3.1287 \(\int x^3 (a+b \tan ^{-1}(c x)) (d+e \log (1+c^2 x^2)) \, dx\)

Optimal. Leaf size=221 \[ \frac{1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )+\frac{e x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}-\frac{e \log \left (c^2 x^2+1\right ) \left (a+b \tan ^{-1}(c x)\right )}{4 c^4}-\frac{1}{8} e x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{b x (2 d-3 e)}{8 c^3}-\frac{b (2 d-3 e) \tan ^{-1}(c x)}{8 c^4}-\frac{b e x^3 \log \left (c^2 x^2+1\right )}{12 c}+\frac{b e x \log \left (c^2 x^2+1\right )}{4 c^3}-\frac{2 b e x}{3 c^3}+\frac{2 b e \tan ^{-1}(c x)}{3 c^4}-\frac{b x^3 (2 d-e)}{24 c}+\frac{b e x^3}{18 c} \]

[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) - (b*(2*d - 3*e)*A
rcTan[c*x])/(8*c^4) + (2*b*e*ArcTan[c*x])/(3*c^4) + (e*x^2*(a + b*ArcTan[c*x]))/(4*c^2) - (e*x^4*(a + b*ArcTan
[c*x]))/8 + (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*ArcTan[c*x])*Log[
1 + c^2*x^2])/(4*c^4) + (x^4*(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/4

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Rubi [A]  time = 0.243993, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.423, Rules used = {2454, 2395, 43, 5019, 459, 321, 203, 2471, 2448, 2455, 302} \[ \frac{1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )+\frac{e x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}-\frac{e \log \left (c^2 x^2+1\right ) \left (a+b \tan ^{-1}(c x)\right )}{4 c^4}-\frac{1}{8} e x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{b x (2 d-3 e)}{8 c^3}-\frac{b (2 d-3 e) \tan ^{-1}(c x)}{8 c^4}-\frac{b e x^3 \log \left (c^2 x^2+1\right )}{12 c}+\frac{b e x \log \left (c^2 x^2+1\right )}{4 c^3}-\frac{2 b e x}{3 c^3}+\frac{2 b e \tan ^{-1}(c x)}{3 c^4}-\frac{b x^3 (2 d-e)}{24 c}+\frac{b e x^3}{18 c} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*ArcTan[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) - (b*(2*d - 3*e)*A
rcTan[c*x])/(8*c^4) + (2*b*e*ArcTan[c*x])/(3*c^4) + (e*x^2*(a + b*ArcTan[c*x]))/(4*c^2) - (e*x^4*(a + b*ArcTan
[c*x]))/8 + (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*ArcTan[c*x])*Log[
1 + c^2*x^2])/(4*c^4) + (x^4*(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/4

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 2395

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 43

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 5019

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_.) + Log[(f_.) + (g_.)*(x_)^2]*(e_.))*(x_)^(m_.), x_Symbol] :> With
[{u = IntHide[x^m*(d + e*Log[f + g*x^2]), x]}, Dist[a + b*ArcTan[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 459

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 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 203

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

Rule 2471

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 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 2455

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 302

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]

