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

Optimal. Leaf size=140 \[ -\frac{e \left (1-c^2 x^2\right ) \log \left (1-c^2 x^2\right ) \left (a+b \tanh ^{-1}(c x)\right )}{2 c^2}+\frac{1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac{b (d-e) \tanh ^{-1}(c x)}{2 c^2}+\frac{b e x \log \left (1-c^2 x^2\right )}{2 c}+\frac{b e \tanh ^{-1}(c x)}{c^2}+\frac{b x (d-e)}{2 c}-\frac{b e x}{c} \]

[Out]

(b*(d - e)*x)/(2*c) - (b*e*x)/c - (b*(d - e)*ArcTanh[c*x])/(2*c^2) + (b*e*ArcTanh[c*x])/c^2 + (d*x^2*(a + b*Ar
cTanh[c*x]))/2 - (e*x^2*(a + b*ArcTanh[c*x]))/2 + (b*e*x*Log[1 - c^2*x^2])/(2*c) - (e*(1 - c^2*x^2)*(a + b*Arc
Tanh[c*x])*Log[1 - c^2*x^2])/(2*c^2)

________________________________________________________________________________________

Rubi [A]  time = 0.118822, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {2454, 2389, 2295, 6083, 321, 207, 2448, 206} \[ -\frac{e \left (1-c^2 x^2\right ) \log \left (1-c^2 x^2\right ) \left (a+b \tanh ^{-1}(c x)\right )}{2 c^2}+\frac{1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac{b (d-e) \tanh ^{-1}(c x)}{2 c^2}+\frac{b e x \log \left (1-c^2 x^2\right )}{2 c}+\frac{b e \tanh ^{-1}(c x)}{c^2}+\frac{b x (d-e)}{2 c}-\frac{b e x}{c} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]),x]

[Out]

(b*(d - e)*x)/(2*c) - (b*e*x)/c - (b*(d - e)*ArcTanh[c*x])/(2*c^2) + (b*e*ArcTanh[c*x])/c^2 + (d*x^2*(a + b*Ar
cTanh[c*x]))/2 - (e*x^2*(a + b*ArcTanh[c*x]))/2 + (b*e*x*Log[1 - c^2*x^2])/(2*c) - (e*(1 - c^2*x^2)*(a + b*Arc
Tanh[c*x])*Log[1 - c^2*x^2])/(2*c^2)

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
 q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
 IGtQ[m, 0])

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 6083

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_.) + Log[(f_.) + (g_.)*(x_)^2]*(e_.))*(x_)^(m_.), x_Symbol] :> Wit
h[{u = IntHide[x^m*(d + e*Log[f + g*x^2]), x]}, Dist[a + b*ArcTanh[c*x], u, x] - Dist[b*c, Int[ExpandIntegrand
[u/(1 - c^2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[(m + 1)/2, 0]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
 c, n, m, p, x]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
 FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

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])

Rubi steps

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

Mathematica [A]  time = 0.104307, size = 129, normalized size = 0.92 \[ \frac{2 e \log \left (1-c^2 x^2\right ) \left (c x (a c x+b)+b \left (c^2 x^2-1\right ) \tanh ^{-1}(c x)\right )+\log (1-c x) (b (d-3 e)-2 a e)-\log (c x+1) (2 a e+b (d-3 e))+2 a c^2 x^2 (d-e)+2 b c^2 x^2 (d-e) \tanh ^{-1}(c x)+2 b c x (d-3 e)}{4 c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]),x]

[Out]

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

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Maple [C]  time = 0.639, size = 2951, normalized size = 21.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*arctanh(c*x))*(d+e*ln(-c^2*x^2+1)),x)

[Out]

