3.33 \(\int (d+e x) (a+b \tanh ^{-1}(c x^3)) \, dx\)
Optimal. Leaf size=285 \[ \frac{(d+e x)^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{2 e}+\frac{b d \log \left (1-c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac{b d \log \left (c^{4/3} x^4+c^{2/3} x^2+1\right )}{4 \sqrt [3]{c}}+\frac{\sqrt{3} b d \tan ^{-1}\left (\frac{2 c^{2/3} x^2+1}{\sqrt{3}}\right )}{2 \sqrt [3]{c}}+\frac{b e \log \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )}{8 c^{2/3}}-\frac{b e \log \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )}{8 c^{2/3}}-\frac{\sqrt{3} b e \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{c} x}{\sqrt{3}}\right )}{4 c^{2/3}}+\frac{\sqrt{3} b e \tan ^{-1}\left (\frac{2 \sqrt [3]{c} x}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right )}{4 c^{2/3}}-\frac{b e \tanh ^{-1}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac{b d^2 \tanh ^{-1}\left (c x^3\right )}{2 e} \]
[Out]
-(Sqrt[3]*b*e*ArcTan[1/Sqrt[3] - (2*c^(1/3)*x)/Sqrt[3]])/(4*c^(2/3)) + (Sqrt[3]*b*e*ArcTan[1/Sqrt[3] + (2*c^(1
/3)*x)/Sqrt[3]])/(4*c^(2/3)) + (Sqrt[3]*b*d*ArcTan[(1 + 2*c^(2/3)*x^2)/Sqrt[3]])/(2*c^(1/3)) - (b*e*ArcTanh[c^
(1/3)*x])/(2*c^(2/3)) - (b*d^2*ArcTanh[c*x^3])/(2*e) + ((d + e*x)^2*(a + b*ArcTanh[c*x^3]))/(2*e) + (b*d*Log[1
- c^(2/3)*x^2])/(2*c^(1/3)) + (b*e*Log[1 - c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3)) - (b*e*Log[1 + c^(1/3)*x + c
^(2/3)*x^2])/(8*c^(2/3)) - (b*d*Log[1 + c^(2/3)*x^2 + c^(4/3)*x^4])/(4*c^(1/3))
________________________________________________________________________________________
Rubi [A] time = 0.451595, antiderivative size = 285, normalized size of antiderivative =
1., number of steps used = 23, number of rules used = 13, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) =
0.812, Rules used = {6273, 12, 1831, 275, 206, 292, 31, 634, 617, 204, 628, 296, 618}
\[ \frac{(d+e x)^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{2 e}+\frac{b d \log \left (1-c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac{b d \log \left (c^{4/3} x^4+c^{2/3} x^2+1\right )}{4 \sqrt [3]{c}}+\frac{\sqrt{3} b d \tan ^{-1}\left (\frac{2 c^{2/3} x^2+1}{\sqrt{3}}\right )}{2 \sqrt [3]{c}}+\frac{b e \log \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )}{8 c^{2/3}}-\frac{b e \log \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )}{8 c^{2/3}}-\frac{\sqrt{3} b e \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{c} x}{\sqrt{3}}\right )}{4 c^{2/3}}+\frac{\sqrt{3} b e \tan ^{-1}\left (\frac{2 \sqrt [3]{c} x}{\sqrt{3}}+\frac{1}{\sqrt{3}}\right )}{4 c^{2/3}}-\frac{b e \tanh ^{-1}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac{b d^2 \tanh ^{-1}\left (c x^3\right )}{2 e} \]
Antiderivative was successfully verified.
