\(\int x (c+a^2 c x^2)^3 \arctan (a x)^3 \, dx\) [381]

Optimal result

Integrand size = 20, antiderivative size = 308 \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=-\frac {19 c^3 x}{140 a}-\frac {19}{840} a c^3 x^3-\frac {1}{280} a^3 c^3 x^5+\frac {3 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}{35 a^2}+\frac {9 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}{280 a^2}+\frac {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)}{56 a^2}-\frac {6 i c^3 \arctan (a x)^2}{35 a^2}-\frac {6 c^3 x \arctan (a x)^2}{35 a}-\frac {3 c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^2}{35 a}-\frac {9 c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{140 a}-\frac {3 c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^2}{56 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \arctan (a x)^3}{8 a^2}-\frac {12 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{35 a^2}-\frac {6 i c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{35 a^2} \]

[Out]

-19/140*c^3*x/a-19/840*a*c^3*x^3-1/280*a^3*c^3*x^5+3/35*c^3*(a^2*x^2+1)*arctan(a*x)/a^2+9/280*c^3*(a^2*x^2+1)^
2*arctan(a*x)/a^2+1/56*c^3*(a^2*x^2+1)^3*arctan(a*x)/a^2-6/35*I*c^3*arctan(a*x)^2/a^2-6/35*c^3*x*arctan(a*x)^2
/a-3/35*c^3*x*(a^2*x^2+1)*arctan(a*x)^2/a-9/140*c^3*x*(a^2*x^2+1)^2*arctan(a*x)^2/a-3/56*c^3*x*(a^2*x^2+1)^3*a
rctan(a*x)^2/a+1/8*c^3*(a^2*x^2+1)^4*arctan(a*x)^3/a^2-12/35*c^3*arctan(a*x)*ln(2/(1+I*a*x))/a^2-6/35*I*c^3*po
lylog(2,1-2/(1+I*a*x))/a^2

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {5050, 5000, 4930, 5040, 4964, 2449, 2352, 8, 200} \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=-\frac {1}{280} a^3 c^3 x^5-\frac {3 c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^2}{56 a}-\frac {9 c^3 x \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{140 a}-\frac {3 c^3 x \left (a^2 x^2+1\right ) \arctan (a x)^2}{35 a}+\frac {c^3 \left (a^2 x^2+1\right )^4 \arctan (a x)^3}{8 a^2}+\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)}{56 a^2}+\frac {9 c^3 \left (a^2 x^2+1\right )^2 \arctan (a x)}{280 a^2}+\frac {3 c^3 \left (a^2 x^2+1\right ) \arctan (a x)}{35 a^2}-\frac {6 i c^3 \arctan (a x)^2}{35 a^2}-\frac {12 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{35 a^2}-\frac {6 i c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{35 a^2}-\frac {6 c^3 x \arctan (a x)^2}{35 a}-\frac {19}{840} a c^3 x^3-\frac {19 c^3 x}{140 a} \]

[In]

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

[Out]

(-19*c^3*x)/(140*a) - (19*a*c^3*x^3)/840 - (a^3*c^3*x^5)/280 + (3*c^3*(1 + a^2*x^2)*ArcTan[a*x])/(35*a^2) + (9
*c^3*(1 + a^2*x^2)^2*ArcTan[a*x])/(280*a^2) + (c^3*(1 + a^2*x^2)^3*ArcTan[a*x])/(56*a^2) - (((6*I)/35)*c^3*Arc
Tan[a*x]^2)/a^2 - (6*c^3*x*ArcTan[a*x]^2)/(35*a) - (3*c^3*x*(1 + a^2*x^2)*ArcTan[a*x]^2)/(35*a) - (9*c^3*x*(1
+ a^2*x^2)^2*ArcTan[a*x]^2)/(140*a) - (3*c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x]^2)/(56*a) + (c^3*(1 + a^2*x^2)^4*Ar
cTan[a*x]^3)/(8*a^2) - (12*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(35*a^2) - (((6*I)/35)*c^3*PolyLog[2, 1 - 2/(1
+ I*a*x)])/a^2

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

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 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 5000

