Integrand size = 22, antiderivative size = 191 \[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=-\frac {5 c^2 x^2}{504 a^2}+\frac {c^2 x^4}{84}+\frac {1}{168} a^2 c^2 x^6+\frac {c^2 x \arctan (a x)}{12 a^3}-\frac {c^2 x^3 \arctan (a x)}{36 a}-\frac {1}{12} a c^2 x^5 \arctan (a x)-\frac {1}{28} a^3 c^2 x^7 \arctan (a x)-\frac {c^2 \arctan (a x)^2}{24 a^4}+\frac {1}{4} c^2 x^4 \arctan (a x)^2+\frac {1}{3} a^2 c^2 x^6 \arctan (a x)^2+\frac {1}{8} a^4 c^2 x^8 \arctan (a x)^2-\frac {2 c^2 \log \left (1+a^2 x^2\right )}{63 a^4} \] Output:
-5/504*c^2*x^2/a^2+1/84*c^2*x^4+1/168*a^2*c^2*x^6+1/12*c^2*x*arctan(a*x)/a ^3-1/36*c^2*x^3*arctan(a*x)/a-1/12*a*c^2*x^5*arctan(a*x)-1/28*a^3*c^2*x^7* arctan(a*x)-1/24*c^2*arctan(a*x)^2/a^4+1/4*c^2*x^4*arctan(a*x)^2+1/3*a^2*c ^2*x^6*arctan(a*x)^2+1/8*a^4*c^2*x^8*arctan(a*x)^2-2/63*c^2*ln(a^2*x^2+1)/ a^4
Time = 0.04 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.58 \[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=\frac {c^2 \left (-5 a^2 x^2+6 a^4 x^4+3 a^6 x^6-2 a x \left (-21+7 a^2 x^2+21 a^4 x^4+9 a^6 x^6\right ) \arctan (a x)+21 \left (1+a^2 x^2\right )^3 \left (-1+3 a^2 x^2\right ) \arctan (a x)^2-16 \log \left (1+a^2 x^2\right )\right )}{504 a^4} \] Input:
Integrate[x^3*(c + a^2*c*x^2)^2*ArcTan[a*x]^2,x]
Output:
(c^2*(-5*a^2*x^2 + 6*a^4*x^4 + 3*a^6*x^6 - 2*a*x*(-21 + 7*a^2*x^2 + 21*a^4 *x^4 + 9*a^6*x^6)*ArcTan[a*x] + 21*(1 + a^2*x^2)^3*(-1 + 3*a^2*x^2)*ArcTan [a*x]^2 - 16*Log[1 + a^2*x^2]))/(504*a^4)
Time = 1.01 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.
| \(\displaystyle \int x^3 \arctan (a x)^2 \left (a^2 c x^2+c\right )^2 \, dx\) |
| \(\Big \downarrow \) 5483 |
| \(\displaystyle \int \left (a^4 c^2 x^7 \arctan (a x)^2+2 a^2 c^2 x^5 \arctan (a x)^2+c^2 x^3 \arctan (a x)^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{8} a^4 c^2 x^8 \arctan (a x)^2-\frac {c^2 \arctan (a x)^2}{24 a^4}-\frac {1}{28} a^3 c^2 x^7 \arctan (a x)+\frac {c^2 x \arctan (a x)}{12 a^3}+\frac {1}{3} a^2 c^2 x^6 \arctan (a x)^2+\frac {1}{168} a^2 c^2 x^6-\frac {5 c^2 x^2}{504 a^2}-\frac {2 c^2 \log \left (a^2 x^2+1\right )}{63 a^4}-\frac {1}{12} a c^2 x^5 \arctan (a x)+\frac {1}{4} c^2 x^4 \arctan (a x)^2-\frac {c^2 x^3 \arctan (a x)}{36 a}+\frac {c^2 x^4}{84}\) |
Input:
Int[x^3*(c + a^2*c*x^2)^2*ArcTan[a*x]^2,x]
Output:
(-5*c^2*x^2)/(504*a^2) + (c^2*x^4)/84 + (a^2*c^2*x^6)/168 + (c^2*x*ArcTan[ a*x])/(12*a^3) - (c^2*x^3*ArcTan[a*x])/(36*a) - (a*c^2*x^5*ArcTan[a*x])/12 - (a^3*c^2*x^7*ArcTan[a*x])/28 - (c^2*ArcTan[a*x]^2)/(24*a^4) + (c^2*x^4* ArcTan[a*x]^2)/4 + (a^2*c^2*x^6*ArcTan[a*x]^2)/3 + (a^4*c^2*x^8*ArcTan[a*x ]^2)/8 - (2*c^2*Log[1 + a^2*x^2])/(63*a^4)
