {(a+b \text {arctanh}(c+d x))^2}{4 (c+d x)^4}}{d e^5}\) |
\(\Big \downarrow \) 6544 |
\(\displaystyle \frac {\frac {1}{2} b \left (\int \frac {a+b \text {arctanh}(c+d x)}{(c+d x)^2}d(c+d x)+\int \frac {a+b \text {arctanh}(c+d x)}{1-(c+d x)^2}d(c+d x)-\frac {a+b \text {arctanh}(c+d x)}{3 (c+d x)^3}+\frac {1}{6} b \left (-\frac {1}{(c+d x)^2}-\log (-c-d x+1)+\log \left ((c+d x)^2\right )\right )\right )-\frac {(a+b \text {arctanh}(c+d x))^2}{4 (c+d x)^4}}{d e^5}\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle \frac {\frac {1}{2} b \left (\int \frac {a+b \text {arctanh}(c+d x)}{1-(c+d x)^2}d(c+d x)+b \int \frac {1}{(c+d x) \left (1-(c+d x)^2\right )}d(c+d x)-\frac {a+b \text {arctanh}(c+d x)}{c+d x}-\frac {a+b \text {arctanh}(c+d x)}{3 (c+d x)^3}+\frac {1}{6} b \left (-\frac {1}{(c+d x)^2}-\log (-c-d x+1)+\log \left ((c+d x)^2\right )\right )\right )-\frac {(a+b \text {arctanh}(c+d x))^2}{4 (c+d x)^4}}{d e^5}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {\frac {1}{2} b \left (\int \frac {a+b \text {arctanh}(c+d x)}{1-(c+d x)^2}d(c+d x)+\frac {1}{2} b \int \frac {1}{(-c-d x+1) (c+d x)^2}d(c+d x)^2-\frac {a+b \text {arctanh}(c+d x)}{c+d x}-\frac {a+b \text {arctanh}(c+d x)}{3 (c+d x)^3}+\frac {1}{6} b \left (-\frac {1}{(c+d x)^2}-\log (-c-d x+1)+\log \left ((c+d x)^2\right )\right )\right )-\frac {(a+b \text {arctanh}(c+d x))^2}{4 (c+d x)^4}}{d e^5}\) |
\(\Big \downarrow \) 47 |
\(\displaystyle \frac {\frac {1}{2} b \left (\int \frac {a+b \text {arctanh}(c+d x)}{1-(c+d x)^2}d(c+d x)+\frac {1}{2} b \left (\int \frac {1}{-c-d x+1}d(c+d x)^2+\int \frac {1}{(c+d x)^2}d(c+d x)^2\right )-\frac {a+b \text {arctanh}(c+d x)}{c+d x}-\frac {a+b \text {arctanh}(c+d x)}{3 (c+d x)^3}+\frac {1}{6} b \left (-\frac {1}{(c+d x)^2}-\log (-c-d x+1)+\log \left ((c+d x)^2\right )\right )\right )-\frac {(a+b \text {arctanh}(c+d x))^2}{4 (c+d x)^4}}{d e^5}\) |
\(\Big \downarrow \) 14 |
\(\displaystyle \frac {\frac {1}{2} b \left (\int \frac {a+b \text {arctanh}(c+d x)}{1-(c+d x)^2}d(c+d x)+\frac {1}{2} b \left (\int \frac {1}{-c-d x+1}d(c+d x)^2+\log \left ((c+d x)^2\right )\right )-\frac {a+b \text {arctanh}(c+d x)}{c+d x}-\frac {a+b \text {arctanh}(c+d x)}{3 (c+d x)^3}+\frac {1}{6} b \left (-\frac {1}{(c+d x)^2}-\log (-c-d x+1)+\log \left ((c+d x)^2\right )\right )\right )-\frac {(a+b \text {arctanh}(c+d x))^2}{4 (c+d x)^4}}{d e^5}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {\frac {1}{2} b \left (\int \frac {a+b \text {arctanh}(c+d x)}{1-(c+d x)^2}d(c+d x)-\frac {a+b \text {arctanh}(c+d x)}{c+d x}-\frac {a+b \text {arctanh}(c+d x)}{3 (c+d x)^3}+\frac {1}{2} b \left (\log \left ((c+d x)^2\right )-\log (-c-d x+1)\right )+\frac {1}{6} b \left (-\frac {1}{(c+d x)^2}-\log (-c-d x+1)+\log \left ((c+d x)^2\right )\right )\right )-\frac {(a+b \text {arctanh}(c+d x))^2}{4 (c+d x)^4}}{d e^5}\) |
\(\Big \downarrow \) 6510 |
\(\displaystyle \frac {\frac {1}{2} b \left (\frac {(a+b \text {arctanh}(c+d x))^2}{2 b}-\frac {a+b \text {arctanh}(c+d x)}{c+d x}-\frac {a+b \text {arctanh}(c+d x)}{3 (c+d x)^3}+\frac {1}{2} b \left (\log \left ((c+d x)^2\right )-\log (-c-d x+1)\right )+\frac {1}{6} b \left (-\frac {1}{(c+d x)^2}-\log (-c-d x+1)+\log \left ((c+d x)^2\right )\right )\right )-\frac {(a+b \text {arctanh}(c+d x))^2}{4 (c+d x)^4}}{d e^5}\) |
(-1/4*(a + b*ArcTanh[c + d*x])^2/(c + d*x)^4 + (b*(-1/3*(a + b*ArcTanh[c + d*x])/(c + d*x)^3 - (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 + (b*(-(c + d*x)^(-2) - Log[1 - c - d*x] + Log[(c + d*x)^2]))/6))/2)/(d*e^5)
