3.1.3 \(\int (d+e x)^2 (a+b \tanh ^{-1}(c x)) \, dx\) [3]

Optimal. Leaf size=96 \[ \frac {b d e x}{c}+\frac {b e^2 x^2}{6 c}+\frac {(d+e x)^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 e}+\frac {b (c d+e)^3 \log (1-c x)}{6 c^3 e}-\frac {b (c d-e)^3 \log (1+c x)}{6 c^3 e} \]

[Out]

b*d*e*x/c+1/6*b*e^2*x^2/c+1/3*(e*x+d)^3*(a+b*arctanh(c*x))/e+1/6*b*(c*d+e)^3*ln(-c*x+1)/c^3/e-1/6*b*(c*d-e)^3*
ln(c*x+1)/c^3/e

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Rubi [A]
time = 0.09, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6063, 716, 647, 31} \begin {gather*} \frac {(d+e x)^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 e}-\frac {b (c d-e)^3 \log (c x+1)}{6 c^3 e}+\frac {b (c d+e)^3 \log (1-c x)}{6 c^3 e}+\frac {b d e x}{c}+\frac {b e^2 x^2}{6 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 31

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

Rule 647

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Dist[e/2 + c*(d/(2*q)),
Int[1/(-q + c*x), x], x] + Dist[e/2 - c*(d/(2*q)), Int[1/(q + c*x), x], x]] /; FreeQ[{a, c, d, e}, x] && NiceS
qrtQ[(-a)*c]

Rule 716

Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[(d + e*x)^m, a + c*x^2,
x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[m, 1] && (NeQ[d, 0] || GtQ[m, 2])

Rule 6063

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

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Mathematica [A]
time = 0.05, size = 129, normalized size = 1.34 \begin {gather*} \frac {1}{6} \left (\frac {6 d (a c d+b e) x}{c}+\frac {e (6 a c d+b e) x^2}{c}+2 a e^2 x^3+2 b x \left (3 d^2+3 d e x+e^2 x^2\right ) \tanh ^{-1}(c x)+\frac {b \left (3 c^2 d^2+3 c d e+e^2\right ) \log (1-c x)}{c^3}+\frac {b \left (3 c^2 d^2-3 c d e+e^2\right ) \log (1+c x)}{c^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(203\) vs. \(2(88)=176\).
time = 0.14, size = 204, normalized size = 2.12

method	result	size
derivativedivides	\(\frac {\frac {\left (c e x +d c \right )^{3} a}{3 c^{2} e}+\frac {b c \arctanh \left (c x \right ) d^{3}}{3 e}+b \arctanh \left (c x \right ) d^{2} c x +b c e \arctanh \left (c x \right ) d \,x^{2}+\frac {b c \,e^{2} \arctanh \left (c x \right ) x^{3}}{3}+b e d x +\frac {b \,e^{2} x^{2}}{6}+\frac {b c \ln \left (c x -1\right ) d^{3}}{6 e}+\frac {b \ln \left (c x -1\right ) d^{2}}{2}+\frac {b e \ln \left (c x -1\right ) d}{2 c}+\frac {b \,e^{2} \ln \left (c x -1\right )}{6 c^{2}}-\frac {b c \ln \left (c x +1\right ) d^{3}}{6 e}+\frac {b \ln \left (c x +1\right ) d^{2}}{2}-\frac {b e \ln \left (c x +1\right ) d}{2 c}+\frac {b \,e^{2} \ln \left (c x +1\right )}{6 c^{2}}}{c}\)	\(204\)
default	\(\frac {\frac {\left (c e x +d c \right )^{3} a}{3 c^{2} e}+\frac {b c \arctanh \left (c x \right ) d^{3}}{3 e}+b \arctanh \left (c x \right ) d^{2} c x +b c e \arctanh \left (c x \right ) d \,x^{2}+\frac {b c \,e^{2} \arctanh \left (c x \right ) x^{3}}{3}+b e d x +\frac {b \,e^{2} x^{2}}{6}+\frac {b c \ln \left (c x -1\right ) d^{3}}{6 e}+\frac {b \ln \left (c x -1\right ) d^{2}}{2}+\frac {b e \ln \left (c x -1\right ) d}{2 c}+\frac {b \,e^{2} \ln \left (c x -1\right )}{6 c^{2}}-\frac {b c \ln \left (c x +1\right ) d^{3}}{6 e}+\frac {b \ln \left (c x +1\right ) d^{2}}{2}-\frac {b e \ln \left (c x +1\right ) d}{2 c}+\frac {b \,e^{2} \ln \left (c x +1\right )}{6 c^{2}}}{c}\)	\(204\)
risch	\(\frac {\left (e x +d \right )^{3} b \ln \left (c x +1\right )}{6 e}-\frac {e^{2} b \,x^{3} \ln \left (-c x +1\right )}{6}-\frac {e b d \,x^{2} \ln \left (-c x +1\right )}{2}+\frac {e^{2} a \,x^{3}}{3}-\frac {b \,d^{2} x \ln \left (-c x +1\right )}{2}+e a d \,x^{2}-\frac {\ln \left (c x +1\right ) b \,d^{3}}{6 e}+a \,d^{2} x +\frac {b \,e^{2} x^{2}}{6 c}+\frac {\ln \left (c x +1\right ) b \,d^{2}}{2 c}+\frac {\ln \left (-c x +1\right ) b \,d^{2}}{2 c}+\frac {b d e x}{c}-\frac {e \ln \left (c x +1\right ) b d}{2 c^{2}}+\frac {e \ln \left (-c x +1\right ) b d}{2 c^{2}}+\frac {e^{2} \ln \left (c x +1\right ) b}{6 c^{3}}+\frac {e^{2} \ln \left (-c x +1\right ) b}{6 c^{3}}\)	\(214\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2*(a+b*arctanh(c*x)),x,method=_RETURNVERBOSE)