Rubi steps

\begin{align*} \int x^3 \left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right ) \, dx &=\frac{e x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}-\frac{1}{8} e x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac{e \left (a+b \tan ^{-1}(c x)\right ) \log \left (1+c^2 x^2\right )}{4 c^4}+\frac{1}{4} x^4 \left (a+b \tan ^{-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 \tan ^{-1}(c x)\right )}{4 c^2}-\frac{1}{8} e x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac{e \left (a+b \tan ^{-1}(c x)\right ) \log \left (1+c^2 x^2\right )}{4 c^4}+\frac{1}{4} x^4 \left (a+b \tan ^{-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 \tan ^{-1}(c x)\right )}{4 c^2}-\frac{1}{8} e x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac{e \left (a+b \tan ^{-1}(c x)\right ) \log \left (1+c^2 x^2\right )}{4 c^4}+\frac{1}{4} x^4 \left (a+b \tan ^{-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 \tan ^{-1}(c x)\right )}{4 c^2}-\frac{1}{8} e x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac{e \left (a+b \tan ^{-1}(c x)\right ) \log \left (1+c^2 x^2\right )}{4 c^4}+\frac{1}{4} x^4 \left (a+b \tan ^{-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{b (2 d-3 e) \tan ^{-1}(c x)}{8 c^4}+\frac{e x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}-\frac{1}{8} e x^4 \left (a+b \tan ^{-1}(c x)\right )+\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 \tan ^{-1}(c x)\right ) \log \left (1+c^2 x^2\right )}{4 c^4}+\frac{1}{4} x^4 \left (a+b \tan ^{-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{b (2 d-3 e) \tan ^{-1}(c x)}{8 c^4}+\frac{e x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}-\frac{1}{8} e x^4 \left (a+b \tan ^{-1}(c x)\right )+\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 \tan ^{-1}(c x)\right ) \log \left (1+c^2 x^2\right )}{4 c^4}+\frac{1}{4} x^4 \left (a+b \tan ^{-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{b (2 d-3 e) \tan ^{-1}(c x)}{8 c^4}+\frac{b e \tan ^{-1}(c x)}{2 c^4}+\frac{e x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}-\frac{1}{8} e x^4 \left (a+b \tan ^{-1}(c x)\right )+\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 \tan ^{-1}(c x)\right ) \log \left (1+c^2 x^2\right )}{4 c^4}+\frac{1}{4} x^4 \left (a+b \tan ^{-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{b (2 d-3 e) \tan ^{-1}(c x)}{8 c^4}+\frac{2 b e \tan ^{-1}(c x)}{3 c^4}+\frac{e x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}-\frac{1}{8} e x^4 \left (a+b \tan ^{-1}(c x)\right )+\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 \tan ^{-1}(c x)\right ) \log \left (1+c^2 x^2\right )}{4 c^4}+\frac{1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.152217, size = 164, normalized size = 0.74 \[ \frac{c x \left (18 a c^3 d x^3-9 a c e x \left (c^2 x^2-2\right )-6 b d \left (c^2 x^2-3\right )+b e \left (7 c^2 x^2-75\right )\right )-6 e \log \left (c^2 x^2+1\right ) \left (a \left (3-3 c^4 x^4\right )+b c x \left (c^2 x^2-3\right )\right )+3 b \tan ^{-1}(c x) \left (6 d \left (c^4 x^4-1\right )+e \left (-3 c^4 x^4+6 c^2 x^2+25\right )+6 e \left (c^4 x^4-1\right ) \log \left (c^2 x^2+1\right )\right )}{72 c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x*(18*a*c^3*d*x^3 - 6*b*d*(-3 + c^2*x^2) - 9*a*c*e*x*(-2 + c^2*x^2) + b*e*(-75 + 7*c^2*x^2)) - 6*e*(b*c*x*(
-3 + c^2*x^2) + a*(3 - 3*c^4*x^4))*Log[1 + c^2*x^2] + 3*b*ArcTan[c*x]*(e*(25 + 6*c^2*x^2 - 3*c^4*x^4) + 6*d*(-
1 + c^4*x^4) + 6*e*(-1 + c^4*x^4)*Log[1 + c^2*x^2]))/(72*c^4)

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Maple [C]  time = 1.831, size = 3897, normalized size = 17.6 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*arctan(c*x))*(d+e*ln(c^2*x^2+1)),x)

[Out]