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 = 1.18 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.79
| method | result | size |
| derivativedivides | \(\frac {\frac {c^{2} \arctan \left (a x \right )^{2} a^{8} x^{8}}{8}+\frac {c^{2} \arctan \left (a x \right )^{2} a^{6} x^{6}}{3}+\frac {a^{4} c^{2} x^{4} \arctan \left (a x \right )^{2}}{4}-\frac {c^{2} \left (\frac {3 \arctan \left (a x \right ) a^{7} x^{7}}{7}+\arctan \left (a x \right ) a^{5} x^{5}+\frac {\arctan \left (a x \right ) x^{3} a^{3}}{3}-\arctan \left (a x \right ) a x +\frac {\arctan \left (a x \right )^{2}}{2}-\frac {a^{6} x^{6}}{14}-\frac {a^{4} x^{4}}{7}+\frac {5 a^{2} x^{2}}{42}+\frac {8 \ln \left (a^{2} x^{2}+1\right )}{21}\right )}{12}}{a^{4}}\) | \(150\) |
| default | \(\frac {\frac {c^{2} \arctan \left (a x \right )^{2} a^{8} x^{8}}{8}+\frac {c^{2} \arctan \left (a x \right )^{2} a^{6} x^{6}}{3}+\frac {a^{4} c^{2} x^{4} \arctan \left (a x \right )^{2}}{4}-\frac {c^{2} \left (\frac {3 \arctan \left (a x \right ) a^{7} x^{7}}{7}+\arctan \left (a x \right ) a^{5} x^{5}+\frac {\arctan \left (a x \right ) x^{3} a^{3}}{3}-\arctan \left (a x \right ) a x +\frac {\arctan \left (a x \right )^{2}}{2}-\frac {a^{6} x^{6}}{14}-\frac {a^{4} x^{4}}{7}+\frac {5 a^{2} x^{2}}{42}+\frac {8 \ln \left (a^{2} x^{2}+1\right )}{21}\right )}{12}}{a^{4}}\) | \(150\) |
| parts | \(\frac {a^{4} c^{2} x^{8} \arctan \left (a x \right )^{2}}{8}+\frac {a^{2} c^{2} x^{6} \arctan \left (a x \right )^{2}}{3}+\frac {c^{2} x^{4} \arctan \left (a x \right )^{2}}{4}-\frac {c^{2} \left (\frac {3 a^{3} \arctan \left (a x \right ) x^{7}}{7}+a \arctan \left (a x \right ) x^{5}+\frac {\arctan \left (a x \right ) x^{3}}{3 a}-\frac {\arctan \left (a x \right ) x}{a^{3}}+\frac {\arctan \left (a x \right )^{2}}{a^{4}}-\frac {\frac {3 a^{6} x^{6}}{2}+3 a^{4} x^{4}-\frac {5 a^{2} x^{2}}{2}-8 \ln \left (a^{2} x^{2}+1\right )+\frac {21 \arctan \left (a x \right )^{2}}{2}}{21 a^{4}}\right )}{12}\) | \(159\) |
| parallelrisch | \(-\frac {-63 c^{2} \arctan \left (a x \right )^{2} a^{8} x^{8}+18 a^{7} c^{2} \arctan \left (a x \right ) x^{7}-168 c^{2} \arctan \left (a x \right )^{2} a^{6} x^{6}-3 a^{6} c^{2} x^{6}+42 a^{5} c^{2} \arctan \left (a x \right ) x^{5}-126 a^{4} c^{2} x^{4} \arctan \left (a x \right )^{2}-6 a^{4} c^{2} x^{4}+14 a^{3} c^{2} x^{3} \arctan \left (a x \right )+5 x^{2} c^{2} a^{2}-42 a \,c^{2} x \arctan \left (a x \right )+21 c^{2} \arctan \left (a x \right )^{2}+16 c^{2} \ln \left (a^{2} x^{2}+1\right )-5 c^{2}}{504 a^{4}}\) | \(178\) |
