3.22 \(\int \frac {(a+b \tanh ^{-1}(c+d x))^2}{(c e+d e x)^5} \, dx\)

Optimal. Leaf size=172 \[ -\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )}{2 d e^5 (c+d x)}-\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )}{6 d e^5 (c+d x)^3}-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d e^5 (c+d x)^4}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d e^5}-\frac {b^2}{12 d e^5 (c+d x)^2}+\frac {2 b^2 \log (c+d x)}{3 d e^5}-\frac {b^2 \log \left (1-(c+d x)^2\right )}{3 d e^5} \]

[Out]

-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

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Rubi [A]  time = 0.26, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {6107, 12, 5916, 5982, 266, 44, 36, 31, 29, 5948} \[ -\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )}{2 d e^5 (c+d x)}-\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )}{6 d e^5 (c+d x)^3}-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d e^5 (c+d x)^4}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d e^5}-\frac {b^2}{12 d e^5 (c+d x)^2}+\frac {2 b^2 \log (c+d x)}{3 d e^5}-\frac {b^2 \log \left (1-(c+d x)^2\right )}{3 d e^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c + d*x])^2/(c*e + d*e*x)^5,x]

[Out]

-b^2/(12*d*e^5*(c + d*x)^2) - (b*(a + b*ArcTanh[c + d*x]))/(6*d*e^5*(c + d*x)^3) - (b*(a + b*ArcTanh[c + d*x])
)/(2*d*e^5*(c + d*x)) + (a + b*ArcTanh[c + d*x])^2/(4*d*e^5) - (a + b*ArcTanh[c + d*x])^2/(4*d*e^5*(c + d*x)^4
) + (2*b^2*Log[c + d*x])/(3*d*e^5) - (b^2*Log[1 - (c + d*x)^2])/(3*d*e^5)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 5916

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcT
anh[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTanh[c*x])^(p - 1))/(1 -
c^2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 5948

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> 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]

Rule 5982

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/d
, Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x], x] - Dist[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]

Rule 6107

Int[((a_.) + ArcTanh[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[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]

Rubi steps

\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{(c e+d e x)^5} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right )^2}{e^5 x^5} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right )^2}{x^5} \, dx,x,c+d x\right )}{d e^5}\\ &=-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d e^5 (c+d x)^4}+\frac {b \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(x)}{x^4 \left (1-x^2\right )} \, dx,x,c+d x\right )}{2 d e^5}\\ &=-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d e^5 (c+d x)^4}+\frac {b \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(x)}{x^4} \, dx,x,c+d x\right )}{2 d e^5}+\frac {b \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(x)}{x^2 \left (1-x^2\right )} \, dx,x,c+d x\right )}{2 d e^5}\\ &=-\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )}{6 d e^5 (c+d x)^3}-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d e^5 (c+d x)^4}+\frac {b \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(x)}{x^2} \, dx,x,c+d x\right )}{2 d e^5}+\frac {b \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(x)}{1-x^2} \, dx,x,c+d x\right )}{2 d e^5}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{x^3 \left (1-x^2\right )} \, dx,x,c+d x\right )}{6 d e^5}\\ &=-\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )}{6 d e^5 (c+d x)^3}-\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )}{2 d e^5 (c+d x)}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d e^5}-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d e^5 (c+d x)^4}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{(1-x) x^2} \, dx,x,(c+d x)^2\right )}{12 d e^5}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{x \left (1-x^2\right )} \, dx,x,c+d x\right )}{2 d e^5}\\ &=-\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )}{6 d e^5 (c+d x)^3}-\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )}{2 d e^5 (c+d x)}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d e^5}-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d e^5 (c+d x)^4}+\frac {b^2 \operatorname {Subst}\left (\int \left (\frac {1}{1-x}+\frac {1}{x^2}+\frac {1}{x}\right ) \, dx,x,(c+d x)^2\right )}{12 d e^5}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{(1-x) x} \, dx,x,(c+d x)^2\right )}{4 d e^5}\\ &=-\frac {b^2}{12 d e^5 (c+d x)^2}-\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )}{6 d e^5 (c+d x)^3}-\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )}{2 d e^5 (c+d x)}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d e^5}-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d e^5 (c+d x)^4}+\frac {b^2 \log (c+d x)}{6 d e^5}-\frac {b^2 \log \left (1-(c+d x)^2\right )}{12 d e^5}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,(c+d x)^2\right )}{4 d e^5}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,(c+d x)^2\right )}{4 d e^5}\\ &=-\frac {b^2}{12 d e^5 (c+d x)^2}-\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )}{6 d e^5 (c+d x)^3}-\frac {b \left (a+b \tanh ^{-1}(c+d x)\right )}{2 d e^5 (c+d x)}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^2}{4 d e^5}-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^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}\\ \end {align*}