-1/8*I/c^4*b*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^2*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*a
rctan(c*x)*Pi*e+1/8*I/c^4*b*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*csgn(I*(1+I*c*x)/(c^2*x^2+1)^(1/2))^2*arctan(c*x)*
Pi*e-1/8*I/c^4*b*arctan(c*x)*Pi*e*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1))^2*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)+
1/4*I/c^4*b*arctan(c*x)*Pi*e*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^2-1/24*
I/c*b*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^2*Pi
*x^3*e-1/12*I/c*b*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))^2*csgn(I*(1+I*c*x)/(c^2*x^2+1)^(1/2))*Pi*x^3*e+1/4*b*d*x/c^3
-1/12*b*d*x^3/c-1/4*b*d*arctan(c*x)/c^4-25/24*b*e*x/c^3+41/24*b*e*arctan(c*x)/c^4+7/72*b*e*x^3/c-1/24*I/c*b*cs
gn(I*(1+I*c*x)^2/(c^2*x^2+1))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^2*Pi*x^3*e+1/24*I/
c*b*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*csgn(I*(1+I*c*x)/(c^2*x^2+1)^(1/2))^2*Pi*x^3*e+1/8*I*b*arctan(c*x)*csgn(I/
((1+I*c*x)^2/(c^2*x^2+1)+1)^2)*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^2*Pi*x^4*e+1/4*I*
b*arctan(c*x)*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))^2*csgn(I*(1+I*c*x)/(c^2*x^2+1)^(1/2))*Pi*x^4*e+1/8*I*b*arctan(c*
x)*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^2*Pi*x^4*e-1/
8*I*b*arctan(c*x)*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*csgn(I*(1+I*c*x)/(c^2*x^2+1)^(1/2))^2*Pi*x^4*e+1/8*I*b*arcta
n(c*x)*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1))^2*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)*Pi*x^4*e-1/4*I*b*arctan(c*x
)*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^2*Pi*x^4*e-1/24*I/c*b*csgn(I*((1+I
*c*x)^2/(c^2*x^2+1)+1))^2*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)*Pi*x^3*e+1/12*I/c*b*csgn(I*((1+I*c*x)^2/(c^2*x
^2+1)+1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^2*Pi*x^3*e+1/8*I/c^3*b*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)*c
sgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^2*Pi*x*e+1/4*I/c^3*b*csgn(I*(1+I*c*x)^2/(c^2*x^2+
1))^2*csgn(I*(1+I*c*x)/(c^2*x^2+1)^(1/2))*Pi*x*e+1/8*I/c^3*b*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*csgn(I*(1+I*c*x)^
2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^2*Pi*x*e-1/8*I/c^3*b*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*csgn(I*(1+I*
c*x)/(c^2*x^2+1)^(1/2))^2*Pi*x*e+1/8*I/c^3*b*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1))^2*csgn(I*((1+I*c*x)^2/(c^2*x^
2+1)+1)^2)*Pi*x*e-1/4*I/c^3*b*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^2*Pi*x
*e-1/8*I/c^4*b*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^2*arctan(c*x)*Pi*e*csgn(I/((1+I*c
*x)^2/(c^2*x^2+1)+1)^2)-1/4*I/c^4*b*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))^2*csgn(I*(1+I*c*x)/(c^2*x^2+1)^(1/2))*arct
an(c*x)*Pi*e-1/8*a*e*x^4-1/8*I*b*arctan(c*x)*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)*csgn(I*(1+I*c*x)^2/(c^2*x^2
+1))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)*Pi*x^4*e+1/24*I/c*b*csgn(I/((1+I*c*x)^2/(c^
2*x^2+1)+1)^2)*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)*P
i*x^3*e-1/8*I/c^3*b*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*csgn(I*(1+I*c*x)^2/(
c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)*Pi*x*e+1/8*I/c^4*b*arctan(c*x)*Pi*e*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+
1)^2)*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)+1/4*a*e/c^
2*x^2-1/4*a*e/c^4*ln(c^2*x^2+1)+1/4*b*arctan(c*x)*x^4*d-1/8*b*arctan(c*x)*x^4*e-1/2/c^3*b*ln((1+I*c*x)^2/(c^2*
x^2+1)+1)*x*e+1/2/c^3*b*ln(2)*x*e+1/2/c^4*b*ln((1+I*c*x)^2/(c^2*x^2+1)+1)*e*arctan(c*x)-1/2/c^4*b*ln(2)*e*arct
an(c*x)+1/6/c^4*b*e*Pi*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))^3+1/6/c^4*b*e*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)*
csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)+1/24*I/c*b*csgn(
I*(1+I*c*x)^2/(c^2*x^2+1))^3*Pi*x^3*e+1/24*I/c*b*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)
^3*Pi*x^3*e-1/24*I/c*b*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^3*Pi*x^3*e-1/8*I/c^3*b*csgn(I*(1+I*c*x)^2/(c^2*x^
2+1))^3*Pi*x*e-1/8*I/c^3*b*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^3*Pi*x*e+1/8*I/c^3*b*
csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^3*Pi*x*e+1/8*I/c^4*b*arctan(c*x)*Pi*e*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))^3+
1/8*I/c^4*b*arctan(c*x)*Pi*e*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^3-1/8*I/c^4*b*csgn(
I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^3*arctan(c*x)*Pi*e-1/8*I*b*arctan(c*x)*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))^3*Pi*x
^4*e-1/8*I*b*arctan(c*x)*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^3*Pi*x^4*e+1/8*I*b*arct
an(c*x)*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^3*Pi*x^4*e+1/6/c^4*b*e*Pi*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c
*x)^2/(c^2*x^2+1)+1)^2)^3-1/6/c^4*b*e*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^3-1/2*b*arctan(c*x)*ln((1+I*c*x
)^2/(c^2*x^2+1)+1)*x^4*e+1/2*b*arctan(c*x)*ln(2)*x^4*e+1/6/c*b*ln((1+I*c*x)^2/(c^2*x^2+1)+1)*x^3*e-1/6/c*b*ln(
2)*x^3*e+2/3*I/c^4*b*ln(2)*e+1/4/c^2*b*arctan(c*x)*x^2*e+1/4*x^4*a*e*ln(c^2*x^2+1)+1/3*I/c^4*b*d-41/36*I/c^4*b
*e+1/4*x^4*a*d+1/6/c^4*b*e*(3*arctan(c*x)*x^3*c^3-3*I*arctan(c*x)*x^2*c^2-c^2*x^2-3*arctan(c*x)*x*c+I*c*x+3*I*
arctan(c*x)+4)*(c*x+I)*ln((1+I*c*x)/(c^2*x^2+1)^(1/2))-1/6/c^4*b*e*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)*cs
gn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^2-1/3/c^4*b*e*Pi*csgn(I*(1+I*c*x)/(c^2*x^2+1)^(1/2
))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))^2-1/6/c^4*b*e*Pi*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*csgn(I*(1+I*c*x)^2/(c^2*x^
2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^2+1/6/c^4*b*e*Pi*csgn(I*(1+I*c*x)/(c^2*x^2+1)^(1/2))^2*csgn(I*(1+I*c*x)^2/
(c^2*x^2+1))-1/6/c^4*b*e*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1))^2*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)+1/3/c^
4*b*e*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^2

