\(\int x^3 (c+a^2 c x^2)^2 \arctan (a x)^3 \, dx\) [371]

Optimal result

Integrand size = 22, antiderivative size = 313 \[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\frac {c^2 x}{21 a^3}-\frac {c^2 x^3}{168 a}-\frac {1}{280} a c^2 x^5-\frac {c^2 \arctan (a x)}{21 a^4}-\frac {5 c^2 x^2 \arctan (a x)}{168 a^2}+\frac {1}{28} c^2 x^4 \arctan (a x)+\frac {1}{56} a^2 c^2 x^6 \arctan (a x)+\frac {2 i c^2 \arctan (a x)^2}{21 a^4}+\frac {c^2 x \arctan (a x)^2}{8 a^3}-\frac {c^2 x^3 \arctan (a x)^2}{24 a}-\frac {1}{8} a c^2 x^5 \arctan (a x)^2-\frac {3}{56} a^3 c^2 x^7 \arctan (a x)^2-\frac {c^2 \arctan (a x)^3}{24 a^4}+\frac {1}{4} c^2 x^4 \arctan (a x)^3+\frac {1}{3} a^2 c^2 x^6 \arctan (a x)^3+\frac {1}{8} a^4 c^2 x^8 \arctan (a x)^3+\frac {4 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{21 a^4}+\frac {2 i c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{21 a^4} \]

[Out]

1/21*c^2*x/a^3-1/168*c^2*x^3/a-1/280*a*c^2*x^5-1/21*c^2*arctan(a*x)/a^4-5/168*c^2*x^2*arctan(a*x)/a^2+1/28*c^2
*x^4*arctan(a*x)+1/56*a^2*c^2*x^6*arctan(a*x)+2/21*I*c^2*polylog(2,1-2/(1+I*a*x))/a^4+1/8*c^2*x*arctan(a*x)^2/
a^3-1/24*c^2*x^3*arctan(a*x)^2/a-1/8*a*c^2*x^5*arctan(a*x)^2-3/56*a^3*c^2*x^7*arctan(a*x)^2-1/24*c^2*arctan(a*
x)^3/a^4+1/4*c^2*x^4*arctan(a*x)^3+1/3*a^2*c^2*x^6*arctan(a*x)^3+1/8*a^4*c^2*x^8*arctan(a*x)^3+4/21*c^2*arctan
(a*x)*ln(2/(1+I*a*x))/a^4+2/21*I*c^2*arctan(a*x)^2/a^4

Rubi [A] (verified)

Time = 1.65 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 106, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {5068, 4946, 5036, 327, 209, 5040, 4964, 2449, 2352, 4930, 5004, 308} \[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\frac {1}{8} a^4 c^2 x^8 \arctan (a x)^3-\frac {c^2 \arctan (a x)^3}{24 a^4}+\frac {2 i c^2 \arctan (a x)^2}{21 a^4}-\frac {c^2 \arctan (a x)}{21 a^4}+\frac {4 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{21 a^4}+\frac {2 i c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{21 a^4}-\frac {3}{56} a^3 c^2 x^7 \arctan (a x)^2+\frac {c^2 x \arctan (a x)^2}{8 a^3}+\frac {c^2 x}{21 a^3}+\frac {1}{3} a^2 c^2 x^6 \arctan (a x)^3+\frac {1}{56} a^2 c^2 x^6 \arctan (a x)-\frac {5 c^2 x^2 \arctan (a x)}{168 a^2}-\frac {1}{8} a c^2 x^5 \arctan (a x)^2+\frac {1}{4} c^2 x^4 \arctan (a x)^3+\frac {1}{28} c^2 x^4 \arctan (a x)-\frac {c^2 x^3 \arctan (a x)^2}{24 a}-\frac {1}{280} a c^2 x^5-\frac {c^2 x^3}{168 a} \]

[In]

Int[x^3*(c + a^2*c*x^2)^2*ArcTan[a*x]^3,x]

[Out]

