\(\int x^3 (c+a^2 c x^2)^3 \arctan (a x) \, dx\) [165]

Optimal result

Integrand size = 20, antiderivative size = 141 \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=\frac {c^3 x}{40 a^3}-\frac {c^3 x^3}{120 a}-\frac {9}{200} a c^3 x^5-\frac {11}{280} a^3 c^3 x^7-\frac {1}{90} a^5 c^3 x^9-\frac {c^3 \arctan (a x)}{40 a^4}+\frac {1}{4} c^3 x^4 \arctan (a x)+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x) \]

[Out]

1/40*c^3*x/a^3-1/120*c^3*x^3/a-9/200*a*c^3*x^5-11/280*a^3*c^3*x^7-1/90*a^5*c^3*x^9-1/40*c^3*arctan(a*x)/a^4+1/
4*c^3*x^4*arctan(a*x)+1/2*a^2*c^3*x^6*arctan(a*x)+3/8*a^4*c^3*x^8*arctan(a*x)+1/10*a^6*c^3*x^10*arctan(a*x)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5068, 4946, 308, 209} \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)-\frac {1}{90} a^5 c^3 x^9+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)-\frac {c^3 \arctan (a x)}{40 a^4}-\frac {11}{280} a^3 c^3 x^7+\frac {c^3 x}{40 a^3}+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)+\frac {1}{4} c^3 x^4 \arctan (a x)-\frac {9}{200} a c^3 x^5-\frac {c^3 x^3}{120 a} \]

[In]

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

[Out]

(c^3*x)/(40*a^3) - (c^3*x^3)/(120*a) - (9*a*c^3*x^5)/200 - (11*a^3*c^3*x^7)/280 - (a^5*c^3*x^9)/90 - (c^3*ArcT
an[a*x])/(40*a^4) + (c^3*x^4*ArcTan[a*x])/4 + (a^2*c^3*x^6*ArcTan[a*x])/2 + (3*a^4*c^3*x^8*ArcTan[a*x])/8 + (a
^6*c^3*x^10*ArcTan[a*x])/10

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 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 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^3 x^3 \arctan (a x)+3 a^2 c^3 x^5 \arctan (a x)+3 a^4 c^3 x^7 \arctan (a x)+a^6 c^3 x^9 \arctan (a x)\right ) \, dx \\ & = c^3 \int x^3 \arctan (a x) \, dx+\left (3 a^2 c^3\right ) \int x^5 \arctan (a x) \, dx+\left (3 a^4 c^3\right ) \int x^7 \arctan (a x) \, dx+\left (a^6 c^3\right ) \int x^9 \arctan (a x) \, dx \\ & = \frac {1}{4} c^3 x^4 \arctan (a x)+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)-\frac {1}{4} \left (a c^3\right ) \int \frac {x^4}{1+a^2 x^2} \, dx-\frac {1}{2} \left (a^3 c^3\right ) \int \frac {x^6}{1+a^2 x^2} \, dx-\frac {1}{8} \left (3 a^5 c^3\right ) \int \frac {x^8}{1+a^2 x^2} \, dx-\frac {1}{10} \left (a^7 c^3\right ) \int \frac {x^{10}}{1+a^2 x^2} \, dx \\ & = \frac {1}{4} c^3 x^4 \arctan (a x)+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)-\frac {1}{4} \left (a c^3\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}{2} \left (a^3 c^3\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 {1}{8} \left (3 a^5 c^3\right ) \int \left (-\frac {1}{a^8}+\frac {x^2}{a^6}-\frac {x^4}{a^4}+\frac {x^6}{a^2}+\frac {1}{a^8 \left (1+a^2 x^2\right )}\right ) \, dx-\frac {1}{10} \left (a^7 c^3\right ) \int \left (\frac {1}{a^{10}}-\frac {x^2}{a^8}+\frac {x^4}{a^6}-\frac {x^6}{a^4}+\frac {x^8}{a^2}-\frac {1}{a^{10} \left (1+a^2 x^2\right )}\right ) \, dx \\ & = \frac {c^3 x}{40 a^3}-\frac {c^3 x^3}{120 a}-\frac {9}{200} a c^3 x^5-\frac {11}{280} a^3 c^3 x^7-\frac {1}{90} a^5 c^3 x^9+\frac {1}{4} c^3 x^4 \arctan (a x)+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)+\frac {c^3 \int \frac {1}{1+a^2 x^2} \, dx}{10 a^3}-\frac {c^3 \int \frac {1}{1+a^2 x^2} \, dx}{4 a^3}-\frac {\left (3 c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx}{8 a^3}+\frac {c^3 \int \frac {1}{1+a^2 x^2} \, dx}{2 a^3} \\ & = \frac {c^3 x}{40 a^3}-\frac {c^3 x^3}{120 a}-\frac {9}{200} a c^3 x^5-\frac {11}{280} a^3 c^3 x^7-\frac {1}{90} a^5 c^3 x^9-\frac {c^3 \arctan (a x)}{40 a^4}+\frac {1}{4} c^3 x^4 \arctan (a x)+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00 \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=\frac {c^3 x}{40 a^3}-\frac {c^3 x^3}{120 a}-\frac {9}{200} a c^3 x^5-\frac {11}{280} a^3 c^3 x^7-\frac {1}{90} a^5 c^3 x^9-\frac {c^3 \arctan (a x)}{40 a^4}+\frac {1}{4} c^3 x^4 \arctan (a x)+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x) \]