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Maxima [A]  time = 1.49209, size = 302, normalized size = 1.37 \begin{align*} \frac{1}{4} \, a d x^{4} + \frac{1}{72} \, b c e{\left (\frac{7 \, c^{2} x^{3} - 6 \,{\left (c^{2} x^{3} - 3 \, x\right )} \log \left (c^{2} x^{2} + 1\right ) - 75 \, x}{c^{4}} + \frac{75 \, \arctan \left (c x\right )}{c^{5}}\right )} + \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 \arctan \left (c x\right ) + \frac{1}{12} \,{\left (3 \, x^{4} \arctan \left (c x\right ) - c{\left (\frac{c^{2} x^{3} - 3 \, x}{c^{4}} + \frac{3 \, \arctan \left (c x\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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*d*x^4 + 1/72*b*c*e*((7*c^2*x^3 - 6*(c^2*x^3 - 3*x)*log(c^2*x^2 + 1) - 75*x)/c^4 + 75*arctan(c*x)/c^5) +
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*arctan(c*x) + 1/12*(3*
x^4*arctan(c*x) - c*((c^2*x^3 - 3*x)/c^4 + 3*arctan(c*x)/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

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Fricas [A]  time = 1.38576, size = 405, normalized size = 1.83 \begin{align*} \frac{18 \, a c^{2} e x^{2} + 9 \,{\left (2 \, a c^{4} d - a c^{4} e\right )} x^{4} -{\left (6 \, b c^{3} d - 7 \, b c^{3} e\right )} x^{3} + 3 \,{\left (6 \, b c d - 25 \, b c e\right )} x + 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\right )} \arctan \left (c x\right ) + 6 \,{\left (3 \, a c^{4} e x^{4} - b c^{3} e x^{3} + 3 \, b c e x - 3 \, a e + 3 \,{\left (b c^{4} e x^{4} - b e\right )} \arctan \left (c x\right )\right )} \log \left (c^{2} x^{2} + 1\right )}{72 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/72*(18*a*c^2*e*x^2 + 9*(2*a*c^4*d - a*c^4*e)*x^4 - (6*b*c^3*d - 7*b*c^3*e)*x^3 + 3*(6*b*c*d - 25*b*c*e)*x +
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)*arctan(c*x) + 6*(3*a*c^4*e*x^4 - b*c^3*e*x^3
+ 3*b*c*e*x - 3*a*e + 3*(b*c^4*e*x^4 - b*e)*arctan(c*x))*log(c^2*x^2 + 1))/c^4

