\(\int x^3 (a+b \coth ^{-1}(c x)) (d+e \log (1-c^2 x^2)) \, dx\) [267]

Optimal result

Integrand size = 27, antiderivative size = 225 \[ \int x^3 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\frac {b (2 d-3 e) x}{8 c^3}-\frac {2 b e x}{3 c^3}+\frac {b (2 d-e) x^3}{24 c}-\frac {b e x^3}{18 c}-\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \coth ^{-1}(c x)\right )-\frac {b (2 d-3 e) \text {arctanh}(c x)}{8 c^4}+\frac {2 b e \text {arctanh}(c x)}{3 c^4}+\frac {b e x \log \left (1-c^2 x^2\right )}{4 c^3}+\frac {b e x^3 \log \left (1-c^2 x^2\right )}{12 c}-\frac {e \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \]

[Out]

1/8*b*(2*d-3*e)*x/c^3-2/3*b*e*x/c^3+1/24*b*(2*d-e)*x^3/c-1/18*b*e*x^3/c-1/4*e*x^2*(a+b*arccoth(c*x))/c^2-1/8*e
*x^4*(a+b*arccoth(c*x))-1/8*b*(2*d-3*e)*arctanh(c*x)/c^4+2/3*b*e*arctanh(c*x)/c^4+1/4*b*e*x*ln(-c^2*x^2+1)/c^3
+1/12*b*e*x^3*ln(-c^2*x^2+1)/c-1/4*e*(a+b*arccoth(c*x))*ln(-c^2*x^2+1)/c^4+1/4*x^4*(a+b*arccoth(c*x))*(d+e*ln(
-c^2*x^2+1))

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {2504, 2442, 45, 6231, 470, 327, 212, 2521, 2498, 2505, 308} \[ \int x^3 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\frac {1}{4} x^4 \left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )-\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{4 c^2}-\frac {e \log \left (1-c^2 x^2\right ) \left (a+b \coth ^{-1}(c x)\right )}{4 c^4}-\frac {1}{8} e x^4 \left (a+b \coth ^{-1}(c x)\right )-\frac {b (2 d-3 e) \text {arctanh}(c x)}{8 c^4}+\frac {2 b e \text {arctanh}(c x)}{3 c^4}+\frac {b x (2 d-3 e)}{8 c^3}-\frac {2 b e x}{3 c^3}+\frac {b e x^3 \log \left (1-c^2 x^2\right )}{12 c}+\frac {b e x \log \left (1-c^2 x^2\right )}{4 c^3}+\frac {b x^3 (2 d-e)}{24 c}-\frac {b e x^3}{18 c} \]

[In]

Int[x^3*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]),x]

[Out]

(b*(2*d - 3*e)*x)/(8*c^3) - (2*b*e*x)/(3*c^3) + (b*(2*d - e)*x^3)/(24*c) - (b*e*x^3)/(18*c) - (e*x^2*(a + b*Ar
cCoth[c*x]))/(4*c^2) - (e*x^4*(a + b*ArcCoth[c*x]))/8 - (b*(2*d - 3*e)*ArcTanh[c*x])/(8*c^4) + (2*b*e*ArcTanh[
c*x])/(3*c^4) + (b*e*x*Log[1 - c^2*x^2])/(4*c^3) + (b*e*x^3*Log[1 - c^2*x^2])/(12*c) - (e*(a + b*ArcCoth[c*x])
*Log[1 - c^2*x^2])/(4*c^4) + (x^4*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]))/4

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 212

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

Rule 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 327

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 470

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

Rule 2442

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f + g*
x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/(g*(q + 1))), x] - Dist[b*e*(n/(g*(q + 1))), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 2498

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 2504

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 2505

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

Rule 2521

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol]
:> With[{t = ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, (f + g*x^s)^r, x]}, Int[t, x] /; SumQ[t]] /; Free
Q[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && IntegerQ[n] && IGtQ[q, 0] && IntegerQ[r] && IntegerQ[s] && (EqQ[
q, 1] || (GtQ[r, 0] && GtQ[s, 1]) || (LtQ[s, 0] && LtQ[r, 0]))

