Integrand size = 20, antiderivative size = 141 \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=\frac {c^3 x}{40 a^3}-\frac {c^3 x^3}{120 a}-\frac {9}{200} a c^3 x^5-\frac {11}{280} a^3 c^3 x^7-\frac {1}{90} a^5 c^3 x^9-\frac {c^3 \arctan (a x)}{40 a^4}+\frac {1}{4} c^3 x^4 \arctan (a x)+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x) \]
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)
Time = 0.05 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00 \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=\frac {c^3 x}{40 a^3}-\frac {c^3 x^3}{120 a}-\frac {9}{200} a c^3 x^5-\frac {11}{280} a^3 c^3 x^7-\frac {1}{90} a^5 c^3 x^9-\frac {c^3 \arctan (a x)}{40 a^4}+\frac {1}{4} c^3 x^4 \arctan (a x)+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x) \]
(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*ArcTan[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
Time = 0.37 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {5483, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^3 \arctan (a x) \left (a^2 c x^2+c\right )^3 \, dx\) |
\(\Big \downarrow \) 5483 |
\(\displaystyle \int \left (a^6 c^3 x^9 \arctan (a x)+3 a^4 c^3 x^7 \arctan (a x)+3 a^2 c^3 x^5 \arctan (a x)+c^3 x^3 \arctan (a x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{10} a^6 c^3 x^{10} \arctan (a x)-\frac {1}{90} a^5 c^3 x^9+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)-\frac {c^3 \arctan (a x)}{40 a^4}-\frac {11}{280} a^3 c^3 x^7+\frac {c^3 x}{40 a^3}+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)+\frac {1}{4} c^3 x^4 \arctan (a x)-\frac {9}{200} a c^3 x^5-\frac {c^3 x^3}{120 a}\) |
(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*ArcTan[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
3.2.65.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(q_), x_Symbol] :> Int[ExpandIntegrand[(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])
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 | \(\frac {c^{3} \left (-\frac {2 x a \left (385 a^{8} x^{8}-495 a^{6} x^{6}+693 a^{4} x^{4}-1155 a^{2} x^{2}+3465\right )}{17325}+\frac {2 x a \left (11 a^{10} x^{10}+11\right ) \arctan \left (\sqrt {a^{2} x^{2}}\right )}{55 \sqrt {a^{2} x^{2}}}\right )}{4 a^{4}}+\frac {3 c^{3} \left (\frac {x a \left (-45 a^{6} x^{6}+63 a^{4} x^{4}-105 a^{2} x^{2}+315\right )}{630}-\frac {x a \left (-9 a^{8} x^{8}+9\right ) \arctan \left (\sqrt {a^{2} x^{2}}\right )}{18 \sqrt {a^{2} x^{2}}}\right )}{4 a^{4}}+\frac {3 c^{3} \left (-\frac {2 a x \left (21 a^{4} x^{4}-35 a^{2} x^{2}+105\right )}{315}+\frac {2 a x \left (7 a^{6} x^{6}+7\right ) \arctan \left (\sqrt {a^{2} x^{2}}\right )}{21 \sqrt {a^{2} x^{2}}}\right )}{4 a^{4}}+\frac {c^{3} \left (\frac {a x \left (-5 a^{2} x^{2}+15\right )}{15}-\frac {a x \left (-5 a^{4} x^{4}+5\right ) \arctan \left (\sqrt {a^{2} x^{2}}\right )}{5 \sqrt {a^{2} x^{2}}}\right )}{4 a^{4}}\) | \(274\) |
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)))
Time = 0.26 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.80 \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=-\frac {140 \, a^{9} c^{3} x^{9} + 495 \, a^{7} c^{3} x^{7} + 567 \, a^{5} c^{3} x^{5} + 105 \, a^{3} c^{3} x^{3} - 315 \, a c^{3} x - 315 \, {\left (4 \, a^{10} c^{3} x^{10} + 15 \, a^{8} c^{3} x^{8} + 20 \, a^{6} c^{3} x^{6} + 10 \, a^{4} c^{3} x^{4} - c^{3}\right )} \arctan \left (a x\right )}{12600 \, a^{4}} \]
-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
Time = 0.58 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.98 \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=\begin {cases} \frac {a^{6} c^{3} x^{10} \operatorname {atan}{\left (a x \right )}}{10} - \frac {a^{5} c^{3} x^{9}}{90} + \frac {3 a^{4} c^{3} x^{8} \operatorname {atan}{\left (a x \right )}}{8} - \frac {11 a^{3} c^{3} x^{7}}{280} + \frac {a^{2} c^{3} x^{6} \operatorname {atan}{\left (a x \right )}}{2} - \frac {9 a c^{3} x^{5}}{200} + \frac {c^{3} x^{4} \operatorname {atan}{\left (a x \right )}}{4} - \frac {c^{3} x^{3}}{120 a} + \frac {c^{3} x}{40 a^{3}} - \frac {c^{3} \operatorname {atan}{\left (a x \right )}}{40 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
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))
Time = 0.29 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.85 \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=-\frac {1}{12600} \, a {\left (\frac {315 \, c^{3} \arctan \left (a x\right )}{a^{5}} + \frac {140 \, a^{8} c^{3} x^{9} + 495 \, a^{6} c^{3} x^{7} + 567 \, a^{4} c^{3} x^{5} + 105 \, a^{2} c^{3} x^{3} - 315 \, c^{3} x}{a^{4}}\right )} + \frac {1}{40} \, {\left (4 \, a^{6} c^{3} x^{10} + 15 \, a^{4} c^{3} x^{8} + 20 \, a^{2} c^{3} x^{6} + 10 \, c^{3} x^{4}\right )} \arctan \left (a x\right ) \]
-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^1 0 + 15*a^4*c^3*x^8 + 20*a^2*c^3*x^6 + 10*c^3*x^4)*arctan(a*x)
\[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} x^{3} \arctan \left (a x\right ) \,d x } \]
Time = 0.50 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.79 \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=\mathrm {atan}\left (a\,x\right )\,\left (\frac {a^6\,c^3\,x^{10}}{10}+\frac {3\,a^4\,c^3\,x^8}{8}+\frac {a^2\,c^3\,x^6}{2}+\frac {c^3\,x^4}{4}\right )+\frac {c^3\,x}{40\,a^3}-\frac {9\,a\,c^3\,x^5}{200}-\frac {c^3\,\mathrm {atan}\left (a\,x\right )}{40\,a^4}-\frac {c^3\,x^3}{120\,a}-\frac {11\,a^3\,c^3\,x^7}{280}-\frac {a^5\,c^3\,x^9}{90} \]