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Sympy [A]  time = 22.3741, size = 279, normalized size = 1.26 \begin{align*} \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{atan}{\left (c x \right )}}{4} + \frac{b e x^{4} \log{\left (c^{2} x^{2} + 1 \right )} \operatorname{atan}{\left (c x \right )}}{4} - \frac{b e x^{4} \operatorname{atan}{\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{atan}{\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{atan}{\left (c x \right )}}{4 c^{4}} - \frac{b e \log{\left (c^{2} x^{2} + 1 \right )} \operatorname{atan}{\left (c x \right )}}{4 c^{4}} + \frac{25 b e \operatorname{atan}{\left (c x \right )}}{24 c^{4}} & \text{for}\: c \neq 0 \\\frac{a d x^{4}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*atan(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*atan(c*x)/4 + b*e*x**4*log(c**2*x**2 + 1)*atan(c*x)/4 - b*e*x**4*atan(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*atan(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*atan(c*x)/(4*c**4) - b*e*log(c**2*x**2 + 1)*at
an(c*x)/(4*c**4) + 25*b*e*atan(c*x)/(24*c**4), Ne(c, 0)), (a*d*x**4/4, True))

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Giac [A]  time = 1.54895, size = 479, normalized size = 2.17 \begin{align*} \frac{18 \, \pi b c^{4} x^{4} e \log \left (c^{2} x^{2} + 1\right ) \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (x\right ) - 9 \, \pi b c^{4} x^{4} e \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (x\right ) - 36 \, b c^{4} x^{4} \arctan \left (\frac{1}{c x}\right ) e \log \left (c^{2} x^{2} + 1\right ) + 36 \, b c^{4} d x^{4} \arctan \left (c x\right ) + 18 \, b c^{4} x^{4} \arctan \left (\frac{1}{c x}\right ) e + 36 \, a c^{4} x^{4} e \log \left (c^{2} x^{2} + 1\right ) + 36 \, a c^{4} d x^{4} - 18 \, a c^{4} x^{4} e - 12 \, b c^{3} x^{3} e \log \left (c^{2} x^{2} + 1\right ) + 18 \, \pi b c^{2} x^{2} e \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (x\right ) - 12 \, b c^{3} d x^{3} + 14 \, b c^{3} x^{3} e - 36 \, b c^{2} x^{2} \arctan \left (\frac{1}{c x}\right ) e + 36 \, a c^{2} x^{2} e - 18 \, \pi b e \log \left (c^{2} x^{2} + 1\right ) \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (x\right ) + 36 \, b c x e \log \left (c^{2} x^{2} + 1\right ) - 150 \, \pi b e \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (x\right ) + 36 \, b c d x - 150 \, b c x e + 36 \, b \arctan \left (\frac{1}{c x}\right ) e \log \left (c^{2} x^{2} + 1\right ) - 36 \, b d \arctan \left (c x\right ) + 150 \, b \arctan \left (c x\right ) e - 36 \, a e \log \left (c^{2} x^{2} + 1\right )}{144 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/144*(18*pi*b*c^4*x^4*e*log(c^2*x^2 + 1)*sgn(c)*sgn(x) - 9*pi*b*c^4*x^4*e*sgn(c)*sgn(x) - 36*b*c^4*x^4*arctan
(1/(c*x))*e*log(c^2*x^2 + 1) + 36*b*c^4*d*x^4*arctan(c*x) + 18*b*c^4*x^4*arctan(1/(c*x))*e + 36*a*c^4*x^4*e*lo
g(c^2*x^2 + 1) + 36*a*c^4*d*x^4 - 18*a*c^4*x^4*e - 12*b*c^3*x^3*e*log(c^2*x^2 + 1) + 18*pi*b*c^2*x^2*e*sgn(c)*
sgn(x) - 12*b*c^3*d*x^3 + 14*b*c^3*x^3*e - 36*b*c^2*x^2*arctan(1/(c*x))*e + 36*a*c^2*x^2*e - 18*pi*b*e*log(c^2
*x^2 + 1)*sgn(c)*sgn(x) + 36*b*c*x*e*log(c^2*x^2 + 1) - 150*pi*b*e*sgn(c)*sgn(x) + 36*b*c*d*x - 150*b*c*x*e +
36*b*arctan(1/(c*x))*e*log(c^2*x^2 + 1) - 36*b*d*arctan(c*x) + 150*b*arctan(c*x)*e - 36*a*e*log(c^2*x^2 + 1))/
c^4