Rule 6231

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \coth ^{-1}(c x)\right )-\frac {e \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )-(b c) \int \left (\frac {x^2 \left (-2 e+c^2 (2 d-e) x^2\right )}{8 c^2 \left (1-c^2 x^2\right )}-\frac {e \left (1+c^2 x^2\right ) \log \left (1-c^2 x^2\right )}{4 c^4}\right ) \, dx \\ & = -\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \coth ^{-1}(c x)\right )-\frac {e \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )-\frac {b \int \frac {x^2 \left (-2 e+c^2 (2 d-e) x^2\right )}{1-c^2 x^2} \, dx}{8 c}+\frac {(b e) \int \left (1+c^2 x^2\right ) \log \left (1-c^2 x^2\right ) \, dx}{4 c^3} \\ & = \frac {b (2 d-e) x^3}{24 c}-\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \coth ^{-1}(c x)\right )-\frac {e \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )-\frac {(b (2 d-3 e)) \int \frac {x^2}{1-c^2 x^2} \, dx}{8 c}+\frac {(b e) \int \left (\log \left (1-c^2 x^2\right )+c^2 x^2 \log \left (1-c^2 x^2\right )\right ) \, dx}{4 c^3} \\ & = \frac {b (2 d-3 e) x}{8 c^3}+\frac {b (2 d-e) x^3}{24 c}-\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \coth ^{-1}(c x)\right )-\frac {e \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )-\frac {(b (2 d-3 e)) \int \frac {1}{1-c^2 x^2} \, dx}{8 c^3}+\frac {(b e) \int \log \left (1-c^2 x^2\right ) \, dx}{4 c^3}+\frac {(b e) \int x^2 \log \left (1-c^2 x^2\right ) \, dx}{4 c} \\ & = \frac {b (2 d-3 e) x}{8 c^3}+\frac {b (2 d-e) x^3}{24 c}-\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \coth ^{-1}(c x)\right )-\frac {b (2 d-3 e) \text {arctanh}(c x)}{8 c^4}+\frac {b e x \log \left (1-c^2 x^2\right )}{4 c^3}+\frac {b e x^3 \log \left (1-c^2 x^2\right )}{12 c}-\frac {e \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac {(b e) \int \frac {x^2}{1-c^2 x^2} \, dx}{2 c}+\frac {1}{6} (b c e) \int \frac {x^4}{1-c^2 x^2} \, dx \\ & = \frac {b (2 d-3 e) x}{8 c^3}-\frac {b e x}{2 c^3}+\frac {b (2 d-e) x^3}{24 c}-\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \coth ^{-1}(c x)\right )-\frac {b (2 d-3 e) \text {arctanh}(c x)}{8 c^4}+\frac {b e x \log \left (1-c^2 x^2\right )}{4 c^3}+\frac {b e x^3 \log \left (1-c^2 x^2\right )}{12 c}-\frac {e \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac {(b e) \int \frac {1}{1-c^2 x^2} \, dx}{2 c^3}+\frac {1}{6} (b c e) \int \left (-\frac {1}{c^4}-\frac {x^2}{c^2}+\frac {1}{c^4 \left (1-c^2 x^2\right )}\right ) \, dx \\ & = \frac {b (2 d-3 e) x}{8 c^3}-\frac {2 b e x}{3 c^3}+\frac {b (2 d-e) x^3}{24 c}-\frac {b e x^3}{18 c}-\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \coth ^{-1}(c x)\right )-\frac {b (2 d-3 e) \text {arctanh}(c x)}{8 c^4}+\frac {b e \text {arctanh}(c x)}{2 c^4}+\frac {b e x \log \left (1-c^2 x^2\right )}{4 c^3}+\frac {b e x^3 \log \left (1-c^2 x^2\right )}{12 c}-\frac {e \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac {(b e) \int \frac {1}{1-c^2 x^2} \, dx}{6 c^3} \\ & = \frac {b (2 d-3 e) x}{8 c^3}-\frac {2 b e x}{3 c^3}+\frac {b (2 d-e) x^3}{24 c}-\frac {b e x^3}{18 c}-\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{4 c^2}-\frac {1}{8} e x^4 \left (a+b \coth ^{-1}(c x)\right )-\frac {b (2 d-3 e) \text {arctanh}(c x)}{8 c^4}+\frac {2 b e \text {arctanh}(c x)}{3 c^4}+\frac {b e x \log \left (1-c^2 x^2\right )}{4 c^3}+\frac {b e x^3 \log \left (1-c^2 x^2\right )}{12 c}-\frac {e \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{4 c^4}+\frac {1}{4} x^4 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.85 \[ \int x^3 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\frac {6 b c (6 d-25 e) x-36 a c^2 e x^2+2 b c^3 (6 d-7 e) x^3+18 a c^4 (2 d-e) x^4-18 b c^2 x^2 \left (-2 c^2 d x^2+e \left (2+c^2 x^2\right )\right ) \coth ^{-1}(c x)+3 (6 b d-12 a e-25 b e) \log (1-c x)-3 (6 b d+12 a e-25 b e) \log (1+c x)+12 e \left (3 a c^4 x^4+b c x \left (3+c^2 x^2\right )+3 b \left (-1+c^4 x^4\right ) \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{144 c^4} \]