-1/4*I/c^2*b*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3*Pi*e*arctanh(c*x)-1/4*I/c^2*b*e*Pi*csgn(I/((c*x+1)^2/(-c^2*x^2+1)
+1)^2)*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1)^2)^2+1/4*I/c^2*b*e*Pi*csgn(I*(c*x+1)^2/(c^2*x^2
-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1)^2)^2-1/4*I/c^2*b*e*Pi*csgn(I*((c*x+1)^2/(-c^2*x^2
+1)+1))^2*csgn(I*((c*x+1)^2/(-c^2*x^2+1)+1)^2)+1/2*I/c^2*b*e*Pi*csgn(I*((c*x+1)^2/(-c^2*x^2+1)+1))*csgn(I*((c*
x+1)^2/(-c^2*x^2+1)+1)^2)^2-1/2*I/c^2*b*Pi*e*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))*csgn(I*(c*x+1)^2/(c^2*x^2-1))^
2-1/4*I/c^2*b*Pi*e*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))-1/2*a*e/c^2*ln(-c^2*x^2+
1)+1/2*x^2*a*e*ln(-c^2*x^2+1)-1/2/c^2*b*d+5/2*b*e*arctanh(c*x)/c^2+1/2*b*d*x/c+1/2*d*a*x^2-1/2/c^2*b*d*arctanh
(c*x)+1/2*b*arctanh(c*x)*x^2*d-1/2*b*arctanh(c*x)*x^2*e-3/2*b*e*x/c-1/4*I*b*arctanh(c*x)*Pi*csgn(I*(c*x+1)^2/(
c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1)^2)*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1)^2)*csgn(I*(c*x+1)^2/(c^2*x^2-1))*x^
2*e-1/4*I/c*b*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1)^2)*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1)^
2)*csgn(I*(c*x+1)^2/(c^2*x^2-1))*x*e+1/4*I/c^2*b*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1)^2)*cs
gn(I/((c*x+1)^2/(-c^2*x^2+1)+1)^2)*csgn(I*(c*x+1)^2/(c^2*x^2-1))*Pi*e*arctanh(c*x)+1/4*I/c*b*Pi*csgn(I*(c*x+1)
^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1)^2)^2*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1)^2)*x*e-1/4*I/c*b*Pi*csgn(I*(c
*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1)^2)^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))*x*e+1/4*I/c*b*Pi*csgn(I*((c*
x+1)^2/(-c^2*x^2+1)+1))^2*csgn(I*((c*x+1)^2/(-c^2*x^2+1)+1)^2)*x*e-1/2*I/c*b*Pi*csgn(I*((c*x+1)^2/(-c^2*x^2+1)
+1))*csgn(I*((c*x+1)^2/(-c^2*x^2+1)+1)^2)^2*x*e+1/2*I/c*b*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))^2*csgn(I*(c*x+1)/(-
c^2*x^2+1)^(1/2))*x*e+1/4*I/c*b*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))^2*x*e-1/4*
I/c^2*b*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1)^2)^2*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1)^2)*Pi*e
*arctanh(c*x)+1/4*I/c^2*b*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1)^2)^2*csgn(I*(c*x+1)^2/(c^2*x
^2-1))*Pi*e*arctanh(c*x)-1/4*I/c^2*b*csgn(I*((c*x+1)^2/(-c^2*x^2+1)+1)^2)*csgn(I*((c*x+1)^2/(-c^2*x^2+1)+1))^2
*Pi*e*arctanh(c*x)+1/2*I/c^2*b*csgn(I*((c*x+1)^2/(-c^2*x^2+1)+1)^2)^2*csgn(I*((c*x+1)^2/(-c^2*x^2+1)+1))*Pi*e*
arctanh(c*x)-1/2*I/c^2*b*csgn(I*(c*x+1)^2/(c^2*x^2-1))^2*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))*Pi*e*arctanh(c*x)-
1/c^2*b*e*ln(2)+3/2/c^2*b*e-1/2*a*e*x^2-b*arctanh(c*x)*ln((c*x+1)^2/(-c^2*x^2+1)+1)*x^2*e+b*arctanh(c*x)*ln(2)
*x^2*e-1/c*b*ln((c*x+1)^2/(-c^2*x^2+1)+1)*x*e+1/c*b*ln(2)*x*e+1/c^2*b*ln((c*x+1)^2/(-c^2*x^2+1)+1)*e*arctanh(c
*x)-1/c^2*b*ln(2)*e*arctanh(c*x)+1/2*a*e/c^2-1/4*I/c^2*b*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)/(-c^2*x^
2+1)^(1/2))^2*Pi*e*arctanh(c*x)+1/4*I/c^2*b*Pi*e*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+
1)^2)*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1)^2)+1/4*I*b*arctanh(c*x)*Pi*csgn(I*(c*x+1)^2/(c^2
*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1)^2)^2*csgn(I/((c*x+1)^2/(-c^2*x^2+1)+1)^2)*x^2*e-1/4*I*b*arctanh(c*x)*Pi*csg
n(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1)^2)^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))*x^2*e+1/4*I*b*arctanh(
c*x)*Pi*csgn(I*((c*x+1)^2/(-c^2*x^2+1)+1))^2*csgn(I*((c*x+1)^2/(-c^2*x^2+1)+1)^2)*x^2*e-1/2*I*b*arctanh(c*x)*P
i*csgn(I*((c*x+1)^2/(-c^2*x^2+1)+1))*csgn(I*((c*x+1)^2/(-c^2*x^2+1)+1)^2)^2*x^2*e+1/2*I*b*arctanh(c*x)*Pi*csgn
(I*(c*x+1)^2/(c^2*x^2-1))^2*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))*x^2*e+1/4*I*b*arctanh(c*x)*Pi*csgn(I*(c*x+1)^2/
(c^2*x^2-1))*csgn(I*(c*x+1)/(-c^2*x^2+1)^(1/2))^2*x^2*e+1/4*I*b*arctanh(c*x)*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(
(c*x+1)^2/(-c^2*x^2+1)+1)^2)^3*x^2*e+1/4*I*b*arctanh(c*x)*Pi*csgn(I*((c*x+1)^2/(-c^2*x^2+1)+1)^2)^3*x^2*e+1/4*
I*b*arctanh(c*x)*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3*x^2*e+1/4*I/c*b*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2
/(-c^2*x^2+1)+1)^2)^3*x*e+1/4*I/c*b*Pi*csgn(I*((c*x+1)^2/(-c^2*x^2+1)+1)^2)^3*x*e+1/4*I/c*b*Pi*csgn(I*(c*x+1)^
2/(c^2*x^2-1))^3*x*e-1/4*I/c^2*b*csgn(I*(c*x+1)^2/(c^2*x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1)^2)^3*Pi*e*arctanh(c*x
)-1/4*I/c^2*b*csgn(I*((c*x+1)^2/(-c^2*x^2+1)+1)^2)^3*Pi*e*arctanh(c*x)-1/4*I/c^2*b*Pi*e*csgn(I*(c*x+1)^2/(c^2*
x^2-1)/((c*x+1)^2/(-c^2*x^2+1)+1)^2)^3-1/4*I/c^2*b*e*Pi*csgn(I*((c*x+1)^2/(-c^2*x^2+1)+1)^2)^3-1/4*I/c^2*b*Pi*
e*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3+1/c^2*b*e*(arctanh(c*x)*x*c+arctanh(c*x)+1)*(c*x-1)*ln((c*x+1)/(-c^2*x^2+1)^
(1/2))