(c^2*x)/(21*a^3) - (c^2*x^3)/(168*a) - (a*c^2*x^5)/280 - (c^2*ArcTan[a*x])/(21*a^4) - (5*c^2*x^2*ArcTan[a*x])/
(168*a^2) + (c^2*x^4*ArcTan[a*x])/28 + (a^2*c^2*x^6*ArcTan[a*x])/56 + (((2*I)/21)*c^2*ArcTan[a*x]^2)/a^4 + (c^
2*x*ArcTan[a*x]^2)/(8*a^3) - (c^2*x^3*ArcTan[a*x]^2)/(24*a) - (a*c^2*x^5*ArcTan[a*x]^2)/8 - (3*a^3*c^2*x^7*Arc
Tan[a*x]^2)/56 - (c^2*ArcTan[a*x]^3)/(24*a^4) + (c^2*x^4*ArcTan[a*x]^3)/4 + (a^2*c^2*x^6*ArcTan[a*x]^3)/3 + (a
^4*c^2*x^8*ArcTan[a*x]^3)/8 + (4*c^2*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(21*a^4) + (((2*I)/21)*c^2*PolyLog[2, 1 -
2/(1 + I*a*x)])/a^4

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[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 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 4930

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

Rule 4946

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

Rule 4964

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(
Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 + c^2*x
^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]

Rule 5004

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTan[c*x])^(p +
1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]

Rule 5036

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2/
e, Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x])^p, x], x] - Dist[d*(f^2/e), Int[(f*x)^(m - 2)*((a + b*ArcTan[c*x])^p/
(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]

Rule 5040

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*((a + b*ArcT
an[c*x])^(p + 1)/(b*e*(p + 1))), x] - Dist[1/(c*d), Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b
, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rule 5068

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Int[Ex
pandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e,
c^2*d] && IGtQ[p, 0] && IGtQ[q, 1] && (EqQ[p, 1] || IntegerQ[m])