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Mathematica [A]  time = 0.26, size = 218, normalized size = 1.27 \[ -\frac {\frac {3 a^2}{(c+d x)^4}+\frac {2 b \tanh ^{-1}(c+d x) \left (3 a+b \left (3 c^3+9 c^2 d x+9 c d^2 x^2+c+3 d^3 x^3+d x\right )\right )}{(c+d x)^4}+\frac {6 a b}{c+d x}+\frac {2 a b}{(c+d x)^3}+b (3 a+4 b) \log (-c-d x+1)-b (3 a-4 b) \log (c+d x+1)-\frac {3 b^2 \left (c^4+4 c^3 d x+6 c^2 d^2 x^2+4 c d^3 x^3+d^4 x^4-1\right ) \tanh ^{-1}(c+d x)^2}{(c+d x)^4}+\frac {b^2}{(c+d x)^2}-8 b^2 \log (c+d x)}{12 d e^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c + d*x])^2/(c*e + d*e*x)^5,x]

[Out]

-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)

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fricas [B]  time = 0.52, size = 547, normalized size = 3.18 \[ -\frac {24 \, a b d^{3} x^{3} + 24 \, a b c^{3} + 4 \, {\left (18 \, a b c + b^{2}\right )} d^{2} x^{2} + 4 \, b^{2} c^{2} + 8 \, a b c + 8 \, {\left (9 \, a b c^{2} + b^{2} c + a b\right )} d x - 3 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4} - b^{2}\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )^{2} + 12 \, a^{2} - 4 \, {\left ({\left (3 \, a b - 4 \, b^{2}\right )} d^{4} x^{4} + 4 \, {\left (3 \, a b - 4 \, b^{2}\right )} c d^{3} x^{3} + 6 \, {\left (3 \, a b - 4 \, b^{2}\right )} c^{2} d^{2} x^{2} + 4 \, {\left (3 \, a b - 4 \, b^{2}\right )} c^{3} d x + {\left (3 \, a b - 4 \, b^{2}\right )} c^{4}\right )} \log \left (d x + c + 1\right ) - 32 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \left (d x + c\right ) + 4 \, {\left ({\left (3 \, a b + 4 \, b^{2}\right )} d^{4} x^{4} + 4 \, {\left (3 \, a b + 4 \, b^{2}\right )} c d^{3} x^{3} + 6 \, {\left (3 \, a b + 4 \, b^{2}\right )} c^{2} d^{2} x^{2} + 4 \, {\left (3 \, a b + 4 \, b^{2}\right )} c^{3} d x + {\left (3 \, a b + 4 \, b^{2}\right )} c^{4}\right )} \log \left (d x + c - 1\right ) + 4 \, {\left (3 \, b^{2} d^{3} x^{3} + 9 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{3} + b^{2} c + {\left (9 \, b^{2} c^{2} + b^{2}\right )} d x + 3 \, a b\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{48 \, {\left (d^{5} e^{5} x^{4} + 4 \, c d^{4} e^{5} x^{3} + 6 \, c^{2} d^{3} e^{5} x^{2} + 4 \, c^{3} d^{2} e^{5} x + c^{4} d e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(d*x+c))^2/(d*e*x+c*e)^5,x, algorithm="fricas")

[Out]

-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)