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Maxima [A]  time = 0.977084, size = 231, normalized size = 1.65 \begin{align*} \frac{1}{2} \, a d x^{2} + \frac{1}{4} \,{\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 - \frac{{\left (c^{2} x^{2} -{\left (c^{2} x^{2} - 1\right )} \log \left (-c^{2} x^{2} + 1\right ) - 1\right )} b e \operatorname{artanh}\left (c x\right )}{2 \, c^{2}} - \frac{{\left (c^{2} x^{2} -{\left (c^{2} x^{2} - 1\right )} \log \left (-c^{2} x^{2} + 1\right ) - 1\right )} a e}{2 \, c^{2}} - \frac{{\left (3 \, c x -{\left (c x + 1\right )} \log \left (c x + 1\right ) -{\left (c x - 1\right )} \log \left (-c x + 1\right )\right )} b e}{2 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arctanh(c*x))*(d+e*log(-c^2*x^2+1)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.11355, size = 298, normalized size = 2.13 \begin{align*} \frac{2 \,{\left (a c^{2} d - a c^{2} e\right )} x^{2} + 2 \,{\left (b c d - 3 \, b c e\right )} x + 2 \,{\left (a c^{2} e x^{2} + b c e x - a e\right )} \log \left (-c^{2} x^{2} + 1\right ) +{\left ({\left (b c^{2} d - b c^{2} e\right )} x^{2} - b d + 3 \, b e +{\left (b c^{2} e x^{2} - b e\right )} \log \left (-c^{2} x^{2} + 1\right )\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )}{4 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arctanh(c*x))*(d+e*log(-c^2*x^2+1)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 6.93077, size = 202, normalized size = 1.44 \begin{align*} \begin{cases} \frac{a d x^{2}}{2} + \frac{a e x^{2} \log{\left (- c^{2} x^{2} + 1 \right )}}{2} - \frac{a e x^{2}}{2} - \frac{a e \log{\left (- c^{2} x^{2} + 1 \right )}}{2 c^{2}} + \frac{b d x^{2} \operatorname{atanh}{\left (c x \right )}}{2} + \frac{b e x^{2} \log{\left (- c^{2} x^{2} + 1 \right )} \operatorname{atanh}{\left (c x \right )}}{2} - \frac{b e x^{2} \operatorname{atanh}{\left (c x \right )}}{2} + \frac{b d x}{2 c} + \frac{b e x \log{\left (- c^{2} x^{2} + 1 \right )}}{2 c} - \frac{3 b e x}{2 c} - \frac{b d \operatorname{atanh}{\left (c x \right )}}{2 c^{2}} - \frac{b e \log{\left (- c^{2} x^{2} + 1 \right )} \operatorname{atanh}{\left (c x \right )}}{2 c^{2}} + \frac{3 b e \operatorname{atanh}{\left (c x \right )}}{2 c^{2}} & \text{for}\: c \neq 0 \\\frac{a d x^{2}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*atanh(c*x))*(d+e*ln(-c**2*x**2+1)),x)