[In]

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

[Out]

(c^3*x)/(40*a^3) - (c^3*x^3)/(120*a) - (9*a*c^3*x^5)/200 - (11*a^3*c^3*x^7)/280 - (a^5*c^3*x^9)/90 - (c^3*ArcT
an[a*x])/(40*a^4) + (c^3*x^4*ArcTan[a*x])/4 + (a^2*c^3*x^6*ArcTan[a*x])/2 + (3*a^4*c^3*x^8*ArcTan[a*x])/8 + (a
^6*c^3*x^10*ArcTan[a*x])/10

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.79

method	result	size
derivativedivides	\(\frac {\frac {c^{3} \arctan \left (a x \right ) a^{10} x^{10}}{10}+\frac {3 c^{3} \arctan \left (a x \right ) a^{8} x^{8}}{8}+\frac {a^{6} c^{3} x^{6} \arctan \left (a x \right )}{2}+\frac {a^{4} c^{3} x^{4} \arctan \left (a x \right )}{4}-\frac {c^{3} \left (\frac {4 a^{9} x^{9}}{9}+\frac {11 a^{7} x^{7}}{7}+\frac {9 a^{5} x^{5}}{5}+\frac {a^{3} x^{3}}{3}-a x +\arctan \left (a x \right )\right )}{40}}{a^{4}}\)	\(112\)
default	\(\frac {\frac {c^{3} \arctan \left (a x \right ) a^{10} x^{10}}{10}+\frac {3 c^{3} \arctan \left (a x \right ) a^{8} x^{8}}{8}+\frac {a^{6} c^{3} x^{6} \arctan \left (a x \right )}{2}+\frac {a^{4} c^{3} x^{4} \arctan \left (a x \right )}{4}-\frac {c^{3} \left (\frac {4 a^{9} x^{9}}{9}+\frac {11 a^{7} x^{7}}{7}+\frac {9 a^{5} x^{5}}{5}+\frac {a^{3} x^{3}}{3}-a x +\arctan \left (a x \right )\right )}{40}}{a^{4}}\)	\(112\)
parts	\(\frac {a^{6} c^{3} x^{10} \arctan \left (a x \right )}{10}+\frac {3 a^{4} c^{3} x^{8} \arctan \left (a x \right )}{8}+\frac {a^{2} c^{3} x^{6} \arctan \left (a x \right )}{2}+\frac {c^{3} x^{4} \arctan \left (a x \right )}{4}-\frac {c^{3} a \left (\frac {\frac {4}{9} a^{8} x^{9}+\frac {11}{7} a^{6} x^{7}+\frac {9}{5} a^{4} x^{5}+\frac {1}{3} a^{2} x^{3}-x}{a^{4}}+\frac {\arctan \left (a x \right )}{a^{5}}\right )}{40}\)	\(114\)
parallelrisch	\(\frac {1260 c^{3} \arctan \left (a x \right ) a^{10} x^{10}-140 a^{9} c^{3} x^{9}+4725 c^{3} \arctan \left (a x \right ) a^{8} x^{8}-495 a^{7} c^{3} x^{7}+6300 a^{6} c^{3} x^{6} \arctan \left (a x \right )-567 a^{5} c^{3} x^{5}+3150 a^{4} c^{3} x^{4} \arctan \left (a x \right )-105 a^{3} c^{3} x^{3}+315 a \,c^{3} x -315 c^{3} \arctan \left (a x \right )}{12600 a^{4}}\)	\(127\)