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[(-b)*p*(d + e*x^2)^
q*((a + b*ArcTan[c*x])^(p - 1)/(2*c*q*(2*q + 1))), x] + (Dist[2*d*(q/(2*q + 1)), Int[(d + e*x^2)^(q - 1)*(a +
b*ArcTan[c*x])^p, x], x] + Dist[b^2*d*p*((p - 1)/(2*q*(2*q + 1))), Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])
^(p - 2), x], x] + Simp[x*(d + e*x^2)^q*((a + b*ArcTan[c*x])^p/(2*q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] &&
EqQ[e, c^2*d] && GtQ[q, 0] && GtQ[p, 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 5050

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

Rubi steps \begin{align*} \text {integral}& = \frac {c^3 \left (1+a^2 x^2\right )^4 \arctan (a x)^3}{8 a^2}-\frac {3 \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx}{8 a} \\ & = \frac {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)}{56 a^2}-\frac {3 c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^2}{56 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \arctan (a x)^3}{8 a^2}-\frac {c \int \left (c+a^2 c x^2\right )^2 \, dx}{56 a}-\frac {(9 c) \int \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx}{28 a} \\ & = \frac {9 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}{280 a^2}+\frac {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)}{56 a^2}-\frac {9 c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{140 a}-\frac {3 c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^2}{56 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \arctan (a x)^3}{8 a^2}-\frac {c \int \left (c^2+2 a^2 c^2 x^2+a^4 c^2 x^4\right ) \, dx}{56 a}-\frac {\left (9 c^2\right ) \int \left (c+a^2 c x^2\right ) \, dx}{280 a}-\frac {\left (9 c^2\right ) \int \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx}{35 a} \\ & = -\frac {c^3 x}{20 a}-\frac {19}{840} a c^3 x^3-\frac {1}{280} a^3 c^3 x^5+\frac {3 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}{35 a^2}+\frac {9 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}{280 a^2}+\frac {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)}{56 a^2}-\frac {3 c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^2}{35 a}-\frac {9 c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{140 a}-\frac {3 c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^2}{56 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \arctan (a x)^3}{8 a^2}-\frac {\left (3 c^3\right ) \int 1 \, dx}{35 a}-\frac {\left (6 c^3\right ) \int \arctan (a x)^2 \, dx}{35 a} \\ & = -\frac {19 c^3 x}{140 a}-\frac {19}{840} a c^3 x^3-\frac {1}{280} a^3 c^3 x^5+\frac {3 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}{35 a^2}+\frac {9 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}{280 a^2}+\frac {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)}{56 a^2}-\frac {6 c^3 x \arctan (a x)^2}{35 a}-\frac {3 c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^2}{35 a}-\frac {9 c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{140 a}-\frac {3 c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^2}{56 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \arctan (a x)^3}{8 a^2}+\frac {1}{35} \left (12 c^3\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx \\ & = -\frac {19 c^3 x}{140 a}-\frac {19}{840} a c^3 x^3-\frac {1}{280} a^3 c^3 x^5+\frac {3 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}{35 a^2}+\frac {9 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}{280 a^2}+\frac {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)}{56 a^2}-\frac {6 i c^3 \arctan (a x)^2}{35 a^2}-\frac {6 c^3 x \arctan (a x)^2}{35 a}-\frac {3 c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^2}{35 a}-\frac {9 c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{140 a}-\frac {3 c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^2}{56 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \arctan (a x)^3}{8 a^2}-\frac {\left (12 c^3\right ) \int \frac {\arctan (a x)}{i-a x} \, dx}{35 a} \\ & = -\frac {19 c^3 x}{140 a}-\frac {19}{840} a c^3 x^3-\frac {1}{280} a^3 c^3 x^5+\frac {3 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}{35 a^2}+\frac {9 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}{280 a^2}+\frac {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)}{56 a^2}-\frac {6 i c^3 \arctan (a x)^2}{35 a^2}-\frac {6 c^3 x \arctan (a x)^2}{35 a}-\frac {3 c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^2}{35 a}-\frac {9 c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{140 a}-\frac {3 c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^2}{56 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \arctan (a x)^3}{8 a^2}-\frac {12 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{35 a^2}+\frac {\left (12 c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{35 a} \\ & = -\frac {19 c^3 x}{140 a}-\frac {19}{840} a c^3 x^3-\frac {1}{280} a^3 c^3 x^5+\frac {3 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}{35 a^2}+\frac {9 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}{280 a^2}+\frac {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)}{56 a^2}-\frac {6 i c^3 \arctan (a x)^2}{35 a^2}-\frac {6 c^3 x \arctan (a x)^2}{35 a}-\frac {3 c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^2}{35 a}-\frac {9 c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{140 a}-\frac {3 c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^2}{56 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \arctan (a x)^3}{8 a^2}-\frac {12 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{35 a^2}-\frac {\left (12 i c^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{35 a^2} \\ & = -\frac {19 c^3 x}{140 a}-\frac {19}{840} a c^3 x^3-\frac {1}{280} a^3 c^3 x^5+\frac {3 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}{35 a^2}+\frac {9 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}{280 a^2}+\frac {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)}{56 a^2}-\frac {6 i c^3 \arctan (a x)^2}{35 a^2}-\frac {6 c^3 x \arctan (a x)^2}{35 a}-\frac {3 c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^2}{35 a}-\frac {9 c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{140 a}-\frac {3 c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^2}{56 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \arctan (a x)^3}{8 a^2}-\frac {12 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{35 a^2}-\frac {6 i c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{35 a^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.10 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.51 \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\frac {c^3 \left (-a x \left (114+19 a^2 x^2+3 a^4 x^4\right )-9 \left (-16 i+35 a x+35 a^3 x^3+21 a^5 x^5+5 a^7 x^7\right ) \arctan (a x)^2+105 \left (1+a^2 x^2\right )^4 \arctan (a x)^3+3 \arctan (a x) \left (38+57 a^2 x^2+24 a^4 x^4+5 a^6 x^6-96 \log \left (1+e^{2 i \arctan (a x)}\right )\right )+144 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )\right )}{840 a^2} \]