[Out]

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

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Giac [B]  time = 1.25823, size = 412, normalized size = 2.94 \begin{align*} \frac{b c^{2} x^{2} e \log \left (c x + 1\right )^{2} - b c^{2} x^{2} e \log \left (-c x + 1\right )^{2} + 2 \, a c^{2} x^{2} e \log \left (c x + 1\right ) - b c^{2} x^{2} e \log \left (c x + 1\right ) + 2 \, a c^{2} x^{2} e \log \left (-c x + 1\right ) + b c^{2} x^{2} e \log \left (-c x + 1\right ) + b c^{2} d x^{2} \log \left (-\frac{c x + 1}{c x - 1}\right ) + 2 \, a c^{2} d x^{2} - 2 \, a c^{2} x^{2} e + 2 \, b c x e \log \left (c x + 1\right ) + 2 \, b c x e \log \left (-c x + 1\right ) + 2 \, b c d x - 6 \, b c x e - b e \log \left (c x + 1\right )^{2} - b e \log \left (c x - 1\right )^{2} + 2 \, b e \log \left (c x - 1\right ) \log \left (-c x + 1\right ) - b d \log \left (c x + 1\right ) - 2 \, a e \log \left (c x + 1\right ) + 3 \, b e \log \left (c x + 1\right ) + b d \log \left (c x - 1\right ) - 2 \, a e \log \left (c x - 1\right ) - 3 \, b e \log \left (c x - 1\right )}{4 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arctanh(c*x))*(d+e*log(-c^2*x^2+1)),x, algorithm="giac")

[Out]

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