3.8 \(\int \frac {a+b \tanh ^{-1}(c x)}{(d+e x)^4} \, dx\)
Optimal. Leaf size=175 \[ -\frac {a+b \tanh ^{-1}(c x)}{3 e (d+e x)^3}-\frac {b c^3 \log (1-c x)}{6 e (c d+e)^3}+\frac {b c^3 \log (c x+1)}{6 e (c d-e)^3}+\frac {b c}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {2 b c^3 d}{3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {b c^3 \left (3 c^2 d^2+e^2\right ) \log (d+e x)}{3 (c d-e)^3 (c d+e)^3} \]
[Out]
1/6*b*c/(c^2*d^2-e^2)/(e*x+d)^2+2/3*b*c^3*d/(c^2*d^2-e^2)^2/(e*x+d)+1/3*(-a-b*arctanh(c*x))/e/(e*x+d)^3-1/6*b*
c^3*ln(-c*x+1)/e/(c*d+e)^3+1/6*b*c^3*ln(c*x+1)/(c*d-e)^3/e-1/3*b*c^3*(3*c^2*d^2+e^2)*ln(e*x+d)/(c^2*d^2-e^2)^3
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Rubi [A] time = 0.19, antiderivative size = 175, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used =
{5926, 710, 801} \[ -\frac {a+b \tanh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {2 b c^3 d}{3 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac {b c}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {b c^3 \left (3 c^2 d^2+e^2\right ) \log (d+e x)}{3 (c d-e)^3 (c d+e)^3}-\frac {b c^3 \log (1-c x)}{6 e (c d+e)^3}+\frac {b c^3 \log (c x+1)}{6 e (c d-e)^3} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*ArcTanh[c*x])/(d + e*x)^4,x]
[Out]
(b*c)/(6*(c^2*d^2 - e^2)*(d + e*x)^2) + (2*b*c^3*d)/(3*(c^2*d^2 - e^2)^2*(d + e*x)) - (a + b*ArcTanh[c*x])/(3*
e*(d + e*x)^3) - (b*c^3*Log[1 - c*x])/(6*e*(c*d + e)^3) + (b*c^3*Log[1 + c*x])/(6*(c*d - e)^3*e) - (b*c^3*(3*c
^2*d^2 + e^2)*Log[d + e*x])/(3*(c*d - e)^3*(c*d + e)^3)
Rule 710
Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 +
a*e^2)), x] + Dist[c/(c*d^2 + a*e^2), Int[((d + e*x)^(m + 1)*(d - e*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d,
e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]
Rule 801
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]
Rule 5926
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1)*(a + b
*ArcTanh[c*x]))/(e*(q + 1)), x] - Dist[(b*c)/(e*(q + 1)), Int[(d + e*x)^(q + 1)/(1 - c^2*x^2), x], x] /; FreeQ
[{a, b, c, d, e, q}, x] && NeQ[q, -1]
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}(c x)}{(d+e x)^4} \, dx &=-\frac {a+b \tanh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {(b c) \int \frac {1}{(d+e x)^3 \left (1-c^2 x^2\right )} \, dx}{3 e}\\ &=\frac {b c}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {a+b \tanh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {\left (b c^3\right ) \int \frac {d-e x}{(d+e x)^2 \left (1-c^2 x^2\right )} \, dx}{3 e \left (c^2 d^2-e^2\right )}\\ &=\frac {b c}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {a+b \tanh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {\left (b c^3\right ) \int \left (-\frac {c (c d-e)}{2 (c d+e)^2 (-1+c x)}+\frac {c (c d+e)}{2 (c d-e)^2 (1+c x)}+\frac {2 d e^2}{(-c d+e) (c d+e) (d+e x)^2}-\frac {e^2 \left (3 c^2 d^2+e^2\right )}{(-c d+e)^2 (c d+e)^2 (d+e x)}\right ) \, dx}{3 e \left (c^2 d^2-e^2\right )}\\ &=\frac {b c}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {2 b c^3 d}{3 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {a+b \tanh ^{-1}(c x)}{3 e (d+e x)^3}-\frac {b c^3 \log (1-c x)}{6 e (c d+e)^3}+\frac {b c^3 \log (1+c x)}{6 (c d-e)^3 e}-\frac {b c^3 \left (3 c^2 d^2+e^2\right ) \log (d+e x)}{3 (c d-e)^3 (c d+e)^3}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 173, normalized size = 0.99 \[ \frac {1}{6} \left (-\frac {2 a}{e (d+e x)^3}-\frac {b c^3 \log (1-c x)}{e (c d+e)^3}+\frac {b c^3 \log (c x+1)}{e (c d-e)^3}+\frac {b c}{\left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {4 b c^3 d}{\left (e^2-c^2 d^2\right )^2 (d+e x)}-\frac {2 b c^3 \left (3 c^2 d^2+e^2\right ) \log (d+e x)}{\left (c^2 d^2-e^2\right )^3}-\frac {2 b \tanh ^{-1}(c x)}{e (d+e x)^3}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*ArcTanh[c*x])/(d + e*x)^4,x]