3.1.22.3.1 Defintions of rubi rules used
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[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
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]
Time = 1.09 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.59
method | result | size |
derivativedivides | \(\frac {-\frac {a^{2}}{4 e^{5} \left (d x +c \right )^{4}}+\frac {b^{2} \left (-\frac {\operatorname {arctanh}\left (d x +c \right )^{2}}{4 \left (d x +c \right )^{4}}-\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right )}{4}-\frac {\operatorname {arctanh}\left (d x +c \right )}{6 \left (d x +c \right )^{3}}-\frac {\operatorname {arctanh}\left (d x +c \right )}{2 \left (d x +c \right )}+\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right )}{4}+\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{8}-\frac {\ln \left (d x +c -1\right )^{2}}{16}-\frac {\ln \left (d x +c +1\right )^{2}}{16}+\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{8}-\frac {\ln \left (d x +c -1\right )}{3}-\frac {1}{12 \left (d x +c \right )^{2}}+\frac {2 \ln \left (d x +c \right )}{3}-\frac {\ln \left (d x +c +1\right )}{3}\right )}{e^{5}}+\frac {2 a b \left (-\frac {\operatorname {arctanh}\left (d x +c \right )}{4 \left (d x +c \right )^{4}}-\frac {\ln \left (d x +c -1\right )}{8}-\frac {1}{12 \left (d x +c \right )^{3}}-\frac {1}{4 \left (d x +c \right )}+\frac {\ln \left (d x +c +1\right )}{8}\right )}{e^{5}}}{d}\) | \(273\) |
default | \(\frac {-\frac {a^{2}}{4 e^{5} \left (d x +c \right )^{4}}+\frac {b^{2} \left (-\frac {\operatorname {arctanh}\left (d x +c \right )^{2}}{4 \left (d x +c \right )^{4}}-\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right )}{4}-\frac {\operatorname {arctanh}\left (d x +c \right )}{6 \left (d x +c \right )^{3}}-\frac {\operatorname {arctanh}\left (d x +c \right )}{2 \left (d x +c \right )}+\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right )}{4}+\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{8}-\frac {\ln \left (d x +c -1\right )^{2}}{16}-\frac {\ln \left (d x +c +1\right )^{2}}{16}+\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{8}-\frac {\ln \left (d x +c -1\right )}{3}-\frac {1}{12 \left (d x +c \right )^{2}}+\frac {2 \ln \left (d x +c \right )}{3}-\frac {\ln \left (d x +c +1\right )}{3}\right )}{e^{5}}+\frac {2 a b \left (-\frac {\operatorname {arctanh}\left (d x +c \right )}{4 \left (d x +c \right )^{4}}-\frac {\ln \left (d x +c -1\right )}{8}-\frac {1}{12 \left (d x +c \right )^{3}}-\frac {1}{4 \left (d x +c \right )}+\frac {\ln \left (d x +c +1\right )}{8}\right )}{e^{5}}}{d}\) | \(273\) |
parts | \(-\frac {a^{2}}{4 e^{5} \left (d x +c \right )^{4} d}+\frac {b^{2} \left (-\frac {\operatorname {arctanh}\left (d x +c \right )^{2}}{4 \left (d x +c \right )^{4}}-\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right )}{4}-\frac {\operatorname {arctanh}\left (d x +c \right )}{6 \left (d x +c \right )^{3}}-\frac {\operatorname {arctanh}\left (d x +c \right )}{2 \left (d x +c \right )}+\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right )}{4}+\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{8}-\frac {\ln \left (d x +c -1\right )^{2}}{16}-\frac {\ln \left (d x +c +1\right )^{2}}{16}+\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{8}-\frac {\ln \left (d x +c -1\right )}{3}-\frac {1}{12 \left (d x +c \right )^{2}}+\frac {2 \ln \left (d x +c \right )}{3}-\frac {\ln \left (d x +c +1\right )}{3}\right )}{e^{5} d}+\frac {2 a b \left (-\frac {\operatorname {arctanh}\left (d x +c \right )}{4 \left (d x +c \right )^{4}}-\frac {\ln \left (d x +c -1\right )}{8}-\frac {1}{12 \left (d x +c \right )^{3}}-\frac {1}{4 \left (d x +c \right )}+\frac {\ln \left (d x +c +1\right )}{8}\right )}{e^{5} d}\) | \(278\) |
