3.74 \(\int x^7 (a+b \tan ^{-1}(c x^2))^2 \, dx\)

Optimal. Leaf size=124 \[ \frac{a b x^2}{4 c^3}-\frac{\left (a+b \tan ^{-1}\left (c x^2\right )\right )^2}{8 c^4}+\frac{1}{8} x^8 \left (a+b \tan ^{-1}\left (c x^2\right )\right )^2-\frac{b x^6 \left (a+b \tan ^{-1}\left (c x^2\right )\right )}{12 c}+\frac{b^2 x^4}{24 c^2}-\frac{b^2 \log \left (c^2 x^4+1\right )}{6 c^4}+\frac{b^2 x^2 \tan ^{-1}\left (c x^2\right )}{4 c^3} \]

[Out]

(a*b*x^2)/(4*c^3) + (b^2*x^4)/(24*c^2) + (b^2*x^2*ArcTan[c*x^2])/(4*c^3) - (b*x^6*(a + b*ArcTan[c*x^2]))/(12*c
) - (a + b*ArcTan[c*x^2])^2/(8*c^4) + (x^8*(a + b*ArcTan[c*x^2])^2)/8 - (b^2*Log[1 + c^2*x^4])/(6*c^4)

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Rubi [C]  time = 1.62698, antiderivative size = 731, normalized size of antiderivative = 5.9, number of steps used = 62, number of rules used = 19, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.187, Rules used = {5035, 2454, 2398, 2411, 43, 2334, 12, 14, 2301, 2395, 2439, 2416, 2389, 2295, 2394, 2393, 2391, 2410, 2390} \[ -\frac{b^2 \text{PolyLog}\left (2,\frac{1}{2} \left (1-i c x^2\right )\right )}{16 c^4}-\frac{b^2 \text{PolyLog}\left (2,\frac{1}{2} \left (1+i c x^2\right )\right )}{16 c^4}+\frac{a b x^2}{8 c^3}-\frac{b x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right )}{32 c^2}+\frac{1}{192} i b \left (-\frac{3 \left (1-i c x^2\right )^4}{c^4}+\frac{16 \left (1-i c x^2\right )^3}{c^4}-\frac{36 \left (1-i c x^2\right )^2}{c^4}+\frac{48 \left (1-i c x^2\right )}{c^4}-\frac{12 \log \left (1-i c x^2\right )}{c^4}\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )+\frac{b \log \left (\frac{1}{2} \left (1+i c x^2\right )\right ) \left (2 i a-b \log \left (1-i c x^2\right )\right )}{16 c^4}+\frac{1}{32} x^8 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2+\frac{1}{64} b x^8 \left (2 i a-b \log \left (1-i c x^2\right )\right )-\frac{1}{16} b x^8 \log \left (1+i c x^2\right ) \left (2 i a-b \log \left (1-i c x^2\right )\right )+\frac{i b x^6 \left (2 i a-b \log \left (1-i c x^2\right )\right )}{48 c}+\frac{b^2 x^4}{128 c^2}-\frac{23 i b^2 x^2}{192 c^3}-\frac{b^2 \left (1-i c x^2\right )^4}{256 c^4}+\frac{b^2 \left (1-i c x^2\right )^3}{36 c^4}-\frac{3 b^2 \left (1-i c x^2\right )^2}{32 c^4}-\frac{b^2 \log ^2\left (1-i c x^2\right )}{32 c^4}+\frac{b^2 \log ^2\left (1+i c x^2\right )}{32 c^4}-\frac{b^2 \log \left (-c x^2+i\right )}{24 c^4}-\frac{b^2 \left (1-i c x^2\right ) \log \left (1-i c x^2\right )}{16 c^4}-\frac{b^2 \left (1+i c x^2\right ) \log \left (1+i c x^2\right )}{8 c^4}-\frac{b^2 \log \left (\frac{1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{16 c^4}+\frac{5 b^2 \log \left (c x^2+i\right )}{192 c^4}-\frac{7 i b^2 x^6}{576 c}-\frac{1}{32} b^2 x^8 \log ^2\left (1+i c x^2\right )+\frac{i b^2 x^6 \log \left (1+i c x^2\right )}{24 c}+\frac{b^2 x^8}{256} \]

Warning: Unable to verify antiderivative.