[Out]
((-2*a)/(e*(d + e*x)^3) + (b*c)/((c^2*d^2 - e^2)*(d + e*x)^2) + (4*b*c^3*d)/((-(c^2*d^2) + e^2)^2*(d + e*x)) -
(2*b*ArcTanh[c*x])/(e*(d + e*x)^3) - (b*c^3*Log[1 - c*x])/(e*(c*d + e)^3) + (b*c^3*Log[1 + c*x])/((c*d - e)^3
*e) - (2*b*c^3*(3*c^2*d^2 + e^2)*Log[d + e*x])/(c^2*d^2 - e^2)^3)/6
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fricas [B] time = 1.70, size = 859, normalized size = 4.91 \[ -\frac {2 \, a c^{6} d^{6} - 5 \, b c^{5} d^{5} e - 6 \, a c^{4} d^{4} e^{2} + 6 \, b c^{3} d^{3} e^{3} + 6 \, a c^{2} d^{2} e^{4} - b c d e^{5} - 2 \, a e^{6} - 4 \, {\left (b c^{5} d^{3} e^{3} - b c^{3} d e^{5}\right )} x^{2} - {\left (9 \, b c^{5} d^{4} e^{2} - 10 \, b c^{3} d^{2} e^{4} + b c e^{6}\right )} x - {\left (b c^{6} d^{6} + 3 \, b c^{5} d^{5} e + 3 \, b c^{4} d^{4} e^{2} + b c^{3} d^{3} e^{3} + {\left (b c^{6} d^{3} e^{3} + 3 \, b c^{5} d^{2} e^{4} + 3 \, b c^{4} d e^{5} + b c^{3} e^{6}\right )} x^{3} + 3 \, {\left (b c^{6} d^{4} e^{2} + 3 \, b c^{5} d^{3} e^{3} + 3 \, b c^{4} d^{2} e^{4} + b c^{3} d e^{5}\right )} x^{2} + 3 \, {\left (b c^{6} d^{5} e + 3 \, b c^{5} d^{4} e^{2} + 3 \, b c^{4} d^{3} e^{3} + b c^{3} d^{2} e^{4}\right )} x\right )} \log \left (c x + 1\right ) + {\left (b c^{6} d^{6} - 3 \, b c^{5} d^{5} e + 3 \, b c^{4} d^{4} e^{2} - b c^{3} d^{3} e^{3} + {\left (b c^{6} d^{3} e^{3} - 3 \, b c^{5} d^{2} e^{4} + 3 \, b c^{4} d e^{5} - b c^{3} e^{6}\right )} x^{3} + 3 \, {\left (b c^{6} d^{4} e^{2} - 3 \, b c^{5} d^{3} e^{3} + 3 \, b c^{4} d^{2} e^{4} - b c^{3} d e^{5}\right )} x^{2} + 3 \, {\left (b c^{6} d^{5} e - 3 \, b c^{5} d^{4} e^{2} + 3 \, b c^{4} d^{3} e^{3} - b c^{3} d^{2} e^{4}\right )} x\right )} \log \left (c x - 1\right ) + 2 \, {\left (3 \, b c^{5} d^{5} e + b c^{3} d^{3} e^{3} + {\left (3 \, b c^{5} d^{2} e^{4} + b c^{3} e^{6}\right )} x^{3} + 3 \, {\left (3 \, b c^{5} d^{3} e^{3} + b c^{3} d e^{5}\right )} x^{2} + 3 \, {\left (3 \, b c^{5} d^{4} e^{2} + b c^{3} d^{2} e^{4}\right )} x\right )} \log \left (e x + d\right ) + {\left (b c^{6} d^{6} - 3 \, b c^{4} d^{4} e^{2} + 3 \, b c^{2} d^{2} e^{4} - b e^{6}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{6 \, {\left (c^{6} d^{9} e - 3 \, c^{4} d^{7} e^{3} + 3 \, c^{2} d^{5} e^{5} - d^{3} e^{7} + {\left (c^{6} d^{6} e^{4} - 3 \, c^{4} d^{4} e^{6} + 3 \, c^{2} d^{2} e^{8} - e^{10}\right )} x^{3} + 3 \, {\left (c^{6} d^{7} e^{3} - 3 \, c^{4} d^{5} e^{5} + 3 \, c^{2} d^{3} e^{7} - d e^{9}\right )} x^{2} + 3 \, {\left (c^{6} d^{8} e^{2} - 3 \, c^{4} d^{6} e^{4} + 3 \, c^{2} d^{4} e^{6} - d^{2} e^{8}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*arctanh(c*x))/(e*x+d)^4,x, algorithm="fricas")
[Out]
-1/6*(2*a*c^6*d^6 - 5*b*c^5*d^5*e - 6*a*c^4*d^4*e^2 + 6*b*c^3*d^3*e^3 + 6*a*c^2*d^2*e^4 - b*c*d*e^5 - 2*a*e^6
- 4*(b*c^5*d^3*e^3 - b*c^3*d*e^5)*x^2 - (9*b*c^5*d^4*e^2 - 10*b*c^3*d^2*e^4 + b*c*e^6)*x - (b*c^6*d^6 + 3*b*c^
5*d^5*e + 3*b*c^4*d^4*e^2 + b*c^3*d^3*e^3 + (b*c^6*d^3*e^3 + 3*b*c^5*d^2*e^4 + 3*b*c^4*d*e^5 + b*c^3*e^6)*x^3
+ 3*(b*c^6*d^4*e^2 + 3*b*c^5*d^3*e^3 + 3*b*c^4*d^2*e^4 + b*c^3*d*e^5)*x^2 + 3*(b*c^6*d^5*e + 3*b*c^5*d^4*e^2 +
3*b*c^4*d^3*e^3 + b*c^3*d^2*e^4)*x)*log(c*x + 1) + (b*c^6*d^6 - 3*b*c^5*d^5*e + 3*b*c^4*d^4*e^2 - b*c^3*d^3*e
^3 + (b*c^6*d^3*e^3 - 3*b*c^5*d^2*e^4 + 3*b*c^4*d*e^5 - b*c^3*e^6)*x^3 + 3*(b*c^6*d^4*e^2 - 3*b*c^5*d^3*e^3 +
3*b*c^4*d^2*e^4 - b*c^3*d*e^5)*x^2 + 3*(b*c^6*d^5*e - 3*b*c^5*d^4*e^2 + 3*b*c^4*d^3*e^3 - b*c^3*d^2*e^4)*x)*lo
g(c*x - 1) + 2*(3*b*c^5*d^5*e + b*c^3*d^3*e^3 + (3*b*c^5*d^2*e^4 + b*c^3*e^6)*x^3 + 3*(3*b*c^5*d^3*e^3 + b*c^3
*d*e^5)*x^2 + 3*(3*b*c^5*d^4*e^2 + b*c^3*d^2*e^4)*x)*log(e*x + d) + (b*c^6*d^6 - 3*b*c^4*d^4*e^2 + 3*b*c^2*d^2
*e^4 - b*e^6)*log(-(c*x + 1)/(c*x - 1)))/(c^6*d^9*e - 3*c^4*d^7*e^3 + 3*c^2*d^5*e^5 - d^3*e^7 + (c^6*d^6*e^4 -
