3.115 \(\int x^5 (a+b \tan ^{-1}(c x^3))^2 \, dx\)

Optimal. Leaf size=90 \[ \frac{\left (a+b \tan ^{-1}\left (c x^3\right )\right )^2}{6 c^2}-\frac{a b x^3}{3 c}+\frac{1}{6} x^6 \left (a+b \tan ^{-1}\left (c x^3\right )\right )^2+\frac{b^2 \log \left (c^2 x^6+1\right )}{6 c^2}-\frac{b^2 x^3 \tan ^{-1}\left (c x^3\right )}{3 c} \]

[Out]

-(a*b*x^3)/(3*c) - (b^2*x^3*ArcTan[c*x^3])/(3*c) + (a + b*ArcTan[c*x^3])^2/(6*c^2) + (x^6*(a + b*ArcTan[c*x^3]
)^2)/6 + (b^2*Log[1 + c^2*x^6])/(6*c^2)

________________________________________________________________________________________

Rubi [C]  time = 1.05174, antiderivative size = 612, normalized size of antiderivative = 6.8, number of steps used = 44, number of rules used = 16, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {5035, 2454, 2401, 2389, 2296, 2295, 2390, 2305, 2304, 2395, 43, 2439, 2416, 2394, 2393, 2391} \[ \frac{b^2 \text{PolyLog}\left (2,\frac{1}{2} \left (1-i c x^3\right )\right )}{12 c^2}+\frac{b^2 \text{PolyLog}\left (2,\frac{1}{2} \left (1+i c x^3\right )\right )}{12 c^2}-\frac{\left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{24 c^2}+\frac{\left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{12 c^2}+\frac{i b \left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )}{24 c^2}-\frac{b \log \left (\frac{1}{2} \left (1+i c x^3\right )\right ) \left (2 i a-b \log \left (1-i c x^3\right )\right )}{12 c^2}-\frac{a b x^3}{2 c}+\frac{1}{24} b x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right )-\frac{1}{12} b x^6 \log \left (1+i c x^3\right ) \left (2 i a-b \log \left (1-i c x^3\right )\right )+\frac{b^2 \left (1-i c x^3\right )^2}{48 c^2}+\frac{b^2 \left (1+i c x^3\right )^2}{48 c^2}+\frac{b^2 \left (1+i c x^3\right )^2 \log ^2\left (1+i c x^3\right )}{24 c^2}-\frac{b^2 \left (1+i c x^3\right ) \log ^2\left (1+i c x^3\right )}{12 c^2}-\frac{b^2 \log \left (-c x^3+i\right )}{24 c^2}+\frac{b^2 \left (1-i c x^3\right ) \log \left (1-i c x^3\right )}{4 c^2}-\frac{b^2 \left (1+i c x^3\right )^2 \log \left (1+i c x^3\right )}{24 c^2}+\frac{b^2 \left (1+i c x^3\right ) \log \left (1+i c x^3\right )}{4 c^2}+\frac{b^2 \log \left (\frac{1}{2} \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{12 c^2}-\frac{b^2 \log \left (c x^3+i\right )}{24 c^2}-\frac{1}{24} b^2 x^6 \log \left (1+i c x^3\right )+\frac{b^2 x^6}{24} \]

Warning: Unable to verify antiderivative.

[In]

Int[x^5*(a + b*ArcTan[c*x^3])^2,x]

[Out]

-(a*b*x^3)/(2*c) + (b^2*x^6)/24 + (b^2*(1 - I*c*x^3)^2)/(48*c^2) + (b^2*(1 + I*c*x^3)^2)/(48*c^2) - (b^2*Log[I
- c*x^3])/(24*c^2) + (b^2*(1 - I*c*x^3)*Log[1 - I*c*x^3])/(4*c^2) + (b*x^6*((2*I)*a - b*Log[1 - I*c*x^3]))/24
+ ((I/24)*b*(1 - I*c*x^3)^2*(2*a + I*b*Log[1 - I*c*x^3]))/c^2 + ((1 - I*c*x^3)*(2*a + I*b*Log[1 - I*c*x^3])^2
)/(12*c^2) - ((1 - I*c*x^3)^2*(2*a + I*b*Log[1 - I*c*x^3])^2)/(24*c^2) - (b*((2*I)*a - b*Log[1 - I*c*x^3])*Log
[(1 + I*c*x^3)/2])/(12*c^2) - (b^2*x^6*Log[1 + I*c*x^3])/24 + (b^2*(1 + I*c*x^3)*Log[1 + I*c*x^3])/(4*c^2) - (
b^2*(1 + I*c*x^3)^2*Log[1 + I*c*x^3])/(24*c^2) + (b^2*Log[(1 - I*c*x^3)/2]*Log[1 + I*c*x^3])/(12*c^2) - (b*x^6
*((2*I)*a - b*Log[1 - I*c*x^3])*Log[1 + I*c*x^3])/12 - (b^2*(1 + I*c*x^3)*Log[1 + I*c*x^3]^2)/(12*c^2) + (b^2*
(1 + I*c*x^3)^2*Log[1 + I*c*x^3]^2)/(24*c^2) - (b^2*Log[I + c*x^3])/(24*c^2) + (b^2*PolyLog[2, (1 - I*c*x^3)/2
])/(12*c^2) + (b^2*PolyLog[2, (1 + I*c*x^3)/2])/(12*c^2)

