Integrand size = 20, antiderivative size = 116 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x^4} \, dx=-\frac {a c^3}{6 x^2}-\frac {1}{6} a^5 c^3 x^2-\frac {c^3 \arctan (a x)}{3 x^3}-\frac {3 a^2 c^3 \arctan (a x)}{x}+3 a^4 c^3 x \arctan (a x)+\frac {1}{3} a^6 c^3 x^3 \arctan (a x)+\frac {8}{3} a^3 c^3 \log (x)-\frac {8}{3} a^3 c^3 \log \left (1+a^2 x^2\right ) \] Output:
-1/6*a*c^3/x^2-1/6*a^5*c^3*x^2-1/3*c^3*arctan(a*x)/x^3-3*a^2*c^3*arctan(a* x)/x+3*a^4*c^3*x*arctan(a*x)+1/3*a^6*c^3*x^3*arctan(a*x)+8/3*a^3*c^3*ln(x) -8/3*a^3*c^3*ln(a^2*x^2+1)
Time = 0.03 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.72 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x^4} \, dx=\frac {c^3 \left (2 \left (-1-9 a^2 x^2+9 a^4 x^4+a^6 x^6\right ) \arctan (a x)-a x \left (1+a^4 x^4-16 a^2 x^2 \log (x)+16 a^2 x^2 \log \left (1+a^2 x^2\right )\right )\right )}{6 x^3} \] Input:
Integrate[((c + a^2*c*x^2)^3*ArcTan[a*x])/x^4,x]
Output:
(c^3*(2*(-1 - 9*a^2*x^2 + 9*a^4*x^4 + a^6*x^6)*ArcTan[a*x] - a*x*(1 + a^4* x^4 - 16*a^2*x^2*Log[x] + 16*a^2*x^2*Log[1 + a^2*x^2])))/(6*x^3)
Time = 0.36 (sec) , antiderivative size = 116, 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.
\(\displaystyle \int \frac {\arctan (a x) \left (a^2 c x^2+c\right )^3}{x^4} \, dx\) |
\(\Big \downarrow \) 5483 |
\(\displaystyle \int \left (a^6 c^3 x^2 \arctan (a x)+3 a^4 c^3 \arctan (a x)+\frac {3 a^2 c^3 \arctan (a x)}{x^2}+\frac {c^3 \arctan (a x)}{x^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} a^6 c^3 x^3 \arctan (a x)-\frac {1}{6} a^5 c^3 x^2+3 a^4 c^3 x \arctan (a x)+\frac {8}{3} a^3 c^3 \log (x)-\frac {3 a^2 c^3 \arctan (a x)}{x}-\frac {8}{3} a^3 c^3 \log \left (a^2 x^2+1\right )-\frac {c^3 \arctan (a x)}{3 x^3}-\frac {a c^3}{6 x^2}\) |
Input:
Int[((c + a^2*c*x^2)^3*ArcTan[a*x])/x^4,x]
Output:
-1/6*(a*c^3)/x^2 - (a^5*c^3*x^2)/6 - (c^3*ArcTan[a*x])/(3*x^3) - (3*a^2*c^ 3*ArcTan[a*x])/x + 3*a^4*c^3*x*ArcTan[a*x] + (a^6*c^3*x^3*ArcTan[a*x])/3 + (8*a^3*c^3*Log[x])/3 - (8*a^3*c^3*Log[1 + a^2*x^2])/3
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 = 0.57 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.85
method | result | size |
parts | \(\frac {a^{6} c^{3} x^{3} \arctan \left (a x \right )}{3}+3 a^{4} c^{3} x \arctan \left (a x \right )-\frac {c^{3} \arctan \left (a x \right )}{3 x^{3}}-\frac {3 a^{2} c^{3} \arctan \left (a x \right )}{x}-\frac {a \,c^{3} \left (\frac {x^{2} a^{4}}{2}+8 a^{2} \ln \left (a^{2} x^{2}+1\right )+\frac {1}{2 x^{2}}-8 a^{2} \ln \left (x \right )\right )}{3}\) | \(99\) |
derivativedivides | \(a^{3} \left (\frac {a^{3} c^{3} x^{3} \arctan \left (a x \right )}{3}+3 a \,c^{3} x \arctan \left (a x \right )-\frac {3 c^{3} \arctan \left (a x \right )}{a x}-\frac {c^{3} \arctan \left (a x \right )}{3 a^{3} x^{3}}-\frac {c^{3} \left (\frac {a^{2} x^{2}}{2}+8 \ln \left (a^{2} x^{2}+1\right )+\frac {1}{2 a^{2} x^{2}}-8 \ln \left (a x \right )\right )}{3}\right )\) | \(102\) |
default | \(a^{3} \left (\frac {a^{3} c^{3} x^{3} \arctan \left (a x \right )}{3}+3 a \,c^{3} x \arctan \left (a x \right )-\frac {3 c^{3} \arctan \left (a x \right )}{a x}-\frac {c^{3} \arctan \left (a x \right )}{3 a^{3} x^{3}}-\frac {c^{3} \left (\frac {a^{2} x^{2}}{2}+8 \ln \left (a^{2} x^{2}+1\right )+\frac {1}{2 a^{2} x^{2}}-8 \ln \left (a x \right )\right )}{3}\right )\) | \(102\) |