| risch | \(-\frac {c^{2} \left (3 a^{8} x^{8}+8 a^{6} x^{6}+6 a^{4} x^{4}-1\right ) \ln \left (i a x +1\right )^{2}}{96 a^{4}}+\frac {c^{2} \left (63 a^{8} x^{8} \ln \left (-i a x +1\right )+18 i a^{7} x^{7}+168 a^{6} x^{6} \ln \left (-i a x +1\right )+42 i a^{5} x^{5}+126 x^{4} \ln \left (-i a x +1\right ) a^{4}+14 i a^{3} x^{3}-42 i a x -21 \ln \left (-i a x +1\right )\right ) \ln \left (i a x +1\right )}{1008 a^{4}}-\frac {c^{2} a^{4} x^{8} \ln \left (-i a x +1\right )^{2}}{32}-\frac {i c^{2} a^{3} x^{7} \ln \left (-i a x +1\right )}{56}-\frac {c^{2} a^{2} x^{6} \ln \left (-i a x +1\right )^{2}}{12}-\frac {i c^{2} a \,x^{5} \ln \left (-i a x +1\right )}{24}+\frac {a^{2} c^{2} x^{6}}{168}-\frac {c^{2} x^{4} \ln \left (-i a x +1\right )^{2}}{16}-\frac {i c^{2} x^{3} \ln \left (-i a x +1\right )}{72 a}+\frac {c^{2} x^{4}}{84}+\frac {i c^{2} x \ln \left (-i a x +1\right )}{24 a^{3}}-\frac {5 c^{2} x^{2}}{504 a^{2}}+\frac {c^{2} \ln \left (-i a x +1\right )^{2}}{96 a^{4}}-\frac {2 c^{2} \ln \left (-a^{2} x^{2}-1\right )}{63 a^{4}}+\frac {c^{2}}{567 a^{4}}\) | \(364\) |
Input:
int(x^3*(a^2*c*x^2+c)^2*arctan(a*x)^2,x,method=_RETURNVERBOSE)
Output:
1/a^4*(1/8*c^2*arctan(a*x)^2*a^8*x^8+1/3*c^2*arctan(a*x)^2*a^6*x^6+1/4*a^4 *c^2*x^4*arctan(a*x)^2-1/12*c^2*(3/7*arctan(a*x)*a^7*x^7+arctan(a*x)*a^5*x ^5+1/3*arctan(a*x)*x^3*a^3-arctan(a*x)*a*x+1/2*arctan(a*x)^2-1/14*a^6*x^6- 1/7*a^4*x^4+5/42*a^2*x^2+8/21*ln(a^2*x^2+1)))
Time = 0.11 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.77 \[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=\frac {3 \, a^{6} c^{2} x^{6} + 6 \, a^{4} c^{2} x^{4} - 5 \, a^{2} c^{2} x^{2} + 21 \, {\left (3 \, a^{8} c^{2} x^{8} + 8 \, a^{6} c^{2} x^{6} + 6 \, a^{4} c^{2} x^{4} - c^{2}\right )} \arctan \left (a x\right )^{2} - 16 \, c^{2} \log \left (a^{2} x^{2} + 1\right ) - 2 \, {\left (9 \, a^{7} c^{2} x^{7} + 21 \, a^{5} c^{2} x^{5} + 7 \, a^{3} c^{2} x^{3} - 21 \, a c^{2} x\right )} \arctan \left (a x\right )}{504 \, a^{4}} \] Input:
integrate(x^3*(a^2*c*x^2+c)^2*arctan(a*x)^2,x, algorithm="fricas")
Output:
1/504*(3*a^6*c^2*x^6 + 6*a^4*c^2*x^4 - 5*a^2*c^2*x^2 + 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)^2 - 16*c^2*log(a^2*x^2 + 1) - 2*(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)*arct an(a*x))/a^4