3*c^4*d^4*e^6 + 3*c^2*d^2*e^8 - e^10)*x^3 + 3*(c^6*d^7*e^3 - 3*c^4*d^5*e^5 + 3*c^2*d^3*e^7 - d*e^9)*x^2 + 3*(
c^6*d^8*e^2 - 3*c^4*d^6*e^4 + 3*c^2*d^4*e^6 - d^2*e^8)*x)
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giac [B] time = 0.24, size = 3320, normalized size = 18.97 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*arctanh(c*x))/(e*x+d)^4,x, algorithm="giac")
[Out]
-1/3*(3*(c*x + 1)^3*b*c^7*d^5*log(-(c*x + 1)*c*d/(c*x - 1) + c*d - (c*x + 1)*e/(c*x - 1) - e)/(c*x - 1)^3 - 9*
(c*x + 1)^2*b*c^7*d^5*log(-(c*x + 1)*c*d/(c*x - 1) + c*d - (c*x + 1)*e/(c*x - 1) - e)/(c*x - 1)^2 + 9*(c*x + 1
)*b*c^7*d^5*log(-(c*x + 1)*c*d/(c*x - 1) + c*d - (c*x + 1)*e/(c*x - 1) - e)/(c*x - 1) - 3*b*c^7*d^5*log(-(c*x
+ 1)*c*d/(c*x - 1) + c*d - (c*x + 1)*e/(c*x - 1) - e) - 3*(c*x + 1)^3*b*c^7*d^5*log(-(c*x + 1)/(c*x - 1))/(c*x
- 1)^3 + 6*(c*x + 1)^2*b*c^7*d^5*log(-(c*x + 1)/(c*x - 1))/(c*x - 1)^2 - 3*(c*x + 1)*b*c^7*d^5*log(-(c*x + 1)
/(c*x - 1))/(c*x - 1) - 6*(c*x + 1)^2*a*c^7*d^5/(c*x - 1)^2 + 12*(c*x + 1)*a*c^7*d^5/(c*x - 1) - 6*a*c^7*d^5 +
9*(c*x + 1)^3*b*c^6*d^4*e*log(-(c*x + 1)*c*d/(c*x - 1) + c*d - (c*x + 1)*e/(c*x - 1) - e)/(c*x - 1)^3 - 9*(c*
x + 1)^2*b*c^6*d^4*e*log(-(c*x + 1)*c*d/(c*x - 1) + c*d - (c*x + 1)*e/(c*x - 1) - e)/(c*x - 1)^2 - 9*(c*x + 1)
*b*c^6*d^4*e*log(-(c*x + 1)*c*d/(c*x - 1) + c*d - (c*x + 1)*e/(c*x - 1) - e)/(c*x - 1) + 9*b*c^6*d^4*e*log(-(c
*x + 1)*c*d/(c*x - 1) + c*d - (c*x + 1)*e/(c*x - 1) - e) - 9*(c*x + 1)^3*b*c^6*d^4*e*log(-(c*x + 1)/(c*x - 1))
/(c*x - 1)^3 + 12*(c*x + 1)^2*b*c^6*d^4*e*log(-(c*x + 1)/(c*x - 1))/(c*x - 1)^2 - 3*(c*x + 1)*b*c^6*d^4*e*log(
-(c*x + 1)/(c*x - 1))/(c*x - 1) + 6*(c*x + 1)^2*a*c^6*d^4*e/(c*x - 1)^2 - 24*(c*x + 1)*a*c^6*d^4*e/(c*x - 1) +
18*a*c^6*d^4*e + 6*(c*x + 1)^2*b*c^6*d^4*e/(c*x - 1)^2 - 12*(c*x + 1)*b*c^6*d^4*e/(c*x - 1) + 6*b*c^6*d^4*e +
10*(c*x + 1)^3*b*c^5*d^3*e^2*log(-(c*x + 1)*c*d/(c*x - 1) + c*d - (c*x + 1)*e/(c*x - 1) - e)/(c*x - 1)^3 + 6*
(c*x + 1)^2*b*c^5*d^3*e^2*log(-(c*x + 1)*c*d/(c*x - 1) + c*d - (c*x + 1)*e/(c*x - 1) - e)/(c*x - 1)^2 - 6*(c*x
+ 1)*b*c^5*d^3*e^2*log(-(c*x + 1)*c*d/(c*x - 1) + c*d - (c*x + 1)*e/(c*x - 1) - e)/(c*x - 1) - 10*b*c^5*d^3*e
^2*log(-(c*x + 1)*c*d/(c*x - 1) + c*d - (c*x + 1)*e/(c*x - 1) - e) - 10*(c*x + 1)^3*b*c^5*d^3*e^2*log(-(c*x +
1)/(c*x - 1))/(c*x - 1)^3 + 6*(c*x + 1)*b*c^5*d^3*e^2*log(-(c*x + 1)/(c*x - 1))/(c*x - 1) + 12*(c*x + 1)^2*a*c
^5*d^3*e^2/(c*x - 1)^2 - 20*a*c^5*d^3*e^2 + 4*(c*x + 1)^2*b*c^5*d^3*e^2/(c*x - 1)^2 + 14*(c*x + 1)*b*c^5*d^3*e
^2/(c*x - 1) - 18*b*c^5*d^3*e^2 + 6*(c*x + 1)^3*b*c^4*d^2*e^3*log(-(c*x + 1)*c*d/(c*x - 1) + c*d - (c*x + 1)*e
/(c*x - 1) - e)/(c*x - 1)^3 + 6*(c*x + 1)^2*b*c^4*d^2*e^3*log(-(c*x + 1)*c*d/(c*x - 1) + c*d - (c*x + 1)*e/(c*
x - 1) - e)/(c*x - 1)^2 + 6*(c*x + 1)*b*c^4*d^2*e^3*log(-(c*x + 1)*c*d/(c*x - 1) + c*d - (c*x + 1)*e/(c*x - 1)
- e)/(c*x - 1) + 6*b*c^4*d^2*e^3*log(-(c*x + 1)*c*d/(c*x - 1) + c*d - (c*x + 1)*e/(c*x - 1) - e) - 6*(c*x + 1
)^3*b*c^4*d^2*e^3*log(-(c*x + 1)/(c*x - 1))/(c*x - 1)^3 - 12*(c*x + 1)^2*b*c^4*d^2*e^3*log(-(c*x + 1)/(c*x - 1
))/(c*x - 1)^2 + 6*(c*x + 1)*b*c^4*d^2*e^3*log(-(c*x + 1)/(c*x - 1))/(c*x - 1) - 12*(c*x + 1)^2*a*c^4*d^2*e^3/
(c*x - 1)^2 + 24*(c*x + 1)*a*c^4*d^2*e^3/(c*x - 1) + 12*a*c^4*d^2*e^3 - 8*(c*x + 1)^2*b*c^4*d^2*e^3/(c*x - 1)^
2 + 10*(c*x + 1)*b*c^4*d^2*e^3/(c*x - 1) + 18*b*c^4*d^2*e^3 + 3*(c*x + 1)^3*b*c^3*d*e^4*log(-(c*x + 1)*c*d/(c*
x - 1) + c*d - (c*x + 1)*e/(c*x - 1) - e)/(c*x - 1)^3 + 3*(c*x + 1)^2*b*c^3*d*e^4*log(-(c*x + 1)*c*d/(c*x - 1)
+ c*d - (c*x + 1)*e/(c*x - 1) - e)/(c*x - 1)^2 - 3*(c*x + 1)*b*c^3*d*e^4*log(-(c*x + 1)*c*d/(c*x - 1) + c*d -
(c*x + 1)*e/(c*x - 1) - e)/(c*x - 1) - 3*b*c^3*d*e^4*log(-(c*x + 1)*c*d/(c*x - 1) + c*d - (c*x + 1)*e/(c*x -
1) - e) - 3*(c*x + 1)^3*b*c^3*d*e^4*log(-(c*x + 1)/(c*x - 1))/(c*x - 1)^3 - 6*(c*x + 1)^2*b*c^3*d*e^4*log(-(c*
