\(\int x^3 (c+a^2 c x^2) \arctan (a x)^3 \, dx\) [363]

Optimal result

Integrand size = 20, antiderivative size = 219 \[ \int x^3 \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=\frac {c x}{15 a^3}-\frac {c x^3}{60 a}-\frac {c \arctan (a x)}{15 a^4}-\frac {c x^2 \arctan (a x)}{60 a^2}+\frac {1}{20} c x^4 \arctan (a x)+\frac {7 i c \arctan (a x)^2}{30 a^4}+\frac {c x \arctan (a x)^2}{4 a^3}-\frac {c x^3 \arctan (a x)^2}{12 a}-\frac {1}{10} a c x^5 \arctan (a x)^2-\frac {c \arctan (a x)^3}{12 a^4}+\frac {1}{4} c x^4 \arctan (a x)^3+\frac {1}{6} a^2 c x^6 \arctan (a x)^3+\frac {7 c \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^4}+\frac {7 i c \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{30 a^4} \]

[Out]

1/15*c*x/a^3-1/60*c*x^3/a-1/15*c*arctan(a*x)/a^4-1/60*c*x^2*arctan(a*x)/a^2+1/20*c*x^4*arctan(a*x)+7/30*I*c*ar
ctan(a*x)^2/a^4+1/4*c*x*arctan(a*x)^2/a^3-1/12*c*x^3*arctan(a*x)^2/a-1/10*a*c*x^5*arctan(a*x)^2-1/12*c*arctan(
a*x)^3/a^4+1/4*c*x^4*arctan(a*x)^3+1/6*a^2*c*x^6*arctan(a*x)^3+7/15*c*arctan(a*x)*ln(2/(1+I*a*x))/a^4+7/30*I*c
*polylog(2,1-2/(1+I*a*x))/a^4

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 52, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5070, 4946, 5036, 327, 209, 5040, 4964, 2449, 2352, 4930, 5004, 308} \[ \int x^3 \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=-\frac {c \arctan (a x)^3}{12 a^4}+\frac {7 i c \arctan (a x)^2}{30 a^4}-\frac {c \arctan (a x)}{15 a^4}+\frac {7 c \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^4}+\frac {7 i c \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{30 a^4}+\frac {c x \arctan (a x)^2}{4 a^3}+\frac {c x}{15 a^3}+\frac {1}{6} a^2 c x^6 \arctan (a x)^3-\frac {c x^2 \arctan (a x)}{60 a^2}-\frac {1}{10} a c x^5 \arctan (a x)^2+\frac {1}{4} c x^4 \arctan (a x)^3+\frac {1}{20} c x^4 \arctan (a x)-\frac {c x^3 \arctan (a x)^2}{12 a}-\frac {c x^3}{60 a} \]

[In]

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

[Out]

(c*x)/(15*a^3) - (c*x^3)/(60*a) - (c*ArcTan[a*x])/(15*a^4) - (c*x^2*ArcTan[a*x])/(60*a^2) + (c*x^4*ArcTan[a*x]
)/20 + (((7*I)/30)*c*ArcTan[a*x]^2)/a^4 + (c*x*ArcTan[a*x]^2)/(4*a^3) - (c*x^3*ArcTan[a*x]^2)/(12*a) - (a*c*x^
5*ArcTan[a*x]^2)/10 - (c*ArcTan[a*x]^3)/(12*a^4) + (c*x^4*ArcTan[a*x]^3)/4 + (a^2*c*x^6*ArcTan[a*x]^3)/6 + (7*
c*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(15*a^4) + (((7*I)/30)*c*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 5070