[In]

Integrate[x^3*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]),x]

[Out]

(6*b*c*(6*d - 25*e)*x - 36*a*c^2*e*x^2 + 2*b*c^3*(6*d - 7*e)*x^3 + 18*a*c^4*(2*d - e)*x^4 - 18*b*c^2*x^2*(-2*c
^2*d*x^2 + e*(2 + c^2*x^2))*ArcCoth[c*x] + 3*(6*b*d - 12*a*e - 25*b*e)*Log[1 - c*x] - 3*(6*b*d + 12*a*e - 25*b
*e)*Log[1 + c*x] + 12*e*(3*a*c^4*x^4 + b*c*x*(3 + c^2*x^2) + 3*b*(-1 + c^4*x^4)*ArcCoth[c*x])*Log[1 - c^2*x^2]
)/(144*c^4)

Maple [A] (verified)

Time = 2.51 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.11

method	result	size
parallelrisch	\(\frac {18 b e \ln \left (-c^{2} x^{2}+1\right ) \operatorname {arccoth}\left (c x \right ) x^{4} c^{4}-18 \,\operatorname {arccoth}\left (c x \right ) b d -18 \ln \left (-c^{2} x^{2}+1\right ) a e -18 \,\operatorname {arccoth}\left (c x \right ) \ln \left (-c^{2} x^{2}+1\right ) b e -18 a \,c^{2} e \,x^{2}+18 b c d x -18 \,\operatorname {arccoth}\left (c x \right ) b \,c^{2} e \,x^{2}+18 \ln \left (-c^{2} x^{2}+1\right ) b c e x -75 b x e c +6 b \,c^{3} d \,x^{3}-7 b e \,x^{3} c^{3}-18 a e -9 e b \,\operatorname {arccoth}\left (c x \right ) x^{4} c^{4}+6 e b \,x^{3} \ln \left (-c^{2} x^{2}+1\right ) c^{3}+18 a e \ln \left (-c^{2} x^{2}+1\right ) x^{4} c^{4}+18 b \,\operatorname {arccoth}\left (c x \right ) x^{4} c^{4} d +75 \,\operatorname {arccoth}\left (c x \right ) b e +18 a \,x^{4} d \,c^{4}-9 a e \,x^{4} c^{4}}{72 c^{4}}\)	\(249\)
default	\(\text {Expression too large to display}\)	\(3170\)
parts	\(\text {Expression too large to display}\)	\(3170\)
risch	\(\text {Expression too large to display}\)	\(7050\)

[In]

int(x^3*(a+b*arccoth(c*x))*(d+e*ln(-c^2*x^2+1)),x,method=_RETURNVERBOSE)

[Out]

