\(\int x^5 (a+b \coth ^{-1}(c x)) (d+e \log (1-c^2 x^2)) \, dx\) [266]

Optimal result

Integrand size = 27, antiderivative size = 297 \[ \int x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\frac {b (3 d-e) x}{18 c^5}-\frac {137 b e x}{180 c^5}+\frac {b (3 d-e) x^3}{54 c^3}-\frac {47 b e x^3}{540 c^3}+\frac {b (3 d-e) x^5}{90 c}-\frac {b e x^5}{75 c}-\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{6 c^4}-\frac {e x^4 \left (a+b \coth ^{-1}(c x)\right )}{12 c^2}-\frac {1}{18} e x^6 \left (a+b \coth ^{-1}(c x)\right )-\frac {b (3 d-e) \text {arctanh}(c x)}{18 c^6}+\frac {137 b e \text {arctanh}(c x)}{180 c^6}+\frac {b e x \log \left (1-c^2 x^2\right )}{6 c^5}+\frac {b e x^3 \log \left (1-c^2 x^2\right )}{18 c^3}+\frac {b e x^5 \log \left (1-c^2 x^2\right )}{30 c}-\frac {e \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{6 c^6}+\frac {1}{6} x^6 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \]

[Out]

1/36*b*(6*d-11*e)*x/c^5-23/45*b*e*x/c^5+1/108*b*(6*d-5*e)*x^3/c^3-8/135*b*e*x^3/c^3+1/90*b*(3*d-e)*x^5/c-1/75*
b*e*x^5/c-1/6*e*x^2*(a+b*arccoth(c*x))/c^4-1/12*e*x^4*(a+b*arccoth(c*x))/c^2-1/18*e*x^6*(a+b*arccoth(c*x))-1/3
6*b*(6*d-11*e)*arctanh(c*x)/c^6+23/45*b*e*arctanh(c*x)/c^6+1/6*b*e*x*ln(-c^2*x^2+1)/c^5+1/18*b*e*x^3*ln(-c^2*x
^2+1)/c^3+1/30*b*e*x^5*ln(-c^2*x^2+1)/c-1/6*e*(a+b*arccoth(c*x))*ln(-c^2*x^2+1)/c^6+1/6*x^6*(a+b*arccoth(c*x))
*(d+e*ln(-c^2*x^2+1))

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {2504, 2442, 45, 6231, 327, 213, 308, 2608, 2498, 212, 2505} \[ \int x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=-\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{6 c^4}+\frac {1}{6} x^6 \left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )-\frac {e x^4 \left (a+b \coth ^{-1}(c x)\right )}{12 c^2}-\frac {e \log \left (1-c^2 x^2\right ) \left (a+b \coth ^{-1}(c x)\right )}{6 c^6}-\frac {1}{18} e x^6 \left (a+b \coth ^{-1}(c x)\right )-\frac {b (3 d-e) \text {arctanh}(c x)}{18 c^6}+\frac {137 b e \text {arctanh}(c x)}{180 c^6}+\frac {b x (3 d-e)}{18 c^5}-\frac {137 b e x}{180 c^5}+\frac {b x^3 (3 d-e)}{54 c^3}-\frac {47 b e x^3}{540 c^3}+\frac {b e x^5 \log \left (1-c^2 x^2\right )}{30 c}+\frac {b e x \log \left (1-c^2 x^2\right )}{6 c^5}+\frac {b e x^3 \log \left (1-c^2 x^2\right )}{18 c^3}+\frac {b x^5 (3 d-e)}{90 c}-\frac {b e x^5}{75 c} \]

[In]

Int[x^5*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]),x]

[Out]