[In]

Int[x^7*(a + b*ArcTan[c*x^2])^2,x]

[Out]

(a*b*x^2)/(8*c^3) - (((23*I)/192)*b^2*x^2)/c^3 + (b^2*x^4)/(128*c^2) - (((7*I)/576)*b^2*x^6)/c + (b^2*x^8)/256
- (3*b^2*(1 - I*c*x^2)^2)/(32*c^4) + (b^2*(1 - I*c*x^2)^3)/(36*c^4) - (b^2*(1 - I*c*x^2)^4)/(256*c^4) - (b^2*
Log[I - c*x^2])/(24*c^4) - (b^2*(1 - I*c*x^2)*Log[1 - I*c*x^2])/(16*c^4) - (b^2*Log[1 - I*c*x^2]^2)/(32*c^4) -
(b*x^4*((2*I)*a - b*Log[1 - I*c*x^2]))/(32*c^2) + ((I/48)*b*x^6*((2*I)*a - b*Log[1 - I*c*x^2]))/c + (b*x^8*((
2*I)*a - b*Log[1 - I*c*x^2]))/64 + (x^8*(2*a + I*b*Log[1 - I*c*x^2])^2)/32 + (I/192)*b*(2*a + I*b*Log[1 - I*c*
x^2])*((48*(1 - I*c*x^2))/c^4 - (36*(1 - I*c*x^2)^2)/c^4 + (16*(1 - I*c*x^2)^3)/c^4 - (3*(1 - I*c*x^2)^4)/c^4
- (12*Log[1 - I*c*x^2])/c^4) + (b*((2*I)*a - b*Log[1 - I*c*x^2])*Log[(1 + I*c*x^2)/2])/(16*c^4) + ((I/24)*b^2*
x^6*Log[1 + I*c*x^2])/c - (b^2*(1 + I*c*x^2)*Log[1 + I*c*x^2])/(8*c^4) - (b^2*Log[(1 - I*c*x^2)/2]*Log[1 + I*c
*x^2])/(16*c^4) - (b*x^8*((2*I)*a - b*Log[1 - I*c*x^2])*Log[1 + I*c*x^2])/16 + (b^2*Log[1 + I*c*x^2]^2)/(32*c^
4) - (b^2*x^8*Log[1 + I*c*x^2]^2)/32 + (5*b^2*Log[I + c*x^2])/(192*c^4) - (b^2*PolyLog[2, (1 - I*c*x^2)/2])/(1
6*c^4) - (b^2*PolyLog[2, (1 + I*c*x^2)/2])/(16*c^4)

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 2398

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((
f + g*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^p)/(g*(q + 1)), x] - Dist[(b*e*n*p)/(g*(q + 1)), Int[((f + g*x)^(q
+ 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1))/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*
f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

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 2334

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]
] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

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 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 2389

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

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 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]

Rule 2410

Int[(Log[(c_.)*((d_) + (e_.)*(x_))]*(x_)^(m_.))/((f_) + (g_.)*(x_)), x_Symbol] :> Int[ExpandIntegrand[Log[c*(d
+ e*x)], x^m/(f + g*x), x], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[e*f - d*g, 0] && EqQ[c*d, 1] && IntegerQ[m
]

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]