x + 1)/(c*x - 1))/(c*x - 1)^2 - 3*(c*x + 1)*b*c^3*d*e^4*log(-(c*x + 1)/(c*x - 1))/(c*x - 1) - 6*(c*x + 1)^2*a*
c^3*d*e^4/(c*x - 1)^2 - 12*(c*x + 1)*a*c^3*d*e^4/(c*x - 1) - 6*a*c^3*d*e^4 - 4*(c*x + 1)^2*b*c^3*d*e^4/(c*x -
1)^2 - 14*(c*x + 1)*b*c^3*d*e^4/(c*x - 1) - 6*b*c^3*d*e^4 + (c*x + 1)^3*b*c^2*e^5*log(-(c*x + 1)*c*d/(c*x - 1)
+ c*d - (c*x + 1)*e/(c*x - 1) - e)/(c*x - 1)^3 + 3*(c*x + 1)^2*b*c^2*e^5*log(-(c*x + 1)*c*d/(c*x - 1) + c*d -
(c*x + 1)*e/(c*x - 1) - e)/(c*x - 1)^2 + 3*(c*x + 1)*b*c^2*e^5*log(-(c*x + 1)*c*d/(c*x - 1) + c*d - (c*x + 1)
*e/(c*x - 1) - e)/(c*x - 1) + b*c^2*e^5*log(-(c*x + 1)*c*d/(c*x - 1) + c*d - (c*x + 1)*e/(c*x - 1) - e) - (c*x
+ 1)^3*b*c^2*e^5*log(-(c*x + 1)/(c*x - 1))/(c*x - 1)^3 - 3*(c*x + 1)*b*c^2*e^5*log(-(c*x + 1)/(c*x - 1))/(c*x
- 1) + 6*(c*x + 1)^2*a*c^2*e^5/(c*x - 1)^2 + 2*a*c^2*e^5 + 2*(c*x + 1)^2*b*c^2*e^5/(c*x - 1)^2 + 2*(c*x + 1)*
b*c^2*e^5/(c*x - 1))*c/((c*x + 1)^3*c^9*d^9/(c*x - 1)^3 - 3*(c*x + 1)^2*c^9*d^9/(c*x - 1)^2 + 3*(c*x + 1)*c^9*
d^9/(c*x - 1) - c^9*d^9 + 3*(c*x + 1)^3*c^8*d^8*e/(c*x - 1)^3 - 3*(c*x + 1)^2*c^8*d^8*e/(c*x - 1)^2 - 3*(c*x +
1)*c^8*d^8*e/(c*x - 1) + 3*c^8*d^8*e + 12*(c*x + 1)^2*c^7*d^7*e^2/(c*x - 1)^2 - 12*(c*x + 1)*c^7*d^7*e^2/(c*x
- 1) - 8*(c*x + 1)^3*c^6*d^6*e^3/(c*x - 1)^3 + 12*(c*x + 1)^2*c^6*d^6*e^3/(c*x - 1)^2 + 12*(c*x + 1)*c^6*d^6*
e^3/(c*x - 1) - 8*c^6*d^6*e^3 - 6*(c*x + 1)^3*c^5*d^5*e^4/(c*x - 1)^3 - 18*(c*x + 1)^2*c^5*d^5*e^4/(c*x - 1)^2
+ 18*(c*x + 1)*c^5*d^5*e^4/(c*x - 1) + 6*c^5*d^5*e^4 + 6*(c*x + 1)^3*c^4*d^4*e^5/(c*x - 1)^3 - 18*(c*x + 1)^2
*c^4*d^4*e^5/(c*x - 1)^2 - 18*(c*x + 1)*c^4*d^4*e^5/(c*x - 1) + 6*c^4*d^4*e^5 + 8*(c*x + 1)^3*c^3*d^3*e^6/(c*x
- 1)^3 + 12*(c*x + 1)^2*c^3*d^3*e^6/(c*x - 1)^2 - 12*(c*x + 1)*c^3*d^3*e^6/(c*x - 1) - 8*c^3*d^3*e^6 + 12*(c*
x + 1)^2*c^2*d^2*e^7/(c*x - 1)^2 + 12*(c*x + 1)*c^2*d^2*e^7/(c*x - 1) - 3*(c*x + 1)^3*c*d*e^8/(c*x - 1)^3 - 3*
(c*x + 1)^2*c*d*e^8/(c*x - 1)^2 + 3*(c*x + 1)*c*d*e^8/(c*x - 1) + 3*c*d*e^8 - (c*x + 1)^3*e^9/(c*x - 1)^3 - 3*
(c*x + 1)^2*e^9/(c*x - 1)^2 - 3*(c*x + 1)*e^9/(c*x - 1) - e^9)
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maple [A] time = 0.04, size = 223, normalized size = 1.27 \[ -\frac {c^{3} a}{3 \left (c x e +c d \right )^{3} e}-\frac {c^{3} b \arctanh \left (c x \right )}{3 \left (c x e +c d \right )^{3} e}+\frac {c^{3} b}{6 \left (c d +e \right ) \left (c d -e \right ) \left (c x e +c d \right )^{2}}-\frac {c^{5} b \ln \left (c x e +c d \right ) d^{2}}{\left (c d +e \right )^{3} \left (c d -e \right )^{3}}-\frac {c^{3} b \,e^{2} \ln \left (c x e +c d \right )}{3 \left (c d +e \right )^{3} \left (c d -e \right )^{3}}+\frac {2 c^{4} b d}{3 \left (c d +e \right )^{2} \left (c d -e \right )^{2} \left (c x e +c d \right )}-\frac {c^{3} b \ln \left (c x -1\right )}{6 e \left (c d +e \right )^{3}}+\frac {b \,c^{3} \ln \left (c x +1\right )}{6 \left (c d -e \right )^{3} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*arctanh(c*x))/(e*x+d)^4,x)
[Out]
-1/3*c^3*a/(c*e*x+c*d)^3/e-1/3*c^3*b/(c*e*x+c*d)^3/e*arctanh(c*x)+1/6*c^3*b/(c*d+e)/(c*d-e)/(c*e*x+c*d)^2-c^5*
b/(c*d+e)^3/(c*d-e)^3*ln(c*e*x+c*d)*d^2-1/3*c^3*b*e^2/(c*d+e)^3/(c*d-e)^3*ln(c*e*x+c*d)+2/3*c^4*b*d/(c*d+e)^2/
(c*d-e)^2/(c*e*x+c*d)-1/6*c^3*b/e/(c*d+e)^3*ln(c*x-1)+1/6*b*c^3*ln(c*x+1)/(c*d-e)^3/e
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maxima [B] time = 0.34, size = 339, normalized size = 1.94 \[ \frac {1}{6} \, {\left ({\left (\frac {c^{2} \log \left (c x + 1\right )}{c^{3} d^{3} e - 3 \, c^{2} d^{2} e^{2} + 3 \, c d e^{3} - e^{4}} - \frac {c^{2} \log \left (c x - 1\right )}{c^{3} d^{3} e + 3 \, c^{2} d^{2} e^{2} + 3 \, c d e^{3} + e^{4}} - \frac {2 \, {\left (3 \, c^{4} d^{2} + c^{2} e^{2}\right )} \log \left (e x + d\right )}{c^{6} d^{6} - 3 \, c^{4} d^{4} e^{2} + 3 \, c^{2} d^{2} e^{4} - e^{6}} + \frac {4 \, c^{2} d e x + 5 \, c^{2} d^{2} - e^{2}}{c^{4} d^{6} - 2 \, c^{2} d^{4} e^{2} + d^{2} e^{4} + {\left (c^{4} d^{4} e^{2} - 2 \, c^{2} d^{2} e^{4} + e^{6}\right )} x^{2} + 2 \, {\left (c^{4} d^{5} e - 2 \, c^{2} d^{3} e^{3} + d e^{5}\right )} x}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x\right )}{e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e}\right )} b - \frac {a}{3 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*arctanh(c*x))/(e*x+d)^4,x, algorithm="maxima")