parallelrisch | \(\frac {2 a^{6} c^{3} x^{6} \arctan \left (a x \right )-a^{5} c^{3} x^{5}+18 a^{4} c^{3} x^{4} \arctan \left (a x \right )+16 a^{3} c^{3} \ln \left (x \right ) x^{3}-16 a^{3} c^{3} \ln \left (a^{2} x^{2}+1\right ) x^{3}+c^{3} x^{3} a^{3}-18 a^{2} c^{3} x^{2} \arctan \left (a x \right )-a \,c^{3} x -2 c^{3} \arctan \left (a x \right )}{6 x^{3}}\) | \(123\) |
risch | \(-\frac {i c^{3} \left (a^{6} x^{6}+9 a^{4} x^{4}-9 a^{2} x^{2}-1\right ) \ln \left (i a x +1\right )}{6 x^{3}}+\frac {i c^{3} \left (a^{6} x^{6} \ln \left (-i a x +1\right )+i a^{5} x^{5}+9 x^{4} \ln \left (-i a x +1\right ) a^{4}-16 i \ln \left (x \right ) a^{3} x^{3}+16 i \ln \left (2 a^{2} x^{2}+2\right ) a^{3} x^{3}-9 a^{2} x^{2} \ln \left (-i a x +1\right )+i a x -\ln \left (-i a x +1\right )\right )}{6 x^{3}}\) | \(156\) |
meijerg | \(\frac {a^{3} c^{3} \left (-\frac {2 a^{2} x^{2}}{3}+\frac {4 a^{4} x^{4} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{3 \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (a^{2} x^{2}+1\right )}{3}\right )}{4}+\frac {3 a^{3} c^{3} \left (\frac {4 a^{2} x^{2} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}-2 \ln \left (a^{2} x^{2}+1\right )\right )}{4}+\frac {3 a^{3} c^{3} \left (4 \ln \left (x \right )+4 \ln \left (a \right )-\frac {4 \arctan \left (\sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}-2 \ln \left (a^{2} x^{2}+1\right )\right )}{4}+\frac {a^{3} c^{3} \left (-\frac {2}{a^{2} x^{2}}+\frac {4}{9}-\frac {4 \ln \left (x \right )}{3}-\frac {4 \ln \left (a \right )}{3}+\frac {-\frac {4 a^{2} x^{2}}{9}+\frac {4}{3}}{a^{2} x^{2}}-\frac {4 \arctan \left (\sqrt {a^{2} x^{2}}\right )}{3 a^{2} x^{2} \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (a^{2} x^{2}+1\right )}{3}\right )}{4}\) | \(239\) |
Input:
int((a^2*c*x^2+c)^3*arctan(a*x)/x^4,x,method=_RETURNVERBOSE)
Output:
1/3*a^6*c^3*x^3*arctan(a*x)+3*a^4*c^3*x*arctan(a*x)-1/3*c^3*arctan(a*x)/x^ 3-3*a^2*c^3*arctan(a*x)/x-1/3*a*c^3*(1/2*x^2*a^4+8*a^2*ln(a^2*x^2+1)+1/2/x ^2-8*a^2*ln(x))
Time = 0.13 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.86 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x^4} \, dx=-\frac {a^{5} c^{3} x^{5} + 16 \, a^{3} c^{3} x^{3} \log \left (a^{2} x^{2} + 1\right ) - 16 \, a^{3} c^{3} x^{3} \log \left (x\right ) + a c^{3} x - 2 \, {\left (a^{6} c^{3} x^{6} + 9 \, a^{4} c^{3} x^{4} - 9 \, a^{2} c^{3} x^{2} - c^{3}\right )} \arctan \left (a x\right )}{6 \, x^{3}} \] Input:
integrate((a^2*c*x^2+c)^3*arctan(a*x)/x^4,x, algorithm="fricas")
Output:
-1/6*(a^5*c^3*x^5 + 16*a^3*c^3*x^3*log(a^2*x^2 + 1) - 16*a^3*c^3*x^3*log(x ) + a*c^3*x - 2*(a^6*c^3*x^6 + 9*a^4*c^3*x^4 - 9*a^2*c^3*x^2 - c^3)*arctan (a*x))/x^3
Time = 0.50 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.01 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x^4} \, dx=\begin {cases} \frac {a^{6} c^{3} x^{3} \operatorname {atan}{\left (a x \right )}}{3} - \frac {a^{5} c^{3} x^{2}}{6} + 3 a^{4} c^{3} x \operatorname {atan}{\left (a x \right )} + \frac {8 a^{3} c^{3} \log {\left (x \right )}}{3} - \frac {8 a^{3} c^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{3} - \frac {3 a^{2} c^{3} \operatorname {atan}{\left (a x \right )}}{x} - \frac {a c^{3}}{6 x^{2}} - \frac {c^{3} \operatorname {atan}{\left (a x \right )}}{3 x^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate((a**2*c*x**2+c)**3*atan(a*x)/x**4,x)