Rubi steps

\begin{align*} \int x^7 \left (a+b \tan ^{-1}\left (c x^2\right )\right )^2 \, dx &=\int \left (\frac{1}{4} x^7 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2+\frac{1}{2} b x^7 \left (-2 i a+b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )-\frac{1}{4} b^2 x^7 \log ^2\left (1+i c x^2\right )\right ) \, dx\\ &=\frac{1}{4} \int x^7 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2 \, dx+\frac{1}{2} b \int x^7 \left (-2 i a+b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right ) \, dx-\frac{1}{4} b^2 \int x^7 \log ^2\left (1+i c x^2\right ) \, dx\\ &=\frac{1}{8} \operatorname{Subst}\left (\int x^3 (2 a+i b \log (1-i c x))^2 \, dx,x,x^2\right )+\frac{1}{4} b \operatorname{Subst}\left (\int x^3 (-2 i a+b \log (1-i c x)) \log (1+i c x) \, dx,x,x^2\right )-\frac{1}{8} b^2 \operatorname{Subst}\left (\int x^3 \log ^2(1+i c x) \, dx,x,x^2\right )\\ &=\frac{1}{32} x^8 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2-\frac{1}{16} b x^8 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )-\frac{1}{32} b^2 x^8 \log ^2\left (1+i c x^2\right )-\frac{1}{16} (i b c) \operatorname{Subst}\left (\int \frac{x^4 (-2 i a+b \log (1-i c x))}{1+i c x} \, dx,x,x^2\right )-\frac{1}{16} (b c) \operatorname{Subst}\left (\int \frac{x^4 (2 a+i b \log (1-i c x))}{1-i c x} \, dx,x,x^2\right )+\frac{1}{16} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^4 \log (1+i c x)}{1-i c x} \, dx,x,x^2\right )+\frac{1}{16} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^4 \log (1+i c x)}{1+i c x} \, dx,x,x^2\right )\\ &=\frac{1}{32} x^8 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2-\frac{1}{16} b x^8 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )-\frac{1}{32} b^2 x^8 \log ^2\left (1+i c x^2\right )-\frac{1}{16} (i b) \operatorname{Subst}\left (\int \frac{\left (-\frac{i}{c}+\frac{i x}{c}\right )^4 (2 a+i b \log (x))}{x} \, dx,x,1-i c x^2\right )-\frac{1}{16} (i b c) \operatorname{Subst}\left (\int \left (-\frac{-2 i a+b \log (1-i c x)}{c^4}+\frac{i x (-2 i a+b \log (1-i c x))}{c^3}+\frac{x^2 (-2 i a+b \log (1-i c x))}{c^2}-\frac{i x^3 (-2 i a+b \log (1-i c x))}{c}-\frac{i (-2 i a+b \log (1-i c x))}{c^4 (-i+c x)}\right ) \, dx,x,x^2\right )+\frac{1}{16} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \left (-\frac{\log (1+i c x)}{c^4}+\frac{i x \log (1+i c x)}{c^3}+\frac{x^2 \log (1+i c x)}{c^2}-\frac{i x^3 \log (1+i c x)}{c}-\frac{i \log (1+i c x)}{c^4 (-i+c x)}\right ) \, dx,x,x^2\right )+\frac{1}{16} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \left (-\frac{\log (1+i c x)}{c^4}-\frac{i x \log (1+i c x)}{c^3}+\frac{x^2 \log (1+i c x)}{c^2}+\frac{i x^3 \log (1+i c x)}{c}+\frac{i \log (1+i c x)}{c^4 (i+c x)}\right ) \, dx,x,x^2\right )\\ &=\frac{1}{32} x^8 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2+\frac{1}{192} i b \left (2 a+i b \log \left (1-i c x^2\right )\right ) \left (\frac{48 \left (1-i c x^2\right )}{c^4}-\frac{36 \left (1-i c x^2\right )^2}{c^4}+\frac{16 \left (1-i c x^2\right )^3}{c^4}-\frac{3 \left (1-i c x^2\right )^4}{c^4}-\frac{12 \log \left (1-i c x^2\right )}{c^4}\right )-\frac{1}{16} b x^8 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )-\frac{1}{32} b^2 x^8 \log ^2\left (1+i c x^2\right )-\frac{1}{16} b \operatorname{Subst}\left (\int x^3 (-2 i a+b \log (1-i c x)) \, dx,x,x^2\right )-\frac{1}{16} b^2 \operatorname{Subst}\left (\int \frac{x \left (-48+36 x-16 x^2+3 x^3\right )+12 \log (x)}{12 c^4 x} \, dx,x,1-i c x^2\right )+\frac{(i b) \operatorname{Subst}\left (\int (-2 i a+b \log (1-i c x)) \, dx,x,x^2\right )}{16 c^3}-\frac{b \operatorname{Subst}\left (\int \frac{-2 i a+b \log (1-i c x)}{-i+c x} \, dx,x,x^2\right )}{16 c^3}-2 \frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \log (1+i c x) \, dx,x,x^2\right )}{16 c^3}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{-i+c x} \, dx,x,x^2\right )}{16 c^3}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{i+c x} \, dx,x,x^2\right )}{16 c^3}+\frac{b \operatorname{Subst}\left (\int x (-2 i a+b \log (1-i c x)) \, dx,x,x^2\right )}{16 c^2}-\frac{(i b) \operatorname{Subst}\left (\int x^2 (-2 i a+b \log (1-i c x)) \, dx,x,x^2\right )}{16 c}+2 \frac{\left (i b^2\right ) \operatorname{Subst}\left (\int x^2 \log (1+i c x) \, dx,x,x^2\right )}{16 c}\\ &=\frac{a b x^2}{8 c^3}-\frac{b x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right )}{32 c^2}+\frac{i b x^6 \left (2 i a-b \log \left (1-i c x^2\right )\right )}{48 c}+\frac{1}{64} b x^8 \left (2 i a-b \log \left (1-i c x^2\right )\right )+\frac{1}{32} x^8 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2+\frac{1}{192} i b \left (2 a+i b \log \left (1-i c x^2\right )\right ) \left (\frac{48 \left (1-i c x^2\right )}{c^4}-\frac{36 \left (1-i c x^2\right )^2}{c^4}+\frac{16 \left (1-i c x^2\right )^3}{c^4}-\frac{3 \left (1-i c x^2\right )^4}{c^4}-\frac{12 \log \left (1-i c x^2\right )}{c^4}\right )+\frac{b \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{16 c^4}-\frac{b^2 \log \left (\frac{1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{16 c^4}-\frac{1}{16} b x^8 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )-\frac{1}{32} b^2 x^8 \log ^2\left (1+i c x^2\right )+\frac{1}{48} b^2 \operatorname{Subst}\left (\int \frac{x^3}{1-i c x} \, dx,x,x^2\right )+2 \left (\frac{i b^2 x^6 \log \left (1+i c x^2\right )}{48 c}+\frac{1}{48} b^2 \operatorname{Subst}\left (\int \frac{x^3}{1+i c x} \, dx,x,x^2\right )\right )-\frac{b^2 \operatorname{Subst}\left (\int \frac{x \left (-48+36 x-16 x^2+3 x^3\right )+12 \log (x)}{x} \, dx,x,1-i c x^2\right )}{192 c^4}-2 \frac{b^2 \operatorname{Subst}\left (\int \log (x) \, dx,x,1+i c x^2\right )}{16 c^4}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+i c x^2\right )}{16 c^4}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \log (1-i c x) \, dx,x,x^2\right )}{16 c^3}-\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^2\right )}{16 c^3}+\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^2\right )}{16 