Rubi steps \begin{align*} \text {integral}& = \int \left (c^2 x^3 \arctan (a x)^3+2 a^2 c^2 x^5 \arctan (a x)^3+a^4 c^2 x^7 \arctan (a x)^3\right ) \, dx \\ & = c^2 \int x^3 \arctan (a x)^3 \, dx+\left (2 a^2 c^2\right ) \int x^5 \arctan (a x)^3 \, dx+\left (a^4 c^2\right ) \int x^7 \arctan (a x)^3 \, dx \\ & = \frac {1}{4} c^2 x^4 \arctan (a x)^3+\frac {1}{3} a^2 c^2 x^6 \arctan (a x)^3+\frac {1}{8} a^4 c^2 x^8 \arctan (a x)^3-\frac {1}{4} \left (3 a c^2\right ) \int \frac {x^4 \arctan (a x)^2}{1+a^2 x^2} \, dx-\left (a^3 c^2\right ) \int \frac {x^6 \arctan (a x)^2}{1+a^2 x^2} \, dx-\frac {1}{8} \left (3 a^5 c^2\right ) \int \frac {x^8 \arctan (a x)^2}{1+a^2 x^2} \, dx \\ & = \frac {1}{4} c^2 x^4 \arctan (a x)^3+\frac {1}{3} a^2 c^2 x^6 \arctan (a x)^3+\frac {1}{8} a^4 c^2 x^8 \arctan (a x)^3-\frac {\left (3 c^2\right ) \int x^2 \arctan (a x)^2 \, dx}{4 a}+\frac {\left (3 c^2\right ) \int \frac {x^2 \arctan (a x)^2}{1+a^2 x^2} \, dx}{4 a}-\left (a c^2\right ) \int x^4 \arctan (a x)^2 \, dx+\left (a c^2\right ) \int \frac {x^4 \arctan (a x)^2}{1+a^2 x^2} \, dx-\frac {1}{8} \left (3 a^3 c^2\right ) \int x^6 \arctan (a x)^2 \, dx+\frac {1}{8} \left (3 a^3 c^2\right ) \int \frac {x^6 \arctan (a x)^2}{1+a^2 x^2} \, dx \\ & = -\frac {c^2 x^3 \arctan (a x)^2}{4 a}-\frac {1}{5} a c^2 x^5 \arctan (a x)^2-\frac {3}{56} a^3 c^2 x^7 \arctan (a x)^2+\frac {1}{4} c^2 x^4 \arctan (a x)^3+\frac {1}{3} a^2 c^2 x^6 \arctan (a x)^3+\frac {1}{8} a^4 c^2 x^8 \arctan (a x)^3+\frac {1}{2} c^2 \int \frac {x^3 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {\left (3 c^2\right ) \int \arctan (a x)^2 \, dx}{4 a^3}-\frac {\left (3 c^2\right ) \int \frac {\arctan (a x)^2}{1+a^2 x^2} \, dx}{4 a^3}+\frac {c^2 \int x^2 \arctan (a x)^2 \, dx}{a}-\frac {c^2 \int \frac {x^2 \arctan (a x)^2}{1+a^2 x^2} \, dx}{a}+\frac {1}{8} \left (3 a c^2\right ) \int x^4 \arctan (a x)^2 \, dx-\frac {1}{8} \left (3 a c^2\right ) \int \frac {x^4 \arctan (a x)^2}{1+a^2 x^2} \, dx+\frac {1}{5} \left (2 a^2 c^2\right ) \int \frac {x^5 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{28} \left (3 a^4 c^2\right ) \int \frac {x^7 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = \frac {3 c^2 x \arctan (a x)^2}{4 a^3}+\frac {c^2 x^3 \arctan (a x)^2}{12 a}-\frac {1}{8} a c^2 x^5 \arctan (a x)^2-\frac {3}{56} a^3 c^2 x^7 \arctan (a x)^2-\frac {c^2 \arctan (a x)^3}{4 a^4}+\frac {1}{4} c^2 x^4 \arctan (a x)^3+\frac {1}{3} a^2 c^2 x^6 \arctan (a x)^3+\frac {1}{8} a^4 c^2 x^8 \arctan (a x)^3+\frac {1}{5} \left (2 c^2\right ) \int x^3 \arctan (a x) \, dx-\frac {1}{5} \left (2 c^2\right ) \int \frac {x^3 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{3} \left (2 c^2\right ) \int \frac {x^3 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {c^2 \int \arctan (a x)^2 \, dx}{a^3}+\frac {c^2 \int \frac {\arctan (a x)^2}{1+a^2 x^2} \, dx}{a^3}+\frac {c^2 \int x \arctan (a x) \, dx}{2 a^2}-\frac {c^2 \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx}{2 a^2}-\frac {\left (3 c^2\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx}{2 a^2}-\frac {\left (3 c^2\right ) \int x^2 \arctan (a x)^2 \, dx}{8 a}+\frac {\left (3 c^2\right ) \int \frac {x^2 \arctan (a x)^2}{1+a^2 x^2} \, dx}{8 a}+\frac {1}{28} \left (3 a^2 c^2\right ) \int x^5 \arctan (a x) \, dx-\frac {1}{28} \left (3 a^2 c^2\right ) \int \frac {x^5 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{20} \left (3 a^2 c^2\right ) \int \frac {x^5 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = \frac {c^2 x^2 \arctan (a x)}{4 a^2}+\frac {1}{10} c^2 x^4 \arctan (a x)+\frac {1}{56} a^2 c^2 x^6 \arctan (a x)+\frac {i c^2 \arctan (a x)^2}{a^4}-\frac {c^2 x \arctan (a x)^2}{4 a^3}-\frac {c^2 x^3 \arctan (a x)^2}{24 a}-\frac {1}{8} a c^2 x^5 \arctan (a x)^2-\frac {3}{56} a^3 c^2 x^7 \arctan (a x)^2+\frac {c^2 \arctan (a x)^3}{12 a^4}+\frac {1}{4} c^2 x^4 \arctan (a x)^3+\frac {1}{3} a^2 c^2 x^6 \arctan (a x)^3+\frac {1}{8} a^4 c^2 x^8 \arctan (a x)^3-\frac {1}{28} \left (3 c^2\right ) \int x^3 \arctan (a x) \, dx+\frac {1}{28} \left (3 c^2\right ) \int \frac {x^3 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{20} \left (3 c^2\right ) \int x^3 \arctan (a x) \, dx+\frac {1}{20} \left (3 c^2\right ) \int \frac {x^3 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{4} c^2 \int \frac {x^3 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {\left (3 c^2\right ) \int \arctan (a x)^2 \, dx}{8 a^3}-\frac {\left (3 c^2\right ) \int \frac {\arctan (a x)^2}{1+a^2 x^2} \, dx}{8 a^3}+\frac {c^2 \int \frac {\arctan (a x)}{i-a x} \, dx}{2 a^3}+\frac {\left (3 c^2\right ) \int \frac {\arctan (a x)}{i-a x} \, dx}{2 a^3}-\frac {\left (2 c^2\right ) \int x \arctan (a x) \, dx}{5 a^2}+\frac {\left (2 c^2\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx}{5 a^2}-\frac {\left (2 c^2\right ) \int x \arctan (a x) \, dx}{3 a^2}+\frac {\left (2 c^2\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx}{3 a^2}+\frac {\left (2 c^2\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx}{a^2}-\frac {c^2 \int \frac {x^2}{1+a^2 x^2} \, dx}{4 a}-\frac {1}{10} \left (a c^2\right ) \int \frac {x^4}{1+a^2 x^2} \, dx-\frac {1}{56} \left (a^3 c^2\right ) \int \frac {x^6}{1+a^2 x^2} \, dx \\ & = -\frac {c^2 x}{4 a^3}-\frac {17 c^2 x^2 \arctan (a x)}{60 a^2}+\frac {1}{28} c^2 x^4 \arctan (a x)+\frac {1}{56} a^2 c^2 x^6 \arctan (a x)-\frac {8 i c^2 \arctan (a x)^2}{15 a^4}+\frac {c^2 x \arctan (a x)^2}{8 a^3}-\frac {c^2 x^3 \arctan (a x)^2}{24 a}-\frac {1}{8} a c^2 x^5 \arctan (a x)^2-\frac {3}{56} a^3 c^2 x^7 \arctan (a x)^2-\frac {c^2 \arctan (a x)^3}{24 a^4}+\frac {1}{4} c^2 x^4 \arctan (a x)^3+\frac {1}{3} a^2 c^2 x^6 \arctan (a x)^3+\frac {1}{8} a^4 c^2 x^8 \arctan (a x)^3+\frac {2 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a^4}+\frac {c^2 \int \frac {1}{1+a^2 x^2} \, dx}{4 a^3}-\frac {\left (2 c^2\right ) \int \frac {\arctan (a x)}{i-a x} \, dx}{5 a^3}-\frac {c^2 \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{2 a^3}-\frac {\left (2 c^2\right ) \int \frac {\arctan (a x)}{i-a x} \, dx}{3 a^3}-\frac {\left (3 c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{2 a^3}-\frac {\left (2 c^2\right ) \int \frac {\arctan (a x)}{i-a x} \, dx}{a^3}+\frac {\left (3 c^2\right ) \int x \arctan (a x) \, dx}{28 a^2}-\frac {\left (3 c^2\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx}{28 a^2}+\frac {\left (3 c^2\right ) \int x \arctan (a x) \, dx}{20 a^2}-\frac {\left (3 c^2\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx}{20 a^2}+\frac {c^2 \int x \arctan (a x) \, dx}{4 a^2}-\frac {c^2 \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx}{4 