[Out]
1/6*((c^2*log(c*x + 1)/(c^3*d^3*e - 3*c^2*d^2*e^2 + 3*c*d*e^3 - e^4) - c^2*log(c*x - 1)/(c^3*d^3*e + 3*c^2*d^2
*e^2 + 3*c*d*e^3 + e^4) - 2*(3*c^4*d^2 + c^2*e^2)*log(e*x + d)/(c^6*d^6 - 3*c^4*d^4*e^2 + 3*c^2*d^2*e^4 - e^6)
+ (4*c^2*d*e*x + 5*c^2*d^2 - e^2)/(c^4*d^6 - 2*c^2*d^4*e^2 + d^2*e^4 + (c^4*d^4*e^2 - 2*c^2*d^2*e^4 + e^6)*x^
2 + 2*(c^4*d^5*e - 2*c^2*d^3*e^3 + d*e^5)*x))*c - 2*arctanh(c*x)/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e)
)*b - 1/3*a/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e)
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mupad [B] time = 2.29, size = 418, normalized size = 2.39 \[ \ln \left (d+e\,x\right )\,\left (\frac {b\,c^3}{6\,e\,{\left (e+c\,d\right )}^3}+\frac {b\,c^3}{6\,e\,{\left (e-c\,d\right )}^3}\right )-\frac {\frac {2\,a\,c^4\,d^4-5\,b\,c^3\,d^3\,e-4\,a\,c^2\,d^2\,e^2+b\,c\,d\,e^3+2\,a\,e^4}{2\,\left (c^4\,d^4-2\,c^2\,d^2\,e^2+e^4\right )}+\frac {x\,\left (b\,c\,e^4-9\,b\,c^3\,d^2\,e^2\right )}{2\,\left (c^4\,d^4-2\,c^2\,d^2\,e^2+e^4\right )}-\frac {2\,b\,c^3\,d\,e^3\,x^2}{c^4\,d^4-2\,c^2\,d^2\,e^2+e^4}}{3\,d^3\,e+9\,d^2\,e^2\,x+9\,d\,e^3\,x^2+3\,e^4\,x^3}-\frac {b\,c^3\,\ln \left (c\,x-1\right )}{6\,c^3\,d^3\,e+18\,c^2\,d^2\,e^2+18\,c\,d\,e^3+6\,e^4}-\frac {b\,c^3\,\ln \left (c\,x+1\right )}{-6\,c^3\,d^3\,e+18\,c^2\,d^2\,e^2-18\,c\,d\,e^3+6\,e^4}-\frac {b\,\ln \left (c\,x+1\right )}{6\,e\,\left (d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3\right )}+\frac {b\,\ln \left (1-c\,x\right )}{3\,e\,\left (2\,d^3+6\,d^2\,e\,x+6\,d\,e^2\,x^2+2\,e^3\,x^3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*atanh(c*x))/(d + e*x)^4,x)
[Out]
log(d + e*x)*((b*c^3)/(6*e*(e + c*d)^3) + (b*c^3)/(6*e*(e - c*d)^3)) - ((2*a*e^4 + 2*a*c^4*d^4 - 4*a*c^2*d^2*e
^2 + b*c*d*e^3 - 5*b*c^3*d^3*e)/(2*(e^4 + c^4*d^4 - 2*c^2*d^2*e^2)) + (x*(b*c*e^4 - 9*b*c^3*d^2*e^2))/(2*(e^4
+ c^4*d^4 - 2*c^2*d^2*e^2)) - (2*b*c^3*d*e^3*x^2)/(e^4 + c^4*d^4 - 2*c^2*d^2*e^2))/(3*d^3*e + 3*e^4*x^3 + 9*d^
2*e^2*x + 9*d*e^3*x^2) - (b*c^3*log(c*x - 1))/(6*e^4 + 6*c^3*d^3*e + 18*c^2*d^2*e^2 + 18*c*d*e^3) - (b*c^3*log
(c*x + 1))/(6*e^4 - 6*c^3*d^3*e + 18*c^2*d^2*e^2 - 18*c*d*e^3) - (b*log(c*x + 1))/(6*e*(d^3 + e^3*x^3 + 3*d*e^
2*x^2 + 3*d^2*e*x)) + (b*log(1 - c*x))/(3*e*(2*d^3 + 2*e^3*x^3 + 6*d*e^2*x^2 + 6*d^2*e*x))
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sympy [A] time = 12.91, size = 10946, normalized size = 62.55 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*atanh(c*x))/(e*x+d)**4,x)
[Out]
Piecewise((a*x/d**4, Eq(c, 0) & Eq(e, 0)), (-24*a*d**3/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*
d**3*e**4*x**3) + 21*b*d**3*atanh(e*x/d)/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3
) + 10*b*d**3/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) - 9*b*d**2*e*x*atanh(e*x/
d)/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) + 9*b*d**2*e*x/(72*d**6*e + 216*d**5
*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) - 9*b*d*e**2*x**2*atanh(e*x/d)/(72*d**6*e + 216*d**5*e**2*x
+ 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) + 3*b*d*e**2*x**2/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2
+ 72*d**3*e**4*x**3) - 3*b*e**3*x**3*atanh(e*x/d)/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*
e**4*x**3), Eq(c, -e/d)), (-24*a*d**3/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) -
21*b*d**3*atanh(e*x/d)/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) - 10*b*d**3/(72
