3.268 \(\int x (a+b \coth ^{-1}(c x)) (d+e \log (1-c^2 x^2)) \, dx\)
Optimal. Leaf size=140 \[ -\frac{e \left (1-c^2 x^2\right ) \log \left (1-c^2 x^2\right ) \left (a+b \coth ^{-1}(c x)\right )}{2 c^2}+\frac{1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac{b (d-e) \tanh ^{-1}(c x)}{2 c^2}+\frac{b e x \log \left (1-c^2 x^2\right )}{2 c}+\frac{b e \tanh ^{-1}(c x)}{c^2}+\frac{b x (d-e)}{2 c}-\frac{b e x}{c} \]
[Out]
(b*(d - e)*x)/(2*c) - (b*e*x)/c + (d*x^2*(a + b*ArcCoth[c*x]))/2 - (e*x^2*(a + b*ArcCoth[c*x]))/2 - (b*(d - e)
*ArcTanh[c*x])/(2*c^2) + (b*e*ArcTanh[c*x])/c^2 + (b*e*x*Log[1 - c^2*x^2])/(2*c) - (e*(1 - c^2*x^2)*(a + b*Arc
Coth[c*x])*Log[1 - c^2*x^2])/(2*c^2)
________________________________________________________________________________________
Rubi [A] time = 0.118628, 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, 6084, 321, 207, 2448, 206} \[ -\frac{e \left (1-c^2 x^2\right ) \log \left (1-c^2 x^2\right ) \left (a+b \coth ^{-1}(c x)\right )}{2 c^2}+\frac{1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac{b (d-e) \tanh ^{-1}(c x)}{2 c^2}+\frac{b e x \log \left (1-c^2 x^2\right )}{2 c}+\frac{b e \tanh ^{-1}(c x)}{c^2}+\frac{b x (d-e)}{2 c}-\frac{b e x}{c} \]
Antiderivative was successfully verified.
[In]
Int[x*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]),x]
[Out]
(b*(d - e)*x)/(2*c) - (b*e*x)/c + (d*x^2*(a + b*ArcCoth[c*x]))/2 - (e*x^2*(a + b*ArcCoth[c*x]))/2 - (b*(d - e)
*ArcTanh[c*x])/(2*c^2) + (b*e*ArcTanh[c*x])/c^2 + (b*e*x*Log[1 - c^2*x^2])/(2*c) - (e*(1 - c^2*x^2)*(a + b*Arc
Coth[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 6084
Int[((a_.) + ArcCoth[(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*ArcCoth[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 \coth ^{-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 \coth ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac{e \left (1-c^2 x^2\right ) \left (a+b \coth ^{-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 \coth ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac{e \left (1-c^2 x^2\right ) \left (a+b \coth ^{-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 \coth ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \coth ^{-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 \coth ^{-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{1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \coth ^{-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{e \left (1-c^2 x^2\right ) \left (a+b \coth ^{-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{1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac{b (d-e) \tanh ^{-1}(c x)}{2 c^2}+\frac{b e \tanh ^{-1}(c x)}{c^2}+\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 \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{2 c^2}\\ \end{align*}
Mathematica [A] time = 0.105695, size = 129, normalized size = 0.92 \[ \frac{2 e \log \left (1-c^2 x^2\right ) \left (c x (a c x+b)+b \left (c^2 x^2-1\right ) \coth ^{-1}(c x)\right )+\log (1-c x) (b (d-3 e)-2 a e)-\log (c x+1) (2 a e+b (d-3 e))+2 a c^2 x^2 (d-e)+2 b c^2 x^2 (d-e) \coth ^{-1}(c x)+2 b c x (d-3 e)}{4 c^2} \]
Antiderivative was successfully verified.