a^2}-\frac {\left (3 c^2\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx}{4 a^2}+\frac {c^2 \int \frac {x^2}{1+a^2 x^2} \, dx}{5 a}+\frac {c^2 \int \frac {x^2}{1+a^2 x^2} \, dx}{3 a}+\frac {1}{112} \left (3 a c^2\right ) \int \frac {x^4}{1+a^2 x^2} \, dx+\frac {1}{80} \left (3 a c^2\right ) \int \frac {x^4}{1+a^2 x^2} \, dx-\frac {1}{10} \left (a c^2\right ) \int \left (-\frac {1}{a^4}+\frac {x^2}{a^2}+\frac {1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx-\frac {1}{56} \left (a^3 c^2\right ) \int \left (\frac {1}{a^6}-\frac {x^2}{a^4}+\frac {x^4}{a^2}-\frac {1}{a^6 \left (1+a^2 x^2\right )}\right ) \, dx \\ & = \frac {307 c^2 x}{840 a^3}-\frac {23 c^2 x^3}{840 a}-\frac {1}{280} a c^2 x^5+\frac {c^2 \arctan (a x)}{4 a^4}-\frac {5 c^2 x^2 \arctan (a x)}{168 a^2}+\frac {1}{28} c^2 x^4 \arctan (a x)+\frac {1}{56} a^2 c^2 x^6 \arctan (a x)+\frac {2 i c^2 \arctan (a x)^2}{21 a^4}+\frac {c^2 x \arctan (a x)^2}{8 a^3}-\frac {c^2 x^3 \arctan (a x)^2}{24 a}-\frac {1}{8} a c^2 x^5 \arctan (a x)^2-\frac {3}{56} a^3 c^2 x^7 \arctan (a x)^2-\frac {c^2 \arctan (a x)^3}{24 a^4}+\frac {1}{4} c^2 x^4 \arctan (a x)^3+\frac {1}{3} a^2 c^2 x^6 \arctan (a x)^3+\frac {1}{8} a^4 c^2 x^8 \arctan (a x)^3-\frac {16 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^4}+\frac {\left (i c^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{2 a^4}+\frac {\left (3 i c^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{2 a^4}+\frac {c^2 \int \frac {1}{1+a^2 x^2} \, dx}{56 a^3}-\frac {c^2 \int \frac {1}{1+a^2 x^2} \, dx}{10 a^3}+\frac {\left (3 c^2\right ) \int \frac {\arctan (a x)}{i-a x} \, dx}{28 a^3}+\frac {\left (3 c^2\right ) \int \frac {\arctan (a x)}{i-a x} \, dx}{20 a^3}-\frac {c^2 \int \frac {1}{1+a^2 x^2} \, dx}{5 a^3}+\frac {c^2 \int \frac {\arctan (a x)}{i-a x} \, dx}{4 a^3}-\frac {c^2 \int \frac {1}{1+a^2 x^2} \, dx}{3 a^3}+\frac {\left (2 c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^3}+\frac {\left (2 c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{3 a^3}+\frac {\left (3 c^2\right ) \int \frac {\arctan (a x)}{i-a x} \, dx}{4 a^3}+\frac {\left (2 c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^3}-\frac {\left (3 c^2\right ) \int \frac {x^2}{1+a^2 x^2} \, dx}{56 a}-\frac {\left (3 c^2\right ) \int \frac {x^2}{1+a^2 x^2} \, dx}{40 a}-\frac {c^2 \int \frac {x^2}{1+a^2 x^2} \, dx}{8 a}+\frac {1}{112} \left (3 a c^2\right ) \int \left (-\frac {1}{a^4}+\frac {x^2}{a^2}+\frac {1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx+\frac {1}{80} \left (3 a c^2\right ) \int \left (-\frac {1}{a^4}+\frac {x^2}{a^2}+\frac {1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx \\ & = \frac {c^2 x}{21 a^3}-\frac {c^2 x^3}{168 a}-\frac {1}{280} a c^2 x^5-\frac {307 c^2 \arctan (a x)}{840 a^4}-\frac {5 c^2 x^2 \arctan (a x)}{168 a^2}+\frac {1}{28} c^2 x^4 \arctan (a x)+\frac {1}{56} a^2 c^2 x^6 \arctan (a x)+\frac {2 i c^2 \arctan (a x)^2}{21 a^4}+\frac {c^2 x \arctan (a x)^2}{8 a^3}-\frac {c^2 x^3 \arctan (a x)^2}{24 a}-\frac {1}{8} a c^2 x^5 \arctan (a x)^2-\frac {3}{56} a^3 c^2 x^7 \arctan (a x)^2-\frac {c^2 \arctan (a x)^3}{24 a^4}+\frac {1}{4} c^2 x^4 \arctan (a x)^3+\frac {1}{3} a^2 c^2 x^6 \arctan (a x)^3+\frac {1}{8} a^4 c^2 x^8 \arctan (a x)^3+\frac {4 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{21 