Rule 5035

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^
m*(a + (I*b*Log[1 - I*c*x^n])/2 - (I*b*Log[1 + I*c*x^n])/2)^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[
p, 0] && IntegerQ[m] && IntegerQ[n]

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 2401

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Int[Exp
andIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[
e*f - d*g, 0] && IGtQ[q, 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 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

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 2390

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

Rule 2305

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

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

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 2439

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*(x_)^(r_.), x_Symbol] :> Simp[(x^(r + 1)*(a + b*Log[c*(d + e*x)^n])^p*(f + g*Log[h*(i + j*x)^m]))/(r +
1), x] + (-Dist[(g*j*m)/(r + 1), Int[(x^(r + 1)*(a + b*Log[c*(d + e*x)^n])^p)/(i + j*x), x], x] - Dist[(b*e*n*
p)/(r + 1), Int[(x^(r + 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1)*(f + g*Log[h*(i + j*x)^m]))/(d + e*x), x], x]) /
; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && IGtQ[p, 0] && IntegerQ[r] && (EqQ[p, 1] || GtQ[r, 0]) && N
eQ[r, -1]

Rule 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2394

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

Rule 2393

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

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int x^5 \left (a+b \tan ^{-1}\left (c x^3\right )\right )^2 \, dx &=\int \left (\frac{1}{4} x^5 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2+\frac{1}{2} b x^5 \left (-2 i a+b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )-\frac{1}{4} b^2 x^5 \log ^2\left (1+i c x^3\right )\right ) \, dx\\ &=\frac{1}{4} \int x^5 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2 \, dx+\frac{1}{2} b \int x^5 \left (-2 i a+b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right ) \, dx-\frac{1}{4} b^2 \int x^5 \log ^2\left (1+i c x^3\right ) \, dx\\ &=\frac{1}{12} \operatorname{Subst}\left (\int x (2 a+i b \log (1-i c x))^2 \, dx,x,x^3\right )+\frac{1}{6} b \operatorname{Subst}\left (\int x (-2 i a+b \log (1-i c x)) \log (1+i c x) \, dx,x,x^3\right )-\frac{1}{12} b^2 \operatorname{Subst}\left (\int x \log ^2(1+i c x) \, dx,x,x^3\right )\\ &=-\frac{1}{12} b x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )+\frac{1}{12} \operatorname{Subst}\left (\int \left (-\frac{i (2 a+i b \log (1-i c x))^2}{c}+\frac{i (1-i c x) (2 a+i b \log (1-i c x))^2}{c}\right ) \, dx,x,x^3\right )-\frac{1}{12} b^2 \operatorname{Subst}\left (\int \left (\frac{i \log ^2(1+i c x)}{c}-\frac{i (1+i c x) \log ^2(1+i c x)}{c}\right ) \, dx,x,x^3\right )-\frac{1}{12} (i b c) \operatorname{Subst}\left (\int \frac{x^2 (-2 i a+b \log (1-i c x))}{1+i c x} \, dx,x,x^3\right )+\frac{1}{12} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^2 \log (1+i c x)}{1-i c x} \, dx,x,x^3\right )\\ &=-\frac{1}{12} b x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )-\frac{i \operatorname{Subst}\left (\int (2 a+i b \log (1-i c x))^2 \, dx,x,x^3\right )}{12 c}+\frac{i \operatorname{Subst}\left (\int (1-i c x) (2 a+i b \log (1-i c x))^2 \, dx,x,x^3\right )}{12 c}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \log ^2(1+i c x) \, dx,x,x^3\right )}{12 c}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int (1+i c x) \log ^2(1+i c x) \, dx,x,x^3\right )}{12 c}-\frac{1}{12} (i b c) \operatorname{Subst}\left (\int \left (\frac{-2 i a+b \log (1-i c x)}{c^2}-\frac{i x (-2 i a+b \log (1-i c x))}{c}+\frac{i (-2 i a+b \log (1-i c x))}{c^2 (-i+c x)}\right ) \, dx,x,x^3\right )+\frac{1}{12} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{\log (1+i c x)}{c^2}+\frac{i x \log (1+i c x)}{c}-\frac{i \log (1+i c x)}{c^2 (i+c x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{1}{12} b x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )-\frac{1}{12} b \operatorname{Subst}\left (\int x (-2 i a+b \log (1-i c x)) \, dx,x,x^3\right )-\frac{1}{12} b^2 \operatorname{Subst}\left (\int x \log (1+i c x) \, dx,x,x^3\right )+\frac{\operatorname{Subst}\left (\int (2 a+i b \log (x))^2 \, dx,x,1-i c x^3\right )}{12 c^2}-\frac{\operatorname{Subst}\left (\int x (2 a+i b \log (x))^2 \, dx,x,1-i c x^3\right )}{12 c^2}-\frac{b^2 \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1+i c x^3\right )}{12 c^2}+\frac{b^2 \operatorname{Subst}\left (\int x \log ^2(x) \, dx,x,1+i c x^3\right )}{12 c^2}-\frac{(i b) \operatorname{Subst}\left (\int (-2 i a+b \log (1-i c x)) \, dx,x,x^3\right )}{12 c}+\frac{b \operatorname{Subst}\left (\int \frac{-2 i a+b \log (1-i c x)}{-i+c x} \, dx,x,x^3\right )}{12 c}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \log (1+i c x) \, dx,x,x^3\right )}{12 c}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{i+c x} \, dx,x,x^3\right )}{12 c}\\ &=-\frac{a b x^3}{6 c}+\frac{1}{24} b x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right )+\frac{\left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{12 c^2}-\frac{\left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{24 c^2}-\frac{b \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )}{12 c^2}-\frac{1}{24} b^2 x^6 \log \left (1+i c x^3\right )+\frac{b^2 \log \left (\frac{1}{2} \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{12 c^2}-\frac{1}{12} b x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )-\frac{b^2 \left (1+i c x^3\right ) \log ^2\left (1+i c x^3\right )}{12 c^2}+\frac{b^2 \left (1+i c x^3\right )^2 \log ^2\left (1+i c x^3\right )}{24 c^2}+\frac{(i b) \operatorname{Subst}\left (\int x (2 a+i b \log (x)) \, dx,x,1-i c x^3\right )}{12 c^2}-\frac{(i b) \operatorname{Subst}\left (\int (2 a+i b \log (x)) \, dx,x,1-i c x^3\right )}{6 c^2}+\frac{b^2 \operatorname{Subst}\left (\int \log (x) \, dx,x,1+i c x^3\right )}{12 c^2}-\frac{b^2 \operatorname{Subst}\left (\int x \log (x) \, dx,x,1+i c x^3\right )}{12 c^2}+\frac{b^2 \operatorname{Subst}\left (\int \log (x) \, dx,x,1+i c x^3\right )}{6 c^2}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \log (1-i c x) \, dx,x,x^3\right )}{12 c}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{2} i (-i+c x)\right )}{1-i c x} \, dx,x,x^3\right )}{12 c}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{1}{2} i (i+c x)\right )}{1+i c x} \, dx,x,x^3\right )}{12 c}-\frac{1}{24} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^2}{1-i c x} \, dx,x,x^3\right )+\frac{1}{24} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+i c x} \, dx,x,x^3\right )\\ &=-\frac{a b x^3}{2 c}-\frac{i b^2 x^3}{4 c}+\frac{b^2 \left (1-i c x^3\right )^2}{48 c^2}+\frac{b^2 \left (1+i c x^3\right )^2}{48 c^2}+\frac{1}{24} b x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right )+\frac{i b \left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )}{24 c^2}+\frac{\left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{12 c^2}-\frac{\left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{24 c^2}-\frac{b \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )}{12 c^2}-\frac{1}{24} b^2 x^6 \log \left (1+i c x^3\right )+\frac{b^2 \left (1+i c x^3\right ) \log \left (1+i c x^3\right )}{4 c^2}-\frac{b^2 \left (1+i c x^3\right )^2 \log \left (1+i c x^3\right )}{24 c^2}+\frac{b^2 \log \left (\frac{1}{2} \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{12 c^2}-\frac{1}{12} b x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )-\frac{b^2 \left (1+i c x^3\right ) \log ^2\left (1+i c x^3\right )}{12 c^2}+\frac{b^2 \left (1+i c x^3\right )^2 \log ^2\left (1+i c x^3\right )}{24 c^2}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1-i c x^3\right )}{12 c^2}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1+i c x^3\right )}{12 c^2}+\frac{b^2 \operatorname{Subst}\left (\int \log (x) \, dx,x,1-i c x^3\right )}{12 c^2}+\frac{b^2 \operatorname{Subst}\left (\int \log (x) \, dx,x,1-i c x^3\right )}{6 c^2}+\frac{1}{24} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{i x}{c}+\frac{i}{c^2 (-i+c x)}\right ) \, dx,x,x^3\right )-\frac{1}{24} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}+\frac{i x}{c}-\frac{i}{c^2 (i+c x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{a b x^3}{2 c}+\frac{b^2 x^6}{24}+\frac{b^2 \left (1-i c x^3\right )^2}{48 c^2}+\frac{b^2 \left (1+i c x^3\right )^2}{48 c^2}-\frac{b^2 \log \left (i-c x^3\right )}{24 c^2}+\frac{b^2 \left (1-i c x^3\right ) \log \left (1-i c x^3\right )}{4 c^2}+\frac{1}{24} b x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right )+\frac{i b \left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )}{24 c^2}+\frac{\left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{12 c^2}-\frac{\left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{24 c^2}-\frac{b \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )}{12 c^2}-\frac{1}{24} b^2 x^6 \log \left (1+i c x^3\right )+\frac{b^2 \left (1+i c x^3\right ) \log \left (1+i c x^3\right )}{4 c^2}-\frac{b^2 \left (1+i c x^3\right )^2 \log \left (1+i c x^3\right )}{24 c^2}+\frac{b^2 \log \left (\frac{1}{2} \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{12 c^2}-\frac{1}{12} b x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )-\frac{b^2 \left (1+i c x^3\right ) \log ^2\left (1+i c x^3\right )}{12 c^2}+\frac{b^2 \left (1+i c x^3\right )^2 \log ^2\left (1+i c x^3\right )}{24 c^2}-\frac{b^2 \log \left (i+c x^3\right )}{24 c^2}+\frac{b^2 \text{Li}_2\left (\frac{1}{2} \left (1-i c x^3\right )\right )}{12 c^2}+\frac{b^2 \text{Li}_2\left (\frac{1}{2} \left (1+i c x^3\right )\right )}{12 c^2}\\ \end{align*}