Output:
Piecewise((a**6*c**3*x**3*atan(a*x)/3 - a**5*c**3*x**2/6 + 3*a**4*c**3*x*a tan(a*x) + 8*a**3*c**3*log(x)/3 - 8*a**3*c**3*log(x**2 + a**(-2))/3 - 3*a* *2*c**3*atan(a*x)/x - a*c**3/(6*x**2) - c**3*atan(a*x)/(3*x**3), Ne(a, 0)) , (0, True))
Time = 0.03 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.83 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x^4} \, dx=-\frac {1}{6} \, {\left (a^{4} c^{3} x^{2} + 16 \, a^{2} c^{3} \log \left (a^{2} x^{2} + 1\right ) - 16 \, a^{2} c^{3} \log \left (x\right ) + \frac {c^{3}}{x^{2}}\right )} a + \frac {1}{3} \, {\left (a^{6} c^{3} x^{3} + 9 \, a^{4} c^{3} x - \frac {9 \, a^{2} c^{3} x^{2} + c^{3}}{x^{3}}\right )} \arctan \left (a x\right ) \] Input:
integrate((a^2*c*x^2+c)^3*arctan(a*x)/x^4,x, algorithm="maxima")
Output:
-1/6*(a^4*c^3*x^2 + 16*a^2*c^3*log(a^2*x^2 + 1) - 16*a^2*c^3*log(x) + c^3/ x^2)*a + 1/3*(a^6*c^3*x^3 + 9*a^4*c^3*x - (9*a^2*c^3*x^2 + c^3)/x^3)*arcta n(a*x)
Time = 0.12 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.95 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x^4} \, dx=-\frac {1}{6} \, a^{5} c^{3} x^{2} - \frac {8}{3} \, a^{3} c^{3} \log \left (a^{2} x^{2} + 1\right ) + \frac {4}{3} \, a^{3} c^{3} \log \left (x^{2}\right ) + \frac {1}{3} \, {\left (a^{6} c^{3} x^{3} + 9 \, a^{4} c^{3} x - \frac {9 \, a^{2} c^{3} x^{2} + c^{3}}{x^{3}}\right )} \arctan \left (a x\right ) - \frac {8 \, a^{3} c^{3} x^{2} + a c^{3}}{6 \, x^{2}} \] Input:
integrate((a^2*c*x^2+c)^3*arctan(a*x)/x^4,x, algorithm="giac")
Output:
-1/6*a^5*c^3*x^2 - 8/3*a^3*c^3*log(a^2*x^2 + 1) + 4/3*a^3*c^3*log(x^2) + 1 /3*(a^6*c^3*x^3 + 9*a^4*c^3*x - (9*a^2*c^3*x^2 + c^3)/x^3)*arctan(a*x) - 1 /6*(8*a^3*c^3*x^2 + a*c^3)/x^2
Time = 0.79 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.84 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x^4} \, dx=-\frac {c^3\,\left (2\,\mathrm {atan}\left (a\,x\right )+a\,x-a^3\,x^3+a^5\,x^5+18\,a^2\,x^2\,\mathrm {atan}\left (a\,x\right )-18\,a^4\,x^4\,\mathrm {atan}\left (a\,x\right )-2\,a^6\,x^6\,\mathrm {atan}\left (a\,x\right )+16\,a^3\,x^3\,\ln \left (a^2\,x^2+1\right )-16\,a^3\,x^3\,\ln \left (x\right )\right )}{6\,x^3} \] Input:
int((atan(a*x)*(c + a^2*c*x^2)^3)/x^4,x)
Output:
-(c^3*(2*atan(a*x) + a*x - a^3*x^3 + a^5*x^5 + 18*a^2*x^2*atan(a*x) - 18*a ^4*x^4*atan(a*x) - 2*a^6*x^6*atan(a*x) + 16*a^3*x^3*log(a^2*x^2 + 1) - 16* a^3*x^3*log(x)))/(6*x^3)
Time = 0.21 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.78 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x^4} \, dx=\frac {c^{3} \left (2 \mathit {atan} \left (a x \right ) a^{6} x^{6}+18 \mathit {atan} \left (a x \right ) a^{4} x^{4}-18 \mathit {atan} \left (a x \right ) a^{2} x^{2}-2 \mathit {atan} \left (a x \right )-16 \,\mathrm {log}\left (a^{2} x^{2}+1\right ) a^{3} x^{3}+16 \,\mathrm {log}\left (x \right ) a^{3} x^{3}-a^{5} x^{5}-a x \right )}{6 x^{3}} \] Input:
int((a^2*c*x^2+c)^3*atan(a*x)/x^4,x)
Output:
(c**3*(2*atan(a*x)*a**6*x**6 + 18*atan(a*x)*a**4*x**4 - 18*atan(a*x)*a**2* x**2 - 2*atan(a*x) - 16*log(a**2*x**2 + 1)*a**3*x**3 + 16*log(x)*a**3*x**3 - a**5*x**5 - a*x))/(6*x**3)