[Out]

1/c*(1/3*(c*e*x+c*d)^3*a/c^2/e+1/3*b*c/e*arctanh(c*x)*d^3+b*arctanh(c*x)*d^2*c*x+b*c*e*arctanh(c*x)*d*x^2+1/3*
b*c*e^2*arctanh(c*x)*x^3+b*e*d*x+1/6*b*e^2*x^2+1/6*b*c/e*ln(c*x-1)*d^3+1/2*b*ln(c*x-1)*d^2+1/2*b/c*e*ln(c*x-1)
*d+1/6*b/c^2*e^2*ln(c*x-1)-1/6*b*c/e*ln(c*x+1)*d^3+1/2*b*ln(c*x+1)*d^2-1/2*b/c*e*ln(c*x+1)*d+1/6*b/c^2*e^2*ln(
c*x+1))

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Maxima [A]
time = 0.26, size = 137, normalized size = 1.43 \begin {gather*} \frac {1}{3} \, a x^{3} e^{2} + a d x^{2} e + a d^{2} x + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, x}{c^{2}} - \frac {\log \left (c x + 1\right )}{c^{3}} + \frac {\log \left (c x - 1\right )}{c^{3}}\right )}\right )} b d e + \frac {{\left (2 \, c x \operatorname {artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} b d^{2}}{2 \, c} + \frac {1}{6} \, {\left (2 \, x^{3} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {x^{2}}{c^{2}} + \frac {\log \left (c^{2} x^{2} - 1\right )}{c^{4}}\right )}\right )} b e^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*x^3*e^2 + a*d*x^2*e + a*d^2*x + 1/2*(2*x^2*arctanh(c*x) + c*(2*x/c^2 - log(c*x + 1)/c^3 + log(c*x - 1)/c
^3))*b*d*e + 1/2*(2*c*x*arctanh(c*x) + log(-c^2*x^2 + 1))*b*d^2/c + 1/6*(2*x^3*arctanh(c*x) + c*(x^2/c^2 + log
(c^2*x^2 - 1)/c^4))*b*e^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (88) = 176\).
time = 0.39, size = 316, normalized size = 3.29 \begin {gather*} \frac {6 \, a c^{3} d^{2} x + {\left (2 \, a c^{3} x^{3} + b c^{2} x^{2}\right )} \cosh \left (1\right )^{2} + {\left (2 \, a c^{3} x^{3} + b c^{2} x^{2}\right )} \sinh \left (1\right )^{2} + 6 \, {\left (a c^{3} d x^{2} + b c^{2} d x\right )} \cosh \left (1\right ) + {\left (3 \, b c^{2} d^{2} - 3 \, b c d \cosh \left (1\right ) + b \cosh \left (1\right )^{2} + b \sinh \left (1\right )^{2} - {\left (3 \, b c d - 2 \, b \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (c x + 1\right ) + {\left (3 \, b c^{2} d^{2} + 3 \, b c d \cosh \left (1\right ) + b \cosh \left (1\right )^{2} + b \sinh \left (1\right )^{2} + {\left (3 \, b c d + 2 \, b \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (c x - 1\right ) + {\left (b c^{3} x^{3} \cosh \left (1\right )^{2} + b c^{3} x^{3} \sinh \left (1\right )^{2} + 3 \, b c^{3} d x^{2} \cosh \left (1\right ) + 3 \, b c^{3} d^{2} x + {\left (2 \, b c^{3} x^{3} \cosh \left (1\right ) + 3 \, b c^{3} d x^{2}\right )} \sinh \left (1\right )\right )} \log \left (-\frac {c x + 1}{c x - 1}\right ) + 2 \, {\left (3 \, a c^{3} d x^{2} + 3 \, b c^{2} d x + {\left (2 \, a c^{3} x^{3} + b c^{2} x^{2}\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )}{6 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(6*a*c^3*d^2*x + (2*a*c^3*x^3 + b*c^2*x^2)*cosh(1)^2 + (2*a*c^3*x^3 + b*c^2*x^2)*sinh(1)^2 + 6*(a*c^3*d*x^
2 + b*c^2*d*x)*cosh(1) + (3*b*c^2*d^2 - 3*b*c*d*cosh(1) + b*cosh(1)^2 + b*sinh(1)^2 - (3*b*c*d - 2*b*cosh(1))*
sinh(1))*log(c*x + 1) + (3*b*c^2*d^2 + 3*b*c*d*cosh(1) + b*cosh(1)^2 + b*sinh(1)^2 + (3*b*c*d + 2*b*cosh(1))*s
inh(1))*log(c*x - 1) + (b*c^3*x^3*cosh(1)^2 + b*c^3*x^3*sinh(1)^2 + 3*b*c^3*d*x^2*cosh(1) + 3*b*c^3*d^2*x + (2
*b*c^3*x^3*cosh(1) + 3*b*c^3*d*x^2)*sinh(1))*log(-(c*x + 1)/(c*x - 1)) + 2*(3*a*c^3*d*x^2 + 3*b*c^2*d*x + (2*a
*c^3*x^3 + b*c^2*x^2)*cosh(1))*sinh(1))/c^3

