Integrand size = 20, antiderivative size = 111 \[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x) \, dx=\frac {c^2 x}{24 a^3}-\frac {c^2 x^3}{72 a}-\frac {1}{24} a c^2 x^5-\frac {1}{56} a^3 c^2 x^7-\frac {c^2 \arctan (a x)}{24 a^4}+\frac {1}{4} c^2 x^4 \arctan (a x)+\frac {1}{3} a^2 c^2 x^6 \arctan (a x)+\frac {1}{8} a^4 c^2 x^8 \arctan (a x) \]
1/24*c^2*x/a^3-1/72*c^2*x^3/a-1/24*a*c^2*x^5-1/56*a^3*c^2*x^7-1/24*c^2*arc tan(a*x)/a^4+1/4*c^2*x^4*arctan(a*x)+1/3*a^2*c^2*x^6*arctan(a*x)+1/8*a^4*c ^2*x^8*arctan(a*x)
Time = 0.04 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00 \[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x) \, dx=\frac {c^2 x}{24 a^3}-\frac {c^2 x^3}{72 a}-\frac {1}{24} a c^2 x^5-\frac {1}{56} a^3 c^2 x^7-\frac {c^2 \arctan (a x)}{24 a^4}+\frac {1}{4} c^2 x^4 \arctan (a x)+\frac {1}{3} a^2 c^2 x^6 \arctan (a x)+\frac {1}{8} a^4 c^2 x^8 \arctan (a x) \]
(c^2*x)/(24*a^3) - (c^2*x^3)/(72*a) - (a*c^2*x^5)/24 - (a^3*c^2*x^7)/56 - (c^2*ArcTan[a*x])/(24*a^4) + (c^2*x^4*ArcTan[a*x])/4 + (a^2*c^2*x^6*ArcTan [a*x])/3 + (a^4*c^2*x^8*ArcTan[a*x])/8
Time = 0.32 (sec) , antiderivative size = 111, 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.
\(\displaystyle \int x^3 \arctan (a x) \left (a^2 c x^2+c\right )^2 \, dx\) |
\(\Big \downarrow \) 5483 |
\(\displaystyle \int \left (a^4 c^2 x^7 \arctan (a x)+2 a^2 c^2 x^5 \arctan (a x)+c^2 x^3 \arctan (a x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{8} a^4 c^2 x^8 \arctan (a x)-\frac {c^2 \arctan (a x)}{24 a^4}-\frac {1}{56} a^3 c^2 x^7+\frac {c^2 x}{24 a^3}+\frac {1}{3} a^2 c^2 x^6 \arctan (a x)+\frac {1}{4} c^2 x^4 \arctan (a x)-\frac {1}{24} a c^2 x^5-\frac {c^2 x^3}{72 a}\) |
(c^2*x)/(24*a^3) - (c^2*x^3)/(72*a) - (a*c^2*x^5)/24 - (a^3*c^2*x^7)/56 - (c^2*ArcTan[a*x])/(24*a^4) + (c^2*x^4*ArcTan[a*x])/4 + (a^2*c^2*x^6*ArcTan [a*x])/3 + (a^4*c^2*x^8*ArcTan[a*x])/8
3.2.57.3.1 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 = 88, normalized size of antiderivative = 0.79
method | result | size |
derivativedivides | \(\frac {\frac {c^{2} \arctan \left (a x \right ) a^{8} x^{8}}{8}+\frac {c^{2} \arctan \left (a x \right ) a^{6} x^{6}}{3}+\frac {a^{4} c^{2} x^{4} \arctan \left (a x \right )}{4}-\frac {c^{2} \left (\frac {3 a^{7} x^{7}}{7}+a^{5} x^{5}+\frac {a^{3} x^{3}}{3}-a x +\arctan \left (a x \right )\right )}{24}}{a^{4}}\) | \(88\) |
default | \(\frac {\frac {c^{2} \arctan \left (a x \right ) a^{8} x^{8}}{8}+\frac {c^{2} \arctan \left (a x \right ) a^{6} x^{6}}{3}+\frac {a^{4} c^{2} x^{4} \arctan \left (a x \right )}{4}-\frac {c^{2} \left (\frac {3 a^{7} x^{7}}{7}+a^{5} x^{5}+\frac {a^{3} x^{3}}{3}-a x +\arctan \left (a x \right )\right )}{24}}{a^{4}}\) | \(88\) |
parts | \(\frac {a^{4} c^{2} x^{8} \arctan \left (a x \right )}{8}+\frac {a^{2} c^{2} x^{6} \arctan \left (a x \right )}{3}+\frac {c^{2} x^{4} \arctan \left (a x \right )}{4}-\frac {c^{2} a \left (\frac {\frac {3}{7} a^{6} x^{7}+a^{4} x^{5}+\frac {1}{3} a^{2} x^{3}-x}{a^{4}}+\frac {\arctan \left (a x \right )}{a^{5}}\right )}{24}\) | \(90\) |
parallelrisch | \(\frac {63 c^{2} \arctan \left (a x \right ) a^{8} x^{8}-9 a^{7} c^{2} x^{7}+168 c^{2} \arctan \left (a x \right ) a^{6} x^{6}-21 a^{5} c^{2} x^{5}+126 a^{4} c^{2} x^{4} \arctan \left (a x \right )-7 a^{3} c^{2} x^{3}+21 a \,c^{2} x -21 c^{2} \arctan \left (a x \right )}{504 a^{4}}\) | \(101\) |