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giac [B]  time = 0.20, size = 758, normalized size = 4.41 \[ \frac {{\left (\frac {3 \, {\left (d x + c + 1\right )}^{3} b^{2} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )^{2}}{{\left (d x + c - 1\right )}^{3}} + \frac {3 \, {\left (d x + c + 1\right )} b^{2} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )^{2}}{d x + c - 1} + \frac {4 \, {\left (d x + c + 1\right )}^{4} b^{2} \log \left (-\frac {d x + c + 1}{d x + c - 1} - 1\right )}{{\left (d x + c - 1\right )}^{4}} + \frac {16 \, {\left (d x + c + 1\right )}^{3} b^{2} \log \left (-\frac {d x + c + 1}{d x + c - 1} - 1\right )}{{\left (d x + c - 1\right )}^{3}} + \frac {24 \, {\left (d x + c + 1\right )}^{2} b^{2} \log \left (-\frac {d x + c + 1}{d x + c - 1} - 1\right )}{{\left (d x + c - 1\right )}^{2}} + \frac {16 \, {\left (d x + c + 1\right )} b^{2} \log \left (-\frac {d x + c + 1}{d x + c - 1} - 1\right )}{d x + c - 1} + 4 \, b^{2} \log \left (-\frac {d x + c + 1}{d x + c - 1} - 1\right ) + \frac {12 \, {\left (d x + c + 1\right )}^{3} a b \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{{\left (d x + c - 1\right )}^{3}} + \frac {12 \, {\left (d x + c + 1\right )} a b \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{d x + c - 1} - \frac {4 \, {\left (d x + c + 1\right )}^{4} b^{2} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{{\left (d x + c - 1\right )}^{4}} - \frac {10 \, {\left (d x + c + 1\right )}^{3} b^{2} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{{\left (d x + c - 1\right )}^{3}} - \frac {12 \, {\left (d x + c + 1\right )}^{2} b^{2} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{{\left (d x + c - 1\right )}^{2}} - \frac {6 \, {\left (d x + c + 1\right )} b^{2} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{d x + c - 1} + \frac {12 \, {\left (d x + c + 1\right )}^{3} a^{2}}{{\left (d x + c - 1\right )}^{3}} + \frac {12 \, {\left (d x + c + 1\right )} a^{2}}{d x + c - 1} + \frac {12 \, {\left (d x + c + 1\right )}^{3} a b}{{\left (d x + c - 1\right )}^{3}} + \frac {24 \, {\left (d x + c + 1\right )}^{2} a b}{{\left (d x + c - 1\right )}^{2}} + \frac {20 \, {\left (d x + c + 1\right )} a b}{d x + c - 1} + 8 \, a b + \frac {2 \, {\left (d x + c + 1\right )}^{3} b^{2}}{{\left (d x + c - 1\right )}^{3}} + \frac {4 \, {\left (d x + c + 1\right )}^{2} b^{2}}{{\left (d x + c - 1\right )}^{2}} + \frac {2 \, {\left (d x + c + 1\right )} b^{2}}{d x + c - 1}\right )} {\left ({\left (c + 1\right )} d - {\left (c - 1\right )} d\right )}}{12 \, {\left (\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}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(d*x+c))^2/(d*e*x+c*e)^5,x, algorithm="giac")

[Out]

1/12*(3*(d*x + c + 1)^3*b^2*log(-(d*x + c + 1)/(d*x + c - 1))^2/(d*x + c - 1)^3 + 3*(d*x + c + 1)*b^2*log(-(d*
x + c + 1)/(d*x + c - 1))^2/(d*x + c - 1) + 4*(d*x + c + 1)^4*b^2*log(-(d*x + c + 1)/(d*x + c - 1) - 1)/(d*x +
c - 1)^4 + 16*(d*x + c + 1)^3*b^2*log(-(d*x + c + 1)/(d*x + c - 1) - 1)/(d*x + c - 1)^3 + 24*(d*x + c + 1)^2*
b^2*log(-(d*x + c + 1)/(d*x + c - 1) - 1)/(d*x + c - 1)^2 + 16*(d*x + c + 1)*b^2*log(-(d*x + c + 1)/(d*x + c -
1) - 1)/(d*x + c - 1) + 4*b^2*log(-(d*x + c + 1)/(d*x + c - 1) - 1) + 12*(d*x + c + 1)^3*a*b*log(-(d*x + c +
1)/(d*x + c - 1))/(d*x + c - 1)^3 + 12*(d*x + c + 1)*a*b*log(-(d*x + c + 1)/(d*x + c - 1))/(d*x + c - 1) - 4*(
d*x + c + 1)^4*b^2*log(-(d*x + c + 1)/(d*x + c - 1))/(d*x + c - 1)^4 - 10*(d*x + c + 1)^3*b^2*log(-(d*x + c +
1)/(d*x + c - 1))/(d*x + c - 1)^3 - 12*(d*x + c + 1)^2*b^2*log(-(d*x + c + 1)/(d*x + c - 1))/(d*x + c - 1)^2 -
6*(d*x + c + 1)*b^2*log(-(d*x + c + 1)/(d*x + c - 1))/(d*x + c - 1) + 12*(d*x + c + 1)^3*a^2/(d*x + c - 1)^3
+ 12*(d*x + c + 1)*a^2/(d*x + c - 1) + 12*(d*x + c + 1)^3*a*b/(d*x + c - 1)^3 + 24*(d*x + c + 1)^2*a*b/(d*x +
c - 1)^2 + 20*(d*x + c + 1)*a*b/(d*x + c - 1) + 8*a*b + 2*(d*x + c + 1)^3*b^2/(d*x + c - 1)^3 + 4*(d*x + c + 1
)^2*b^2/(d*x + c - 1)^2 + 2*(d*x + c + 1)*b^2/(d*x + c - 1))*((c + 1)*d - (c - 1)*d)/((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)