[In]
Int[(d + e*x)*(a + b*ArcTanh[c*x^3]),x]
[Out]
-(Sqrt[3]*b*e*ArcTan[1/Sqrt[3] - (2*c^(1/3)*x)/Sqrt[3]])/(4*c^(2/3)) + (Sqrt[3]*b*e*ArcTan[1/Sqrt[3] + (2*c^(1
/3)*x)/Sqrt[3]])/(4*c^(2/3)) + (Sqrt[3]*b*d*ArcTan[(1 + 2*c^(2/3)*x^2)/Sqrt[3]])/(2*c^(1/3)) - (b*e*ArcTanh[c^
(1/3)*x])/(2*c^(2/3)) - (b*d^2*ArcTanh[c*x^3])/(2*e) + ((d + e*x)^2*(a + b*ArcTanh[c*x^3]))/(2*e) + (b*d*Log[1
- c^(2/3)*x^2])/(2*c^(1/3)) + (b*e*Log[1 - c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3)) - (b*e*Log[1 + c^(1/3)*x + c
^(2/3)*x^2])/(8*c^(2/3)) - (b*d*Log[1 + c^(2/3)*x^2 + c^(4/3)*x^4])/(4*c^(1/3))
Rule 6273
Int[((a_.) + ArcTanh[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(a + b*ArcTan
h[u]))/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/(1 - u^2), x],
x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && !FunctionOfQ[(c + d*x)^(m
+ 1), u, x] && FalseQ[PowerVariableExpn[u, m + 1, x]]
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 1831
Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[((c*x)^(m + ii)*(Coeff[Pq,
x, ii] + Coeff[Pq, x, n/2 + ii]*x^(n/2)))/(c^ii*(a + b*x^n)), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; Fr
eeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && Expon[Pq, x] < n
Rule 275
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 292
Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Rule 31
Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]
Rule 634
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && !NiceSqrtQ[b^2 - 4*a*c]
Rule 617
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 628
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Rule 296
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator[Rt[-(a/b), n]], s = Denominator[Rt
[-(a/b), n]], k, u}, Simp[u = Int[(r*Cos[(2*k*m*Pi)/n] - s*Cos[(2*k*(m + 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi
)/n]*x + s^2*x^2), x] + Int[(r*Cos[(2*k*m*Pi)/n] + s*Cos[(2*k*(m + 1)*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x
+ s^2*x^2), x]; (2*r^(m + 2)*Int[1/(r^2 - s^2*x^2), x])/(a*n*s^m) + Dist[(2*r^(m + 1))/(a*n*s^m), Sum[u, {k,
1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && NegQ[a/b]
Rule 618
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rubi steps
\begin{align*} \int (d+e x) \left (a+b \tanh ^{-1}\left (c x^3\right )\right ) \, dx &=\frac{(d+e x)^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{2 e}-\frac{b \int \frac{3 c x^2 (d+e x)^2}{1-c^2 x^6} \, dx}{2 e}\\ &=\frac{(d+e x)^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{2 e}-\frac{(3 b c) \int \frac{x^2 (d+e x)^2}{1-c^2 x^6} \, dx}{2 e}\\ &=\frac{(d+e x)^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{2 e}-\frac{(3 b c) \int \left (\frac{d^2 x^2}{1-c^2 x^6}+\frac{2 d e x^3}{1-c^2 x^6}+\frac{e^2 x^4}{1-c^2 x^6}\right ) \, dx}{2 e}\\ &=\frac{(d+e x)^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{2 e}-(3 b c d) \int \frac{x^3}{1-c^2 x^6} \, dx-\frac{\left (3 b c d^2\right ) \int \frac{x^2}{1-c^2 x^6} \, dx}{2 e}-\frac{1}{2} (3 b c e) \int \frac{x^4}{1-c^2 x^6} \, dx\\ &=\frac{(d+e x)^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{2 e}-\frac{1}{2} (3 b c d) \operatorname{Subst}\left (\int \frac{x}{1-c^2 x^3} \, dx,x,x^2\right )-\frac{\left (b c d^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x^2} \, dx,x,x^3\right )}{2 e}-\frac{(b e) \int \frac{1}{1-c^{2/3} x^2} \, dx}{2 \sqrt [3]{c}}-\frac{(b e) \int \frac{-\frac{1}{2}-\frac{\sqrt [3]{c} x}{2}}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{2 \sqrt [3]{c}}-\frac{(b e) \int \frac{-\frac{1}{2}+\frac{\sqrt [3]{c} x}{2}}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{2 \sqrt [3]{c}}\\ &=-\frac{b e \tanh ^{-1}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac{b d^2 \tanh ^{-1}\left (c x^3\right )}{2 e}+\frac{(d+e x)^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{2 e}-\frac{1}{2} \left (b \sqrt [3]{c} d\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^{2/3} x} \, dx,x,x^2\right )+\frac{1}{2} \left (b \sqrt [3]{c} d\right ) \operatorname{Subst}\left (\int \frac{1-c^{2/3} x}{1+c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )+\frac{(b e) \int \frac{-\sqrt [3]{c}+2 c^{2/3} x}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{8 c^{2/3}}-\frac{(b e) \int \frac{\sqrt [3]{c}+2 c^{2/3} x}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{8 c^{2/3}}+\frac{(3 b e) \int \frac{1}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{8 \sqrt [3]{c}}+\frac{(3 b e) \int \frac{1}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{8 \sqrt [3]{c}}\\ &=-\frac{b e \tanh ^{-1}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac{b d^2 \tanh ^{-1}\left (c x^3\right )}{2 e}+\frac{(d+e x)^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{2 e}+\frac{b d \log \left (1-c^{2/3} x^2\right )}{2 \sqrt [3]{c}}+\frac{b e \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}-\frac{b e \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}-\frac{(b d) \operatorname{Subst}\left (\int \frac{c^{2/3}+2 c^{4/3} x}{1+c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )}{4 \sqrt [3]{c}}+\frac{1}{4} \left (3 b \sqrt [3]{c} d\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )+\frac{(3 b e) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac{(3 b e) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{c} x\right )}{4 c^{2/3}}\\ &=-\frac{\sqrt{3} b e \tan ^{-1}\left (\frac{1-2 \sqrt [3]{c} x}{\sqrt{3}}\right )}{4 c^{2/3}}+\frac{\sqrt{3} b e \tan ^{-1}\left (\frac{1+2 \sqrt [3]{c} x}{\sqrt{3}}\right )}{4 c^{2/3}}-\frac{b e \tanh ^{-1}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac{b d^2 \tanh ^{-1}\left (c x^3\right )}{2 e}+\frac{(d+e x)^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{2 e}+\frac{b d \log \left (1-c^{2/3} x^2\right )}{2 \sqrt [3]{c}}+\frac{b e \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}-\frac{b e \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}-\frac{b d \log \left (1+c^{2/3} x^2+c^{4/3} x^4\right )}{4 \sqrt [3]{c}}-\frac{(3 b d) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 c^{2/3} x^2\right )}{2 \sqrt [3]{c}}\\ &=-\frac{\sqrt{3} b e \tan ^{-1}\left (\frac{1-2 \sqrt [3]{c} x}{\sqrt{3}}\right )}{4 c^{2/3}}+\frac{\sqrt{3} b e \tan ^{-1}\left (\frac{1+2 \sqrt [3]{c} x}{\sqrt{3}}\right )}{4 c^{2/3}}+\frac{\sqrt{3} b d \tan ^{-1}\left (\frac{1+2 c^{2/3} x^2}{\sqrt{3}}\right )}{2 \sqrt [3]{c}}-\frac{b e \tanh ^{-1}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}-\frac{b d^2 \tanh ^{-1}\left (c x^3\right )}{2 e}+\frac{(d+e x)^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{2 e}+\frac{b d \log \left (1-c^{2/3} x^2\right )}{2 \sqrt [3]{c}}+\frac{b e \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}-\frac{b e \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}-\frac{b d \log \left (1+c^{2/3} x^2+c^{4/3} x^4\right )}{4 \sqrt [3]{c}}\\ \end{align*}
Mathematica [A] time = 0.102132, size = 333, normalized size = 1.17 \[ a d x+\frac{1}{2} a e x^2-\frac{b d \left (\log \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )+\log \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )-2 \log \left (1-\sqrt [3]{c} x\right )-2 \log \left (\sqrt [3]{c} x+1\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{c} x-1}{\sqrt{3}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{c} x+1}{\sqrt{3}}\right )\right )}{4 \sqrt [3]{c}}+\frac{b e \log \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )}{8 c^{2/3}}-\frac{b e \log \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )}{8 c^{2/3}}+\frac{b e \log \left (1-\sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac{b e \log \left (\sqrt [3]{c} x+1\right )}{4 c^{2/3}}+\frac{\sqrt{3} b e \tan ^{-1}\left (\frac{2 \sqrt [3]{c} x-1}{\sqrt{3}}\right )}{4 c^{2/3}}+\frac{\sqrt{3} b e \tan ^{-1}\left (\frac{2 \sqrt [3]{c} x+1}{\sqrt{3}}\right )}{4 c^{2/3}}+b d x \tanh ^{-1}\left (c x^3\right )+\frac{1}{2} b e x^2 \tanh ^{-1}\left (c x^3\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x)*(a + b*ArcTanh[c*x^3]),x]
[Out]
a*d*x + (a*e*x^2)/2 + (Sqrt[3]*b*e*ArcTan[(-1 + 2*c^(1/3)*x)/Sqrt[3]])/(4*c^(2/3)) + (Sqrt[3]*b*e*ArcTan[(1 +
2*c^(1/3)*x)/Sqrt[3]])/(4*c^(2/3)) + b*d*x*ArcTanh[c*x^3] + (b*e*x^2*ArcTanh[c*x^3])/2 + (b*e*Log[1 - c^(1/3)*
x])/(4*c^(2/3)) - (b*e*Log[1 + c^(1/3)*x])/(4*c^(2/3)) + (b*e*Log[1 - c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3)) -
(b*e*Log[1 + c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3)) - (b*d*(-2*Sqrt[3]*ArcTan[(-1 + 2*c^(1/3)*x)/Sqrt[3]] + 2*S
qrt[3]*ArcTan[(1 + 2*c^(1/3)*x)/Sqrt[3]] - 2*Log[1 - c^(1/3)*x] - 2*Log[1 + c^(1/3)*x] + Log[1 - c^(1/3)*x + c
^(2/3)*x^2] + Log[1 + c^(1/3)*x + c^(2/3)*x^2]))/(4*c^(1/3))
________________________________________________________________________________________
Maple [A] time = 0.031, size = 362, normalized size = 1.3 \begin{align*}{\frac{a{x}^{2}e}{2}}+adx+{\frac{b{\it Artanh} \left ( c{x}^{3} \right ){x}^{2}e}{2}}+b{\it Artanh} \left ( c{x}^{3} \right ) dx+{\frac{bd}{2\,c}\ln \left ( x-\sqrt [3]{{c}^{-1}} \right ) \left ({c}^{-1} \right ) ^{-{\frac{2}{3}}}}-{\frac{bd}{4\,c}\ln \left ({x}^{2}+\sqrt [3]{{c}^{-1}}x+ \left ({c}^{-1} \right ) ^{{\frac{2}{3}}} \right ) \left ({c}^{-1} \right ) ^{-{\frac{2}{3}}}}-{\frac{bd\sqrt{3}}{2\,c}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{x}{\sqrt [3]{{c}^{-1}}}}+1 \right ) } \right ) \left ({c}^{-1} \right ) ^{-{\frac{2}{3}}}}+{\frac{be}{4\,c}\ln \left ( x-\sqrt [3]{{c}^{-1}} \right ){\frac{1}{\sqrt [3]{{c}^{-1}}}}}-{\frac{be}{8\,c}\ln \left ({x}^{2}+\sqrt [3]{{c}^{-1}}x+ \left ({c}^{-1} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{c}^{-1}}}}}+{\frac{be\sqrt{3}}{4\,c}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{x}{\sqrt [3]{{c}^{-1}}}}+1 \right ) } \right ){\frac{1}{\sqrt [3]{{c}^{-1}}}}}+{\frac{bd}{2\,c}\ln \left ( x+\sqrt [3]{{c}^{-1}} \right ) \left ({c}^{-1} \right ) ^{-{\frac{2}{3}}}}-{\frac{bd}{4\,c}\ln \left ({x}^{2}-\sqrt [3]{{c}^{-1}}x+ \left ({c}^{-1} \right ) ^{{\frac{2}{3}}} \right ) \left ({c}^{-1} \right ) ^{-{\frac{2}{3}}}}+{\frac{bd\sqrt{3}}{2\,c}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{x}{\sqrt [3]{{c}^{-1}}}}-1 \right ) } \right ) \left ({c}^{-1} \right ) ^{-{\frac{2}{3}}}}-{\frac{be}{4\,c}\ln \left ( x+\sqrt [3]{{c}^{-1}} \right ){\frac{1}{\sqrt [3]{{c}^{-1}}}}}+{\frac{be}{8\,c}\ln \left ({x}^{2}-\sqrt [3]{{c}^{-1}}x+ \left ({c}^{-1} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{c}^{-1}}}}}+{\frac{be\sqrt{3}}{4\,c}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{x}{\sqrt [3]{{c}^{-1}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{c}^{-1}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x+d)*(a+b*arctanh(c*x^3)),x)
[Out]
1/2*a*x^2*e+a*d*x+1/2*b*arctanh(c*x^3)*x^2*e+b*arctanh(c*x^3)*d*x+1/2*b*d/c/(1/c)^(2/3)*ln(x-(1/c)^(1/3))-1/4*
b*d/c/(1/c)^(2/3)*ln(x^2+(1/c)^(1/3)*x+(1/c)^(2/3))-1/2*b*d/c/(1/c)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/c)^
(1/3)*x+1))+1/4*b*e/c/(1/c)^(1/3)*ln(x-(1/c)^(1/3))-1/8*b*e/c/(1/c)^(1/3)*ln(x^2+(1/c)^(1/3)*x+(1/c)^(2/3))+1/
4*b*e*3^(1/2)/c/(1/c)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/c)^(1/3)*x+1))+1/2*b*d/c/(1/c)^(2/3)*ln(x+(1/c)^(1/3))-1/
4*b*d/c/(1/c)^(2/3)*ln(x^2-(1/c)^(1/3)*x+(1/c)^(2/3))+1/2*b*d/c/(1/c)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/c
)^(1/3)*x-1))-1/4*b*e/c/(1/c)^(1/3)*ln(x+(1/c)^(1/3))+1/8*b*e/c/(1/c)^(1/3)*ln(x^2-(1/c)^(1/3)*x+(1/c)^(2/3))+
1/4*b*e*3^(1/2)/c/(1/c)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/c)^(1/3)*x-1))
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)*(a+b*arctanh(c*x^3)),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [C] time = 14.7197, size = 9110, normalized size = 31.96 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)*(a+b*arctanh(c*x^3)),x, algorithm="fricas")
[Out]
1/2*a*e*x^2 + a*d*x + 1/8*(4*(1/2)^(2/3)*b^2*d*e*(-I*sqrt(3) + 1)/(((8*c*d^3 + e^3)*b^3/c^2 + (8*c*d^3 - e^3)*
b^3/c^2)^(1/3)*c) - (1/2)^(1/3)*((8*c*d^3 + e^3)*b^3/c^2 + (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*(I*sqrt(3) + 1))*log
(2*(4*(1/2)^(2/3)*b^2*d*e*(-I*sqrt(3) + 1)/(((8*c*d^3 + e^3)*b^3/c^2 + (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*c) - (1/
2)^(1/3)*((8*c*d^3 + e^3)*b^3/c^2 + (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*(I*sqrt(3) + 1))*b*c*d^2 + 4*b^2*d*e^2 + 1/
4*(4*(1/2)^(2/3)*b^2*d*e*(-I*sqrt(3) + 1)/(((8*c*d^3 + e^3)*b^3/c^2 + (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*c) - (1/2
)^(1/3)*((8*c*d^3 + e^3)*b^3/c^2 + (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*(I*sqrt(3) + 1))^2*c*e + (8*b^2*c*d^3 + b^2*
e^3)*x) - 1/16*(4*(1/2)^(2/3)*b^2*d*e*(-I*sqrt(3) + 1)/(((8*c*d^3 + e^3)*b^3/c^2 + (8*c*d^3 - e^3)*b^3/c^2)^(1
/3)*c) - (1/2)^(1/3)*((8*c*d^3 + e^3)*b^3/c^2 + (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*(I*sqrt(3) + 1) - 2*sqrt(3/2)*s