parallelrisch | \(\frac {-3 b^{2} \operatorname {arctanh}\left (d x +c \right )^{2} d^{5}+8 \ln \left (d x +c \right ) x^{4} b^{2} d^{9}-8 \ln \left (d x +c -1\right ) x^{4} b^{2} d^{9}+8 \ln \left (d x +c \right ) b^{2} c^{4} d^{5}-8 \ln \left (d x +c -1\right ) b^{2} c^{4} d^{5}+3 b^{2} d^{9} \operatorname {arctanh}\left (d x +c \right )^{2} x^{4}+3 \operatorname {arctanh}\left (d x +c \right )^{2} b^{2} c^{4} d^{5}-8 x^{4} \operatorname {arctanh}\left (d x +c \right ) b^{2} d^{9}-2 x \,\operatorname {arctanh}\left (d x +c \right ) b^{2} d^{6}-6 x^{3} \operatorname {arctanh}\left (d x +c \right ) b^{2} d^{8}-8 \,\operatorname {arctanh}\left (d x +c \right ) b^{2} c^{4} d^{5}-6 \,\operatorname {arctanh}\left (d x +c \right ) b^{2} c^{3} d^{5}-2 \,\operatorname {arctanh}\left (d x +c \right ) b^{2} c \,d^{5}-6 \,\operatorname {arctanh}\left (d x +c \right ) a b \,d^{5}-2 x \,b^{2} c \,d^{6}-2 x a b \,d^{6}-6 x^{3} a b \,d^{8}-2 c a b \,d^{5}-x^{2} b^{2} d^{7}-3 a^{2} d^{5}-b^{2} c^{2} d^{5}-6 a b \,c^{3} d^{5}+18 x^{2} \operatorname {arctanh}\left (d x +c \right )^{2} b^{2} c^{2} d^{7}+12 x \operatorname {arctanh}\left (d x +c \right )^{2} b^{2} c^{3} d^{6}+6 x^{4} \operatorname {arctanh}\left (d x +c \right ) a b \,d^{9}-32 x \,\operatorname {arctanh}\left (d x +c \right ) b^{2} c^{3} d^{6}-18 x \,\operatorname {arctanh}\left (d x +c \right ) b^{2} c^{2} d^{6}-48 x^{2} \operatorname {arctanh}\left (d x +c \right ) b^{2} c^{2} d^{7}-18 x^{2} \operatorname {arctanh}\left (d x +c \right ) b^{2} c \,d^{7}-32 x^{3} \operatorname {arctanh}\left (d x +c \right ) b^{2} c \,d^{8}+6 \,\operatorname {arctanh}\left (d x +c \right ) a b \,c^{4} d^{5}-18 x a b \,c^{2} d^{6}-18 x^{2} a b c \,d^{7}+12 d^{8} c \,b^{2} x^{3} \operatorname {arctanh}\left (d x +c \right )^{2}+32 \ln \left (d x +c \right ) x^{3} b^{2} c \,d^{8}-32 \ln \left (d x +c -1\right ) x^{3} b^{2} c \,d^{8}+48 \ln \left (d x +c \right ) x^{2} b^{2} c^{2} d^{7}-48 \ln \left (d x +c -1\right ) x^{2} b^{2} c^{2} d^{7}+32 \ln \left (d x +c \right ) x \,b^{2} c^{3} d^{6}-32 \ln \left (d x +c -1\right ) x \,b^{2} c^{3} d^{6}+24 x \,\operatorname {arctanh}\left (d x +c \right ) a b \,c^{3} d^{6}+36 x^{2} \operatorname {arctanh}\left (d x +c \right ) a b \,c^{2} d^{7}+24 x^{3} \operatorname {arctanh}\left (d x +c \right ) a b c \,d^{8}}{12 \left (d x +c \right )^{4} d^{6} e^{5}}\) | \(704\) |
risch | \(\text {Expression too large to display}\) | \(1096\) |
1/d*(-1/4*a^2/e^5/(d*x+c)^4+b^2/e^5*(-1/4/(d*x+c)^4*arctanh(d*x+c)^2-1/4*a rctanh(d*x+c)*ln(d*x+c-1)-1/6/(d*x+c)^3*arctanh(d*x+c)-1/2/(d*x+c)*arctanh (d*x+c)+1/4*arctanh(d*x+c)*ln(d*x+c+1)+1/8*ln(d*x+c-1)*ln(1/2*d*x+1/2*c+1/ 2)-1/16*ln(d*x+c-1)^2-1/16*ln(d*x+c+1)^2+1/8*(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)-1/3*ln(d*x+c-1)-1/12/(d*x+c)^2+2/3*ln(d*x+c) -1/3*ln(d*x+c+1))+2*a*b/e^5*(-1/4/(d*x+c)^4*arctanh(d*x+c)-1/8*ln(d*x+c-1) -1/12/(d*x+c)^3-1/4/(d*x+c)+1/8*ln(d*x+c+1)))
Leaf count of result is larger than twice the leaf count of optimal. 547 vs. \(2 (158) = 316\).