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maple [B]  time = 0.08, size = 431, normalized size = 2.51 \[ -\frac {a^{2}}{4 d \,e^{5} \left (d x +c \right )^{4}}-\frac {b^{2} \arctanh \left (d x +c \right )^{2}}{4 d \,e^{5} \left (d x +c \right )^{4}}-\frac {b^{2} \arctanh \left (d x +c \right )}{6 d \,e^{5} \left (d x +c \right )^{3}}-\frac {b^{2} \arctanh \left (d x +c \right )}{2 d \,e^{5} \left (d x +c \right )}-\frac {b^{2} \arctanh \left (d x +c \right ) \ln \left (d x +c -1\right )}{4 d \,e^{5}}+\frac {b^{2} \arctanh \left (d x +c \right ) \ln \left (d x +c +1\right )}{4 d \,e^{5}}-\frac {b^{2} \ln \left (d x +c -1\right )^{2}}{16 d \,e^{5}}+\frac {b^{2} \ln \left (d x +c -1\right ) \ln \left (\frac {1}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,e^{5}}-\frac {b^{2} \ln \left (d x +c +1\right )^{2}}{16 d \,e^{5}}+\frac {b^{2} \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right ) \ln \left (d x +c +1\right )}{8 d \,e^{5}}-\frac {b^{2} \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,e^{5}}-\frac {b^{2}}{12 d \,e^{5} \left (d x +c \right )^{2}}+\frac {2 b^{2} \ln \left (d x +c \right )}{3 d \,e^{5}}-\frac {b^{2} \ln \left (d x +c -1\right )}{3 d \,e^{5}}-\frac {b^{2} \ln \left (d x +c +1\right )}{3 d \,e^{5}}-\frac {a b \arctanh \left (d x +c \right )}{2 d \,e^{5} \left (d x +c \right )^{4}}-\frac {a b}{6 d \,e^{5} \left (d x +c \right )^{3}}-\frac {a b}{2 d \,e^{5} \left (d x +c \right )}-\frac {a b \ln \left (d x +c -1\right )}{4 d \,e^{5}}+\frac {a b \ln \left (d x +c +1\right )}{4 d \,e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(d*x+c))^2/(d*e*x+c*e)^5,x)

[Out]