Time = 0.54 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.97 \[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=\begin {cases} \frac {a^{4} c^{2} x^{8} \operatorname {atan}^{2}{\left (a x \right )}}{8} - \frac {a^{3} c^{2} x^{7} \operatorname {atan}{\left (a x \right )}}{28} + \frac {a^{2} c^{2} x^{6} \operatorname {atan}^{2}{\left (a x \right )}}{3} + \frac {a^{2} c^{2} x^{6}}{168} - \frac {a c^{2} x^{5} \operatorname {atan}{\left (a x \right )}}{12} + \frac {c^{2} x^{4} \operatorname {atan}^{2}{\left (a x \right )}}{4} + \frac {c^{2} x^{4}}{84} - \frac {c^{2} x^{3} \operatorname {atan}{\left (a x \right )}}{36 a} - \frac {5 c^{2} x^{2}}{504 a^{2}} + \frac {c^{2} x \operatorname {atan}{\left (a x \right )}}{12 a^{3}} - \frac {2 c^{2} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{63 a^{4}} - \frac {c^{2} \operatorname {atan}^{2}{\left (a x \right )}}{24 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate(x**3*(a**2*c*x**2+c)**2*atan(a*x)**2,x)
Output:
Piecewise((a**4*c**2*x**8*atan(a*x)**2/8 - a**3*c**2*x**7*atan(a*x)/28 + a **2*c**2*x**6*atan(a*x)**2/3 + a**2*c**2*x**6/168 - a*c**2*x**5*atan(a*x)/ 12 + c**2*x**4*atan(a*x)**2/4 + c**2*x**4/84 - c**2*x**3*atan(a*x)/(36*a) - 5*c**2*x**2/(504*a**2) + c**2*x*atan(a*x)/(12*a**3) - 2*c**2*log(x**2 + a**(-2))/(63*a**4) - c**2*atan(a*x)**2/(24*a**4), Ne(a, 0)), (0, True))
Time = 0.12 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.88 \[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=-\frac {1}{252} \, a {\left (\frac {21 \, c^{2} \arctan \left (a x\right )}{a^{5}} + \frac {9 \, a^{6} c^{2} x^{7} + 21 \, a^{4} c^{2} x^{5} + 7 \, a^{2} c^{2} x^{3} - 21 \, c^{2} x}{a^{4}}\right )} \arctan \left (a x\right ) + \frac {1}{24} \, {\left (3 \, a^{4} c^{2} x^{8} + 8 \, a^{2} c^{2} x^{6} + 6 \, c^{2} x^{4}\right )} \arctan \left (a x\right )^{2} + \frac {3 \, a^{6} c^{2} x^{6} + 6 \, a^{4} c^{2} x^{4} - 5 \, a^{2} c^{2} x^{2} + 21 \, c^{2} \arctan \left (a x\right )^{2} - 16 \, c^{2} \log \left (a^{2} x^{2} + 1\right )}{504 \, a^{4}} \] Input:
integrate(x^3*(a^2*c*x^2+c)^2*arctan(a*x)^2,x, algorithm="maxima")
Output:
-1/252*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)*arctan(a*x) + 1/24*(3*a^4*c^2*x^8 + 8*a^2*c^2*x^ 6 + 6*c^2*x^4)*arctan(a*x)^2 + 1/504*(3*a^6*c^2*x^6 + 6*a^4*c^2*x^4 - 5*a^ 2*c^2*x^2 + 21*c^2*arctan(a*x)^2 - 16*c^2*log(a^2*x^2 + 1))/a^4
Time = 0.14 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.84 \[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=\frac {1}{24} \, {\left (3 \, a^{4} c^{2} x^{8} + 8 \, a^{2} c^{2} x^{6} + 6 \, c^{2} x^{4}\right )} \arctan \left (a x\right )^{2} - \frac {18 \, a^{7} c^{2} x^{7} \arctan \left (a x\right ) - 3 \, a^{6} c^{2} x^{6} + 42 \, a^{5} c^{2} x^{5} \arctan \left (a x\right ) - 6 \, a^{4} c^{2} x^{4} + 14 \, a^{3} c^{2} x^{3} \arctan \left (a x\right ) + 5 \, a^{2} c^{2} x^{2} - 42 \, a c^{2} x \arctan \left (a x\right ) + 21 \, c^{2} \arctan \left (a x\right )^{2} + 16 \, c^{2} \log \left (a^{2} x^{2} + 1\right )}{504 \, a^{4}} \] Input:
integrate(x^3*(a^2*c*x^2+c)^2*arctan(a*x)^2,x, algorithm="giac")
Output:
1/24*(3*a^4*c^2*x^8 + 8*a^2*c^2*x^6 + 6*c^2*x^4)*arctan(a*x)^2 - 1/504*(18 *a^7*c^2*x^7*arctan(a*x) - 3*a^6*c^2*x^6 + 42*a^5*c^2*x^5*arctan(a*x) - 6* a^4*c^2*x^4 + 14*a^3*c^2*x^3*arctan(a*x) + 5*a^2*c^2*x^2 - 42*a*c^2*x*arct an(a*x) + 21*c^2*arctan(a*x)^2 + 16*c^2*log(a^2*x^2 + 1))/a^4
Time = 0.76 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.76 \[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx={\mathrm {atan}\left (a\,x\right )}^2\,\left (\frac {c^2\,x^4}{4}-\frac {c^2}{24\,a^4}+\frac {a^2\,c^2\,x^6}{3}+\frac {a^4\,c^2\,x^8}{8}\right )+\frac {c^2\,x^4}{84}-a^2\,\mathrm {atan}\left (a\,x\right )\,\left (\frac {a\,c^2\,x^7}{28}-\frac {c^2\,x}{12\,a^5}+\frac {c^2\,x^5}{12\,a}+\frac {c^2\,x^3}{36\,a^3}\right )-\frac {2\,c^2\,\ln \left (a^2\,x^2+1\right )}{63\,a^4}-\frac {5\,c^2\,x^2}{504\,a^2}+\frac {a^2\,c^2\,x^6}{168} \] Input:
int(x^3*atan(a*x)^2*(c + a^2*c*x^2)^2,x)
Output:
atan(a*x)^2*((c^2*x^4)/4 - c^2/(24*a^4) + (a^2*c^2*x^6)/3 + (a^4*c^2*x^8)/ 8) + (c^2*x^4)/84 - a^2*atan(a*x)*((a*c^2*x^7)/28 - (c^2*x)/(12*a^5) + (c^ 2*x^5)/(12*a) + (c^2*x^3)/(36*a^3)) - (2*c^2*log(a^2*x^2 + 1))/(63*a^4) - (5*c^2*x^2)/(504*a^2) + (a^2*c^2*x^6)/168
Time = 0.18 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.73 \[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=\frac {c^{2} \left (63 \mathit {atan} \left (a x \right )^{2} a^{8} x^{8}+168 \mathit {atan} \left (a x \right )^{2} a^{6} x^{6}+126 \mathit {atan} \left (a x \right )^{2} a^{4} x^{4}-21 \mathit {atan} \left (a x \right )^{2}-18 \mathit {atan} \left (a x \right ) a^{7} x^{7}-42 \mathit {atan} \left (a x \right ) a^{5} x^{5}-14 \mathit {atan} \left (a x \right ) a^{3} x^{3}+42 \mathit {atan} \left (a x \right ) a x -16 \,\mathrm {log}\left (a^{2} x^{2}+1\right )+3 a^{6} x^{6}+6 a^{4} x^{4}-5 a^{2} x^{2}\right )}{504 a^{4}} \] Input:
int(x^3*(a^2*c*x^2+c)^2*atan(a*x)^2,x)
Output:
(c**2*(63*atan(a*x)**2*a**8*x**8 + 168*atan(a*x)**2*a**6*x**6 + 126*atan(a *x)**2*a**4*x**4 - 21*atan(a*x)**2 - 18*atan(a*x)*a**7*x**7 - 42*atan(a*x) *a**5*x**5 - 14*atan(a*x)*a**3*x**3 + 42*atan(a*x)*a*x - 16*log(a**2*x**2 + 1) + 3*a**6*x**6 + 6*a**4*x**4 - 5*a**2*x**2))/(504*a**4)