(b*(3*d - e)*x)/(18*c^5) - (137*b*e*x)/(180*c^5) + (b*(3*d - e)*x^3)/(54*c^3) - (47*b*e*x^3)/(540*c^3) + (b*(3
*d - e)*x^5)/(90*c) - (b*e*x^5)/(75*c) - (e*x^2*(a + b*ArcCoth[c*x]))/(6*c^4) - (e*x^4*(a + b*ArcCoth[c*x]))/(
12*c^2) - (e*x^6*(a + b*ArcCoth[c*x]))/18 - (b*(3*d - e)*ArcTanh[c*x])/(18*c^6) + (137*b*e*ArcTanh[c*x])/(180*
c^6) + (b*e*x*Log[1 - c^2*x^2])/(6*c^5) + (b*e*x^3*Log[1 - c^2*x^2])/(18*c^3) + (b*e*x^5*Log[1 - c^2*x^2])/(30
*c) - (e*(a + b*ArcCoth[c*x])*Log[1 - c^2*x^2])/(6*c^6) + (x^6*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]))/
6

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 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[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 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 2608

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 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 )}{6 c^4}-\frac {e x^4 \left (a+b \coth ^{-1}(c x)\right )}{12 c^2}-\frac {1}{18} e x^6 \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 )}{6 c^6}+\frac {1}{6} x^6 \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 {e x^2}{6 c^4 \left (-1+c^2 x^2\right )}+\frac {e x^4}{12 c^2 \left (-1+c^2 x^2\right )}-\frac {d \left (1-\frac {e}{3 d}\right ) x^6}{6 \left (-1+c^2 x^2\right )}-\frac {e \left (1+c^2 x^2+c^4 x^4\right ) \log \left (1-c^2 x^2\right )}{6 c^6}\right ) \, dx \\ & = -\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{6 c^4}-\frac {e x^4 \left (a+b \coth ^{-1}(c x)\right )}{12 c^2}-\frac {1}{18} e x^6 \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 )}{6 c^6}+\frac {1}{6} x^6 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac {1}{18} (b c (3 d-e)) \int \frac {x^6}{-1+c^2 x^2} \, dx+\frac {(b e) \int \left (1+c^2 x^2+c^4 x^4\right ) \log \left (1-c^2 x^2\right ) \, dx}{6 c^5}-\frac {(b e) \int \frac {x^2}{-1+c^2 x^2} \, dx}{6 c^3}-\frac {(b e) \int \frac {x^4}{-1+c^2 x^2} \, dx}{12 c} \\ & = -\frac {b e x}{6 c^5}-\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{6 c^4}-\frac {e x^4 \left (a+b \coth ^{-1}(c x)\right )}{12 c^2}-\frac {1}{18} e x^6 \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 )}{6 c^6}+\frac {1}{6} x^6 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac {1}{18} (b c (3 d-e)) \int \left (\frac {1}{c^6}+\frac {x^2}{c^4}+\frac {x^4}{c^2}+\frac {1}{c^6 \left (-1+c^2 x^2\right )}\right ) \, dx-\frac {(b e) \int \frac {1}{-1+c^2 x^2} \, dx}{6 c^5}+\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 )+c^4 x^4 \log \left (1-c^2 x^2\right )\right ) \, dx}{6 c^5}-\frac {(b 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}{12 c} \\ & = \frac {b (3 d-e) x}{18 c^5}-\frac {b e x}{4 c^5}+\frac {b (3 d-e) x^3}{54 c^3}-\frac {b e x^3}{36 c^3}+\frac {b (3 d-e) x^5}{90 c}-\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{6 c^4}-\frac {e x^4 \left (a+b \coth ^{-1}(c x)\right )}{12 c^2}-\frac {1}{18} e x^6 \left (a+b \coth ^{-1}(c x)\right )+\frac {b e \text {arctanh}(c x)}{6 c^6}-\frac {e \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{6 c^6}+\frac {1}{6} x^6 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac {(b (3 d-e)) \int \frac {1}{-1+c^2 x^2} \, dx}{18 c^5}-\frac {(b e) \int \frac {1}{-1+c^2 x^2} \, dx}{12 c^5}+\frac {(b e) \int \log \left (1-c^2 x^2\right ) \, dx}{6 c^5}+\frac {(b e) \int x^2 \log \left (1-c^2 x^2\right ) \, dx}{6 c^3}+\frac {(b e) \int x^4 \log \left (1-c^2 x^2\right ) \, dx}{6 c} \\ & = \frac {b (3 d-e) x}{18 c^5}-\frac {b e x}{4 c^5}+\frac {b (3 d-e) x^3}{54 c^3}-\frac {b e x^3}{36 c^3}+\frac {b (3 d-e) x^5}{90 c}-\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{6 c^4}-\frac {e x^4 \left (a+b \coth ^{-1}(c x)\right )}{12 c^2}-\frac {1}{18} e x^6 \left (a+b \coth ^{-1}(c x)\right )-\frac {b (3 d-e) \text {arctanh}(c x)}{18 c^6}+\frac {b e \text {arctanh}(c x)}{4 c^6}+\frac {b e x \log \left (1-c^2 x^2\right )}{6 c^5}+\frac {b e x^3 \log \left (1-c^2 x^2\right )}{18 c^3}+\frac {b e x^5 \log \left (1-c^2 x^2\right )}{30 c}-\frac {e \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{6 c^6}+\frac {1}{6} x^6 \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}{3 c^3}+\frac {(b e) \int \frac {x^4}{1-c^2 x^2} \, dx}{9 c}+\frac {1}{15} (b c e) \int \frac {x^6}{1-c^2 x^2} \, dx \\ & = \frac {b (3 d-e) x}{18 c^5}-\frac {7 b e x}{12 c^5}+\frac {b (3 d-e) x^3}{54 c^3}-\frac {b e x^3}{36 c^3}+\frac {b (3 d-e) x^5}{90 c}-\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{6 c^4}-\frac {e x^4 \left (a+b \coth ^{-1}(c x)\right )}{12 c^2}-\frac {1}{18} e x^6 \left (a+b \coth ^{-1}(c x)\right )-\frac {b (3 d-e) \text {arctanh}(c x)}{18 c^6}+\frac {b e \text {arctanh}(c x)}{4 c^6}+\frac {b e x \log \left (1-c^2 x^2\right )}{6 c^5}+\frac {b e x^3 \log \left (1-c^2 x^2\right )}{18 c^3}+\frac {b e x^5 \log \left (1-c^2 x^2\right )}{30 c}-\frac {e \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{6 c^6}+\frac {1}{6} x^6 \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}{3 c^5}+\frac {(b 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}{9 c}+\frac {1}{15} (b c e) \int \left (-\frac {1}{c^6}-\frac {x^2}{c^4}-\frac {x^4}{c^2}+\frac {1}{c^6 \left (1-c^2 x^2\right )}\right ) \, dx \\ & = \frac {b (3 d-e) x}{18 c^5}-\frac {137 b e x}{180 c^5}+\frac {b (3 d-e) x^3}{54 c^3}-\frac {47 b e x^3}{540 c^3}+\frac {b (3 d-e) x^5}{90 c}-\frac {b e x^5}{75 c}-\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{6 c^4}-\frac {e x^4 \left (a+b \coth ^{-1}(c x)\right )}{12 c^2}-\frac {1}{18} e x^6 \left (a+b \coth ^{-1}(c x)\right )-\frac {b (3 d-e) \text {arctanh}(c x)}{18 c^6}+\frac {7 b e \text {arctanh}(c x)}{12 c^6}+\frac {b e x \log \left (1-c^2 x^2\right )}{6 c^5}+\frac {b e x^3 \log \left (1-c^2 x^2\right )}{18 c^3}+\frac {b e x^5 \log \left (1-c^2 x^2\right )}{30 c}-\frac {e \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{6 c^6}+\frac {1}{6} x^6 \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}{15 c^5}+\frac {(b e) \int \frac {1}{1-c^2 x^2} \, dx}{9 c^5} \\ & = \frac {b (3 d-e) x}{18 c^5}-\frac {137 b e x}{180 c^5}+\frac {b (3 d-e) x^3}{54 c^3}-\frac {47 b e x^3}{540 c^3}+\frac {b (3 d-e) x^5}{90 c}-\frac {b e x^5}{75 c}-\frac {e x^2 \left (a+b \coth ^{-1}(c x)\right )}{6 c^4}-\frac {e x^4 \left (a+b \coth ^{-1}(c x)\right )}{12 c^2}-\frac {1}{18} e x^6 \left (a+b \coth ^{-1}(c x)\right )-\frac {b (3 d-e) \text {arctanh}(c x)}{18 c^6}+\frac {137 b e \text {arctanh}(c x)}{180 c^6}+\frac {b e x \log \left (1-c^2 x^2\right )}{6 c^5}+\frac {b e x^3 \log \left (1-c^2 x^2\right )}{18 c^3}+\frac {b e x^5 \log \left (1-c^2 x^2\right )}{30 c}-\frac {e \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{6 c^6}+\frac {1}{6} x^6 \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 = 0.09 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.79 \[ \int x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\frac {30 b c (10 d-49 e) x-300 a c^2 e x^2+10 b c^3 (10 d-19 e) x^3-150 a c^4 e x^4+4 b c^5 (15 d-11 e) x^5+100 a c^6 (3 d-e) x^6-50 b c^2 x^2 \left (-6 c^4 d x^4+e \left (6+3 c^2 x^2+2 c^4 x^4\right )\right ) \coth ^{-1}(c x)+15 (10 b d-20 a e-49 b e) \log (1-c x)-15 (10 b d+20 a e-49 b e) \log (1+c x)+20 e \left (15 a c^6 x^6+b c x \left (15+5 c^2 x^2+3 c^4 x^4\right )+15 b \left (-1+c^6 x^6\right ) \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{1800 c^6} \]