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

Rubi steps \begin{align*} \text {integral}& = c \int x^3 \arctan (a x)^3 \, dx+\left (a^2 c\right ) \int x^5 \arctan (a x)^3 \, dx \\ & = \frac {1}{4} c x^4 \arctan (a x)^3+\frac {1}{6} a^2 c x^6 \arctan (a x)^3-\frac {1}{4} (3 a c) \int \frac {x^4 \arctan (a x)^2}{1+a^2 x^2} \, dx-\frac {1}{2} \left (a^3 c\right ) \int \frac {x^6 \arctan (a x)^2}{1+a^2 x^2} \, dx \\ & = \frac {1}{4} c x^4 \arctan (a x)^3+\frac {1}{6} a^2 c x^6 \arctan (a x)^3-\frac {(3 c) \int x^2 \arctan (a x)^2 \, dx}{4 a}+\frac {(3 c) \int \frac {x^2 \arctan (a x)^2}{1+a^2 x^2} \, dx}{4 a}-\frac {1}{2} (a c) \int x^4 \arctan (a x)^2 \, dx+\frac {1}{2} (a c) \int \frac {x^4 \arctan (a x)^2}{1+a^2 x^2} \, dx \\ & = -\frac {c x^3 \arctan (a x)^2}{4 a}-\frac {1}{10} a c x^5 \arctan (a x)^2+\frac {1}{4} c x^4 \arctan (a x)^3+\frac {1}{6} a^2 c x^6 \arctan (a x)^3+\frac {1}{2} c \int \frac {x^3 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {(3 c) \int \arctan (a x)^2 \, dx}{4 a^3}-\frac {(3 c) \int \frac {\arctan (a x)^2}{1+a^2 x^2} \, dx}{4 a^3}+\frac {c \int x^2 \arctan (a x)^2 \, dx}{2 a}-\frac {c \int \frac {x^2 \arctan (a x)^2}{1+a^2 x^2} \, dx}{2 a}+\frac {1}{5} \left (a^2 c\right ) \int \frac {x^5 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = \frac {3 c x \arctan (a x)^2}{4 a^3}-\frac {c x^3 \arctan (a x)^2}{12 a}-\frac {1}{10} a c x^5 \arctan (a x)^2-\frac {c \arctan (a x)^3}{4 a^4}+\frac {1}{4} c x^4 \arctan (a x)^3+\frac {1}{6} a^2 c x^6 \arctan (a x)^3+\frac {1}{5} c \int x^3 \arctan (a x) \, dx-\frac {1}{5} c \int \frac {x^3 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{3} c \int \frac {x^3 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {c \int \arctan (a x)^2 \, dx}{2 a^3}+\frac {c \int \frac {\arctan (a x)^2}{1+a^2 x^2} \, dx}{2 a^3}+\frac {c \int x \arctan (a x) \, dx}{2 a^2}-\frac {c \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx}{2 a^2}-\frac {(3 c) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx}{2 a^2} \\ & = \frac {c x^2 \arctan (a x)}{4 a^2}+\frac {1}{20} c x^4 \arctan (a x)+\frac {i c \arctan (a x)^2}{a^4}+\frac {c x \arctan (a x)^2}{4 a^3}-\frac {c x^3 \arctan (a x)^2}{12 a}-\frac {1}{10} a c x^5 \arctan (a x)^2-\frac {c \arctan (a x)^3}{12 a^4}+\frac {1}{4} c x^4 \arctan (a x)^3+\frac {1}{6} a^2 c x^6 \arctan (a x)^3+\frac {c \int \frac {\arctan (a x)}{i-a x} \, dx}{2 a^3}+\frac {(3 c) \int \frac {\arctan (a x)}{i-a x} \, dx}{2 a^3}-\frac {c \int x \arctan (a x) \, dx}{5 a^2}+\frac {c \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx}{5 a^2}-\frac {c \int x \arctan (a x) \, dx}{3 a^2}+\frac {c \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx}{3 a^2}+\frac {c \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx}{a^2}-\frac {c \int \frac {x^2}{1+a^2 x^2} \, dx}{4 a}-\frac {1}{20} (a c) \int \frac {x^4}{1+a^2 x^2} \, dx \\ & = -\frac {c x}{4 a^3}-\frac {c x^2 \arctan (a x)}{60 a^2}+\frac {1}{20} c x^4 \arctan (a x)+\frac {7 i c \arctan (a x)^2}{30 a^4}+\frac {c x \arctan (a x)^2}{4 a^3}-\frac {c x^3 \arctan (a x)^2}{12 a}-\frac {1}{10} a c x^5 \arctan (a x)^2-\frac {c \arctan (a x)^3}{12 a^4}+\frac {1}{4} c x^4 \arctan (a x)^3+\frac {1}{6} a^2 c x^6 \arctan (a x)^3+\frac {2 c \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a^4}-\frac {c \int \frac {\arctan (a x)}{i-a x} \, dx}{5 a^3}+\frac {c \int \frac {1}{1+a^2 x^2} \, dx}{4 a^3}-\frac {c \int \frac {\arctan (a x)}{i-a x} \, dx}{3 a^3}-\frac {c \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{2 a^3}-\frac {c \int \frac {\arctan (a x)}{i-a x} \, dx}{a^3}-\frac {(3 c) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{2 a^3}+\frac {c \int \frac {x^2}{1+a^2 x^2} \, dx}{10 a}+\frac {c \int \frac {x^2}{1+a^2 x^2} \, dx}{6 a}-\frac {1}{20} (a c) \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 x}{15 a^3}-\frac {c x^3}{60 a}+\frac {c \arctan (a x)}{4 a^4}-\frac {c x^2 \arctan (a x)}{60 a^2}+\frac {1}{20} c x^4 \arctan (a x)+\frac {7 i c \arctan (a x)^2}{30 a^4}+\frac {c x \arctan (a x)^2}{4 a^3}-\frac {c x^3 \arctan (a x)^2}{12 a}-\frac {1}{10} a c x^5 \arctan (a x)^2-\frac {c \arctan (a x)^3}{12 a^4}+\frac {1}{4} c x^4 \arctan (a x)^3+\frac {1}{6} a^2 c x^6 \arctan (a x)^3+\frac {7 c \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^4}+\frac {(i c) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{2 a^4}+\frac {(3 i c) \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 \int \frac {1}{1+a^2 x^2} \, dx}{20 a^3}-\frac {c \int \frac {1}{1+a^2 x^2} \, dx}{10 a^3}-\frac {c \int \frac {1}{1+a^2 x^2} \, dx}{6 a^3}+\frac {c \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^3}+\frac {c \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{3 a^3}+\frac {c \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^3} \\ & = \frac {c x}{15 a^3}-\frac {c x^3}{60 a}-\frac {c \arctan (a x)}{15 a^4}-\frac {c x^2 \arctan (a x)}{60 a^2}+\frac {1}{20} c x^4 \arctan (a x)+\frac {7 i c \arctan (a x)^2}{30 a^4}+\frac {c x \arctan (a x)^2}{4 a^3}-\frac {c x^3 \arctan (a x)^2}{12 a}-\frac {1}{10} a c x^5 \arctan (a x)^2-\frac {c \arctan (a x)^3}{12 a^4}+\frac {1}{4} c x^4 \arctan (a x)^3+\frac {1}{6} a^2 c x^6 \arctan (a x)^3+\frac {7 c \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^4}+\frac {i c \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^4}-\frac {(i c) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{5 a^4}-\frac {(i c) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{3 a^4}-\frac {(i c) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{a^4} \\ & = \frac {c x}{15 a^3}-\frac {c x^3}{60 a}-\frac {c \arctan (a x)}{15 a^4}-\frac {c x^2 \arctan (a x)}{60 a^2}+\frac {1}{20} c x^4 \arctan (a x)+\frac {7 i c \arctan (a x)^2}{30 a^4}+\frac {c x \arctan (a x)^2}{4 a^3}-\frac {c x^3 \arctan (a x)^2}{12 a}-\frac {1}{10} a c x^5 \arctan (a x)^2-\frac {c \arctan (a x)^3}{12 a^4}+\frac {1}{4} c x^4 \arctan (a x)^3+\frac {1}{6} a^2 c x^6 \arctan (a x)^3+\frac {7 c \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^4}+\frac {7 i c \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{30 a^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.62 \[ \int x^3 \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=\frac {c \left (4 a x-a^3 x^3-\left (14 i-15 a x+5 a^3 x^3+6 a^5 x^5\right ) \arctan (a x)^2+5 \left (-1+3 a^4 x^4+2 a^6 x^6\right ) \arctan (a x)^3+\arctan (a x) \left (-4-a^2 x^2+3 a^4 x^4+28 \log \left (1+e^{2 i \arctan (a x)}\right )\right )-14 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )\right )}{60 a^4} \]