1/72*(18*b*e*ln(-c^2*x^2+1)*arccoth(c*x)*x^4*c^4-18*arccoth(c*x)*b*d-18*ln(-c^2*x^2+1)*a*e-18*arccoth(c*x)*ln(
-c^2*x^2+1)*b*e-18*a*c^2*e*x^2+18*b*c*d*x-18*arccoth(c*x)*b*c^2*e*x^2+18*ln(-c^2*x^2+1)*b*c*e*x-75*b*x*e*c+6*b
*c^3*d*x^3-7*b*e*x^3*c^3-18*a*e-9*e*b*arccoth(c*x)*x^4*c^4+6*e*b*x^3*ln(-c^2*x^2+1)*c^3+18*a*e*ln(-c^2*x^2+1)*
x^4*c^4+18*b*arccoth(c*x)*x^4*c^4*d+75*arccoth(c*x)*b*e+18*a*x^4*d*c^4-9*a*e*x^4*c^4)/c^4

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.87 \[ \int x^3 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=-\frac {36 \, a c^{2} e x^{2} - 18 \, {\left (2 \, a c^{4} d - a c^{4} e\right )} x^{4} - 2 \, {\left (6 \, b c^{3} d - 7 \, b c^{3} e\right )} x^{3} - 6 \, {\left (6 \, b c d - 25 \, b c e\right )} x - 12 \, {\left (3 \, a c^{4} e x^{4} + b c^{3} e x^{3} + 3 \, b c e x - 3 \, a e\right )} \log \left (-c^{2} x^{2} + 1\right ) + 3 \, {\left (6 \, b c^{2} e x^{2} - 3 \, {\left (2 \, b c^{4} d - b c^{4} e\right )} x^{4} + 6 \, b d - 25 \, b e - 6 \, {\left (b c^{4} e x^{4} - b e\right )} \log \left (-c^{2} x^{2} + 1\right )\right )} \log \left (\frac {c x + 1}{c x - 1}\right )}{144 \, c^{4}} \]

[In]

integrate(x^3*(a+b*arccoth(c*x))*(d+e*log(-c^2*x^2+1)),x, algorithm="fricas")

[Out]

-1/144*(36*a*c^2*e*x^2 - 18*(2*a*c^4*d - a*c^4*e)*x^4 - 2*(6*b*c^3*d - 7*b*c^3*e)*x^3 - 6*(6*b*c*d - 25*b*c*e)
*x - 12*(3*a*c^4*e*x^4 + b*c^3*e*x^3 + 3*b*c*e*x - 3*a*e)*log(-c^2*x^2 + 1) + 3*(6*b*c^2*e*x^2 - 3*(2*b*c^4*d
- b*c^4*e)*x^4 + 6*b*d - 25*b*e - 6*(b*c^4*e*x^4 - b*e)*log(-c^2*x^2 + 1))*log((c*x + 1)/(c*x - 1)))/c^4

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.12 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.27 \[ \int x^3 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\begin {cases} \frac {a d x^{4}}{4} + \frac {a e x^{4} \log {\left (- c^{2} x^{2} + 1 \right )}}{4} - \frac {a e x^{4}}{8} - \frac {a e x^{2}}{4 c^{2}} - \frac {a e \log {\left (- c^{2} x^{2} + 1 \right )}}{4 c^{4}} + \frac {b d x^{4} \operatorname {acoth}{\left (c x \right )}}{4} + \frac {b e x^{4} \log {\left (- c^{2} x^{2} + 1 \right )} \operatorname {acoth}{\left (c x \right )}}{4} - \frac {b e x^{4} \operatorname {acoth}{\left (c x \right )}}{8} + \frac {b d x^{3}}{12 c} + \frac {b e x^{3} \log {\left (- c^{2} x^{2} + 1 \right )}}{12 c} - \frac {7 b e x^{3}}{72 c} - \frac {b e x^{2} \operatorname {acoth}{\left (c x \right )}}{4 c^{2}} + \frac {b d x}{4 c^{3}} + \frac {b e x \log {\left (- c^{2} x^{2} + 1 \right )}}{4 c^{3}} - \frac {25 b e x}{24 c^{3}} - \frac {b d \operatorname {acoth}{\left (c x \right )}}{4 c^{4}} - \frac {b e \log {\left (- c^{2} x^{2} + 1 \right )} \operatorname {acoth}{\left (c x \right )}}{4 c^{4}} + \frac {25 b e \operatorname {acoth}{\left (c x \right )}}{24 c^{4}} & \text {for}\: c \neq 0 \\\frac {d x^{4} \left (a + \frac {i \pi b}{2}\right )}{4} & \text {otherwise} \end {cases} \]