a^4}+\frac {i c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^4}-\frac {\left (2 i c^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{5 a^4}-\frac {\left (2 i c^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{3 a^4}-\frac {\left (2 i c^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{a^4}+\frac {\left (3 c^2\right ) \int \frac {1}{1+a^2 x^2} \, dx}{112 a^3}+\frac {\left (3 c^2\right ) \int \frac {1}{1+a^2 x^2} \, dx}{80 a^3}+\frac {\left (3 c^2\right ) \int \frac {1}{1+a^2 x^2} \, dx}{56 a^3}+\frac {\left (3 c^2\right ) \int \frac {1}{1+a^2 x^2} \, dx}{40 a^3}-\frac {\left (3 c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{28 a^3}+\frac {c^2 \int \frac {1}{1+a^2 x^2} \, dx}{8 a^3}-\frac {\left (3 c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{20 a^3}-\frac {c^2 \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{4 a^3}-\frac {\left (3 c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{4 a^3} \\ & = \frac {c^2 x}{21 a^3}-\frac {c^2 x^3}{168 a}-\frac {1}{280} a c^2 x^5-\frac {c^2 \arctan (a x)}{21 a^4}-\frac {5 c^2 x^2 \arctan (a x)}{168 a^2}+\frac {1}{28} c^2 x^4 \arctan (a x)+\frac {1}{56} a^2 c^2 x^6 \arctan (a x)+\frac {2 i c^2 \arctan (a x)^2}{21 a^4}+\frac {c^2 x \arctan (a x)^2}{8 a^3}-\frac {c^2 x^3 \arctan (a x)^2}{24 a}-\frac {1}{8} a c^2 x^5 \arctan (a x)^2-\frac {3}{56} a^3 c^2 x^7 \arctan (a x)^2-\frac {c^2 \arctan (a x)^3}{24 a^4}+\frac {1}{4} c^2 x^4 \arctan (a x)^3+\frac {1}{3} a^2 c^2 x^6 \arctan (a x)^3+\frac {1}{8} a^4 c^2 x^8 \arctan (a x)^3+\frac {4 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{21 a^4}-\frac {8 i c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{15 a^4}+\frac {\left (3 i c^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{28 a^4}+\frac {\left (3 i c^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{20 a^4}+\frac {\left (i c^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{4 a^4}+\frac {\left (3 i c^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{4 a^4} \\ & = \frac {c^2 x}{21 a^3}-\frac {c^2 x^3}{168 a}-\frac {1}{280} a c^2 x^5-\frac {c^2 \arctan (a x)}{21 a^4}-\frac {5 c^2 x^2 \arctan (a x)}{168 a^2}+\frac {1}{28} c^2 x^4 \arctan (a x)+\frac {1}{56} a^2 c^2 x^6 \arctan (a x)+\frac {2 i c^2 \arctan (a x)^2}{21 a^4}+\frac {c^2 x \arctan (a x)^2}{8 a^3}-\frac {c^2 x^3 \arctan (a x)^2}{24 a}-\frac {1}{8} a c^2 x^5 \arctan (a x)^2-\frac {3}{56} a^3 c^2 x^7 \arctan (a x)^2-\frac {c^2 \arctan (a x)^3}{24 a^4}+\frac {1}{4} c^2 x^4 \arctan (a x)^3+\frac {1}{3} a^2 c^2 x^6 \arctan (a x)^3+\frac {1}{8} a^4 c^2 x^8 \arctan (a x)^3+\frac {4 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{21 a^4}+\frac {2 i c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{21 a^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.03 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.53 \[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\frac {c^2 \left (40 a x-5 a^3 x^3-3 a^5 x^5-5 \left (16 i-21 a x+7 a^3 x^3+21 a^5 x^5+9 a^7 x^7\right ) \arctan (a x)^2+35 \left (1+a^2 x^2\right )^3 \left (-1+3 a^2 x^2\right ) \arctan (a x)^3+5 \arctan (a x) \left (-8-5 a^2 x^2+6 a^4 x^4+3 a^6 x^6+32 \log \left (1+e^{2 i \arctan (a x)}\right )\right )-80 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )\right )}{840 a^4} \]