c^3}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{1-i c x} \, dx,x,x^2\right )}{32 c}-\frac{1}{64} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^4}{1-i c x} \, dx,x,x^2\right )\\ &=\frac{a b x^2}{8 c^3}-\frac{b x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right )}{32 c^2}+\frac{i b x^6 \left (2 i a-b \log \left (1-i c x^2\right )\right )}{48 c}+\frac{1}{64} b x^8 \left (2 i a-b \log \left (1-i c x^2\right )\right )+\frac{1}{32} x^8 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2+\frac{1}{192} i b \left (2 a+i b \log \left (1-i c x^2\right )\right ) \left (\frac{48 \left (1-i c x^2\right )}{c^4}-\frac{36 \left (1-i c x^2\right )^2}{c^4}+\frac{16 \left (1-i c x^2\right )^3}{c^4}-\frac{3 \left (1-i c x^2\right )^4}{c^4}-\frac{12 \log \left (1-i c x^2\right )}{c^4}\right )+\frac{b \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{16 c^4}-\frac{b^2 \log \left (\frac{1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{16 c^4}-\frac{1}{16} b x^8 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )+\frac{b^2 \log ^2\left (1+i c x^2\right )}{32 c^4}-\frac{1}{32} b^2 x^8 \log ^2\left (1+i c x^2\right )-2 \left (-\frac{i b^2 x^2}{16 c^3}+\frac{b^2 \left (1+i c x^2\right ) \log \left (1+i c x^2\right )}{16 c^4}\right )+2 \left (\frac{i b^2 x^6 \log \left (1+i c x^2\right )}{48 c}+\frac{1}{48} b^2 \operatorname{Subst}\left (\int \left (\frac{i}{c^3}+\frac{x}{c^2}-\frac{i x^2}{c}-\frac{1}{c^3 (-i+c x)}\right ) \, dx,x,x^2\right )\right )+\frac{1}{48} b^2 \operatorname{Subst}\left (\int \left (-\frac{i}{c^3}+\frac{x}{c^2}+\frac{i x^2}{c}-\frac{1}{c^3 (i+c x)}\right ) \, dx,x,x^2\right )-\frac{b^2 \operatorname{Subst}\left (\int \left (-48+36 x-16 x^2+3 x^3+\frac{12 \log (x)}{x}\right ) \, dx,x,1-i c x^2\right )}{192 c^4}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1-i c x^2\right )}{16 c^4}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1+i c x^2\right )}{16 c^4}-\frac{b^2 \operatorname{Subst}\left (\int \log (x) \, dx,x,1-i c x^2\right )}{16 c^4}+\frac{\left (i b^2\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^2\right )}{32 c}-\frac{1}{64} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^4}-\frac{i x}{c^3}+\frac{x^2}{c^2}+\frac{i x^3}{c}+\frac{i}{c^4 (i+c x)}\right ) \, dx,x,x^2\right )\\ &=\frac{a b x^2}{8 c^3}-\frac{55 i b^2 x^2}{192 c^3}-\frac{5 b^2 x^4}{384 c^2}+\frac{i b^2 x^6}{576 c}+\frac{b^2 x^8}{256}-\frac{3 b^2 \left (1-i c x^2\right )^2}{32 c^4}+\frac{b^2 \left (1-i c x^2\right )^3}{36 c^4}-\frac{b^2 \left (1-i c x^2\right )^4}{256 c^4}-\frac{b^2 \left (1-i c x^2\right ) \log \left (1-i c x^2\right )}{16 c^4}-\frac{b x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right )}{32 c^2}+\frac{i b x^6 \left (2 i a-b \log \left (1-i c x^2\right )\right )}{48 c}+\frac{1}{64} b x^8 \left (2 i a-b \log \left (1-i c x^2\right )\right )+\frac{1}{32} x^8 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2+\frac{1}{192} i b \left (2 a+i b \log \left (1-i c x^2\right )\right ) \left (\frac{48 \left (1-i c