risch	\(-\frac {i c^{3} x^{4} \left (4 a^{6} x^{6}+15 a^{4} x^{4}+20 a^{2} x^{2}+10\right ) \ln \left (i a x +1\right )}{80}+\frac {i c^{3} a^{6} x^{10} \ln \left (-i a x +1\right )}{20}-\frac {a^{5} c^{3} x^{9}}{90}+\frac {3 i c^{3} a^{4} x^{8} \ln \left (-i a x +1\right )}{16}-\frac {11 a^{3} c^{3} x^{7}}{280}+\frac {i c^{3} a^{2} x^{6} \ln \left (-i a x +1\right )}{4}-\frac {9 a \,c^{3} x^{5}}{200}+\frac {i c^{3} x^{4} \ln \left (-i a x +1\right )}{8}-\frac {c^{3} x^{3}}{120 a}+\frac {c^{3} x}{40 a^{3}}-\frac {c^{3} \arctan \left (a x \right )}{40 a^{4}}\)	\(185\)
meijerg	\(\frac {c^{3} \left (-\frac {2 x a \left (385 a^{8} x^{8}-495 a^{6} x^{6}+693 a^{4} x^{4}-1155 a^{2} x^{2}+3465\right )}{17325}+\frac {2 x a \left (11 a^{10} x^{10}+11\right ) \arctan \left (\sqrt {a^{2} x^{2}}\right )}{55 \sqrt {a^{2} x^{2}}}\right )}{4 a^{4}}+\frac {3 c^{3} \left (\frac {x a \left (-45 a^{6} x^{6}+63 a^{4} x^{4}-105 a^{2} x^{2}+315\right )}{630}-\frac {x a \left (-9 a^{8} x^{8}+9\right ) \arctan \left (\sqrt {a^{2} x^{2}}\right )}{18 \sqrt {a^{2} x^{2}}}\right )}{4 a^{4}}+\frac {3 c^{3} \left (-\frac {2 a x \left (21 a^{4} x^{4}-35 a^{2} x^{2}+105\right )}{315}+\frac {2 a x \left (7 a^{6} x^{6}+7\right ) \arctan \left (\sqrt {a^{2} x^{2}}\right )}{21 \sqrt {a^{2} x^{2}}}\right )}{4 a^{4}}+\frac {c^{3} \left (\frac {a x \left (-5 a^{2} x^{2}+15\right )}{15}-\frac {a x \left (-5 a^{4} x^{4}+5\right ) \arctan \left (\sqrt {a^{2} x^{2}}\right )}{5 \sqrt {a^{2} x^{2}}}\right )}{4 a^{4}}\)	\(274\)

[In]

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

[Out]

1/a^4*(1/10*c^3*arctan(a*x)*a^10*x^10+3/8*c^3*arctan(a*x)*a^8*x^8+1/2*a^6*c^3*x^6*arctan(a*x)+1/4*a^4*c^3*x^4*
arctan(a*x)-1/40*c^3*(4/9*a^9*x^9+11/7*a^7*x^7+9/5*a^5*x^5+1/3*a^3*x^3-a*x+arctan(a*x)))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.80 \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=-\frac {140 \, a^{9} c^{3} x^{9} + 495 \, a^{7} c^{3} x^{7} + 567 \, a^{5} c^{3} x^{5} + 105 \, a^{3} c^{3} x^{3} - 315 \, a c^{3} x - 315 \, {\left (4 \, a^{10} c^{3} x^{10} + 15 \, a^{8} c^{3} x^{8} + 20 \, a^{6} c^{3} x^{6} + 10 \, a^{4} c^{3} x^{4} - c^{3}\right )} \arctan \left (a x\right )}{12600 \, a^{4}} \]