*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) + 9*b*d**2*e*x*atanh(e*x/d)/(72*d**6*e + 2
16*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3) - 9*b*d**2*e*x/(72*d**6*e + 216*d**5*e**2*x + 216*d**
4*e**3*x**2 + 72*d**3*e**4*x**3) + 9*b*d*e**2*x**2*atanh(e*x/d)/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x
**2 + 72*d**3*e**4*x**3) - 3*b*d*e**2*x**2/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x*
*3) + 3*b*e**3*x**3*atanh(e*x/d)/(72*d**6*e + 216*d**5*e**2*x + 216*d**4*e**3*x**2 + 72*d**3*e**4*x**3), Eq(c,
e/d)), ((a*x + b*x*atanh(c*x) + b*log(x - 1/c)/c + b*atanh(c*x)/c)/d**4, Eq(e, 0)), (-a/(3*d**3*e + 9*d**2*e*
*2*x + 9*d*e**3*x**2 + 3*e**4*x**3), Eq(c, 0)), (-2*a*c**6*d**6/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6
*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 1
8*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x
**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) + 6*a*c**4*d**4*e**2/(6*c**6*d**9*e + 18*c
**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 5
4*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x
**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) - 6*a*c**2*d**2*e
**4/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3
- 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**
6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**1
0*x**3) + 2*a*e**6/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*
c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 5
4*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**
9*x**2 - 6*e**10*x**3) + 6*b*c**6*d**5*e*x*atanh(c*x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3
*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**
4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d*
*3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) + 6*b*c**6*d**4*e**2*x**2*atanh(c*x)/(6*c**6*d**9*e
+ 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4
*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*
e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) + 2*b*c**6*
d**3*e**3*x**3*atanh(c*x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**
3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e
**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 1
8*d*e**9*x**2 - 6*e**10*x**3) + 6*b*c**5*d**5*e*log(x - 1/c)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d*
*7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c
**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3
- 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) - 6*b*c**5*d**5*e*log(d/e + x)/(6*c**6*d**9*e
+ 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**
4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3
*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) + 6*b*c**5
*d**5*e*atanh(c*x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*
c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 5
4*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**
9*x**2 - 6*e**10*x**3) + 5*b*c**5*d**5*e/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**
6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 +
18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*
d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) + 18*b*c**5*d**4*e**2*x*log(x - 1/c)/(6*c**6*d**9*e + 18*c**6*d**
8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*
d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 1
8*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) - 18*b*c**5*d**4*e**2*x*
log(d/e + x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d
**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2
*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2
- 6*e**10*x**3) + 18*b*c**5*d**4*e**2*x*atanh(c*x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x