-1/4/d*a^2/e^5/(d*x+c)^4-1/4/d*b^2/e^5/(d*x+c)^4*arctanh(d*x+c)^2-1/6/d*b^2/e^5*arctanh(d*x+c)/(d*x+c)^3-1/2/d
*b^2/e^5*arctanh(d*x+c)/(d*x+c)-1/4/d*b^2/e^5*arctanh(d*x+c)*ln(d*x+c-1)+1/4/d*b^2/e^5*arctanh(d*x+c)*ln(d*x+c
+1)-1/16/d*b^2/e^5*ln(d*x+c-1)^2+1/8/d*b^2/e^5*ln(d*x+c-1)*ln(1/2+1/2*d*x+1/2*c)-1/16/d*b^2/e^5*ln(d*x+c+1)^2+
1/8/d*b^2/e^5*ln(-1/2*d*x-1/2*c+1/2)*ln(d*x+c+1)-1/8/d*b^2/e^5*ln(-1/2*d*x-1/2*c+1/2)*ln(1/2+1/2*d*x+1/2*c)-1/
12*b^2/d/e^5/(d*x+c)^2+2/3*b^2*ln(d*x+c)/d/e^5-1/3/d*b^2/e^5*ln(d*x+c-1)-1/3/d*b^2/e^5*ln(d*x+c+1)-1/2/d*a*b/e
^5/(d*x+c)^4*arctanh(d*x+c)-1/6/d*a*b/e^5/(d*x+c)^3-1/2/d*a*b/e^5/(d*x+c)-1/4/d*a*b/e^5*ln(d*x+c-1)+1/4/d*a*b/
e^5*ln(d*x+c+1)

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maxima [B]  time = 0.38, size = 613, normalized size = 3.56 \[ -\frac {1}{12} \, {\left (d {\left (\frac {2 \, {\left (3 \, d^{2} x^{2} + 6 \, c d x + 3 \, c^{2} + 1\right )}}{d^{5} e^{5} x^{3} + 3 \, c d^{4} e^{5} x^{2} + 3 \, c^{2} d^{3} e^{5} x + c^{3} d^{2} e^{5}} - \frac {3 \, \log \left (d x + c + 1\right )}{d^{2} e^{5}} + \frac {3 \, \log \left (d x + c - 1\right )}{d^{2} e^{5}}\right )} + \frac {6 \, \operatorname {artanh}\left (d x + c\right )}{d^{5} e^{5} x^{4} + 4 \, c d^{4} e^{5} x^{3} + 6 \, c^{2} d^{3} e^{5} x^{2} + 4 \, c^{3} d^{2} e^{5} x + c^{4} d e^{5}}\right )} a b - \frac {1}{48} \, {\left (d^{2} {\left (\frac {3 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \log \left (d x + c + 1\right )^{2} + 3 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \log \left (d x + c - 1\right )^{2} + 2 \, {\left (8 \, d^{2} x^{2} + 16 \, c d x + 8 \, c^{2} - 3 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \log \left (d x + c - 1\right )\right )} \log \left (d x + c + 1\right ) + 16 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \log \left (d x + c - 1\right ) + 4}{d^{5} e^{5} x^{2} + 2 \, c d^{4} e^{5} x + c^{2} d^{3} e^{5}} - \frac {32 \, \log \left (d x + c\right )}{d^{3} e^{5}}\right )} + 4 \, d {\left (\frac {2 \, {\left (3 \, d^{2} x^{2} + 6 \, c d x + 3 \, c^{2} + 1\right )}}{d^{5} e^{5} x^{3} + 3 \, c d^{4} e^{5} x^{2} + 3 \, c^{2} d^{3} e^{5} x + c^{3} d^{2} e^{5}} - \frac {3 \, \log \left (d x + c + 1\right )}{d^{2} e^{5}} + \frac {3 \, \log \left (d x + c - 1\right )}{d^{2} e^{5}}\right )} \operatorname {artanh}\left (d x + c\right )\right )} b^{2} - \frac {b^{2} \operatorname {artanh}\left (d x + c\right )^{2}}{4 \, {\left (d^{5} e^{5} x^{4} + 4 \, c d^{4} e^{5} x^{3} + 6 \, c^{2} d^{3} e^{5} x^{2} + 4 \, c^{3} d^{2} e^{5} x + c^{4} d e^{5}\right )}} - \frac {a^{2}}{4 \, {\left (d^{5} e^{5} x^{4} + 4 \, c d^{4} e^{5} x^{3} + 6 \, c^{2} d^{3} e^{5} x^{2} + 4 \, c^{3} d^{2} e^{5} x + c^{4} d e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(d*x+c))^2/(d*e*x+c*e)^5,x, algorithm="maxima")

[Out]

-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)

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mupad [B]  time = 3.42, size = 2746, normalized size = 15.97 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c + d*x))^2/(c*e + d*e*x)^5,x)

[Out]