[In]

Integrate[x^5*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]),x]

[Out]

(30*b*c*(10*d - 49*e)*x - 300*a*c^2*e*x^2 + 10*b*c^3*(10*d - 19*e)*x^3 - 150*a*c^4*e*x^4 + 4*b*c^5*(15*d - 11*
e)*x^5 + 100*a*c^6*(3*d - e)*x^6 - 50*b*c^2*x^2*(-6*c^4*d*x^4 + e*(6 + 3*c^2*x^2 + 2*c^4*x^4))*ArcCoth[c*x] +
15*(10*b*d - 20*a*e - 49*b*e)*Log[1 - c*x] - 15*(10*b*d + 20*a*e - 49*b*e)*Log[1 + c*x] + 20*e*(15*a*c^6*x^6 +
b*c*x*(15 + 5*c^2*x^2 + 3*c^4*x^4) + 15*b*(-1 + c^6*x^6)*ArcCoth[c*x])*Log[1 - c^2*x^2])/(1800*c^6)

Maple [A] (verified)

Time = 4.22 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.06

method	result	size
parallelrisch	\(\frac {150 b e \ln \left (-c^{2} x^{2}+1\right ) \operatorname {arccoth}\left (c x \right ) x^{6} c^{6}-150 \,\operatorname {arccoth}\left (c x \right ) b d -150 \ln \left (-c^{2} x^{2}+1\right ) a e -150 \,\operatorname {arccoth}\left (c x \right ) \ln \left (-c^{2} x^{2}+1\right ) b e -150 a \,c^{2} e \,x^{2}+150 b c d x -150 \,\operatorname {arccoth}\left (c x \right ) b \,c^{2} e \,x^{2}+150 \ln \left (-c^{2} x^{2}+1\right ) b c e x -735 b x e c +50 b \,c^{3} d \,x^{3}-95 b e \,x^{3} c^{3}+30 b \,c^{5} d \,x^{5}-22 b \,c^{5} e \,x^{5}-150 a e +150 a \,c^{6} d \,x^{6}-50 a \,c^{6} e \,x^{6}-75 e b \,\operatorname {arccoth}\left (c x \right ) x^{4} c^{4}+50 e b \,x^{3} \ln \left (-c^{2} x^{2}+1\right ) c^{3}+30 b e \ln \left (-c^{2} x^{2}+1\right ) x^{5} c^{5}+150 a e \ln \left (-c^{2} x^{2}+1\right ) x^{6} c^{6}+150 b \,\operatorname {arccoth}\left (c x \right ) x^{6} c^{6} d -50 b \,\operatorname {arccoth}\left (c x \right ) x^{6} c^{6} e +735 \,\operatorname {arccoth}\left (c x \right ) b e -75 a e \,x^{4} c^{4}}{900 c^{6}}\)	\(314\)
default	\(\text {Expression too large to display}\)	\(3871\)
parts	\(\text {Expression too large to display}\)	\(3871\)
risch	\(\text {Expression too large to display}\)	\(7318\)