[In]

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

[Out]

(c^3*(-(a*x*(114 + 19*a^2*x^2 + 3*a^4*x^4)) - 9*(-16*I + 35*a*x + 35*a^3*x^3 + 21*a^5*x^5 + 5*a^7*x^7)*ArcTan[
a*x]^2 + 105*(1 + a^2*x^2)^4*ArcTan[a*x]^3 + 3*ArcTan[a*x]*(38 + 57*a^2*x^2 + 24*a^4*x^4 + 5*a^6*x^6 - 96*Log[
1 + E^((2*I)*ArcTan[a*x])]) + (144*I)*PolyLog[2, -E^((2*I)*ArcTan[a*x])]))/(840*a^2)

Maple [A] (verified)

Time = 5.19 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.12

method	result	size
parts	\(\frac {c^{3} \arctan \left (a x \right )^{3} a^{6} x^{8}}{8}+\frac {c^{3} \arctan \left (a x \right )^{3} a^{4} x^{6}}{2}+\frac {3 c^{3} \arctan \left (a x \right )^{3} a^{2} x^{4}}{4}+\frac {c^{3} \arctan \left (a x \right )^{3} x^{2}}{2}+\frac {c^{3} \arctan \left (a x \right )^{3}}{8 a^{2}}-\frac {3 c^{3} \left (\frac {\arctan \left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {3 a^{5} \arctan \left (a x \right )^{2} x^{5}}{5}+a^{3} \arctan \left (a x \right )^{2} x^{3}+a \arctan \left (a x \right )^{2} x -\frac {a^{6} \arctan \left (a x \right ) x^{6}}{21}-\frac {8 \arctan \left (a x \right ) a^{4} x^{4}}{35}-\frac {19 a^{2} \arctan \left (a x \right ) x^{2}}{35}-\frac {16 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{35}+\frac {a^{5} x^{5}}{105}+\frac {19 a^{3} x^{3}}{315}+\frac {38 a x}{105}-\frac {38 \arctan \left (a x \right )}{105}-\frac {8 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{35}+\frac {8 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{35}\right )}{8 a^{2}}\)	\(346\)
derivativedivides	\(\frac {\frac {c^{3} \arctan \left (a x \right )^{3} a^{8} x^{8}}{8}+\frac {a^{6} c^{3} x^{6} \arctan \left (a x \right )^{3}}{2}+\frac {3 a^{4} c^{3} x^{4} \arctan \left (a x \right )^{3}}{4}+\frac {a^{2} c^{3} x^{2} \arctan \left (a x \right )^{3}}{2}+\frac {c^{3} \arctan \left (a x \right )^{3}}{8}-\frac {3 c^{3} \left (\frac {\arctan \left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {3 a^{5} \arctan \left (a x \right )^{2} x^{5}}{5}+a^{3} \arctan \left (a x \right )^{2} x^{3}+a \arctan \left (a x \right )^{2} x -\frac {a^{6} \arctan \left (a x \right ) x^{6}}{21}-\frac {8 \arctan \left (a x \right ) a^{4} x^{4}}{35}-\frac {19 a^{2} \arctan \left (a x \right ) x^{2}}{35}-\frac {16 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{35}+\frac {a^{5} x^{5}}{105}+\frac {19 a^{3} x^{3}}{315}+\frac {38 a x}{105}-\frac {38 \arctan \left (a x \right )}{105}-\frac {8 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{35}+\frac {8 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{35}\right )}{8}}{a^{2}}\)	\(347\)
default	\(\frac {\frac {c^{3} \arctan \left (a x \right )^{3} a^{8} x^{8}}{8}+\frac {a^{6} c^{3} x^{6} \arctan \left (a x \right )^{3}}{2}+\frac {3 a^{4} c^{3} x^{4} \arctan \left (a x \right )^{3}}{4}+\frac {a^{2} c^{3} x^{2} \arctan \left (a x \right )^{3}}{2}+\frac {c^{3} \arctan \left (a x \right )^{3}}{8}-\frac {3 c^{3} \left (\frac {\arctan \left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {3 a^{5} \arctan \left (a x \right )^{2} x^{5}}{5}+a^{3} \arctan \left (a x \right )^{2} x^{3}+a \arctan \left (a x \right )^{2} x -\frac {a^{6} \arctan \left (a x \right ) x^{6}}{21}-\frac {8 \arctan \left (a x \right ) a^{4} x^{4}}{35}-\frac {19 a^{2} \arctan \left (a x \right ) x^{2}}{35}-\frac {16 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{35}+\frac {a^{5} x^{5}}{105}+\frac {19 a^{3} x^{3}}{315}+\frac {38 a x}{105}-\frac {38 \arctan \left (a x \right )}{105}-\frac {8 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{35}+\frac {8 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{35}\right )}{8}}{a^{2}}\)	\(347\)

[In]

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

[Out]

1/8*c^3*arctan(a*x)^3*a^6*x^8+1/2*c^3*arctan(a*x)^3*a^4*x^6+3/4*c^3*arctan(a*x)^3*a^2*x^4+1/2*c^3*arctan(a*x)^
3*x^2+1/8*c^3*arctan(a*x)^3/a^2-3/8/a^2*c^3*(1/7*arctan(a*x)^2*a^7*x^7+3/5*a^5*arctan(a*x)^2*x^5+a^3*arctan(a*
x)^2*x^3+a*arctan(a*x)^2*x-1/21*a^6*arctan(a*x)*x^6-8/35*arctan(a*x)*a^4*x^4-19/35*a^2*arctan(a*x)*x^2-16/35*a
rctan(a*x)*ln(a^2*x^2+1)+1/105*a^5*x^5+19/315*a^3*x^3+38/105*a*x-38/105*arctan(a*x)-8/35*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)+8/35*I*(ln(I+a*x)*ln(a^2*x^2+1)-dilo
g(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]

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

[In]

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

[Out]

integral((a^6*c^3*x^7 + 3*a^4*c^3*x^5 + 3*a^2*c^3*x^3 + c^3*x)*arctan(a*x)^3, x)