Mathematica [A]  time = 0.0649459, size = 85, normalized size = 0.94 \[ \frac{2 b \tan ^{-1}\left (c x^3\right ) \left (a c^2 x^6+a-b c x^3\right )+a c x^3 \left (a c x^3-2 b\right )+b^2 \log \left (c^2 x^6+1\right )+b^2 \left (c^2 x^6+1\right ) \tan ^{-1}\left (c x^3\right )^2}{6 c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*ArcTan[c*x^3])^2,x]

[Out]

(a*c*x^3*(-2*b + a*c*x^3) + 2*b*(a - b*c*x^3 + a*c^2*x^6)*ArcTan[c*x^3] + b^2*(1 + c^2*x^6)*ArcTan[c*x^3]^2 +
b^2*Log[1 + c^2*x^6])/(6*c^2)

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Maple [A]  time = 0.037, size = 113, normalized size = 1.3 \begin{align*}{\frac{{x}^{6}{a}^{2}}{6}}+{\frac{{b}^{2}{x}^{6} \left ( \arctan \left ( c{x}^{3} \right ) \right ) ^{2}}{6}}-{\frac{{b}^{2}{x}^{3}\arctan \left ( c{x}^{3} \right ) }{3\,c}}+{\frac{{b}^{2} \left ( \arctan \left ( c{x}^{3} \right ) \right ) ^{2}}{6\,{c}^{2}}}+{\frac{{b}^{2}\ln \left ({c}^{2}{x}^{6}+1 \right ) }{6\,{c}^{2}}}+{\frac{ab{x}^{6}\arctan \left ( c{x}^{3} \right ) }{3}}-{\frac{ab{x}^{3}}{3\,c}}+{\frac{ab\arctan \left ( c{x}^{3} \right ) }{3\,{c}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a+b*arctan(c*x^3))^2,x)