x^2\right )}{c^4}-\frac{36 \left (1-i c x^2\right )^2}{c^4}+\frac{16 \left (1-i c x^2\right )^3}{c^4}-\frac{3 \left (1-i c x^2\right )^4}{c^4}-\frac{12 \log \left (1-i c x^2\right )}{c^4}\right )+\frac{b \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{16 c^4}-\frac{b^2 \log \left (\frac{1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{16 c^4}-\frac{1}{16} b x^8 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )+\frac{b^2 \log ^2\left (1+i c x^2\right )}{32 c^4}-\frac{1}{32} b^2 x^8 \log ^2\left (1+i c x^2\right )+2 \left (\frac{i b^2 x^2}{48 c^3}+\frac{b^2 x^4}{96 c^2}-\frac{i b^2 x^6}{144 c}-\frac{b^2 \log \left (i-c x^2\right )}{48 c^4}+\frac{i b^2 x^6 \log \left (1+i c x^2\right )}{48 c}\right )-2 \left (-\frac{i b^2 x^2}{16 c^3}+\frac{b^2 \left (1+i c x^2\right ) \log \left (1+i c x^2\right )}{16 c^4}\right )+\frac{5 b^2 \log \left (i+c x^2\right )}{192 c^4}-\frac{b^2 \text{Li}_2\left (\frac{1}{2} \left (1-i c x^2\right )\right )}{16 c^4}-\frac{b^2 \text{Li}_2\left (\frac{1}{2} \left (1+i c x^2\right )\right )}{16 c^4}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-i c x^2\right )}{16 c^4}\\ &=\frac{a b x^2}{8 c^3}-\frac{55 i b^2 x^2}{192 c^3}-\frac{5 b^2 x^4}{384 c^2}+\frac{i b^2 x^6}{576 c}+\frac{b^2 x^8}{256}-\frac{3 b^2 \left (1-i c x^2\right )^2}{32 c^4}+\frac{b^2 \left (1-i c x^2\right )^3}{36 c^4}-\frac{b^2 \left (1-i c x^2\right )^4}{256 c^4}-\frac{b^2 \left (1-i c x^2\right ) \log \left (1-i c x^2\right )}{16 c^4}-\frac{b^2 \log ^2\left (1-i c x^2\right )}{32 c^4}-\frac{b x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right )}{32 c^2}+\frac{i b x^6 \left (2 i a-b \log \left (1-i c x^2\right )\right )}{48 c}+\frac{1}{64} b x^8 \left (2 i a-b \log \left (1-i c x^2\right )\right )+\frac{1}{32} x^8 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2+\frac{1}{192} i b \left (2 a+i b \log \left (1-i c x^2\right )\right ) \left (\frac{48 \left (1-i c x^2\right )}{c^4}-\frac{36 \left (1-i c x^2\right )^2}{c^4}+\frac{16 \left (1-i c x^2\right )^3}{c^4}-\frac{3 \left (1-i c x^2\right )^4}{c^4}-\frac{12 \log \left (1-i c x^2\right )}{c^4}\right )+\frac{b \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{16 c^4}-\frac{b^2 \log \left (\frac{1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{16 c^4}-\frac{1}{16} b x^8 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )+\frac{b^2 \log ^2\left (1+i c x^2\right )}{32 c^4}-\frac{1}{32} b^2 x^8 \log ^2\left (1+i c x^2\right )+2 \left (\frac{i b^2 x^2}{48 c^3}+\frac{b^2 x^4}{96 c^2}-\frac{i b^2 x^6}{144 c}-\frac{b^2 \log \left (i-c x^2\right )}{48 c^4}+\frac{i b^2 x^6 \log \left (1+i c x^2\right )}{48 c}\right )-2 \left (-\frac{i b^2 x^2}{16 c^3}+\frac{b^2 \left (1+i c x^2\right ) \log \left (1+i c x^2\right )}{16 c^4}\right )+\frac{5 b^2 \log \left (i+c x^2\right )}{192 c^4}-\frac{b^2 \text{Li}_2\left (\frac{1}{2} \left (1-i c x^2\right )\right )}{16 c^4}-\frac{b^2 \text{Li}_2\left (\frac{1}{2} \left (1+i c x^2\right )\right )}{16 c^4}\\ \end{align*}