[In]

integrate(x**3*(a+b*acoth(c*x))*(d+e*ln(-c**2*x**2+1)),x)

[Out]

Piecewise((a*d*x**4/4 + a*e*x**4*log(-c**2*x**2 + 1)/4 - a*e*x**4/8 - a*e*x**2/(4*c**2) - a*e*log(-c**2*x**2 +
1)/(4*c**4) + b*d*x**4*acoth(c*x)/4 + b*e*x**4*log(-c**2*x**2 + 1)*acoth(c*x)/4 - b*e*x**4*acoth(c*x)/8 + b*d
*x**3/(12*c) + b*e*x**3*log(-c**2*x**2 + 1)/(12*c) - 7*b*e*x**3/(72*c) - b*e*x**2*acoth(c*x)/(4*c**2) + b*d*x/
(4*c**3) + b*e*x*log(-c**2*x**2 + 1)/(4*c**3) - 25*b*e*x/(24*c**3) - b*d*acoth(c*x)/(4*c**4) - b*e*log(-c**2*x
**2 + 1)*acoth(c*x)/(4*c**4) + 25*b*e*acoth(c*x)/(24*c**4), Ne(c, 0)), (d*x**4*(a + I*pi*b/2)/4, True))

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.22 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.20 \[ \int x^3 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\frac {1}{4} \, a d x^{4} + \frac {1}{8} \, {\left (2 \, x^{4} \log \left (-c^{2} x^{2} + 1\right ) - c^{2} {\left (\frac {c^{2} x^{4} + 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )}\right )} b e \operatorname {arcoth}\left (c x\right ) + \frac {1}{24} \, {\left (6 \, x^{4} \operatorname {arcoth}\left (c x\right ) + c {\left (\frac {2 \, {\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4}} - \frac {3 \, \log \left (c x + 1\right )}{c^{5}} + \frac {3 \, \log \left (c x - 1\right )}{c^{5}}\right )}\right )} b d + \frac {1}{8} \, {\left (2 \, x^{4} \log \left (-c^{2} x^{2} + 1\right ) - c^{2} {\left (\frac {c^{2} x^{4} + 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )}\right )} a e - \frac {{\left (2 \, {\left (-6 i \, \pi c^{3} + 7 \, c^{3}\right )} x^{3} + 6 \, {\left (-6 i \, \pi c + 25 \, c\right )} x + 3 \, {\left (6 i \, \pi - 4 \, c^{3} x^{3} - 12 \, c x - 25\right )} \log \left (c x + 1\right ) + 3 \, {\left (-6 i \, \pi - 4 \, c^{3} x^{3} - 12 \, c x + 25\right )} \log \left (c x - 1\right )\right )} b e}{144 \, c^{4}} \]

[In]

integrate(x^3*(a+b*arccoth(c*x))*(d+e*log(-c^2*x^2+1)),x, algorithm="maxima")

[Out]

1/4*a*d*x^4 + 1/8*(2*x^4*log(-c^2*x^2 + 1) - c^2*((c^2*x^4 + 2*x^2)/c^4 + 2*log(c^2*x^2 - 1)/c^6))*b*e*arccoth
(c*x) + 1/24*(6*x^4*arccoth(c*x) + c*(2*(c^2*x^3 + 3*x)/c^4 - 3*log(c*x + 1)/c^5 + 3*log(c*x - 1)/c^5))*b*d +
1/8*(2*x^4*log(-c^2*x^2 + 1) - c^2*((c^2*x^4 + 2*x^2)/c^4 + 2*log(c^2*x^2 - 1)/c^6))*a*e - 1/144*(2*(-6*I*pi*c
^3 + 7*c^3)*x^3 + 6*(-6*I*pi*c + 25*c)*x + 3*(6*I*pi - 4*c^3*x^3 - 12*c*x - 25)*log(c*x + 1) + 3*(-6*I*pi - 4*
c^3*x^3 - 12*c*x + 25)*log(c*x - 1))*b*e/c^4

Giac [F(-2)]

Exception generated. \[ \int x^3 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^3*(a+b*arccoth(c*x))*(d+e*log(-c^2*x^2+1)),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [B] (verification not implemented)