[In]

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

[Out]

(c^2*(40*a*x - 5*a^3*x^3 - 3*a^5*x^5 - 5*(16*I - 21*a*x + 7*a^3*x^3 + 21*a^5*x^5 + 9*a^7*x^7)*ArcTan[a*x]^2 +
35*(1 + a^2*x^2)^3*(-1 + 3*a^2*x^2)*ArcTan[a*x]^3 + 5*ArcTan[a*x]*(-8 - 5*a^2*x^2 + 6*a^4*x^4 + 3*a^6*x^6 + 32
*Log[1 + E^((2*I)*ArcTan[a*x])]) - (80*I)*PolyLog[2, -E^((2*I)*ArcTan[a*x])]))/(840*a^4)

Maple [A] (verified)

Time = 3.50 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.06

method	result	size
derivativedivides	\(\frac {\frac {c^{2} \arctan \left (a x \right )^{3} a^{8} x^{8}}{8}+\frac {c^{2} \arctan \left (a x \right )^{3} a^{6} x^{6}}{3}+\frac {a^{4} c^{2} x^{4} \arctan \left (a x \right )^{3}}{4}-\frac {c^{2} \arctan \left (a x \right )^{3}}{24}-\frac {c^{2} \left (\frac {3 \arctan \left (a x \right )^{2} a^{7} x^{7}}{7}+a^{5} \arctan \left (a x \right )^{2} x^{5}+\frac {a^{3} \arctan \left (a x \right )^{2} x^{3}}{3}-a \arctan \left (a x \right )^{2} x -\frac {a^{6} \arctan \left (a x \right ) x^{6}}{7}-\frac {2 \arctan \left (a x \right ) a^{4} x^{4}}{7}+\frac {5 a^{2} \arctan \left (a x \right ) x^{2}}{21}+\frac {16 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{21}+\frac {a^{5} x^{5}}{35}+\frac {a^{3} x^{3}}{21}-\frac {8 a x}{21}+\frac {8 \arctan \left (a x \right )}{21}+\frac {8 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{21}-\frac {8 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{21}\right )}{8}}{a^{4}}\)	\(331\)
default	\(\frac {\frac {c^{2} \arctan \left (a x \right )^{3} a^{8} x^{8}}{8}+\frac {c^{2} \arctan \left (a x \right )^{3} a^{6} x^{6}}{3}+\frac {a^{4} c^{2} x^{4} \arctan \left (a x \right )^{3}}{4}-\frac {c^{2} \arctan \left (a x \right )^{3}}{24}-\frac {c^{2} \left (\frac {3 \arctan \left (a x \right )^{2} a^{7} x^{7}}{7}+a^{5} \arctan \left (a x \right )^{2} x^{5}+\frac {a^{3} \arctan \left (a x \right )^{2} x^{3}}{3}-a \arctan \left (a x \right )^{2} x -\frac {a^{6} \arctan \left (a x \right ) x^{6}}{7}-\frac {2 \arctan \left (a x \right ) a^{4} x^{4}}{7}+\frac {5 a^{2} \arctan \left (a x \right ) x^{2}}{21}+\frac {16 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{21}+\frac {a^{5} x^{5}}{35}+\frac {a^{3} x^{3}}{21}-\frac {8 a x}{21}+\frac {8 \arctan \left (a x \right )}{21}+\frac {8 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{21}-\frac {8 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{21}\right )}{8}}{a^{4}}\)	\(331\)
parts	\(\frac {a^{4} c^{2} x^{8} \arctan \left (a x \right )^{3}}{8}+\frac {a^{2} c^{2} x^{6} \arctan \left (a x \right )^{3}}{3}+\frac {c^{2} x^{4} \arctan \left (a x \right )^{3}}{4}-\frac {c^{2} \left (\frac {3 a^{3} \arctan \left (a x \right )^{2} x^{7}}{7}+a \arctan \left (a x \right )^{2} x^{5}+\frac {\arctan \left (a x \right )^{2} x^{3}}{3 a}-\frac {\arctan \left (a x \right )^{2} x}{a^{3}}+\frac {\arctan \left (a x \right )^{3}}{a^{4}}-\frac {2 \left (\frac {3 a^{6} \arctan \left (a x \right ) x^{6}}{2}+3 \arctan \left (a x \right ) a^{4} x^{4}-\frac {5 a^{2} \arctan \left (a x \right ) x^{2}}{2}-8 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {3 a^{5} x^{5}}{10}-\frac {a^{3} x^{3}}{2}+4 a x -4 \arctan \left (a x \right )-4 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )+4 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )+7 \arctan \left (a x \right )^{3}\right )}{21 a^{4}}\right )}{8}\)	\(337\)

[In]

int(x^3*(a^2*c*x^2+c)^2*arctan(a*x)^3,x,method=_RETURNVERBOSE)

[Out]

1/a^4*(1/8*c^2*arctan(a*x)^3*a^8*x^8+1/3*c^2*arctan(a*x)^3*a^6*x^6+1/4*a^4*c^2*x^4*arctan(a*x)^3-1/24*c^2*arct
an(a*x)^3-1/8*c^2*(3/7*arctan(a*x)^2*a^7*x^7+a^5*arctan(a*x)^2*x^5+1/3*a^3*arctan(a*x)^2*x^3-a*arctan(a*x)^2*x
-1/7*a^6*arctan(a*x)*x^6-2/7*arctan(a*x)*a^4*x^4+5/21*a^2*arctan(a*x)*x^2+16/21*arctan(a*x)*ln(a^2*x^2+1)+1/35
*a^5*x^5+1/21*a^3*x^3-8/21*a*x+8/21*arctan(a*x)+8/21*I*(ln(a*x-I)*ln(a^2*x^2+1)-dilog(-1/2*I*(I+a*x))-ln(a*x-I
)*ln(-1/2*I*(I+a*x))-1/2*ln(a*x-I)^2)-8/21*I*(ln(I+a*x)*ln(a^2*x^2+1)-dilog(1/2*I*(a*x-I))-ln(I+a*x)*ln(1/2*I*
(a*x-I))-1/2*ln(I+a*x)^2)))