qrt(1/2)*sqrt(-(32*b^2*d*e + (4*(1/2)^(2/3)*b^2*d*e*(-I*sqrt(3) + 1)/(((8*c*d^3 + e^3)*b^3/c^2 + (8*c*d^3 - e^
3)*b^3/c^2)^(1/3)*c) - (1/2)^(1/3)*((8*c*d^3 + e^3)*b^3/c^2 + (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*(I*sqrt(3) + 1))^
2*c)/c))*log(-(4*(1/2)^(2/3)*b^2*d*e*(-I*sqrt(3) + 1)/(((8*c*d^3 + e^3)*b^3/c^2 + (8*c*d^3 - e^3)*b^3/c^2)^(1/
3)*c) - (1/2)^(1/3)*((8*c*d^3 + e^3)*b^3/c^2 + (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*(I*sqrt(3) + 1))*b*c*d^2 - 2*b^2
*d*e^2 - 1/8*(4*(1/2)^(2/3)*b^2*d*e*(-I*sqrt(3) + 1)/(((8*c*d^3 + e^3)*b^3/c^2 + (8*c*d^3 - e^3)*b^3/c^2)^(1/3
)*c) - (1/2)^(1/3)*((8*c*d^3 + e^3)*b^3/c^2 + (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*(I*sqrt(3) + 1))^2*c*e + 1/4*sqrt
(3/2)*sqrt(1/2)*(8*b*c*d^2 - (4*(1/2)^(2/3)*b^2*d*e*(-I*sqrt(3) + 1)/(((8*c*d^3 + e^3)*b^3/c^2 + (8*c*d^3 - e^
3)*b^3/c^2)^(1/3)*c) - (1/2)^(1/3)*((8*c*d^3 + e^3)*b^3/c^2 + (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*(I*sqrt(3) + 1))*
c*e)*sqrt(-(32*b^2*d*e + (4*(1/2)^(2/3)*b^2*d*e*(-I*sqrt(3) + 1)/(((8*c*d^3 + e^3)*b^3/c^2 + (8*c*d^3 - e^3)*b
^3/c^2)^(1/3)*c) - (1/2)^(1/3)*((8*c*d^3 + e^3)*b^3/c^2 + (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*(I*sqrt(3) + 1))^2*c)
/c) + (8*b^2*c*d^3 + b^2*e^3)*x) - 1/16*(4*(1/2)^(2/3)*b^2*d*e*(-I*sqrt(3) + 1)/(((8*c*d^3 + e^3)*b^3/c^2 + (8
*c*d^3 - e^3)*b^3/c^2)^(1/3)*c) - (1/2)^(1/3)*((8*c*d^3 + e^3)*b^3/c^2 + (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*(I*sqr
t(3) + 1) + 2*sqrt(3/2)*sqrt(1/2)*sqrt(-(32*b^2*d*e + (4*(1/2)^(2/3)*b^2*d*e*(-I*sqrt(3) + 1)/(((8*c*d^3 + e^3
)*b^3/c^2 + (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*c) - (1/2)^(1/3)*((8*c*d^3 + e^3)*b^3/c^2 + (8*c*d^3 - e^3)*b^3/c^2
)^(1/3)*(I*sqrt(3) + 1))^2*c)/c))*log(-(4*(1/2)^(2/3)*b^2*d*e*(-I*sqrt(3) + 1)/(((8*c*d^3 + e^3)*b^3/c^2 + (8*
c*d^3 - e^3)*b^3/c^2)^(1/3)*c) - (1/2)^(1/3)*((8*c*d^3 + e^3)*b^3/c^2 + (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*(I*sqrt
(3) + 1))*b*c*d^2 - 2*b^2*d*e^2 - 1/8*(4*(1/2)^(2/3)*b^2*d*e*(-I*sqrt(3) + 1)/(((8*c*d^3 + e^3)*b^3/c^2 + (8*c
*d^3 - e^3)*b^3/c^2)^(1/3)*c) - (1/2)^(1/3)*((8*c*d^3 + e^3)*b^3/c^2 + (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*(I*sqrt(
3) + 1))^2*c*e - 1/4*sqrt(3/2)*sqrt(1/2)*(8*b*c*d^2 - (4*(1/2)^(2/3)*b^2*d*e*(-I*sqrt(3) + 1)/(((8*c*d^3 + e^3
)*b^3/c^2 + (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*c) - (1/2)^(1/3)*((8*c*d^3 + e^3)*b^3/c^2 + (8*c*d^3 - e^3)*b^3/c^2
)^(1/3)*(I*sqrt(3) + 1))*c*e)*sqrt(-(32*b^2*d*e + (4*(1/2)^(2/3)*b^2*d*e*(-I*sqrt(3) + 1)/(((8*c*d^3 + e^3)*b^
3/c^2 + (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*c) - (1/2)^(1/3)*((8*c*d^3 + e^3)*b^3/c^2 + (8*c*d^3 - e^3)*b^3/c^2)^(1
/3)*(I*sqrt(3) + 1))^2*c)/c) + (8*b^2*c*d^3 + b^2*e^3)*x) + 1/16*(4*(1/2)^(2/3)*b^2*d*e*(-I*sqrt(3) + 1)/(((8*
c*d^3 + e^3)*b^3/c^2 - (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*c) + (1/2)^(1/3)*((8*c*d^3 + e^3)*b^3/c^2 - (8*c*d^3 - e
^3)*b^3/c^2)^(1/3)*(I*sqrt(3) + 1) + 2*sqrt(3/2)*sqrt(1/2)*sqrt((32*b^2*d*e - (4*(1/2)^(2/3)*b^2*d*e*(-I*sqrt(