Time = 5.42 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.84 \[ \int x^3 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\ln \left (1-\frac {1}{c\,x}\right )\,\left (\frac {\frac {b\,e\,x^5}{8}-\frac {b\,e\,x^3}{4\,c^2}+\frac {b\,c^2\,e\,x^7}{8}}{2\,\left (c\,x^2+x\right )\,\left (c\,x-1\right )}+\frac {\frac {b\,d\,x^5}{4}-\frac {b\,c^2\,d\,x^7}{4}}{2\,\left (c\,x^2+x\right )\,\left (c\,x-1\right )}+\frac {\ln \left (1-c^2\,x^2\right )\,\left (\frac {b\,e\,x^5}{4}-\frac {b\,c^2\,e\,x^7}{4}\right )}{2\,\left (c\,x^2+x\right )\,\left (c\,x-1\right )}-\frac {b\,e\,\ln \left (1-c^2\,x^2\right )\,\left (x-c^2\,x^3\right )}{8\,c^4\,\left (c\,x^2+x\right )\,\left (c\,x-1\right )}\right )+x\,\left (\frac {b\,\left (6\,d-7\,e\right )}{24\,c^3}-\frac {3\,b\,e}{4\,c^3}\right )+\ln \left (1-c^2\,x^2\right )\,\left (\frac {a\,e\,x^4}{4}+\frac {b\,e\,x}{4\,c^3}+\frac {b\,e\,x^3}{12\,c}\right )-\ln \left (\frac {1}{c\,x}+1\right )\,\left (\ln \left (1-c^2\,x^2\right )\,\left (\frac {b\,e}{8\,c^4}-\frac {b\,e\,x^4}{8}\right )-\frac {b\,d\,x^4}{8}+\frac {b\,e\,x^4}{16}+\frac {b\,e\,x^2}{8\,c^2}\right )+x^2\,\left (\frac {a\,\left (2\,d-e\right )}{4\,c^2}-\frac {a\,d}{2\,c^2}\right )+\frac {a\,x^4\,\left (2\,d-e\right )}{8}-\frac {\ln \left (c\,x-1\right )\,\left (12\,a\,e-6\,b\,d+25\,b\,e\right )}{48\,c^4}-\frac {\ln \left (c\,x+1\right )\,\left (12\,a\,e+6\,b\,d-25\,b\,e\right )}{48\,c^4}+\frac {b\,x^3\,\left (6\,d-7\,e\right )}{72\,c} \]

[In]

int(x^3*(a + b*acoth(c*x))*(d + e*log(1 - c^2*x^2)),x)

[Out]

log(1 - 1/(c*x))*(((b*e*x^5)/8 - (b*e*x^3)/(4*c^2) + (b*c^2*e*x^7)/8)/(2*(x + c*x^2)*(c*x - 1)) + ((b*d*x^5)/4
- (b*c^2*d*x^7)/4)/(2*(x + c*x^2)*(c*x - 1)) + (log(1 - c^2*x^2)*((b*e*x^5)/4 - (b*c^2*e*x^7)/4))/(2*(x + c*x
^2)*(c*x - 1)) - (b*e*log(1 - c^2*x^2)*(x - c^2*x^3))/(8*c^4*(x + c*x^2)*(c*x - 1))) + x*((b*(6*d - 7*e))/(24*
c^3) - (3*b*e)/(4*c^3)) + log(1 - c^2*x^2)*((a*e*x^4)/4 + (b*e*x)/(4*c^3) + (b*e*x^3)/(12*c)) - log(1/(c*x) +
1)*(log(1 - c^2*x^2)*((b*e)/(8*c^4) - (b*e*x^4)/8) - (b*d*x^4)/8 + (b*e*x^4)/16 + (b*e*x^2)/(8*c^2)) + x^2*((a
*(2*d - e))/(4*c^2) - (a*d)/(2*c^2)) + (a*x^4*(2*d - e))/8 - (log(c*x - 1)*(12*a*e - 6*b*d + 25*b*e))/(48*c^4)
- (log(c*x + 1)*(12*a*e + 6*b*d - 25*b*e))/(48*c^4) + (b*x^3*(6*d - 7*e))/(72*c)