[In]
Integrate[x*(a + b*ArcCoth[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*ArcCoth[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)*ArcCoth[c*x])*Log[1 - c^2*x^
2])/(4*c^2)
________________________________________________________________________________________
Maple [C] time = 0.602, size = 2616, normalized size = 18.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*(a+b*arccoth(c*x))*(d+e*ln(-c^2*x^2+1)),x)
[Out]
-1/2*a*e*x^2+1/4*I/c^2*b*arccoth(c*x)*e*Pi*csgn(I*(c*x+1)/(c*x-1))^3+1/4*I*b*arccoth(c*x)*Pi*csgn(I*(c*x+1)/(c
*x-1)/((c*x+1)/(c*x-1)-1)^2)^3*x^2*e+1/4*I*b*arccoth(c*x)*Pi*csgn(I*((c*x+1)/(c*x-1)-1)^2)^3*x^2*e-1/4*I*b*arc
coth(c*x)*Pi*csgn(I*(c*x+1)/(c*x-1))^3*x^2*e-1/2*I*b*arccoth(c*x)*Pi*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1
)^2)^2*x^2*e-1/2*I/c*b*Pi*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^2*x*e+1/2*I/c^2*b*arccoth(c*x)*e*Pi*cs
gn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^2-1/4*I/c^2*b*e*Pi*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^2
*csgn(I*(c*x+1)/(c*x-1))-1/4*I/c^2*b*e*Pi*csgn(I/((c*x+1)/(c*x-1)-1)^2)*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1
)-1)^2)^2+1/2*I/c^2*b*e*Pi*csgn(I*((c*x+1)/(c*x-1)-1)^2)^2*csgn(I*((c*x+1)/(c*x-1)-1))-1/4*I/c^2*b*e*Pi*csgn(I
*((c*x+1)/(c*x-1)-1)^2)*csgn(I*((c*x+1)/(c*x-1)-1))^2+1/2*a*e/c^2+1/2*d*a*x^2+1/2*a*e*ln(-c^2*x^2+1)*x^2-1/2*a
*e/c^2*ln(-c^2*x^2+1)+1/2/c*b*x*d+3/2/c^2*b*e-1/c^2*b*e*ln(2)-3/2*b*e*x/c-1/2/c^2*b*d-1/2/c^2*b*arccoth(c*x)*d
+5/2/c^2*b*arccoth(c*x)*e+1/2*b*arccoth(c*x)*x^2*d-1/2*b*arccoth(c*x)*x^2*e-1/2*I*b*arccoth(c*x)*Pi*csgn(I*((c
*x+1)/(c*x-1)-1)^2)^2*csgn(I*((c*x+1)/(c*x-1)-1))*x^2*e+1/4*I*b*arccoth(c*x)*Pi*csgn(I*((c*x+1)/(c*x-1)-1)^2)*
csgn(I*((c*x+1)/(c*x-1)-1))^2*x^2*e-1/4*I*b*arccoth(c*x)*Pi*csgn(I/((c*x-1)/(c*x+1))^(1/2))^2*csgn(I*(c*x+1)/(
c*x-1))*x^2*e+1/2*I*b*arccoth(c*x)*Pi*csgn(I/((c*x-1)/(c*x+1))^(1/2))*csgn(I*(c*x+1)/(c*x-1))^2*x^2*e+1/4*I/c*
b*Pi*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^2*csgn(I*(c*x+1)/(c*x-1))*x*e+1/4*I/c*b*Pi*csgn(I*(c*x+1)/(
c*x-1)/((c*x+1)/(c*x-1)-1)^2)^2*csgn(I/((c*x+1)/(c*x-1)-1)^2)*x*e-1/2*I/c*b*Pi*csgn(I*((c*x+1)/(c*x-1)-1)^2)^2
*csgn(I*((c*x+1)/(c*x-1)-1))*x*e+1/4*I/c*b*Pi*csgn(I*((c*x+1)/(c*x-1)-1)^2)*csgn(I*((c*x+1)/(c*x-1)-1))^2*x*e-
1/4*I/c*b*Pi*csgn(I/((c*x-1)/(c*x+1))^(1/2))^2*csgn(I*(c*x+1)/(c*x-1))*x*e+1/2*I/c*b*Pi*csgn(I/((c*x-1)/(c*x+1
))^(1/2))*csgn(I*(c*x+1)/(c*x-1))^2*x*e-1/4*I/c^2*b*arccoth(c*x)*e*Pi*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-
1)^2)^2*csgn(I*(c*x+1)/(c*x-1))-1/4*I/c^2*b*arccoth(c*x)*e*Pi*csgn(I/((c*x+1)/(c*x-1)-1)^2)*csgn(I*(c*x+1)/(c*
x-1)/((c*x+1)/(c*x-1)-1)^2)^2+1/2*I/c^2*b*arccoth(c*x)*e*Pi*csgn(I*((c*x+1)/(c*x-1)-1)^2)^2*csgn(I*((c*x+1)/(c
*x-1)-1))-1/4*I/c^2*b*arccoth(c*x)*e*Pi*csgn(I*((c*x+1)/(c*x-1)-1)^2)*csgn(I*((c*x+1)/(c*x-1)-1))^2+1/4*I/c^2*
b*arccoth(c*x)*e*Pi*csgn(I/((c*x-1)/(c*x+1))^(1/2))^2*csgn(I*(c*x+1)/(c*x-1))-1/2*I/c^2*b*arccoth(c*x)*e*Pi*cs
gn(I/((c*x-1)/(c*x+1))^(1/2))*csgn(I*(c*x+1)/(c*x-1))^2+1/4*I/c^2*b*e*Pi*csgn(I/((c*x+1)/(c*x-1)-1)^2)*csgn(I*
(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)*csgn(I*(c*x+1)/(c*x-1))+1/4*I*b*arccoth(c*x)*Pi*csgn(I*(c*x+1)/(c*x-1)/
((c*x+1)/(c*x-1)-1)^2)^2*csgn(I*(c*x+1)/(c*x-1))*x^2*e+1/4*I*b*arccoth(c*x)*Pi*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)
/(c*x-1)-1)^2)^2*csgn(I/((c*x+1)/(c*x-1)-1)^2)*x^2*e-1/2*I/c^2*b*Pi*e-1/2/c^2*b*e*(arccoth(c*x)*x*c+arccoth(c*
x)+1)*(c*x-1)*ln((c*x-1)/(c*x+1))-1/4*I/c^2*b*e*Pi*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^3-1/4*I/c^2*b