Fricas [F]

\[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} x^{3} \arctan \left (a x\right )^{3} \,d x } \]

[In]

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

[Out]

integral((a^4*c^2*x^7 + 2*a^2*c^2*x^5 + c^2*x^3)*arctan(a*x)^3, x)

Sympy [F]

\[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=c^{2} \left (\int x^{3} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int 2 a^{2} x^{5} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{4} x^{7} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]

[In]

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

[Out]

c**2*(Integral(x**3*atan(a*x)**3, x) + Integral(2*a**2*x**5*atan(a*x)**3, x) + Integral(a**4*x**7*atan(a*x)**3
, x))

Maxima [F]

\[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} x^{3} \arctan \left (a x\right )^{3} \,d x } \]

[In]

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

[Out]

1/2688*(28*(129024*a^9*c^2*integrate(1/2688*x^9*arctan(a*x)^3/(a^5*x^2 + a^3), x) - 24192*a^8*c^2*integrate(1/
2688*x^8*arctan(a*x)^2/(a^5*x^2 + a^3), x) - 6048*a^8*c^2*integrate(1/2688*x^8*log(a^2*x^2 + 1)^2/(a^5*x^2 + a
^3), x) - 3456*a^8*c^2*integrate(1/2688*x^8*log(a^2*x^2 + 1)/(a^5*x^2 + a^3), x) + 387072*a^7*c^2*integrate(1/
2688*x^7*arctan(a*x)^3/(a^5*x^2 + a^3), x) + 6912*a^7*c^2*integrate(1/2688*x^7*arctan(a*x)/(a^5*x^2 + a^3), x)
- 64512*a^6*c^2*integrate(1/2688*x^6*arctan(a*x)^2/(a^5*x^2 + a^3), x) - 16128*a^6*c^2*integrate(1/2688*x^6*l
og(a^2*x^2 + 1)^2/(a^5*x^2 + a^3), x) - 8064*a^6*c^2*integrate(1/2688*x^6*log(a^2*x^2 + 1)/(a^5*x^2 + a^3), x)
+ 387072*a^5*c^2*integrate(1/2688*x^5*arctan(a*x)^3/(a^5*x^2 + a^3), x) + 16128*a^5*c^2*integrate(1/2688*x^5*
arctan(a*x)/(a^5*x^2 + a^3), x) - 48384*a^4*c^2*integrate(1/2688*x^4*arctan(a*x)^2/(a^5*x^2 + a^3), x) - 12096
*a^4*c^2*integrate(1/2688*x^4*log(a^2*x^2 + 1)^2/(a^5*x^2 + a^3), x) - 2688*a^4*c^2*integrate(1/2688*x^4*log(a
^2*x^2 + 1)/(a^5*x^2 + a^3), x) + 129024*a^3*c^2*integrate(1/2688*x^3*arctan(a*x)^3/(a^5*x^2 + a^3), x) + 5376
*a^3*c^2*integrate(1/2688*x^3*arctan(a*x)/(a^5*x^2 + a^3), x) + 8064*a^2*c^2*integrate(1/2688*x^2*log(a^2*x^2
+ 1)/(a^5*x^2 + a^3), x) - 16128*a*c^2*integrate(1/2688*x*arctan(a*x)/(a^5*x^2 + a^3), x) + 2016*c^2*integrate
(1/2688*log(a^2*x^2 + 1)^2/(a^5*x^2 + a^3), x) + c^2*arctan(a*x)^3/a^4)*a^4 + 56*(3*a^8*c^2*x^8 + 8*a^6*c^2*x^
6 + 6*a^4*c^2*x^4 - c^2)*arctan(a*x)^3 - 4*(9*a^7*c^2*x^7 + 21*a^5*c^2*x^5 + 7*a^3*c^2*x^3 - 21*a*c^2*x)*arcta
n(a*x)^2 + (9*a^7*c^2*x^7 + 21*a^5*c^2*x^5 + 7*a^3*c^2*x^3 - 21*a*c^2*x)*log(a^2*x^2 + 1)^2)/a^4

Giac [F]

\[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} x^{3} \arctan \left (a x\right )^{3} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3 \, dx=\int x^3\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^2 \,d x \]

[In]

int(x^3*atan(a*x)^3*(c + a^2*c*x^2)^2,x)

[Out]

int(x^3*atan(a*x)^3*(c + a^2*c*x^2)^2, x)