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*(2
4*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 + c*(24*c*e^5 + 6*e^5) + 12*c^2*e^5) + 4*c*e^5 + 2*e^5 + 6*c^2*
e^5 + 8*c^3*e^5) + x*(d*(c*(6*c*e^5 + 2*e^5 + c*(24*c*e^5 + 6*e^5) + 12*c^2*e^5) + 4*c*e^5 + 2*e^5 + 6*c^2*e^5
+ 8*c^3*e^5) + 6*d*e^5 + c*(c*(6*d*e^5 + d*(24*c*e^5 + 6*e^5) + 48*c*d*e^5) + d*(6*c*e^5 + 2*e^5 + c*(24*c*e^
5 + 6*e^5) + 12*c^2*e^5) + 4*d*e^5 + 24*c^2*d*e^5 + 12*c*d*e^5) + 18*c^2*d*e^5 + 24*c^3*d*e^5 + 12*c*d*e^5) +
x^3*(d*(d*(6*d*e^5 + d*(24*c*e^5 + 6*e^5) + 48*c*d*e^5) + 6*d^2*e^5 + 60*c*d^2*e^5) + 6*d^3*e^5 + 68*c*d^3*e^5
) + 6*c*e^5 + 6*e^5 + 6*c^2*e^5 + 6*c^3*e^5 + 6*c^4*e^5 + x^2*(c*(d*(6*d*e^5 + d*(24*c*e^5 + 6*e^5) + 48*c*d*e
^5) + 6*d^2*e^5 + 60*c*d^2*e^5) + d*(c*(6*d*e^5 + d*(24*c*e^5 + 6*e^5) + 48*c*d*e^5) + d*(6*c*e^5 + 2*e^5 + c*
(24*c*e^5 + 6*e^5) + 12*c^2*e^5) + 4*d*e^5 + 24*c^2*d*e^5 + 12*c*d*e^5) + 6*d^2*e^5 + 18*c*d^2*e^5 + 36*c^2*d^
2*e^5) + 50*d^4*e^5*x^4))/(8*d*e^5*(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))) - (x^2*((b^2*d)/2 + 9*a*b*c*d) + x*(a*b + b^2*c + 9*a*b*c^2) + (3*a^2 + b^2*c^2 + 2*a*b*c + 6*a
*b*c^3)/(2*d) + 3*a*b*d^2*x^3)/(6*c^4*e^5 + 6*d^4*e^5*x^4 + 24*c*d^3*e^5*x^3 + 36*c^2*d^2*e^5*x^2 + 24*c^3*d*e
^5*x) - (log(c + d*x + 1)*(x^2*((3*b^2*c + b^2 + 6*b^2*c^2)/(32*e^5) + d*(d*((3*b^2*c + b^2 + 6*b^2*c^2)/(96*d
^2*e^5) + (c*(4*b^2*c + b^2))/(32*d^2*e^5)) + c*((4*b^2*c + b^2)/(16*d*e^5) + (b^2*c)/(8*d*e^5)) + (3*b^2*c +
b^2 + 6*b^2*c^2)/(48*d*e^5)) + c*(d*((4*b^2*c + b^2)/(16*d*e^5) + (b^2*c)/(8*d*e^5)) + (4*b^2*c + b^2)/(32*e^5
) + (3*b^2*c)/(16*e^5)) - (b^2*(d/4 - (3*c*d)/4 + (3*c^2*d)/2 + d*(d*((6*c^2 - 3*c + 1)/(12*d) + (c*(4*c - 1))
/(4*d)) - c/2 + c^2 + c*(3*c - 1/2) + 1/6) + c*((5*c*d)/2 - d/4 + d*(3*c - 1/2))))/(8*d*e^5)) + x*(c*(d*((3*b^
2*c + b^2 + 6*b^2*c^2)/(96*d^2*e^5) + (c*(4*b^2*c + b^2))/(32*d^2*e^5)) + c*((4*b^2*c + b^2)/(16*d*e^5) + (b^2
*c)/(8*d*e^5)) + (3*b^2*c + b^2 + 6*b^2*c^2)/(48*d*e^5)) + d*(c*((3*b^2*c + b^2 + 6*b^2*c^2)/(96*d^2*e^5) + (c
*(4*b^2*c + b^2))/(32*d^2*e^5)) + (2*b^2*c + b^2 + 3*b^2*c^2 + 4*b^2*c^3)/(96*d^2*e^5)) + (2*b^2*c + b^2 + 3*b
^2*c^2 + 4*b^2*c^3)/(32*d*e^5) - (b^2*(c/2 + c*(d*((6*c^2 - 3*c + 1)/(12*d) + (c*(4*c - 1))/(4*d)) - c/2 + c^2
+ c*(3*c - 1/2) + 1/6) - (3*c^2)/4 + c^3 + d*(c*((6*c^2 - 3*c + 1)/(12*d) + (c*(4*c - 1))/(4*d)) + (2*c - 3*c
^2 + 4*c^3 - 1)/(12*d)) - 1/4))/(8*d*e^5)) + x^3*(d*(d*((4*b^2*c + b^2)/(16*d*e^5) + (b^2*c)/(8*d*e^5)) + (4*b
^2*c + b^2)/(32*e^5) + (3*b^2*c)/(16*e^5)) + (b^2*d + 4*b^2*c*d)/(32*e^5) - (b^2*((17*c*d^2)/6 - d^2/4 + d*((5
*c*d)/2 - d/4 + d*(3*c - 1/2))))/(8*d*e^5) + (11*b^2*c*d)/(48*e^5)) + c*(c*((3*b^2*c + b^2 + 6*b^2*c^2)/(96*d^
2*e^5) + (c*(4*b^2*c + b^2))/(32*d^2*e^5)) + (2*b^2*c + b^2 + 3*b^2*c^2 + 4*b^2*c^3)/(96*d^2*e^5)) + (8*a*b +
b^2*c + b^2 + b^2*c^2 + b^2*c^3 + b^2*c^4)/(32*d^2*e^5) - (b^2*(c*(c*((6*c^2 - 3*c + 1)/(12*d) + (c*(4*c - 1))
/(4*d)) + (2*c - 3*c^2 + 4*c^3 - 1)/(12*d)) + (c^2 - c - c^3 + c^4 + 1)/(4*d)))/(8*d*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) + (2*b^2*log(c + d*x))/(3*d*e^5) - (log(c + d*x - 1)*(3*a*b + 4*b^2))/(1
2*d*e^5) + (log(c + d*x + 1)*(3*a*b - 4*b^2))/(12*d*e^5)