Time = 0.30 (sec) , antiderivative size = 547, normalized size of antiderivative = 3.18 \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^5} \, dx=-\frac {24 \, a b d^{3} x^{3} + 24 \, a b c^{3} + 4 \, {\left (18 \, a b c + b^{2}\right )} d^{2} x^{2} + 4 \, b^{2} c^{2} + 8 \, a b c + 8 \, {\left (9 \, a b c^{2} + b^{2} c + a b\right )} d x - 3 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4} - b^{2}\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )^{2} + 12 \, a^{2} - 4 \, {\left ({\left (3 \, a b - 4 \, b^{2}\right )} d^{4} x^{4} + 4 \, {\left (3 \, a b - 4 \, b^{2}\right )} c d^{3} x^{3} + 6 \, {\left (3 \, a b - 4 \, b^{2}\right )} c^{2} d^{2} x^{2} + 4 \, {\left (3 \, a b - 4 \, b^{2}\right )} c^{3} d x + {\left (3 \, a b - 4 \, b^{2}\right )} c^{4}\right )} \log \left (d x + c + 1\right ) - 32 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \left (d x + c\right ) + 4 \, {\left ({\left (3 \, a b + 4 \, b^{2}\right )} d^{4} x^{4} + 4 \, {\left (3 \, a b + 4 \, b^{2}\right )} c d^{3} x^{3} + 6 \, {\left (3 \, a b + 4 \, b^{2}\right )} c^{2} d^{2} x^{2} + 4 \, {\left (3 \, a b + 4 \, b^{2}\right )} c^{3} d x + {\left (3 \, a b + 4 \, b^{2}\right )} c^{4}\right )} \log \left (d x + c - 1\right ) + 4 \, {\left (3 \, b^{2} d^{3} x^{3} + 9 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{3} + b^{2} c + {\left (9 \, b^{2} c^{2} + b^{2}\right )} d x + 3 \, a b\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{48 \, {\left (d^{5} e^{5} x^{4} + 4 \, c d^{4} e^{5} x^{3} + 6 \, c^{2} d^{3} e^{5} x^{2} + 4 \, c^{3} d^{2} e^{5} x + c^{4} d e^{5}\right )}} \]
-1/48*(24*a*b*d^3*x^3 + 24*a*b*c^3 + 4*(18*a*b*c + b^2)*d^2*x^2 + 4*b^2*c^ 2 + 8*a*b*c + 8*(9*a*b*c^2 + b^2*c + a*b)*d*x - 3*(b^2*d^4*x^4 + 4*b^2*c*d ^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b^2*c^4 - b^2)*log(-(d*x + c + 1)/(d*x + c - 1))^2 + 12*a^2 - 4*((3*a*b - 4*b^2)*d^4*x^4 + 4*(3*a*b - 4 *b^2)*c*d^3*x^3 + 6*(3*a*b - 4*b^2)*c^2*d^2*x^2 + 4*(3*a*b - 4*b^2)*c^3*d* x + (3*a*b - 4*b^2)*c^4)*log(d*x + c + 1) - 32*(b^2*d^4*x^4 + 4*b^2*c*d^3* x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b^2*c^4)*log(d*x + c) + 4*((3*a* b + 4*b^2)*d^4*x^4 + 4*(3*a*b + 4*b^2)*c*d^3*x^3 + 6*(3*a*b + 4*b^2)*c^2*d ^2*x^2 + 4*(3*a*b + 4*b^2)*c^3*d*x + (3*a*b + 4*b^2)*c^4)*log(d*x + c - 1) + 4*(3*b^2*d^3*x^3 + 9*b^2*c*d^2*x^2 + 3*b^2*c^3 + b^2*c + (9*b^2*c^2 + b ^2)*d*x + 3*a*b)*log(-(d*x + c + 1)/(d*x + c - 1)))/(d^5*e^5*x^4 + 4*c*d^4 *e^5*x^3 + 6*c^2*d^3*e^5*x^2 + 4*c^3*d^2*e^5*x + c^4*d*e^5)
Leaf count of result is larger than twice the leaf count of optimal. 3516 vs. \(2 (148) = 296\).