[In]

int(x^5*(a+b*arccoth(c*x))*(d+e*ln(-c^2*x^2+1)),x,method=_RETURNVERBOSE)

[Out]

1/900*(150*b*e*ln(-c^2*x^2+1)*arccoth(c*x)*x^6*c^6-150*arccoth(c*x)*b*d-150*ln(-c^2*x^2+1)*a*e-150*arccoth(c*x
)*ln(-c^2*x^2+1)*b*e-150*a*c^2*e*x^2+150*b*c*d*x-150*arccoth(c*x)*b*c^2*e*x^2+150*ln(-c^2*x^2+1)*b*c*e*x-735*b
*x*e*c+50*b*c^3*d*x^3-95*b*e*x^3*c^3+30*b*c^5*d*x^5-22*b*c^5*e*x^5-150*a*e+150*a*c^6*d*x^6-50*a*c^6*e*x^6-75*e
*b*arccoth(c*x)*x^4*c^4+50*e*b*x^3*ln(-c^2*x^2+1)*c^3+30*b*e*ln(-c^2*x^2+1)*x^5*c^5+150*a*e*ln(-c^2*x^2+1)*x^6
*c^6+150*b*arccoth(c*x)*x^6*c^6*d-50*b*arccoth(c*x)*x^6*c^6*e+735*arccoth(c*x)*b*e-75*a*e*x^4*c^4)/c^6

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.83 \[ \int x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=-\frac {150 \, a c^{4} e x^{4} - 100 \, {\left (3 \, a c^{6} d - a c^{6} e\right )} x^{6} + 300 \, a c^{2} e x^{2} - 4 \, {\left (15 \, b c^{5} d - 11 \, b c^{5} e\right )} x^{5} - 10 \, {\left (10 \, b c^{3} d - 19 \, b c^{3} e\right )} x^{3} - 30 \, {\left (10 \, b c d - 49 \, b c e\right )} x - 20 \, {\left (15 \, a c^{6} e x^{6} + 3 \, b c^{5} e x^{5} + 5 \, b c^{3} e x^{3} + 15 \, b c e x - 15 \, a e\right )} \log \left (-c^{2} x^{2} + 1\right ) + 5 \, {\left (15 \, b c^{4} e x^{4} - 10 \, {\left (3 \, b c^{6} d - b c^{6} e\right )} x^{6} + 30 \, b c^{2} e x^{2} + 30 \, b d - 147 \, b e - 30 \, {\left (b c^{6} e x^{6} - b e\right )} \log \left (-c^{2} x^{2} + 1\right )\right )} \log \left (\frac {c x + 1}{c x - 1}\right )}{1800 \, c^{6}} \]

[In]

integrate(x^5*(a+b*arccoth(c*x))*(d+e*log(-c^2*x^2+1)),x, algorithm="fricas")

[Out]

-1/1800*(150*a*c^4*e*x^4 - 100*(3*a*c^6*d - a*c^6*e)*x^6 + 300*a*c^2*e*x^2 - 4*(15*b*c^5*d - 11*b*c^5*e)*x^5 -
10*(10*b*c^3*d - 19*b*c^3*e)*x^3 - 30*(10*b*c*d - 49*b*c*e)*x - 20*(15*a*c^6*e*x^6 + 3*b*c^5*e*x^5 + 5*b*c^3*
e*x^3 + 15*b*c*e*x - 15*a*e)*log(-c^2*x^2 + 1) + 5*(15*b*c^4*e*x^4 - 10*(3*b*c^6*d - b*c^6*e)*x^6 + 30*b*c^2*e
*x^2 + 30*b*d - 147*b*e - 30*(b*c^6*e*x^6 - b*e)*log(-c^2*x^2 + 1))*log((c*x + 1)/(c*x - 1)))/c^6