**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*
e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3
*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) + 9*b*c**5*d**4*e**2*x/(6*c**6*d**9*e + 18*c**6*d**8*e
**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**
5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c
**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) + 18*b*c**5*d**3*e**3*x**2*
log(x - 1/c)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d
**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2
*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2
- 6*e**10*x**3) - 18*b*c**5*d**3*e**3*x**2*log(d/e + x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e
**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*
d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6
*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) + 18*b*c**5*d**3*e**3*x**2*atanh(c*x)/(6*c**6*d**
9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*
e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d
**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) + 4*b*c
**5*d**3*e**3*x**2/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*
c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 5
4*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**
9*x**2 - 6*e**10*x**3) + 6*b*c**5*d**2*e**4*x**3*log(x - 1/c)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d
**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*
c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**
3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) - 6*b*c**5*d**2*e**4*x**3*log(d/e + x)/(6*c*
*6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4
*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*
c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) +
6*b*c**5*d**2*e**4*x**3*atanh(c*x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**
6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c
**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*
e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) + 12*b*c**4*d**4*e**2*atanh(c*x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x
+ 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5
*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d*
*2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) + 18*b*c**4*d**3*e**3*x*atanh(c*x
)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 -
54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*
x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*
x**3) + 18*b*c**4*d**2*e**4*x**2*atanh(c*x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*
c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**
3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 -
18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) + 6*b*c**4*d*e**5*x**3*atanh(c*x)/(6*c**6*d**9*e + 18*c**6*d**
8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*
d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 1
8*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) + 2*b*c**3*d**3*e**3*log
(x - 1/c)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7
*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d*
*4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 -
6*e**10*x**3) - 2*b*c**3*d**3*e**3*log(d/e + x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2
+ 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6
*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**
7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) + 2*b*c**3*d**3*e**3*atanh(c*x)/(6*c**6*d**9*e + 18*c**6*d
**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**
4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 +
18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) - 6*b*c**3*d**3*e**3/(
6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*