risch | \(-\frac {i c^{2} x^{4} \left (3 a^{4} x^{4}+8 a^{2} x^{2}+6\right ) \ln \left (i a x +1\right )}{48}+\frac {i c^{2} a^{4} x^{8} \ln \left (-i a x +1\right )}{16}-\frac {a^{3} c^{2} x^{7}}{56}+\frac {i c^{2} a^{2} x^{6} \ln \left (-i a x +1\right )}{6}-\frac {a \,c^{2} x^{5}}{24}+\frac {i c^{2} x^{4} \ln \left (-i a x +1\right )}{8}-\frac {c^{2} x^{3}}{72 a}+\frac {c^{2} x}{24 a^{3}}-\frac {c^{2} \arctan \left (a x \right )}{24 a^{4}}\) | \(146\) |
meijerg | \(\frac {c^{2} \left (\frac {x a \left (-45 a^{6} x^{6}+63 a^{4} x^{4}-105 a^{2} x^{2}+315\right )}{630}-\frac {x a \left (-9 a^{8} x^{8}+9\right ) \arctan \left (\sqrt {a^{2} x^{2}}\right )}{18 \sqrt {a^{2} x^{2}}}\right )}{4 a^{4}}+\frac {c^{2} \left (-\frac {2 a x \left (21 a^{4} x^{4}-35 a^{2} x^{2}+105\right )}{315}+\frac {2 a x \left (7 a^{6} x^{6}+7\right ) \arctan \left (\sqrt {a^{2} x^{2}}\right )}{21 \sqrt {a^{2} x^{2}}}\right )}{2 a^{4}}+\frac {c^{2} \left (\frac {a x \left (-5 a^{2} x^{2}+15\right )}{15}-\frac {a x \left (-5 a^{4} x^{4}+5\right ) \arctan \left (\sqrt {a^{2} x^{2}}\right )}{5 \sqrt {a^{2} x^{2}}}\right )}{4 a^{4}}\) | \(194\) |
1/a^4*(1/8*c^2*arctan(a*x)*a^8*x^8+1/3*c^2*arctan(a*x)*a^6*x^6+1/4*a^4*c^2 *x^4*arctan(a*x)-1/24*c^2*(3/7*a^7*x^7+a^5*x^5+1/3*a^3*x^3-a*x+arctan(a*x) ))
Time = 0.24 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.82 \[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x) \, dx=-\frac {9 \, a^{7} c^{2} x^{7} + 21 \, a^{5} c^{2} x^{5} + 7 \, a^{3} c^{2} x^{3} - 21 \, a c^{2} x - 21 \, {\left (3 \, a^{8} c^{2} x^{8} + 8 \, a^{6} c^{2} x^{6} + 6 \, a^{4} c^{2} x^{4} - c^{2}\right )} \arctan \left (a x\right )}{504 \, a^{4}} \]
-1/504*(9*a^7*c^2*x^7 + 21*a^5*c^2*x^5 + 7*a^3*c^2*x^3 - 21*a*c^2*x - 21*( 3*a^8*c^2*x^8 + 8*a^6*c^2*x^6 + 6*a^4*c^2*x^4 - c^2)*arctan(a*x))/a^4
Time = 0.43 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.94 \[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x) \, dx=\begin {cases} \frac {a^{4} c^{2} x^{8} \operatorname {atan}{\left (a x \right )}}{8} - \frac {a^{3} c^{2} x^{7}}{56} + \frac {a^{2} c^{2} x^{6} \operatorname {atan}{\left (a x \right )}}{3} - \frac {a c^{2} x^{5}}{24} + \frac {c^{2} x^{4} \operatorname {atan}{\left (a x \right )}}{4} - \frac {c^{2} x^{3}}{72 a} + \frac {c^{2} x}{24 a^{3}} - \frac {c^{2} \operatorname {atan}{\left (a x \right )}}{24 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((a**4*c**2*x**8*atan(a*x)/8 - a**3*c**2*x**7/56 + a**2*c**2*x**6 *atan(a*x)/3 - a*c**2*x**5/24 + c**2*x**4*atan(a*x)/4 - c**2*x**3/(72*a) + c**2*x/(24*a**3) - c**2*atan(a*x)/(24*a**4), Ne(a, 0)), (0, True))
Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.88 \[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x) \, dx=-\frac {1}{504} \, a {\left (\frac {21 \, c^{2} \arctan \left (a x\right )}{a^{5}} + \frac {9 \, a^{6} c^{2} x^{7} + 21 \, a^{4} c^{2} x^{5} + 7 \, a^{2} c^{2} x^{3} - 21 \, c^{2} x}{a^{4}}\right )} + \frac {1}{24} \, {\left (3 \, a^{4} c^{2} x^{8} + 8 \, a^{2} c^{2} x^{6} + 6 \, c^{2} x^{4}\right )} \arctan \left (a x\right ) \]
-1/504*a*(21*c^2*arctan(a*x)/a^5 + (9*a^6*c^2*x^7 + 21*a^4*c^2*x^5 + 7*a^2 *c^2*x^3 - 21*c^2*x)/a^4) + 1/24*(3*a^4*c^2*x^8 + 8*a^2*c^2*x^6 + 6*c^2*x^ 4)*arctan(a*x)
\[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x) \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} x^{3} \arctan \left (a x\right ) \,d x } \]
Time = 0.49 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.80 \[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x) \, dx=\mathrm {atan}\left (a\,x\right )\,\left (\frac {a^4\,c^2\,x^8}{8}+\frac {a^2\,c^2\,x^6}{3}+\frac {c^2\,x^4}{4}\right )+\frac {c^2\,x}{24\,a^3}-\frac {a\,c^2\,x^5}{24}-\frac {c^2\,\mathrm {atan}\left (a\,x\right )}{24\,a^4}-\frac {c^2\,x^3}{72\,a}-\frac {a^3\,c^2\,x^7}{56} \]