Time = 2.44 (sec) , antiderivative size = 3516, normalized size of antiderivative = 20.44 \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^5} \, dx=\text {Too large to display} \]
Piecewise((-3*a**2/(12*c**4*d*e**5 + 48*c**3*d**2*e**5*x + 72*c**2*d**3*e* *5*x**2 + 48*c*d**4*e**5*x**3 + 12*d**5*e**5*x**4) + 6*a*b*c**4*atanh(c + d*x)/(12*c**4*d*e**5 + 48*c**3*d**2*e**5*x + 72*c**2*d**3*e**5*x**2 + 48*c *d**4*e**5*x**3 + 12*d**5*e**5*x**4) + 24*a*b*c**3*d*x*atanh(c + d*x)/(12* c**4*d*e**5 + 48*c**3*d**2*e**5*x + 72*c**2*d**3*e**5*x**2 + 48*c*d**4*e** 5*x**3 + 12*d**5*e**5*x**4) - 6*a*b*c**3/(12*c**4*d*e**5 + 48*c**3*d**2*e* *5*x + 72*c**2*d**3*e**5*x**2 + 48*c*d**4*e**5*x**3 + 12*d**5*e**5*x**4) + 36*a*b*c**2*d**2*x**2*atanh(c + d*x)/(12*c**4*d*e**5 + 48*c**3*d**2*e**5* x + 72*c**2*d**3*e**5*x**2 + 48*c*d**4*e**5*x**3 + 12*d**5*e**5*x**4) - 18 *a*b*c**2*d*x/(12*c**4*d*e**5 + 48*c**3*d**2*e**5*x + 72*c**2*d**3*e**5*x* *2 + 48*c*d**4*e**5*x**3 + 12*d**5*e**5*x**4) + 24*a*b*c*d**3*x**3*atanh(c + d*x)/(12*c**4*d*e**5 + 48*c**3*d**2*e**5*x + 72*c**2*d**3*e**5*x**2 + 4 8*c*d**4*e**5*x**3 + 12*d**5*e**5*x**4) - 18*a*b*c*d**2*x**2/(12*c**4*d*e* *5 + 48*c**3*d**2*e**5*x + 72*c**2*d**3*e**5*x**2 + 48*c*d**4*e**5*x**3 + 12*d**5*e**5*x**4) - 2*a*b*c/(12*c**4*d*e**5 + 48*c**3*d**2*e**5*x + 72*c* *2*d**3*e**5*x**2 + 48*c*d**4*e**5*x**3 + 12*d**5*e**5*x**4) + 6*a*b*d**4* x**4*atanh(c + d*x)/(12*c**4*d*e**5 + 48*c**3*d**2*e**5*x + 72*c**2*d**3*e **5*x**2 + 48*c*d**4*e**5*x**3 + 12*d**5*e**5*x**4) - 6*a*b*d**3*x**3/(12* c**4*d*e**5 + 48*c**3*d**2*e**5*x + 72*c**2*d**3*e**5*x**2 + 48*c*d**4*e** 5*x**3 + 12*d**5*e**5*x**4) - 2*a*b*d*x/(12*c**4*d*e**5 + 48*c**3*d**2*...
Leaf count of result is larger than twice the leaf count of optimal. 613 vs. \(2 (158) = 316\).
Time = 0.21 (sec) , antiderivative size = 613, normalized size of antiderivative = 3.56 \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^5} \, dx=-\frac {1}{12} \, {\left (d {\left (\frac {2 \, {\left (3 \, d^{2} x^{2} + 6 \, c d x + 3 \, c^{2} + 1\right )}}{d^{5} e^{5} x^{3} + 3 \, c d^{4} e^{5} x^{2} + 3 \, c^{2} d^{3} e^{5} x + c^{3} d^{2} e^{5}} - \frac {3 \, \log \left (d x + c + 1\right )}{d^{2} e^{5}} + \frac {3 \, \log \left (d x + c - 1\right )}{d^{2} e^{5}}\right )} + \frac {6 \, \operatorname {artanh}\left (d x + c\right )}{d^{5} e^{5} x^{4} + 4 \, c d^{4} e^{5} x^{3} + 6 \, c^{2} d^{3} e^{5} x^{2} + 4 \, c^{3} d^{2} e^{5} x + c^{4} d e^{5}}\right )} a b - \frac {1}{48} \, {\left (d^{2} {\left (\frac {3 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \log \left (d x + c + 1\right )^{2} + 3 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \log \left (d x + c - 1\right )^{2} + 2 \, {\left (8 \, d^{2} x^{2} + 16 \, c d x + 8 \, c^{2} - 3 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \log \left (d x + c - 1\right )\right )} \log \left (d x + c + 1\right ) + 16 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \log \left (d x + c - 1\right ) + 4}{d^{5} e^{5} x^{2} + 2 \, c d^{4} e^{5} x + c^{2} d^{3} e^{5}} - \frac {32 \, \log \left (d x + c\right )}{d^{3} e^{5}}\right )} + 4 \, d {\left (\frac {2 \, {\left (3 \, d^{2} x^{2} + 6 \, c d x + 3 \, c^{2} + 1\right )}}{d^{5} e^{5} x^{3} + 3 \, c d^{4} e^{5} x^{2} + 3 \, c^{2} d^{3} e^{5} x + c^{3} d^{2} e^{5}} - \frac {3 \, \log \left (d x + c + 1\right )}{d^{2} e^{5}} + \frac {3 \, \log \left (d x + c - 1\right )}{d^{2} e^{5}}\right )} \operatorname {artanh}\left (d x + c\right )\right )} b^{2} - \frac {b^{2} \operatorname {artanh}\left (d x + c\right )^{2}}{4 \, {\left (d^{5} e^{5} x^{4} + 4 \, c d^{4} e^{5} x^{3} + 6 \, c^{2} d^{3} e^{5} x^{2} + 4 \, c^{3} d^{2} e^{5} x + c^{4} d e^{5}\right )}} - \frac {a^{2}}{4 \, {\left (d^{5} e^{5} x^{4} + 4 \, c d^{4} e^{5} x^{3} + 6 \, c^{2} d^{3} e^{5} x^{2} + 4 \, c^{3} d^{2} e^{5} x + c^{4} d e^{5}\right )}} \]
-1/12*(d*(2*(3*d^2*x^2 + 6*c*d*x + 3*c^2 + 1)/(d^5*e^5*x^3 + 3*c*d^4*e^5*x ^2 + 3*c^2*d^3*e^5*x + c^3*d^2*e^5) - 3*log(d*x + c + 1)/(d^2*e^5) + 3*log (d*x + c - 1)/(d^2*e^5)) + 6*arctanh(d*x + c)/(d^5*e^5*x^4 + 4*c*d^4*e^5*x ^3 + 6*c^2*d^3*e^5*x^2 + 4*c^3*d^2*e^5*x + c^4*d*e^5))*a*b - 1/48*(d^2*((3 *(d^2*x^2 + 2*c*d*x + c^2)*log(d*x + c + 1)^2 + 3*(d^2*x^2 + 2*c*d*x + c^2 )*log(d*x + c - 1)^2 + 2*(8*d^2*x^2 + 16*c*d*x + 8*c^2 - 3*(d^2*x^2 + 2*c* d*x + c^2)*log(d*x + c - 1))*log(d*x + c + 1) + 16*(d^2*x^2 + 2*c*d*x + c^ 2)*log(d*x + c - 1) + 4)/(d^5*e^5*x^2 + 2*c*d^4*e^5*x + c^2*d^3*e^5) - 32* log(d*x + c)/(d^3*e^5)) + 4*d*(2*(3*d^2*x^2 + 6*c*d*x + 3*c^2 + 1)/(d^5*e^ 5*x^3 + 3*c*d^4*e^5*x^2 + 3*c^2*d^3*e^5*x + c^3*d^2*e^5) - 3*log(d*x + c + 1)/(d^2*e^5) + 3*log(d*x + c - 1)/(d^2*e^5))*arctanh(d*x + c))*b^2 - 1/4* b^2*arctanh(d*x + c)^2/(d^5*e^5*x^4 + 4*c*d^4*e^5*x^3 + 6*c^2*d^3*e^5*x^2 + 4*c^3*d^2*e^5*x + c^4*d*e^5) - 1/4*a^2/(d^5*e^5*x^4 + 4*c*d^4*e^5*x^3 + 6*c^2*d^3*e^5*x^2 + 4*c^3*d^2*e^5*x + c^4*d*e^5)
Leaf count of result is larger than twice the leaf count of optimal. 730 vs. \(2 (158) = 316\).