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.45 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.22 \[ \int x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\begin {cases} \frac {a d x^{6}}{6} + \frac {a e x^{6} \log {\left (- c^{2} x^{2} + 1 \right )}}{6} - \frac {a e x^{6}}{18} - \frac {a e x^{4}}{12 c^{2}} - \frac {a e x^{2}}{6 c^{4}} - \frac {a e \log {\left (- c^{2} x^{2} + 1 \right )}}{6 c^{6}} + \frac {b d x^{6} \operatorname {acoth}{\left (c x \right )}}{6} + \frac {b e x^{6} \log {\left (- c^{2} x^{2} + 1 \right )} \operatorname {acoth}{\left (c x \right )}}{6} - \frac {b e x^{6} \operatorname {acoth}{\left (c x \right )}}{18} + \frac {b d x^{5}}{30 c} + \frac {b e x^{5} \log {\left (- c^{2} x^{2} + 1 \right )}}{30 c} - \frac {11 b e x^{5}}{450 c} - \frac {b e x^{4} \operatorname {acoth}{\left (c x \right )}}{12 c^{2}} + \frac {b d x^{3}}{18 c^{3}} + \frac {b e x^{3} \log {\left (- c^{2} x^{2} + 1 \right )}}{18 c^{3}} - \frac {19 b e x^{3}}{180 c^{3}} - \frac {b e x^{2} \operatorname {acoth}{\left (c x \right )}}{6 c^{4}} + \frac {b d x}{6 c^{5}} + \frac {b e x \log {\left (- c^{2} x^{2} + 1 \right )}}{6 c^{5}} - \frac {49 b e x}{60 c^{5}} - \frac {b d \operatorname {acoth}{\left (c x \right )}}{6 c^{6}} - \frac {b e \log {\left (- c^{2} x^{2} + 1 \right )} \operatorname {acoth}{\left (c x \right )}}{6 c^{6}} + \frac {49 b e \operatorname {acoth}{\left (c x \right )}}{60 c^{6}} & \text {for}\: c \neq 0 \\\frac {d x^{6} \left (a + \frac {i \pi b}{2}\right )}{6} & \text {otherwise} \end {cases} \]

[In]

integrate(x**5*(a+b*acoth(c*x))*(d+e*ln(-c**2*x**2+1)),x)

[Out]

Piecewise((a*d*x**6/6 + a*e*x**6*log(-c**2*x**2 + 1)/6 - a*e*x**6/18 - a*e*x**4/(12*c**2) - a*e*x**2/(6*c**4)
- a*e*log(-c**2*x**2 + 1)/(6*c**6) + b*d*x**6*acoth(c*x)/6 + b*e*x**6*log(-c**2*x**2 + 1)*acoth(c*x)/6 - b*e*x
**6*acoth(c*x)/18 + b*d*x**5/(30*c) + b*e*x**5*log(-c**2*x**2 + 1)/(30*c) - 11*b*e*x**5/(450*c) - b*e*x**4*aco
th(c*x)/(12*c**2) + b*d*x**3/(18*c**3) + b*e*x**3*log(-c**2*x**2 + 1)/(18*c**3) - 19*b*e*x**3/(180*c**3) - b*e
*x**2*acoth(c*x)/(6*c**4) + b*d*x/(6*c**5) + b*e*x*log(-c**2*x**2 + 1)/(6*c**5) - 49*b*e*x/(60*c**5) - b*d*aco
th(c*x)/(6*c**6) - b*e*log(-c**2*x**2 + 1)*acoth(c*x)/(6*c**6) + 49*b*e*acoth(c*x)/(60*c**6), Ne(c, 0)), (d*x*
*6*(a + I*pi*b/2)/6, True))