Mathematica [A]  time = 0.0909221, size = 121, normalized size = 0.98 \[ \frac{c x^2 \left (3 a^2 c^3 x^6-2 a b c^2 x^4+6 a b+b^2 c x^2\right )-2 b \tan ^{-1}\left (c x^2\right ) \left (a \left (3-3 c^4 x^8\right )+b c x^2 \left (c^2 x^4-3\right )\right )-4 b^2 \log \left (c^2 x^4+1\right )+3 b^2 \left (c^4 x^8-1\right ) \tan ^{-1}\left (c x^2\right )^2}{24 c^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7*(a + b*ArcTan[c*x^2])^2,x]

[Out]

(c*x^2*(6*a*b + b^2*c*x^2 - 2*a*b*c^2*x^4 + 3*a^2*c^3*x^6) - 2*b*(b*c*x^2*(-3 + c^2*x^4) + a*(3 - 3*c^4*x^8))*
ArcTan[c*x^2] + 3*b^2*(-1 + c^4*x^8)*ArcTan[c*x^2]^2 - 4*b^2*Log[1 + c^2*x^4])/(24*c^4)

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Maple [A]  time = 0.036, size = 151, normalized size = 1.2 \begin{align*}{\frac{{x}^{8}{a}^{2}}{8}}+{\frac{{b}^{2}{x}^{8} \left ( \arctan \left ( c{x}^{2} \right ) \right ) ^{2}}{8}}-{\frac{{b}^{2}\arctan \left ( c{x}^{2} \right ){x}^{6}}{12\,c}}+{\frac{{b}^{2}{x}^{2}\arctan \left ( c{x}^{2} \right ) }{4\,{c}^{3}}}-{\frac{{b}^{2} \left ( \arctan \left ( c{x}^{2} \right ) \right ) ^{2}}{8\,{c}^{4}}}+{\frac{{b}^{2}{x}^{4}}{24\,{c}^{2}}}-{\frac{{b}^{2}\ln \left ({c}^{2}{x}^{4}+1 \right ) }{6\,{c}^{4}}}+{\frac{ab{x}^{8}\arctan \left ( c{x}^{2} \right ) }{4}}-{\frac{ab{x}^{6}}{12\,c}}+{\frac{ab{x}^{2}}{4\,{c}^{3}}}-{\frac{ab\arctan \left ( c{x}^{2} \right ) }{4\,{c}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(a+b*arctan(c*x^2))^2,x)

[Out]

1/8*x^8*a^2+1/8*b^2*x^8*arctan(c*x^2)^2-1/12*b^2*arctan(c*x^2)/c*x^6+1/4*b^2*x^2*arctan(c*x^2)/c^3-1/8*b^2/c^4
*arctan(c*x^2)^2+1/24*b^2*x^4/c^2-1/6*b^2*ln(c^2*x^4+1)/c^4+1/4*a*b*x^8*arctan(c*x^2)-1/12*a*b/c*x^6+1/4*a*b*x
^2/c^3-1/4*a*b/c^4*arctan(c*x^2)

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Maxima [A]  time = 1.77938, size = 228, normalized size = 1.84 \begin{align*} \frac{1}{8} \, b^{2} x^{8} \arctan \left (c x^{2}\right )^{2} + \frac{1}{8} \, a^{2} x^{8} + \frac{1}{12} \,{\left (3 \, x^{8} \arctan \left (c x^{2}\right ) - c{\left (\frac{c^{2} x^{6} - 3 \, x^{2}}{c^{4}} + \frac{3 \, \arctan \left (c x^{2}\right )}{c^{5}}\right )}\right )} a b - \frac{1}{24} \,{\left (2 \, c{\left (\frac{c^{2} x^{6} - 3 \, x^{2}}{c^{4}} + \frac{3 \, \arctan \left (c x^{2}\right )}{c^{5}}\right )} \arctan \left (c x^{2}\right ) - \frac{c^{2} x^{4} + 3 \, \arctan \left (c x^{2}\right )^{2} - 3 \, \log \left (12 \, c^{7} x^{4} + 12 \, c^{5}\right ) - \log \left (c^{2} x^{4} + 1\right )}{c^{4}}\right )} b^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(a+b*arctan(c*x^2))^2,x, algorithm="maxima")