Time = 0.30 (sec) , antiderivative size = 730, normalized size of antiderivative = 4.24 \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^5} \, dx=\frac {1}{12} \, {\left ({\left (c + 1\right )} d - {\left (c - 1\right )} d\right )} {\left (\frac {3 \, {\left (\frac {{\left (d x + c + 1\right )}^{3} b^{2}}{{\left (d x + c - 1\right )}^{3}} + \frac {{\left (d x + c + 1\right )} b^{2}}{d x + c - 1}\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )^{2}}{\frac {{\left (d x + c + 1\right )}^{4} d^{2} e^{5}}{{\left (d x + c - 1\right )}^{4}} + \frac {4 \, {\left (d x + c + 1\right )}^{3} d^{2} e^{5}}{{\left (d x + c - 1\right )}^{3}} + \frac {6 \, {\left (d x + c + 1\right )}^{2} d^{2} e^{5}}{{\left (d x + c - 1\right )}^{2}} + \frac {4 \, {\left (d x + c + 1\right )} d^{2} e^{5}}{d x + c - 1} + d^{2} e^{5}} + \frac {2 \, {\left (\frac {6 \, {\left (d x + c + 1\right )}^{3} a b}{{\left (d x + c - 1\right )}^{3}} + \frac {6 \, {\left (d x + c + 1\right )} a b}{d x + c - 1} + \frac {3 \, {\left (d x + c + 1\right )}^{3} b^{2}}{{\left (d x + c - 1\right )}^{3}} + \frac {6 \, {\left (d x + c + 1\right )}^{2} b^{2}}{{\left (d x + c - 1\right )}^{2}} + \frac {5 \, {\left (d x + c + 1\right )} b^{2}}{d x + c - 1} + 2 \, b^{2}\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{\frac {{\left (d x + c + 1\right )}^{4} d^{2} e^{5}}{{\left (d x + c - 1\right )}^{4}} + \frac {4 \, {\left (d x + c + 1\right )}^{3} d^{2} e^{5}}{{\left (d x + c - 1\right )}^{3}} + \frac {6 \, {\left (d x + c + 1\right )}^{2} d^{2} e^{5}}{{\left (d x + c - 1\right )}^{2}} + \frac {4 \, {\left (d x + c + 1\right )} d^{2} e^{5}}{d x + c - 1} + d^{2} e^{5}} + \frac {2 \, {\left (\frac {6 \, {\left (d x + c + 1\right )}^{3} a^{2}}{{\left (d x + c - 1\right )}^{3}} + \frac {6 \, {\left (d x + c + 1\right )} a^{2}}{d x + c - 1} + \frac {6 \, {\left (d x + c + 1\right )}^{3} a b}{{\left (d x + c - 1\right )}^{3}} + \frac {12 \, {\left (d x + c + 1\right )}^{2} a b}{{\left (d x + c - 1\right )}^{2}} + \frac {10 \, {\left (d x + c + 1\right )} a b}{d x + c - 1} + 4 \, a b + \frac {{\left (d x + c + 1\right )}^{3} b^{2}}{{\left (d x + c - 1\right )}^{3}} + \frac {2 \, {\left (d x + c + 1\right )}^{2} b^{2}}{{\left (d x + c - 1\right )}^{2}} + \frac {{\left (d x + c + 1\right )} b^{2}}{d x + c - 1}\right )}}{\frac {{\left (d x + c + 1\right )}^{4} d^{2} e^{5}}{{\left (d x + c - 1\right )}^{4}} + \frac {4 \, {\left (d x + c + 1\right )}^{3} d^{2} e^{5}}{{\left (d x + c - 1\right )}^{3}} + \frac {6 \, {\left (d x + c + 1\right )}^{2} d^{2} e^{5}}{{\left (d x + c - 1\right )}^{2}} + \frac {4 \, {\left (d x + c + 1\right )} d^{2} e^{5}}{d x + c - 1} + d^{2} e^{5}} + \frac {4 \, b^{2} \log \left (-\frac {d x + c + 1}{d x + c - 1} - 1\right )}{d^{2} e^{5}} - \frac {4 \, b^{2} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{d^{2} e^{5}}\right )} \]
