Integrand size = 20, antiderivative size = 108 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x^2} \, dx=-\frac {2}{5} a^3 c^3 x^2-\frac {1}{20} a^5 c^3 x^4-\frac {c^3 \arctan (a x)}{x}+3 a^2 c^3 x \arctan (a x)+a^4 c^3 x^3 \arctan (a x)+\frac {1}{5} a^6 c^3 x^5 \arctan (a x)+a c^3 \log (x)-\frac {8}{5} a c^3 \log \left (1+a^2 x^2\right ) \]
-2/5*a^3*c^3*x^2-1/20*a^5*c^3*x^4-c^3*arctan(a*x)/x+3*a^2*c^3*x*arctan(a*x )+a^4*c^3*x^3*arctan(a*x)+1/5*a^6*c^3*x^5*arctan(a*x)+a*c^3*ln(x)-8/5*a*c^ 3*ln(a^2*x^2+1)
Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.72 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x^2} \, dx=\frac {c^3 \left (4 \left (-5+15 a^2 x^2+5 a^4 x^4+a^6 x^6\right ) \arctan (a x)-a x \left (8 a^2 x^2+a^4 x^4-20 \log (x)+32 \log \left (1+a^2 x^2\right )\right )\right )}{20 x} \]
(c^3*(4*(-5 + 15*a^2*x^2 + 5*a^4*x^4 + a^6*x^6)*ArcTan[a*x] - a*x*(8*a^2*x ^2 + a^4*x^4 - 20*Log[x] + 32*Log[1 + a^2*x^2])))/(20*x)
Time = 0.34 (sec) , antiderivative size = 108, 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.
\(\displaystyle \int \frac {\arctan (a x) \left (a^2 c x^2+c\right )^3}{x^2} \, dx\) |
\(\Big \downarrow \) 5483 |
\(\displaystyle \int \left (a^6 c^3 x^4 \arctan (a x)+3 a^4 c^3 x^2 \arctan (a x)+3 a^2 c^3 \arctan (a x)+\frac {c^3 \arctan (a x)}{x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{5} a^6 c^3 x^5 \arctan (a x)-\frac {1}{20} a^5 c^3 x^4+a^4 c^3 x^3 \arctan (a x)-\frac {2}{5} a^3 c^3 x^2+3 a^2 c^3 x \arctan (a x)-\frac {8}{5} a c^3 \log \left (a^2 x^2+1\right )-\frac {c^3 \arctan (a x)}{x}+a c^3 \log (x)\) |
(-2*a^3*c^3*x^2)/5 - (a^5*c^3*x^4)/20 - (c^3*ArcTan[a*x])/x + 3*a^2*c^3*x* ArcTan[a*x] + a^4*c^3*x^3*ArcTan[a*x] + (a^6*c^3*x^5*ArcTan[a*x])/5 + a*c^ 3*Log[x] - (8*a*c^3*Log[1 + a^2*x^2])/5
3.2.70.3.1 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 = 0.31 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.88
method | result | size |
parts | \(\frac {a^{6} c^{3} x^{5} \arctan \left (a x \right )}{5}+a^{4} c^{3} x^{3} \arctan \left (a x \right )+3 a^{2} c^{3} x \arctan \left (a x \right )-\frac {c^{3} \arctan \left (a x \right )}{x}-\frac {c^{3} a \left (\frac {a^{4} x^{4}}{4}+2 a^{2} x^{2}+8 \ln \left (a^{2} x^{2}+1\right )-5 \ln \left (x \right )\right )}{5}\) | \(95\) |
derivativedivides | \(a \left (\frac {a^{5} c^{3} x^{5} \arctan \left (a x \right )}{5}+a^{3} c^{3} x^{3} \arctan \left (a x \right )+3 a \,c^{3} x \arctan \left (a x \right )-\frac {c^{3} \arctan \left (a x \right )}{a x}-\frac {c^{3} \left (\frac {a^{4} x^{4}}{4}+2 a^{2} x^{2}+8 \ln \left (a^{2} x^{2}+1\right )-5 \ln \left (a x \right )\right )}{5}\right )\) | \(99\) |
default | \(a \left (\frac {a^{5} c^{3} x^{5} \arctan \left (a x \right )}{5}+a^{3} c^{3} x^{3} \arctan \left (a x \right )+3 a \,c^{3} x \arctan \left (a x \right )-\frac {c^{3} \arctan \left (a x \right )}{a x}-\frac {c^{3} \left (\frac {a^{4} x^{4}}{4}+2 a^{2} x^{2}+8 \ln \left (a^{2} x^{2}+1\right )-5 \ln \left (a x \right )\right )}{5}\right )\) | \(99\) |
parallelrisch | \(\frac {4 a^{6} c^{3} x^{6} \arctan \left (a x \right )-a^{5} c^{3} x^{5}+20 a^{4} c^{3} x^{4} \arctan \left (a x \right )-8 a^{3} c^{3} x^{3}+60 a^{2} c^{3} x^{2} \arctan \left (a x \right )+20 c^{3} a \ln \left (x \right ) x -32 c^{3} a \ln \left (a^{2} x^{2}+1\right ) x -20 c^{3} \arctan \left (a x \right )}{20 x}\) | \(109\) |
risch | \(-\frac {i c^{3} \left (a^{6} x^{6}+5 a^{4} x^{4}+15 a^{2} x^{2}-5\right ) \ln \left (i a x +1\right )}{10 x}+\frac {i c^{3} \left (2 a^{6} x^{6} \ln \left (-i a x +1\right )+i a^{5} x^{5}+10 x^{4} \ln \left (-i a x +1\right ) a^{4}+8 i a^{3} x^{3}+30 a^{2} x^{2} \ln \left (-i a x +1\right )-20 i a \ln \left (x \right ) x +32 i a \ln \left (-13 a^{2} x^{2}-13\right ) x -10 \ln \left (-i a x +1\right )\right )}{20 x}\) | \(153\) |