c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x +
54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**
3) + 6*b*c**3*d**2*e**4*x*log(x - 1/c)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*
d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 1
8*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d*
*2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) - 6*b*c**3*d**2*e**4*x*log(d/e + x)/(6*c**6*d**9*e + 18*c**6*d**8*e
**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**
5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c
**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) + 6*b*c**3*d**2*e**4*x*atan
h(c*x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e*
*3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*
e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e
**10*x**3) - 10*b*c**3*d**2*e**4*x/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6
*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c*
*2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e
**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) + 6*b*c**3*d*e**5*x**2*log(x - 1/c)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*
x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e*
*5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*
d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) - 6*b*c**3*d*e**5*x**2*log(d/e
+ x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3
- 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e*
*6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**
10*x**3) + 6*b*c**3*d*e**5*x**2*atanh(c*x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c
**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3
+ 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 1
8*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) - 4*b*c**3*d*e**5*x**2/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 1
8*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x*
*2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*
e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) + 2*b*c**3*e**6*x**3*log(x - 1/c)/(6
*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c
**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x +
54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3
) - 2*b*c**3*e**6*x**3*log(d/e + x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**
6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c
**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*
e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) + 2*b*c**3*e**6*x**3*atanh(c*x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x +
18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*
x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**
2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3) - 6*b*c**2*d**2*e**4*atanh(c*x)/(6
*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c
**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x +
54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3
) + b*c*d*e**5/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4
*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c*
*2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x*
*2 - 6*e**10*x**3) + b*c*e**6*x/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e*
*4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4*d**4*e**6*x**3 + 18*c**2*
d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 - 6*d**3*e**7 - 18*d**2*e**8
*x - 18*d*e**9*x**2 - 6*e**10*x**3) + 2*b*e**6*atanh(c*x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*
e**3*x**2 + 6*c**6*d**6*e**4*x**3 - 18*c**4*d**7*e**3 - 54*c**4*d**6*e**4*x - 54*c**4*d**5*e**5*x**2 - 18*c**4
*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 -
6*d**3*e**7 - 18*d**2*e**8*x - 18*d*e**9*x**2 - 6*e**10*x**3), True))
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