1/12*((c + 1)*d - (c - 1)*d)*(3*((d*x + c + 1)^3*b^2/(d*x + c - 1)^3 + (d* x + c + 1)*b^2/(d*x + c - 1))*log(-(d*x + c + 1)/(d*x + c - 1))^2/((d*x + c + 1)^4*d^2*e^5/(d*x + c - 1)^4 + 4*(d*x + c + 1)^3*d^2*e^5/(d*x + c - 1) ^3 + 6*(d*x + c + 1)^2*d^2*e^5/(d*x + c - 1)^2 + 4*(d*x + c + 1)*d^2*e^5/( d*x + c - 1) + d^2*e^5) + 2*(6*(d*x + c + 1)^3*a*b/(d*x + c - 1)^3 + 6*(d* x + c + 1)*a*b/(d*x + c - 1) + 3*(d*x + c + 1)^3*b^2/(d*x + c - 1)^3 + 6*( d*x + c + 1)^2*b^2/(d*x + c - 1)^2 + 5*(d*x + c + 1)*b^2/(d*x + c - 1) + 2 *b^2)*log(-(d*x + c + 1)/(d*x + c - 1))/((d*x + c + 1)^4*d^2*e^5/(d*x + c - 1)^4 + 4*(d*x + c + 1)^3*d^2*e^5/(d*x + c - 1)^3 + 6*(d*x + c + 1)^2*d^2 *e^5/(d*x + c - 1)^2 + 4*(d*x + c + 1)*d^2*e^5/(d*x + c - 1) + d^2*e^5) + 2*(6*(d*x + c + 1)^3*a^2/(d*x + c - 1)^3 + 6*(d*x + c + 1)*a^2/(d*x + c - 1) + 6*(d*x + c + 1)^3*a*b/(d*x + c - 1)^3 + 12*(d*x + c + 1)^2*a*b/(d*x + c - 1)^2 + 10*(d*x + c + 1)*a*b/(d*x + c - 1) + 4*a*b + (d*x + c + 1)^3*b ^2/(d*x + c - 1)^3 + 2*(d*x + c + 1)^2*b^2/(d*x + c - 1)^2 + (d*x + c + 1) *b^2/(d*x + c - 1))/((d*x + c + 1)^4*d^2*e^5/(d*x + c - 1)^4 + 4*(d*x + c + 1)^3*d^2*e^5/(d*x + c - 1)^3 + 6*(d*x + c + 1)^2*d^2*e^5/(d*x + c - 1)^2 + 4*(d*x + c + 1)*d^2*e^5/(d*x + c - 1) + d^2*e^5) + 4*b^2*log(-(d*x + c + 1)/(d*x + c - 1) - 1)/(d^2*e^5) - 4*b^2*log(-(d*x + c + 1)/(d*x + c - 1) )/(d^2*e^5))
Time = 6.54 (sec) , antiderivative size = 2746, normalized size of antiderivative = 15.97 \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c e+d e x)^5} \, dx=\text {Too large to display} \]
log(1 - d*x - c)^2*(b^2/(16*d*e^5) - b^2/(4*d*(4*c^4*e^5 + 4*d^4*e^5*x^4 + 16*c*d^3*e^5*x^3 + 24*c^2*d^2*e^5*x^2 + 16*c^3*d*e^5*x))) + log(c + d*x + 1)^2*(b^2/(16*d*e^5) - b^2/(16*d^2*e^5*(4*c^3*x + c^4/d + d^3*x^4 + 6*c^2 *d*x^2 + 4*c*d^2*x^3))) + log(1 - d*x - c)*(log(c + d*x + 1)*(b^2/(4*d*(2* c^4*e^5 + 2*d^4*e^5*x^4 + 8*c*d^3*e^5*x^3 + 12*c^2*d^2*e^5*x^2 + 8*c^3*d*e ^5*x)) - (b^2*(c^4 + d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2*x^2 + 4*c^3*d*x))/( 4*d*(2*c^4*e^5 + 2*d^4*e^5*x^4 + 8*c*d^3*e^5*x^3 + 12*c^2*d^2*e^5*x^2 + 8* c^3*d*e^5*x))) + (3*b^2)/(4*d*(24*c^4*e^5 + 24*d^4*e^5*x^4 + 96*c*d^3*e^5* x^3 + 144*c^2*d^2*e^5*x^2 + 96*c^3*d*e^5*x)) + (3*b*(8*a - b))/(4*d*(24*c^ 4*e^5 + 24*d^4*e^5*x^4 + 96*c*d^3*e^5*x^3 + 144*c^2*d^2*e^5*x^2 + 96*c^3*d *e^5*x)) - (b^2*(c*(2*c - 3*c^2 + 4*c^3 + c*(6*c^2 - 3*c + c*(12*c - 3) + 1) - 1) - 3*c + x^2*(d*(2*d - 6*c*d + 12*c^2*d + d*(6*c^2 - 3*c + c*(12*c - 3) + 1) + c*(24*c*d - 3*d + d*(12*c - 3))) - 9*c*d^2 + c*(30*c*d^2 - 3*d ^2 + d*(24*c*d - 3*d + d*(12*c - 3))) + 3*d^2 + 18*c^2*d^2) + x*(d*(2*c - 3*c^2 + 4*c^3 + c*(6*c^2 - 3*c + c*(12*c - 3) + 1) - 1) - 3*d + 6*c*d + c* (2*d - 6*c*d + 12*c^2*d + d*(6*c^2 - 3*c + c*(12*c - 3) + 1) + c*(24*c*d - 3*d + d*(12*c - 3))) - 9*c^2*d + 12*c^3*d) + 3*c^2 - 3*c^3 + 3*c^4 + 25*d ^4*x^4 + x^3*(34*c*d^3 + d*(30*c*d^2 - 3*d^2 + d*(24*c*d - 3*d + d*(12*c - 3))) - 3*d^3) + 3))/(4*d*(24*c^4*e^5 + 24*d^4*e^5*x^4 + 96*c*d^3*e^5*x^3 + 144*c^2*d^2*e^5*x^2 + 96*c^3*d*e^5*x)) + (b^2*(c*(c*(6*c*e^5 + 2*e^5 ...