meijerg | \(\frac {a \,c^{3} \left (\frac {a^{2} x^{2} \left (-3 a^{2} x^{2}+6\right )}{15}+\frac {4 a^{6} x^{6} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{5 \sqrt {a^{2} x^{2}}}-\frac {2 \ln \left (a^{2} x^{2}+1\right )}{5}\right )}{4}+\frac {3 a \,c^{3} \left (-\frac {2 a^{2} x^{2}}{3}+\frac {4 a^{4} x^{4} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{3 \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (a^{2} x^{2}+1\right )}{3}\right )}{4}+\frac {3 a \,c^{3} \left (\frac {4 a^{2} x^{2} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}-2 \ln \left (a^{2} x^{2}+1\right )\right )}{4}+\frac {a \,c^{3} \left (4 \ln \left (x \right )+4 \ln \left (a \right )-\frac {4 \arctan \left (\sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}-2 \ln \left (a^{2} x^{2}+1\right )\right )}{4}\) | \(214\) |
1/5*a^6*c^3*x^5*arctan(a*x)+a^4*c^3*x^3*arctan(a*x)+3*a^2*c^3*x*arctan(a*x )-c^3*arctan(a*x)/x-1/5*c^3*a*(1/4*a^4*x^4+2*a^2*x^2+8*ln(a^2*x^2+1)-5*ln( x))
Time = 0.24 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.90 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x^2} \, dx=-\frac {a^{5} c^{3} x^{5} + 8 \, a^{3} c^{3} x^{3} + 32 \, a c^{3} x \log \left (a^{2} x^{2} + 1\right ) - 20 \, a c^{3} x \log \left (x\right ) - 4 \, {\left (a^{6} c^{3} x^{6} + 5 \, a^{4} c^{3} x^{4} + 15 \, a^{2} c^{3} x^{2} - 5 \, c^{3}\right )} \arctan \left (a x\right )}{20 \, x} \]
-1/20*(a^5*c^3*x^5 + 8*a^3*c^3*x^3 + 32*a*c^3*x*log(a^2*x^2 + 1) - 20*a*c^ 3*x*log(x) - 4*(a^6*c^3*x^6 + 5*a^4*c^3*x^4 + 15*a^2*c^3*x^2 - 5*c^3)*arct an(a*x))/x
Time = 0.48 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.02 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x^2} \, dx=\begin {cases} \frac {a^{6} c^{3} x^{5} \operatorname {atan}{\left (a x \right )}}{5} - \frac {a^{5} c^{3} x^{4}}{20} + a^{4} c^{3} x^{3} \operatorname {atan}{\left (a x \right )} - \frac {2 a^{3} c^{3} x^{2}}{5} + 3 a^{2} c^{3} x \operatorname {atan}{\left (a x \right )} + a c^{3} \log {\left (x \right )} - \frac {8 a c^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{5} - \frac {c^{3} \operatorname {atan}{\left (a x \right )}}{x} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((a**6*c**3*x**5*atan(a*x)/5 - a**5*c**3*x**4/20 + a**4*c**3*x**3 *atan(a*x) - 2*a**3*c**3*x**2/5 + 3*a**2*c**3*x*atan(a*x) + a*c**3*log(x) - 8*a*c**3*log(x**2 + a**(-2))/5 - c**3*atan(a*x)/x, Ne(a, 0)), (0, True))
Time = 0.19 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.86 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x^2} \, dx=-\frac {1}{20} \, {\left (a^{4} c^{3} x^{4} + 8 \, a^{2} c^{3} x^{2} + 32 \, c^{3} \log \left (a^{2} x^{2} + 1\right ) - 20 \, c^{3} \log \left (x\right )\right )} a + \frac {1}{5} \, {\left (a^{6} c^{3} x^{5} + 5 \, a^{4} c^{3} x^{3} + 15 \, a^{2} c^{3} x - \frac {5 \, c^{3}}{x}\right )} \arctan \left (a x\right ) \]
-1/20*(a^4*c^3*x^4 + 8*a^2*c^3*x^2 + 32*c^3*log(a^2*x^2 + 1) - 20*c^3*log( x))*a + 1/5*(a^6*c^3*x^5 + 5*a^4*c^3*x^3 + 15*a^2*c^3*x - 5*c^3/x)*arctan( a*x)
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )}{x^{2}} \,d x } \]
Time = 0.64 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.79 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x^2} \, dx=-\frac {c^3\,\left (\mathrm {atan}\left (a\,x\right )+\frac {2\,a^3\,x^3}{5}+\frac {a^5\,x^5}{20}-a\,x\,\ln \left (x\right )-3\,a^2\,x^2\,\mathrm {atan}\left (a\,x\right )-a^4\,x^4\,\mathrm {atan}\left (a\,x\right )-\frac {a^6\,x^6\,\mathrm {atan}\left (a\,x\right )}{5}+\frac {8\,a\,x\,\ln \left (a^2\,x^2+1\right )}{5}\right )}{x} \]