3) + 1)/(((8*c*d^3 + e^3)*b^3/c^2 - (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*c) + (1/2)^(1/3)*((8*c*d^3 + e^3)*b^3/c^2 -
(8*c*d^3 - e^3)*b^3/c^2)^(1/3)*(I*sqrt(3) + 1))^2*c)/c))*log((4*(1/2)^(2/3)*b^2*d*e*(-I*sqrt(3) + 1)/(((8*c*d
^3 + e^3)*b^3/c^2 - (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*c) + (1/2)^(1/3)*((8*c*d^3 + e^3)*b^3/c^2 - (8*c*d^3 - e^3)
*b^3/c^2)^(1/3)*(I*sqrt(3) + 1))*b*c*d^2 - 2*b^2*d*e^2 + 1/8*(4*(1/2)^(2/3)*b^2*d*e*(-I*sqrt(3) + 1)/(((8*c*d^
3 + e^3)*b^3/c^2 - (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*c) + (1/2)^(1/3)*((8*c*d^3 + e^3)*b^3/c^2 - (8*c*d^3 - e^3)*
b^3/c^2)^(1/3)*(I*sqrt(3) + 1))^2*c*e + 1/4*sqrt(3/2)*sqrt(1/2)*(8*b*c*d^2 - (4*(1/2)^(2/3)*b^2*d*e*(-I*sqrt(3
) + 1)/(((8*c*d^3 + e^3)*b^3/c^2 - (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*c) + (1/2)^(1/3)*((8*c*d^3 + e^3)*b^3/c^2 -
(8*c*d^3 - e^3)*b^3/c^2)^(1/3)*(I*sqrt(3) + 1))*c*e)*sqrt((32*b^2*d*e - (4*(1/2)^(2/3)*b^2*d*e*(-I*sqrt(3) + 1
)/(((8*c*d^3 + e^3)*b^3/c^2 - (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*c) + (1/2)^(1/3)*((8*c*d^3 + e^3)*b^3/c^2 - (8*c*
d^3 - e^3)*b^3/c^2)^(1/3)*(I*sqrt(3) + 1))^2*c)/c) - (8*b^2*c*d^3 - b^2*e^3)*x) + 1/16*(4*(1/2)^(2/3)*b^2*d*e*
(-I*sqrt(3) + 1)/(((8*c*d^3 + e^3)*b^3/c^2 - (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*c) + (1/2)^(1/3)*((8*c*d^3 + e^3)*
b^3/c^2 - (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*(I*sqrt(3) + 1) - 2*sqrt(3/2)*sqrt(1/2)*sqrt((32*b^2*d*e - (4*(1/2)^(
2/3)*b^2*d*e*(-I*sqrt(3) + 1)/(((8*c*d^3 + e^3)*b^3/c^2 - (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*c) + (1/2)^(1/3)*((8*
c*d^3 + e^3)*b^3/c^2 - (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*(I*sqrt(3) + 1))^2*c)/c))*log((4*(1/2)^(2/3)*b^2*d*e*(-I
*sqrt(3) + 1)/(((8*c*d^3 + e^3)*b^3/c^2 - (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*c) + (1/2)^(1/3)*((8*c*d^3 + e^3)*b^3
/c^2 - (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*(I*sqrt(3) + 1))*b*c*d^2 - 2*b^2*d*e^2 + 1/8*(4*(1/2)^(2/3)*b^2*d*e*(-I*
sqrt(3) + 1)/(((8*c*d^3 + e^3)*b^3/c^2 - (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*c) + (1/2)^(1/3)*((8*c*d^3 + e^3)*b^3/
c^2 - (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*(I*sqrt(3) + 1))^2*c*e - 1/4*sqrt(3/2)*sqrt(1/2)*(8*b*c*d^2 - (4*(1/2)^(2
/3)*b^2*d*e*(-I*sqrt(3) + 1)/(((8*c*d^3 + e^3)*b^3/c^2 - (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*c) + (1/2)^(1/3)*((8*c
*d^3 + e^3)*b^3/c^2 - (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*(I*sqrt(3) + 1))*c*e)*sqrt((32*b^2*d*e - (4*(1/2)^(2/3)*b
^2*d*e*(-I*sqrt(3) + 1)/(((8*c*d^3 + e^3)*b^3/c^2 - (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*c) + (1/2)^(1/3)*((8*c*d^3
+ e^3)*b^3/c^2 - (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*(I*sqrt(3) + 1))^2*c)/c) - (8*b^2*c*d^3 - b^2*e^3)*x) - 1/8*(4
*(1/2)^(2/3)*b^2*d*e*(-I*sqrt(3) + 1)/(((8*c*d^3 + e^3)*b^3/c^2 - (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*c) + (1/2)^(1
/3)*((8*c*d^3 + e^3)*b^3/c^2 - (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*(I*sqrt(3) + 1))*log(-2*(4*(1/2)^(2/3)*b^2*d*e*(
-I*sqrt(3) + 1)/(((8*c*d^3 + e^3)*b^3/c^2 - (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*c) + (1/2)^(1/3)*((8*c*d^3 + e^3)*b