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (82) = 164\).
time = 0.31, size = 178, normalized size = 1.85 \begin {gather*} \begin {cases} a d^{2} x + a d e x^{2} + \frac {a e^{2} x^{3}}{3} + b d^{2} x \operatorname {atanh}{\left (c x \right )} + b d e x^{2} \operatorname {atanh}{\left (c x \right )} + \frac {b e^{2} x^{3} \operatorname {atanh}{\left (c x \right )}}{3} + \frac {b d^{2} \log {\left (x - \frac {1}{c} \right )}}{c} + \frac {b d^{2} \operatorname {atanh}{\left (c x \right )}}{c} + \frac {b d e x}{c} + \frac {b e^{2} x^{2}}{6 c} - \frac {b d e \operatorname {atanh}{\left (c x \right )}}{c^{2}} + \frac {b e^{2} \log {\left (x - \frac {1}{c} \right )}}{3 c^{3}} + \frac {b e^{2} \operatorname {atanh}{\left (c x \right )}}{3 c^{3}} & \text {for}\: c \neq 0 \\a \left (d^{2} x + d e x^{2} + \frac {e^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d**2*x + a*d*e*x**2 + a*e**2*x**3/3 + b*d**2*x*atanh(c*x) + b*d*e*x**2*atanh(c*x) + b*e**2*x**3*a
tanh(c*x)/3 + b*d**2*log(x - 1/c)/c + b*d**2*atanh(c*x)/c + b*d*e*x/c + b*e**2*x**2/(6*c) - b*d*e*atanh(c*x)/c
**2 + b*e**2*log(x - 1/c)/(3*c**3) + b*e**2*atanh(c*x)/(3*c**3), Ne(c, 0)), (a*(d**2*x + d*e*x**2 + e**2*x**3/
3), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 532 vs. \(2 (88) = 176\).
time = 0.44, size = 532, normalized size = 5.54 \begin {gather*} \frac {1}{3} \, c {\left (\frac {{\left (\frac {3 \, {\left (c x + 1\right )}^{2} b c^{2} d^{2}}{{\left (c x - 1\right )}^{2}} - \frac {6 \, {\left (c x + 1\right )} b c^{2} d^{2}}{c x - 1} + 3 \, b c^{2} d^{2} + \frac {6 \, {\left (c x + 1\right )}^{2} b c d e}{{\left (c x - 1\right )}^{2}} - \frac {6 \, {\left (c x + 1\right )} b c d e}{c x - 1} + \frac {3 \, {\left (c x + 1\right )}^{2} b e^{2}}{{\left (c x - 1\right )}^{2}} + b e^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{\frac {{\left (c x + 1\right )}^{3} c^{4}}{{\left (c x - 1\right )}^{3}} - \frac {3 \, {\left (c x + 1\right )}^{2} c^{4}}{{\left (c x - 1\right )}^{2}} + \frac {3 \, {\left (c x + 1\right )} c^{4}}{c x - 1} - c^{4}} + \frac {2 \, {\left (\frac {3 \, {\left (c x + 1\right )}^{2} a c^{2} d^{2}}{{\left (c x - 1\right )}^{2}} - \frac {6 \, {\left (c x + 1\right )} a c^{2} d^{2}}{c x - 1} + 3 \, a c^{2} d^{2} + \frac {6 \, {\left (c x + 1\right )}^{2} a c d e}{{\left (c x - 1\right )}^{2}} - \frac {6 \, {\left (c x + 1\right )} a c d e}{c x - 1} + \frac {3 \, {\left (c x + 1\right )}^{2} b c d e}{{\left (c x - 1\right )}^{2}} - \frac {6 \, {\left (c x + 1\right )} b c d e}{c x - 1} + 3 \, b c d e + \frac {3 \, {\left (c x + 1\right )}^{2} a e^{2}}{{\left (c x - 1\right )}^{2}} + a e^{2} + \frac {{\left (c x + 1\right )}^{2} b e^{2}}{{\left (c x - 1\right )}^{2}} - \frac {{\left (c x + 1\right )} b e^{2}}{c x - 1}\right )}}{\frac {{\left (c x + 1\right )}^{3} c^{4}}{{\left (c x - 1\right )}^{3}} - \frac {3 \, {\left (c x + 1\right )}^{2} c^{4}}{{\left (c x - 1\right )}^{2}} + \frac {3 \, {\left (c x + 1\right )} c^{4}}{c x - 1} - c^{4}} - \frac {{\left (3 \, b c^{2} d^{2} + b e^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1} + 1\right )}{c^{4}} + \frac {{\left (3 \, b c^{2} d^{2} + b e^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{c^{4}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*c*((3*(c*x + 1)^2*b*c^2*d^2/(c*x - 1)^2 - 6*(c*x + 1)*b*c^2*d^2/(c*x - 1) + 3*b*c^2*d^2 + 6*(c*x + 1)^2*b*
c*d*e/(c*x - 1)^2 - 6*(c*x + 1)*b*c*d*e/(c*x - 1) + 3*(c*x + 1)^2*b*e^2/(c*x - 1)^2 + b*e^2)*log(-(c*x + 1)/(c
*x - 1))/((c*x + 1)^3*c^4/(c*x - 1)^3 - 3*(c*x + 1)^2*c^4/(c*x - 1)^2 + 3*(c*x + 1)*c^4/(c*x - 1) - c^4) + 2*(
3*(c*x + 1)^2*a*c^2*d^2/(c*x - 1)^2 - 6*(c*x + 1)*a*c^2*d^2/(c*x - 1) + 3*a*c^2*d^2 + 6*(c*x + 1)^2*a*c*d*e/(c
*x - 1)^2 - 6*(c*x + 1)*a*c*d*e/(c*x - 1) + 3*(c*x + 1)^2*b*c*d*e/(c*x - 1)^2 - 6*(c*x + 1)*b*c*d*e/(c*x - 1)
+ 3*b*c*d*e + 3*(c*x + 1)^2*a*e^2/(c*x - 1)^2 + a*e^2 + (c*x + 1)^2*b*e^2/(c*x - 1)^2 - (c*x + 1)*b*e^2/(c*x -
1))/((c*x + 1)^3*c^4/(c*x - 1)^3 - 3*(c*x + 1)^2*c^4/(c*x - 1)^2 + 3*(c*x + 1)*c^4/(c*x - 1) - c^4) - (3*b*c^
2*d^2 + b*e^2)*log(-(c*x + 1)/(c*x - 1) + 1)/c^4 + (3*b*c^2*d^2 + b*e^2)*log(-(c*x + 1)/(c*x - 1))/c^4)