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.21 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.11 \[ \int x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\frac {1}{6} \, a d x^{6} + \frac {1}{36} \, {\left (6 \, x^{6} \log \left (-c^{2} x^{2} + 1\right ) - c^{2} {\left (\frac {2 \, c^{4} x^{6} + 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} + \frac {6 \, \log \left (c^{2} x^{2} - 1\right )}{c^{8}}\right )}\right )} b e \operatorname {arcoth}\left (c x\right ) + \frac {1}{180} \, {\left (30 \, x^{6} \operatorname {arcoth}\left (c x\right ) + c {\left (\frac {2 \, {\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac {15 \, \log \left (c x + 1\right )}{c^{7}} + \frac {15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} b d + \frac {1}{36} \, {\left (6 \, x^{6} \log \left (-c^{2} x^{2} + 1\right ) - c^{2} {\left (\frac {2 \, c^{4} x^{6} + 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} + \frac {6 \, \log \left (c^{2} x^{2} - 1\right )}{c^{8}}\right )}\right )} a e - \frac {{\left (4 \, {\left (-15 i \, \pi c^{5} + 11 \, c^{5}\right )} x^{5} + 10 \, {\left (-10 i \, \pi c^{3} + 19 \, c^{3}\right )} x^{3} + 30 \, {\left (-10 i \, \pi c + 49 \, c\right )} x + 5 \, {\left (30 i \, \pi - 12 \, c^{5} x^{5} - 20 \, c^{3} x^{3} - 60 \, c x - 147\right )} \log \left (c x + 1\right ) + 5 \, {\left (-30 i \, \pi - 12 \, c^{5} x^{5} - 20 \, c^{3} x^{3} - 60 \, c x + 147\right )} \log \left (c x - 1\right )\right )} b e}{1800 \, c^{6}} \]

[In]

integrate(x^5*(a+b*arccoth(c*x))*(d+e*log(-c^2*x^2+1)),x, algorithm="maxima")

[Out]

1/6*a*d*x^6 + 1/36*(6*x^6*log(-c^2*x^2 + 1) - c^2*((2*c^4*x^6 + 3*c^2*x^4 + 6*x^2)/c^6 + 6*log(c^2*x^2 - 1)/c^
8))*b*e*arccoth(c*x) + 1/180*(30*x^6*arccoth(c*x) + c*(2*(3*c^4*x^5 + 5*c^2*x^3 + 15*x)/c^6 - 15*log(c*x + 1)/
c^7 + 15*log(c*x - 1)/c^7))*b*d + 1/36*(6*x^6*log(-c^2*x^2 + 1) - c^2*((2*c^4*x^6 + 3*c^2*x^4 + 6*x^2)/c^6 + 6
*log(c^2*x^2 - 1)/c^8))*a*e - 1/1800*(4*(-15*I*pi*c^5 + 11*c^5)*x^5 + 10*(-10*I*pi*c^3 + 19*c^3)*x^3 + 30*(-10
*I*pi*c + 49*c)*x + 5*(30*I*pi - 12*c^5*x^5 - 20*c^3*x^3 - 60*c*x - 147)*log(c*x + 1) + 5*(-30*I*pi - 12*c^5*x
^5 - 20*c^3*x^3 - 60*c*x + 147)*log(c*x - 1))*b*e/c^6

Giac [F(-2)]