Sympy [F]

\[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=c^{3} \left (\int x \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int 3 a^{2} x^{3} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int 3 a^{4} x^{5} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{6} x^{7} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]

[In]

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

[Out]

c**3*(Integral(x*atan(a*x)**3, x) + Integral(3*a**2*x**3*atan(a*x)**3, x) + Integral(3*a**4*x**5*atan(a*x)**3,
x) + Integral(a**6*x**7*atan(a*x)**3, x))

Maxima [F]

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

[In]

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

[Out]

1/4480*(280*(a^8*c^3*x^8 + 4*a^6*c^3*x^6 + 6*a^4*c^3*x^4 + 4*a^2*c^3*x^2 + c^3)*arctan(a*x)^3 + 140*(71680*a^9
*c^3*integrate(1/4480*x^9*arctan(a*x)^3/(a^3*x^2 + a), x) - 13440*a^8*c^3*integrate(1/4480*x^8*arctan(a*x)^2/(
a^3*x^2 + a), x) - 3360*a^8*c^3*integrate(1/4480*x^8*log(a^2*x^2 + 1)^2/(a^3*x^2 + a), x) - 1920*a^8*c^3*integ
rate(1/4480*x^8*log(a^2*x^2 + 1)/(a^3*x^2 + a), x) + 286720*a^7*c^3*integrate(1/4480*x^7*arctan(a*x)^3/(a^3*x^
2 + a), x) + 3840*a^7*c^3*integrate(1/4480*x^7*arctan(a*x)/(a^3*x^2 + a), x) - 53760*a^6*c^3*integrate(1/4480*
x^6*arctan(a*x)^2/(a^3*x^2 + a), x) - 13440*a^6*c^3*integrate(1/4480*x^6*log(a^2*x^2 + 1)^2/(a^3*x^2 + a), x)
- 8064*a^6*c^3*integrate(1/4480*x^6*log(a^2*x^2 + 1)/(a^3*x^2 + a), x) + 430080*a^5*c^3*integrate(1/4480*x^5*a
rctan(a*x)^3/(a^3*x^2 + a), x) + 16128*a^5*c^3*integrate(1/4480*x^5*arctan(a*x)/(a^3*x^2 + a), x) - 80640*a^4*
c^3*integrate(1/4480*x^4*arctan(a*x)^2/(a^3*x^2 + a), x) - 20160*a^4*c^3*integrate(1/4480*x^4*log(a^2*x^2 + 1)
^2/(a^3*x^2 + a), x) - 13440*a^4*c^3*integrate(1/4480*x^4*log(a^2*x^2 + 1)/(a^3*x^2 + a), x) + 286720*a^3*c^3*
integrate(1/4480*x^3*arctan(a*x)^3/(a^3*x^2 + a), x) + 26880*a^3*c^3*integrate(1/4480*x^3*arctan(a*x)/(a^3*x^2
+ a), x) - 53760*a^2*c^3*integrate(1/4480*x^2*arctan(a*x)^2/(a^3*x^2 + a), x) - 13440*a^2*c^3*integrate(1/448
0*x^2*log(a^2*x^2 + 1)^2/(a^3*x^2 + a), x) - 13440*a^2*c^3*integrate(1/4480*x^2*log(a^2*x^2 + 1)/(a^3*x^2 + a)
, x) + 71680*a*c^3*integrate(1/4480*x*arctan(a*x)^3/(a^3*x^2 + a), x) + 26880*a*c^3*integrate(1/4480*x*arctan(
a*x)/(a^3*x^2 + a), x) - c^3*arctan(a*x)^3/a^2 - 3360*c^3*integrate(1/4480*log(a^2*x^2 + 1)^2/(a^3*x^2 + a), x
))*a^2 - 12*(5*a^7*c^3*x^7 + 21*a^5*c^3*x^5 + 35*a^3*c^3*x^3 + 35*a*c^3*x)*arctan(a*x)^2 + 3*(5*a^7*c^3*x^7 +
21*a^5*c^3*x^5 + 35*a^3*c^3*x^3 + 35*a*c^3*x)*log(a^2*x^2 + 1)^2)/a^2

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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