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Mupad [B]
time = 0.94, size = 127, normalized size = 1.32 \begin {gather*} \frac {a\,e^2\,x^3}{3}+a\,d^2\,x+\frac {b\,d^2\,\ln \left (c^2\,x^2-1\right )}{2\,c}+\frac {b\,e^2\,\ln \left (c^2\,x^2-1\right )}{6\,c^3}+\frac {b\,e^2\,x^2}{6\,c}+a\,d\,e\,x^2+b\,d^2\,x\,\mathrm {atanh}\left (c\,x\right )+\frac {b\,e^2\,x^3\,\mathrm {atanh}\left (c\,x\right )}{3}+\frac {b\,d\,e\,x}{c}-\frac {b\,d\,e\,\mathrm {atanh}\left (c\,x\right )}{c^2}+b\,d\,e\,x^2\,\mathrm {atanh}\left (c\,x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x))*(d + e*x)^2,x)

[Out]

(a*e^2*x^3)/3 + a*d^2*x + (b*d^2*log(c^2*x^2 - 1))/(2*c) + (b*e^2*log(c^2*x^2 - 1))/(6*c^3) + (b*e^2*x^2)/(6*c
) + a*d*e*x^2 + b*d^2*x*atanh(c*x) + (b*e^2*x^3*atanh(c*x))/3 + (b*d*e*x)/c - (b*d*e*atanh(c*x))/c^2 + b*d*e*x
^2*atanh(c*x)

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