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sympy [A]  time = 10.59, size = 3516, normalized size = 20.44 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atanh(d*x+c))**2/(d*e*x+c*e)**5,x)

[Out]

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 + 48*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 + 7
2*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*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*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) + 8*b**2*c**4*log(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) - 8*b**2*c**4*log(c/d + x + 1/d)/(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) + 3*b**2*c**4*atanh(c + d*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)
+ 8*b**2*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) + 32*b**2*c**3*d*x*log(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) - 32*b**2*c**3*d*x*log(c/d + x + 1/d)/(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) + 12*b**2*c**3*d*x*ata
nh(c + d*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) + 32*b**2*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*b**2*c**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 + 48*c*d**4*e**5*x**3 + 12*d**5*e**5*x**4) + 48*b**2*c**2*d**2*x**2*log(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) - 48*b**2*
c**2*d**2*x**2*log(c/d + x + 1/d)/(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*b**2*c**2*d**2*x**2*atanh(c + d*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) + 48*b**2*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*
b**2*c**2*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) - b**2*c**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) + 32*b**2*c*d**3*x**3*log(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) - 32*b**2*c*d**3*x**3*log(c/d + x + 1/d)/(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) + 12*b**2
*c*d**3*x**3*atanh(c + d*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) + 32*b**2*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 + 48*c*d**4*e**5*x**3 + 12*d**5*e**5*x**4) - 18*b**2*c*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) - 2*b**2*c*d*x/(1
2*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*b*
*2*c*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) + 8*b**2*d**4*x**4*log(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) - 8*b**2*d**4*x**4*log(c/d + x + 1/d)/(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) + 3*b**2*d**4*x**4*atanh(c + d*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)
+ 8*b**2*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*b**2*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 + 48*c*d**4*e**5*x**3 + 12*d**5*e**5*x**4) - b**2*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*b**2*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) - 3*b**2*ata
nh(c + d*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), Ne(d, 0)), (x*(a + b*atanh(c))**2/(c**5*e**5), True))

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