\(\int x^2 (c+a^2 c x^2) \arctan (a x)^3 \, dx\) [364]

Optimal result

Integrand size = 20, antiderivative size = 211 \[ \int x^2 \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=-\frac {c x^2}{20 a}+\frac {c x \arctan (a x)}{10 a^2}+\frac {1}{10} c x^3 \arctan (a x)-\frac {c \arctan (a x)^2}{20 a^3}-\frac {c x^2 \arctan (a x)^2}{5 a}-\frac {3}{20} a c x^4 \arctan (a x)^2-\frac {2 i c \arctan (a x)^3}{15 a^3}+\frac {1}{3} c x^3 \arctan (a x)^3+\frac {1}{5} a^2 c x^5 \arctan (a x)^3-\frac {2 c \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{5 a^3}-\frac {2 i c \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{5 a^3}-\frac {c \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{5 a^3} \]

[Out]

-1/20*c*x^2/a+1/10*c*x*arctan(a*x)/a^2+1/10*c*x^3*arctan(a*x)-1/20*c*arctan(a*x)^2/a^3-1/5*c*x^2*arctan(a*x)^2
/a-3/20*a*c*x^4*arctan(a*x)^2-2/15*I*c*arctan(a*x)^3/a^3+1/3*c*x^3*arctan(a*x)^3+1/5*a^2*c*x^5*arctan(a*x)^3-2
/5*c*arctan(a*x)^2*ln(2/(1+I*a*x))/a^3-2/5*I*c*arctan(a*x)*polylog(2,1-2/(1+I*a*x))/a^3-1/5*c*polylog(3,1-2/(1
+I*a*x))/a^3

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5070, 4946, 5036, 4930, 266, 5004, 5040, 4964, 5114, 6745, 272, 45} \[ \int x^2 \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=-\frac {2 i c \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{5 a^3}-\frac {2 i c \arctan (a x)^3}{15 a^3}-\frac {c \arctan (a x)^2}{20 a^3}-\frac {2 c \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{5 a^3}-\frac {c \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{5 a^3}+\frac {1}{5} a^2 c x^5 \arctan (a x)^3+\frac {c x \arctan (a x)}{10 a^2}-\frac {3}{20} a c x^4 \arctan (a x)^2+\frac {1}{3} c x^3 \arctan (a x)^3+\frac {1}{10} c x^3 \arctan (a x)-\frac {c x^2 \arctan (a x)^2}{5 a}-\frac {c x^2}{20 a} \]

[In]

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

[Out]

-1/20*(c*x^2)/a + (c*x*ArcTan[a*x])/(10*a^2) + (c*x^3*ArcTan[a*x])/10 - (c*ArcTan[a*x]^2)/(20*a^3) - (c*x^2*Ar
cTan[a*x]^2)/(5*a) - (3*a*c*x^4*ArcTan[a*x]^2)/20 - (((2*I)/15)*c*ArcTan[a*x]^3)/a^3 + (c*x^3*ArcTan[a*x]^3)/3
+ (a^2*c*x^5*ArcTan[a*x]^3)/5 - (2*c*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/(5*a^3) - (((2*I)/5)*c*ArcTan[a*x]*Pol
yLog[2, 1 - 2/(1 + I*a*x)])/a^3 - (c*PolyLog[3, 1 - 2/(1 + I*a*x)])/(5*a^3)

Rule 45

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 266

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

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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]))

Rule 5114

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

Rule 6745

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

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

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.81 \[ \int x^2 \left (c+a^2 c x^2\right ) \arctan (a x)^3 \, dx=\frac {c \left (-3-3 a^2 x^2+6 a x \arctan (a x)+6 a^3 x^3 \arctan (a x)-3 \arctan (a x)^2-12 a^2 x^2 \arctan (a x)^2-9 a^4 x^4 \arctan (a x)^2+8 i \arctan (a x)^3+20 a^3 x^3 \arctan (a x)^3+12 a^5 x^5 \arctan (a x)^3-24 \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )+24 i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )-12 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )\right )}{60 a^3} \]

[In]

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

[Out]

(c*(-3 - 3*a^2*x^2 + 6*a*x*ArcTan[a*x] + 6*a^3*x^3*ArcTan[a*x] - 3*ArcTan[a*x]^2 - 12*a^2*x^2*ArcTan[a*x]^2 -
9*a^4*x^4*ArcTan[a*x]^2 + (8*I)*ArcTan[a*x]^3 + 20*a^3*x^3*ArcTan[a*x]^3 + 12*a^5*x^5*ArcTan[a*x]^3 - 24*ArcTa
n[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] + (24*I)*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] - 12*PolyLog[3
, -E^((2*I)*ArcTan[a*x])]))/(60*a^3)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 46.49 (sec) , antiderivative size = 900, normalized size of antiderivative = 4.27

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(900\)
default	\(\text {Expression too large to display}\)	\(900\)
parts	\(\text {Expression too large to display}\)	\(902\)

[In]

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

[Out]

1/a^3*(1/5*c*arctan(a*x)^3*a^5*x^5+1/3*c*arctan(a*x)^3*a^3*x^3-1/5*c*(3/4*a^4*arctan(a*x)^2*x^4+x^2*arctan(a*x
)^2*a^2-arctan(a*x)^2*ln(a^2*x^2+1)+2*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))-1/12*I*(-6*Pi*arctan(a*x)^
2*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3+12*Pi*arctan(a*x)^2*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*csgn(I*(
(1+I*a*x)^2/(a^2*x^2+1)+1))-6*Pi*arctan(a*x)^2*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I*((1+I*a*x)^2/(a^2*
x^2+1)+1))^2+6*Pi*arctan(a*x)^2*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))^2*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))-12*Pi*ar
ctan(a*x)^2*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^2+6*Pi*arctan(a*x)^2*csgn(I*(1
+I*a*x)^2/(a^2*x^2+1))^3+6*Pi*arctan(a*x)^2*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)
^2)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)-6*Pi*arctan(a*x)^2*csgn(I*(1+I*a*x)^2/(a^2*x
^2+1))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2-6*Pi*arctan(a*x)^2*csgn(I/((1+I*a*x)^2/
(a^2*x^2+1)+1)^2)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2+6*Pi*arctan(a*x)^2*csgn(I*(1
+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3+8*arctan(a*x)^3+3*I*a^2*x^2-6*I*arctan(a*x)*a*x+3*I*arc
tan(a*x)^2-6*I*arctan(a*x)*a^3*x^3+24*I*ln(2)*arctan(a*x)^2+3*I)-2*I*arctan(a*x)*polylog(2,-(1+I*a*x)^2/(a^2*x
^2+1))+polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/120*(3*a^2*c*x^5 + 5*c*x^3)*arctan(a*x)^3 - 1/160*(3*a^2*c*x^5 + 5*c*x^3)*arctan(a*x)*log(a^2*x^2 + 1)^2 + i
ntegrate(1/160*(140*(a^4*c*x^6 + 2*a^2*c*x^4 + c*x^2)*arctan(a*x)^3 - 4*(3*a^3*c*x^5 + 5*a*c*x^3)*arctan(a*x)^
2 + 4*(3*a^4*c*x^6 + 5*a^2*c*x^4)*arctan(a*x)*log(a^2*x^2 + 1) + (3*a^3*c*x^5 + 5*a*c*x^3 + 15*(a^4*c*x^6 + 2*
a^2*c*x^4 + c*x^2)*arctan(a*x))*log(a^2*x^2 + 1)^2)/(a^2*x^2 + 1), x)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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