*e*Pi*csgn(I*((c*x+1)/(c*x-1)-1)^2)^3+1/4*I/c^2*b*Pi*e*csgn(I*(c*x+1)/(c*x-1))^3+1/2*I*b*arccoth(c*x)*Pi*x^2*e
+1/2*I/c^2*b*e*Pi*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^2+1/2*I/c*b*Pi*x*e-1/2*I/c^2*b*Pi*e*arccoth(c*
x)+b*ln(2)*arccoth(c*x)*x^2*e-b*arccoth(c*x)*ln((c*x+1)/(c*x-1)-1)*x^2*e+1/c*b*ln(2)*x*e-1/c*b*ln((c*x+1)/(c*x
-1)-1)*x*e-1/c^2*b*arccoth(c*x)*e*ln(2)+1/c^2*b*arccoth(c*x)*e*ln((c*x+1)/(c*x-1)-1)+1/4*I/c^2*b*arccoth(c*x)*
e*Pi*csgn(I/((c*x+1)/(c*x-1)-1)^2)*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)*csgn(I*(c*x+1)/(c*x-1))-1/4*I
*b*arccoth(c*x)*Pi*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)*csgn(I*(c*x+1)/(c*x-1))*csgn(I/((c*x+1)/(c*x-
1)-1)^2)*x^2*e-1/4*I/c*b*Pi*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)*csgn(I*(c*x+1)/(c*x-1))*csgn(I/((c*x
+1)/(c*x-1)-1)^2)*x*e+1/4*I/c^2*b*Pi*e*csgn(I/((c*x-1)/(c*x+1))^(1/2))^2*csgn(I*(c*x+1)/(c*x-1))-1/2*I/c^2*b*P
i*e*csgn(I/((c*x-1)/(c*x+1))^(1/2))*csgn(I*(c*x+1)/(c*x-1))^2+1/4*I/c*b*Pi*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*
x-1)-1)^2)^3*x*e+1/4*I/c*b*Pi*csgn(I*((c*x+1)/(c*x-1)-1)^2)^3*x*e-1/4*I/c*b*Pi*csgn(I*(c*x+1)/(c*x-1))^3*x*e-1
/4*I/c^2*b*arccoth(c*x)*e*Pi*csgn(I*(c*x+1)/(c*x-1)/((c*x+1)/(c*x-1)-1)^2)^3-1/4*I/c^2*b*arccoth(c*x)*e*Pi*csg
n(I*((c*x+1)/(c*x-1)-1)^2)^3
________________________________________________________________________________________
Maxima [A] time = 1.02803, size = 231, normalized size = 1.65 \begin{align*} \frac{1}{2} \, a d x^{2} + \frac{1}{4} \,{\left (2 \, x^{2} \operatorname{arcoth}\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{arcoth}\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*arccoth(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*arccoth(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*arccoth(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
________________________________________________________________________________________
Fricas [A] time = 1.62476, size = 297, normalized size = 2.12 \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*arccoth(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
________________________________________________________________________________________
Sympy [A] time = 12.6635, size = 209, normalized size = 1.49 \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{acoth}{\left (c x \right )}}{2} + \frac{b e x^{2} \log{\left (- c^{2} x^{2} + 1 \right )} \operatorname{acoth}{\left (c x \right )}}{2} - \frac{b e x^{2} \operatorname{acoth}{\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{acoth}{\left (c x \right )}}{2 c^{2}} - \frac{b e \log{\left (- c^{2} x^{2} + 1 \right )} \operatorname{acoth}{\left (c x \right )}}{2 c^{2}} + \frac{3 b e \operatorname{acoth}{\left (c x \right )}}{2 c^{2}} & \text{for}\: c \neq 0 \\\frac{d x^{2} \left (a + \frac{i \pi b}{2}\right )}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(a+b*acoth(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*acoth(c*x)/2 + b*e*x**2*log(-c**2*x**2 + 1)*acoth(c*x)/2 - b*e*x**2*acoth(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*acoth(c*x)/(2*c**2) - b*e*log(-c**2*x**2 + 1)*acoth(c*x)/(2*c**2)
+ 3*b*e*acoth(c*x)/(2*c**2), Ne(c, 0)), (d*x**2*(a + I*pi*b/2)/2, True))
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arcoth}\left (c x\right ) + a\right )}{\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(a+b*arccoth(c*x))*(d+e*log(-c^2*x^2+1)),x, algorithm="giac")
[Out]
integrate((b*arccoth(c*x) + a)*(e*log(-c^2*x^2 + 1) + d)*x, x)