Exception generated. \[ \int x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^5*(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 = 5.57 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.72 \[ \int x^5 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\ln \left (1-c^2\,x^2\right )\,\left (\frac {a\,e\,x^6}{6}+\frac {b\,e\,x}{6\,c^5}+\frac {b\,e\,x^5}{30\,c}+\frac {b\,e\,x^3}{18\,c^3}\right )-\ln \left (\frac {1}{c\,x}+1\right )\,\left (\ln \left (1-c^2\,x^2\right )\,\left (\frac {b\,e}{12\,c^6}-\frac {b\,e\,x^6}{12}\right )-\frac {b\,d\,x^6}{12}+\frac {b\,e\,x^6}{36}+\frac {b\,e\,x^4}{24\,c^2}+\frac {b\,e\,x^2}{12\,c^4}\right )+\ln \left (1-\frac {1}{c\,x}\right )\,\left (\frac {\frac {b\,d\,x^7}{6}-\frac {b\,c^2\,d\,x^9}{6}}{2\,\left (c\,x^2+x\right )\,\left (c\,x-1\right )}+\frac {\frac {b\,e\,x^7}{36}+\frac {b\,e\,x^5}{12\,c^2}-\frac {b\,e\,x^3}{6\,c^4}+\frac {b\,c^2\,e\,x^9}{18}}{2\,\left (c\,x^2+x\right )\,\left (c\,x-1\right )}+\frac {\ln \left (1-c^2\,x^2\right )\,\left (\frac {b\,e\,x^7}{6}-\frac {b\,c^2\,e\,x^9}{6}\right )}{2\,\left (c\,x^2+x\right )\,\left (c\,x-1\right )}-\frac {b\,e\,\ln \left (1-c^2\,x^2\right )\,\left (x-c^2\,x^3\right )}{12\,c^6\,\left (c\,x^2+x\right )\,\left (c\,x-1\right )}\right )+x^4\,\left (\frac {a\,\left (3\,d-e\right )}{12\,c^2}-\frac {a\,d}{4\,c^2}\right )+x^3\,\left (\frac {b\,\left (15\,d-11\,e\right )}{270\,c^3}-\frac {7\,b\,e}{108\,c^3}\right )+x\,\left (\frac {\frac {b\,\left (15\,d-11\,e\right )}{90\,c^3}-\frac {7\,b\,e}{36\,c^3}}{c^2}-\frac {b\,e}{2\,c^5}\right )+\frac {a\,x^6\,\left (3\,d-e\right )}{18}+\frac {x^2\,\left (\frac {a\,\left (3\,d-e\right )}{3\,c^2}-\frac {a\,d}{c^2}\right )}{2\,c^2}-\frac {\ln \left (c\,x-1\right )\,\left (20\,a\,e-10\,b\,d+49\,b\,e\right )}{120\,c^6}-\frac {\ln \left (c\,x+1\right )\,\left (20\,a\,e+10\,b\,d-49\,b\,e\right )}{120\,c^6}+\frac {b\,x^5\,\left (15\,d-11\,e\right )}{450\,c} \]

[In]

int(x^5*(a + b*acoth(c*x))*(d + e*log(1 - c^2*x^2)),x)

[Out]

log(1 - c^2*x^2)*((a*e*x^6)/6 + (b*e*x)/(6*c^5) + (b*e*x^5)/(30*c) + (b*e*x^3)/(18*c^3)) - log(1/(c*x) + 1)*(l
og(1 - c^2*x^2)*((b*e)/(12*c^6) - (b*e*x^6)/12) - (b*d*x^6)/12 + (b*e*x^6)/36 + (b*e*x^4)/(24*c^2) + (b*e*x^2)
/(12*c^4)) + log(1 - 1/(c*x))*(((b*d*x^7)/6 - (b*c^2*d*x^9)/6)/(2*(x + c*x^2)*(c*x - 1)) + ((b*e*x^7)/36 + (b*
e*x^5)/(12*c^2) - (b*e*x^3)/(6*c^4) + (b*c^2*e*x^9)/18)/(2*(x + c*x^2)*(c*x - 1)) + (log(1 - c^2*x^2)*((b*e*x^
7)/6 - (b*c^2*e*x^9)/6))/(2*(x + c*x^2)*(c*x - 1)) - (b*e*log(1 - c^2*x^2)*(x - c^2*x^3))/(12*c^6*(x + c*x^2)*
(c*x - 1))) + x^4*((a*(3*d - e))/(12*c^2) - (a*d)/(4*c^2)) + x^3*((b*(15*d - 11*e))/(270*c^3) - (7*b*e)/(108*c
^3)) + x*(((b*(15*d - 11*e))/(90*c^3) - (7*b*e)/(36*c^3))/c^2 - (b*e)/(2*c^5)) + (a*x^6*(3*d - e))/18 + (x^2*(
(a*(3*d - e))/(3*c^2) - (a*d)/c^2))/(2*c^2) - (log(c*x - 1)*(20*a*e - 10*b*d + 49*b*e))/(120*c^6) - (log(c*x +
1)*(20*a*e + 10*b*d - 49*b*e))/(120*c^6) + (b*x^5*(15*d - 11*e))/(450*c)