^3/c^2 - (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*(I*sqrt(3) + 1))*b*c*d^2 + 4*b^2*d*e^2 - 1/4*(4*(1/2)^(2/3)*b^2*d*e*(-
I*sqrt(3) + 1)/(((8*c*d^3 + e^3)*b^3/c^2 - (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*c) + (1/2)^(1/3)*((8*c*d^3 + e^3)*b^
3/c^2 - (8*c*d^3 - e^3)*b^3/c^2)^(1/3)*(I*sqrt(3) + 1))^2*c*e - (8*b^2*c*d^3 - b^2*e^3)*x) + 1/4*(b*e*x^2 + 2*
b*d*x)*log(-(c*x^3 + 1)/(c*x^3 - 1))
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)*(a+b*atanh(c*x**3)),x)
[Out]
Timed out
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Giac [A] time = 2.17951, size = 432, normalized size = 1.52 \begin{align*} -\frac{1}{8} \, b c^{5}{\left (\frac{2 \, \left (-\frac{1}{c}\right )^{\frac{2}{3}} \log \left ({\left | x - \left (-\frac{1}{c}\right )^{\frac{1}{3}} \right |}\right )}{c^{5}} - \frac{2 \, \sqrt{3}{\left | c \right |}^{\frac{4}{3}} \arctan \left (\frac{1}{3} \, \sqrt{3} c^{\frac{1}{3}}{\left (2 \, x + \frac{1}{c^{\frac{1}{3}}}\right )}\right )}{c^{7}} + \frac{{\left | c \right |}^{\frac{4}{3}} \log \left (x^{2} + \frac{x}{c^{\frac{1}{3}}} + \frac{1}{c^{\frac{2}{3}}}\right )}{c^{7}} - \frac{2 \, \log \left ({\left | x - \frac{1}{c^{\frac{1}{3}}} \right |}\right )}{c^{\frac{17}{3}}} + \frac{2 \, \sqrt{3} \left (-c^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{1}{c}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{1}{c}\right )^{\frac{1}{3}}}\right )}{c^{7}} - \frac{\left (-c^{2}\right )^{\frac{2}{3}} \log \left (x^{2} + x \left (-\frac{1}{c}\right )^{\frac{1}{3}} + \left (-\frac{1}{c}\right )^{\frac{2}{3}}\right )}{c^{7}}\right )} e + \frac{1}{4} \, b c^{3} d{\left (\frac{2 \, \sqrt{3}{\left | c \right |}^{\frac{2}{3}} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} + \frac{1}{{\left | c \right |}^{\frac{2}{3}}}\right )}{\left | c \right |}^{\frac{2}{3}}\right )}{c^{4}} - \frac{{\left | c \right |}^{\frac{2}{3}} \log \left (x^{4} + \frac{x^{2}}{{\left | c \right |}^{\frac{2}{3}}} + \frac{1}{{\left | c \right |}^{\frac{4}{3}}}\right )}{c^{4}} + \frac{2 \, \log \left ({\left | x^{2} - \frac{1}{{\left | c \right |}^{\frac{2}{3}}} \right |}\right )}{c^{2}{\left | c \right |}^{\frac{4}{3}}}\right )} + \frac{1}{4} \, b x^{2} e \log \left (-\frac{c x^{3} + 1}{c x^{3} - 1}\right ) + \frac{1}{2} \, a x^{2} e + \frac{1}{2} \, b d x \log \left (-\frac{c x^{3} + 1}{c x^{3} - 1}\right ) + a d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)*(a+b*arctanh(c*x^3)),x, algorithm="giac")
[Out]
-1/8*b*c^5*(2*(-1/c)^(2/3)*log(abs(x - (-1/c)^(1/3)))/c^5 - 2*sqrt(3)*abs(c)^(4/3)*arctan(1/3*sqrt(3)*c^(1/3)*
(2*x + 1/c^(1/3)))/c^7 + abs(c)^(4/3)*log(x^2 + x/c^(1/3) + 1/c^(2/3))/c^7 - 2*log(abs(x - 1/c^(1/3)))/c^(17/3
) + 2*sqrt(3)*(-c^2)^(2/3)*arctan(1/3*sqrt(3)*(2*x + (-1/c)^(1/3))/(-1/c)^(1/3))/c^7 - (-c^2)^(2/3)*log(x^2 +
x*(-1/c)^(1/3) + (-1/c)^(2/3))/c^7)*e + 1/4*b*c^3*d*(2*sqrt(3)*abs(c)^(2/3)*arctan(1/3*sqrt(3)*(2*x^2 + 1/abs(
c)^(2/3))*abs(c)^(2/3))/c^4 - abs(c)^(2/3)*log(x^4 + x^2/abs(c)^(2/3) + 1/abs(c)^(4/3))/c^4 + 2*log(abs(x^2 -
1/abs(c)^(2/3)))/(c^2*abs(c)^(4/3))) + 1/4*b*x^2*e*log(-(c*x^3 + 1)/(c*x^3 - 1)) + 1/2*a*x^2*e + 1/2*b*d*x*log
(-(c*x^3 + 1)/(c*x^3 - 1)) + a*d*x