[Out]

1/6*x^6*a^2+1/6*b^2*x^6*arctan(c*x^3)^2-1/3*b^2*x^3*arctan(c*x^3)/c+1/6*b^2/c^2*arctan(c*x^3)^2+1/6*b^2*ln(c^2
*x^6+1)/c^2+1/3*a*b*x^6*arctan(c*x^3)-1/3*a*b*x^3/c+1/3*a*b/c^2*arctan(c*x^3)

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Maxima [A]  time = 1.87051, size = 170, normalized size = 1.89 \begin{align*} \frac{1}{6} \, b^{2} x^{6} \arctan \left (c x^{3}\right )^{2} + \frac{1}{6} \, a^{2} x^{6} + \frac{1}{3} \,{\left (x^{6} \arctan \left (c x^{3}\right ) - c{\left (\frac{x^{3}}{c^{2}} - \frac{\arctan \left (c x^{3}\right )}{c^{3}}\right )}\right )} a b - \frac{1}{6} \,{\left (2 \, c{\left (\frac{x^{3}}{c^{2}} - \frac{\arctan \left (c x^{3}\right )}{c^{3}}\right )} \arctan \left (c x^{3}\right ) + \frac{\arctan \left (c x^{3}\right )^{2} - \log \left (6 \, c^{5} x^{6} + 6 \, c^{3}\right )}{c^{2}}\right )} b^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arctan(c*x^3))^2,x, algorithm="maxima")

[Out]

1/6*b^2*x^6*arctan(c*x^3)^2 + 1/6*a^2*x^6 + 1/3*(x^6*arctan(c*x^3) - c*(x^3/c^2 - arctan(c*x^3)/c^3))*a*b - 1/
6*(2*c*(x^3/c^2 - arctan(c*x^3)/c^3)*arctan(c*x^3) + (arctan(c*x^3)^2 - log(6*c^5*x^6 + 6*c^3))/c^2)*b^2

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Fricas [A]  time = 2.61903, size = 200, normalized size = 2.22 \begin{align*} \frac{a^{2} c^{2} x^{6} - 2 \, a b c x^{3} +{\left (b^{2} c^{2} x^{6} + b^{2}\right )} \arctan \left (c x^{3}\right )^{2} + b^{2} \log \left (c^{2} x^{6} + 1\right ) + 2 \,{\left (a b c^{2} x^{6} - b^{2} c x^{3} + a b\right )} \arctan \left (c x^{3}\right )}{6 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arctan(c*x^3))^2,x, algorithm="fricas")

[Out]

1/6*(a^2*c^2*x^6 - 2*a*b*c*x^3 + (b^2*c^2*x^6 + b^2)*arctan(c*x^3)^2 + b^2*log(c^2*x^6 + 1) + 2*(a*b*c^2*x^6 -
b^2*c*x^3 + a*b)*arctan(c*x^3))/c^2

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(a+b*atan(c*x**3))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.15752, size = 135, normalized size = 1.5 \begin{align*} \frac{a^{2} c x^{6} + \frac{2 \,{\left (c^{2} x^{6} \arctan \left (c x^{3}\right ) - c x^{3} + \arctan \left (c x^{3}\right )\right )} a b}{c} + \frac{{\left (c^{2} x^{6} \arctan \left (c x^{3}\right )^{2} - 2 \, c x^{3} \arctan \left (c x^{3}\right ) + \arctan \left (c x^{3}\right )^{2} + \log \left (c^{2} x^{6} + 1\right )\right )} b^{2}}{c}}{6 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arctan(c*x^3))^2,x, algorithm="giac")

[Out]

1/6*(a^2*c*x^6 + 2*(c^2*x^6*arctan(c*x^3) - c*x^3 + arctan(c*x^3))*a*b/c + (c^2*x^6*arctan(c*x^3)^2 - 2*c*x^3*
arctan(c*x^3) + arctan(c*x^3)^2 + log(c^2*x^6 + 1))*b^2/c)/c