[In]

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

[Out]

-1/12600*(140*a^9*c^3*x^9 + 495*a^7*c^3*x^7 + 567*a^5*c^3*x^5 + 105*a^3*c^3*x^3 - 315*a*c^3*x - 315*(4*a^10*c^
3*x^10 + 15*a^8*c^3*x^8 + 20*a^6*c^3*x^6 + 10*a^4*c^3*x^4 - c^3)*arctan(a*x))/a^4

Sympy [A] (verification not implemented)

Time = 0.58 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.98 \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=\begin {cases} \frac {a^{6} c^{3} x^{10} \operatorname {atan}{\left (a x \right )}}{10} - \frac {a^{5} c^{3} x^{9}}{90} + \frac {3 a^{4} c^{3} x^{8} \operatorname {atan}{\left (a x \right )}}{8} - \frac {11 a^{3} c^{3} x^{7}}{280} + \frac {a^{2} c^{3} x^{6} \operatorname {atan}{\left (a x \right )}}{2} - \frac {9 a c^{3} x^{5}}{200} + \frac {c^{3} x^{4} \operatorname {atan}{\left (a x \right )}}{4} - \frac {c^{3} x^{3}}{120 a} + \frac {c^{3} x}{40 a^{3}} - \frac {c^{3} \operatorname {atan}{\left (a x \right )}}{40 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a**6*c**3*x**10*atan(a*x)/10 - a**5*c**3*x**9/90 + 3*a**4*c**3*x**8*atan(a*x)/8 - 11*a**3*c**3*x**7
/280 + a**2*c**3*x**6*atan(a*x)/2 - 9*a*c**3*x**5/200 + c**3*x**4*atan(a*x)/4 - c**3*x**3/(120*a) + c**3*x/(40
*a**3) - c**3*atan(a*x)/(40*a**4), Ne(a, 0)), (0, True))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.85 \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=-\frac {1}{12600} \, a {\left (\frac {315 \, c^{3} \arctan \left (a x\right )}{a^{5}} + \frac {140 \, a^{8} c^{3} x^{9} + 495 \, a^{6} c^{3} x^{7} + 567 \, a^{4} c^{3} x^{5} + 105 \, a^{2} c^{3} x^{3} - 315 \, c^{3} x}{a^{4}}\right )} + \frac {1}{40} \, {\left (4 \, a^{6} c^{3} x^{10} + 15 \, a^{4} c^{3} x^{8} + 20 \, a^{2} c^{3} x^{6} + 10 \, c^{3} x^{4}\right )} \arctan \left (a x\right ) \]

[In]

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

[Out]

-1/12600*a*(315*c^3*arctan(a*x)/a^5 + (140*a^8*c^3*x^9 + 495*a^6*c^3*x^7 + 567*a^4*c^3*x^5 + 105*a^2*c^3*x^3 -
315*c^3*x)/a^4) + 1/40*(4*a^6*c^3*x^10 + 15*a^4*c^3*x^8 + 20*a^2*c^3*x^6 + 10*c^3*x^4)*arctan(a*x)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.50 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.79 \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=\mathrm {atan}\left (a\,x\right )\,\left (\frac {a^6\,c^3\,x^{10}}{10}+\frac {3\,a^4\,c^3\,x^8}{8}+\frac {a^2\,c^3\,x^6}{2}+\frac {c^3\,x^4}{4}\right )+\frac {c^3\,x}{40\,a^3}-\frac {9\,a\,c^3\,x^5}{200}-\frac {c^3\,\mathrm {atan}\left (a\,x\right )}{40\,a^4}-\frac {c^3\,x^3}{120\,a}-\frac {11\,a^3\,c^3\,x^7}{280}-\frac {a^5\,c^3\,x^9}{90} \]

[In]

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

[Out]

atan(a*x)*((c^3*x^4)/4 + (a^2*c^3*x^6)/2 + (3*a^4*c^3*x^8)/8 + (a^6*c^3*x^10)/10) + (c^3*x)/(40*a^3) - (9*a*c^
3*x^5)/200 - (c^3*atan(a*x))/(40*a^4) - (c^3*x^3)/(120*a) - (11*a^3*c^3*x^7)/280 - (a^5*c^3*x^9)/90