[In]

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

[Out]

(c*(4*a*x - a^3*x^3 - (14*I - 15*a*x + 5*a^3*x^3 + 6*a^5*x^5)*ArcTan[a*x]^2 + 5*(-1 + 3*a^4*x^4 + 2*a^6*x^6)*A
rcTan[a*x]^3 + ArcTan[a*x]*(-4 - a^2*x^2 + 3*a^4*x^4 + 28*Log[1 + E^((2*I)*ArcTan[a*x])]) - (14*I)*PolyLog[2,
-E^((2*I)*ArcTan[a*x])]))/(60*a^4)

Maple [A] (verified)

Time = 1.84 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.24

method	result	size
derivativedivides	\(\frac {\frac {c \arctan \left (a x \right )^{3} a^{6} x^{6}}{6}+\frac {c \arctan \left (a x \right )^{3} a^{4} x^{4}}{4}-\frac {c \left (\frac {2 a^{5} \arctan \left (a x \right )^{2} x^{5}}{5}+\frac {a^{3} \arctan \left (a x \right )^{2} x^{3}}{3}-a \arctan \left (a x \right )^{2} x +\frac {\arctan \left (a x \right )^{3}}{3}-\frac {\arctan \left (a x \right ) a^{4} x^{4}}{5}+\frac {a^{2} \arctan \left (a x \right ) x^{2}}{15}+\frac {14 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{15}+\frac {a^{3} x^{3}}{15}-\frac {4 a x}{15}+\frac {4 \arctan \left (a x \right )}{15}+\frac {7 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{15}-\frac {7 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{15}\right )}{4}}{a^{4}}\)	\(272\)
default	\(\frac {\frac {c \arctan \left (a x \right )^{3} a^{6} x^{6}}{6}+\frac {c \arctan \left (a x \right )^{3} a^{4} x^{4}}{4}-\frac {c \left (\frac {2 a^{5} \arctan \left (a x \right )^{2} x^{5}}{5}+\frac {a^{3} \arctan \left (a x \right )^{2} x^{3}}{3}-a \arctan \left (a x \right )^{2} x +\frac {\arctan \left (a x \right )^{3}}{3}-\frac {\arctan \left (a x \right ) a^{4} x^{4}}{5}+\frac {a^{2} \arctan \left (a x \right ) x^{2}}{15}+\frac {14 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{15}+\frac {a^{3} x^{3}}{15}-\frac {4 a x}{15}+\frac {4 \arctan \left (a x \right )}{15}+\frac {7 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{15}-\frac {7 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{15}\right )}{4}}{a^{4}}\)	\(272\)
parts	\(\frac {a^{2} c \,x^{6} \arctan \left (a x \right )^{3}}{6}+\frac {c \,x^{4} \arctan \left (a x \right )^{3}}{4}-\frac {c \left (\frac {2 a \arctan \left (a x \right )^{2} x^{5}}{5}+\frac {\arctan \left (a x \right )^{2} x^{3}}{3 a}-\frac {\arctan \left (a x \right )^{2} x}{a^{3}}+\frac {\arctan \left (a x \right )^{3}}{a^{4}}-\frac {2 \left (\frac {3 \arctan \left (a x \right ) a^{4} x^{4}}{2}-\frac {a^{2} \arctan \left (a x \right ) x^{2}}{2}-7 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a^{3} x^{3}}{2}+2 a x -2 \arctan \left (a x \right )-\frac {7 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{2}+\frac {7 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{2}+5 \arctan \left (a x \right )^{3}\right )}{15 a^{4}}\right )}{4}\)	\(281\)

[In]

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

[Out]

1/a^4*(1/6*c*arctan(a*x)^3*a^6*x^6+1/4*c*arctan(a*x)^3*a^4*x^4-1/4*c*(2/5*a^5*arctan(a*x)^2*x^5+1/3*a^3*arctan
(a*x)^2*x^3-a*arctan(a*x)^2*x+1/3*arctan(a*x)^3-1/5*arctan(a*x)*a^4*x^4+1/15*a^2*arctan(a*x)*x^2+14/15*arctan(
a*x)*ln(a^2*x^2+1)+1/15*a^3*x^3-4/15*a*x+4/15*arctan(a*x)+7/15*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)-7/15*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]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/960*(20*(23040*a^7*c*integrate(1/960*x^7*arctan(a*x)^3/(a^5*x^2 + a^3), x) - 5760*a^6*c*integrate(1/960*x^6*
arctan(a*x)^2/(a^5*x^2 + a^3), x) - 1440*a^6*c*integrate(1/960*x^6*log(a^2*x^2 + 1)^2/(a^5*x^2 + a^3), x) - 11
52*a^6*c*integrate(1/960*x^6*log(a^2*x^2 + 1)/(a^5*x^2 + a^3), x) + 46080*a^5*c*integrate(1/960*x^5*arctan(a*x
)^3/(a^5*x^2 + a^3), x) + 2304*a^5*c*integrate(1/960*x^5*arctan(a*x)/(a^5*x^2 + a^3), x) - 8640*a^4*c*integrat
e(1/960*x^4*arctan(a*x)^2/(a^5*x^2 + a^3), x) - 2160*a^4*c*integrate(1/960*x^4*log(a^2*x^2 + 1)^2/(a^5*x^2 + a
^3), x) - 960*a^4*c*integrate(1/960*x^4*log(a^2*x^2 + 1)/(a^5*x^2 + a^3), x) + 23040*a^3*c*integrate(1/960*x^3
*arctan(a*x)^3/(a^5*x^2 + a^3), x) + 1920*a^3*c*integrate(1/960*x^3*arctan(a*x)/(a^5*x^2 + a^3), x) + 2880*a^2
*c*integrate(1/960*x^2*log(a^2*x^2 + 1)/(a^5*x^2 + a^3), x) - 5760*a*c*integrate(1/960*x*arctan(a*x)/(a^5*x^2
+ a^3), x) + 720*c*integrate(1/960*log(a^2*x^2 + 1)^2/(a^5*x^2 + a^3), x) + c*arctan(a*x)^3/a^4)*a^4 + 40*(2*a
^6*c*x^6 + 3*a^4*c*x^4 - c)*arctan(a*x)^3 - 4*(6*a^5*c*x^5 + 5*a^3*c*x^3 - 15*a*c*x)*arctan(a*x)^2 + (6*a^5*c*
x^5 + 5*a^3*c*x^3 - 15*a*c*x)*log(a^2*x^2 + 1)^2)/a^4

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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