[Out]

1/8*b^2*x^8*arctan(c*x^2)^2 + 1/8*a^2*x^8 + 1/12*(3*x^8*arctan(c*x^2) - c*((c^2*x^6 - 3*x^2)/c^4 + 3*arctan(c*
x^2)/c^5))*a*b - 1/24*(2*c*((c^2*x^6 - 3*x^2)/c^4 + 3*arctan(c*x^2)/c^5)*arctan(c*x^2) - (c^2*x^4 + 3*arctan(c
*x^2)^2 - 3*log(12*c^7*x^4 + 12*c^5) - log(c^2*x^4 + 1))/c^4)*b^2

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Fricas [A]  time = 2.99301, size = 301, normalized size = 2.43 \begin{align*} \frac{3 \, a^{2} c^{4} x^{8} - 2 \, a b c^{3} x^{6} + b^{2} c^{2} x^{4} + 6 \, a b c x^{2} + 3 \,{\left (b^{2} c^{4} x^{8} - b^{2}\right )} \arctan \left (c x^{2}\right )^{2} + 6 \, a b \arctan \left (\frac{1}{c x^{2}}\right ) - 4 \, b^{2} \log \left (c^{2} x^{4} + 1\right ) + 2 \,{\left (3 \, a b c^{4} x^{8} - b^{2} c^{3} x^{6} + 3 \, b^{2} c x^{2}\right )} \arctan \left (c x^{2}\right )}{24 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(a+b*arctan(c*x^2))^2,x, algorithm="fricas")

[Out]

1/24*(3*a^2*c^4*x^8 - 2*a*b*c^3*x^6 + b^2*c^2*x^4 + 6*a*b*c*x^2 + 3*(b^2*c^4*x^8 - b^2)*arctan(c*x^2)^2 + 6*a*
b*arctan(1/(c*x^2)) - 4*b^2*log(c^2*x^4 + 1) + 2*(3*a*b*c^4*x^8 - b^2*c^3*x^6 + 3*b^2*c*x^2)*arctan(c*x^2))/c^
4

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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**7*(a+b*atan(c*x**2))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.19591, size = 196, normalized size = 1.58 \begin{align*} \frac{3 \, a^{2} c x^{8} + 2 \,{\left (3 \, c x^{8} \arctan \left (c x^{2}\right ) - \frac{3 \, \arctan \left (c x^{2}\right )}{c^{3}} - \frac{c^{9} x^{6} - 3 \, c^{7} x^{2}}{c^{9}}\right )} a b +{\left (3 \, c x^{8} \arctan \left (c x^{2}\right )^{2} - \frac{2 \, c^{3} x^{6} \arctan \left (c x^{2}\right ) - c^{2} x^{4} - 6 \, c x^{2} \arctan \left (c x^{2}\right ) + 3 \, \arctan \left (c x^{2}\right )^{2} + 4 \, \log \left (c^{2} x^{4} + 1\right )}{c^{3}}\right )} b^{2}}{24 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(a+b*arctan(c*x^2))^2,x, algorithm="giac")

[Out]

1/24*(3*a^2*c*x^8 + 2*(3*c*x^8*arctan(c*x^2) - 3*arctan(c*x^2)/c^3 - (c^9*x^6 - 3*c^7*x^2)/c^9)*a*b + (3*c*x^8
*arctan(c*x^2)^2 - (2*c^3*x^6*arctan(c*x^2) - c^2*x^4 - 6*c*x^2*arctan(c*x^2) + 3*arctan(c*x^2)^2 + 4*log(c^2*
x^4 + 1))/c^3)*b^2)/c