Integrand size = 20, antiderivative size = 138 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x^3} \, dx=-\frac {a c^3}{2 x}-\frac {5}{4} a^3 c^3 x-\frac {1}{12} a^5 c^3 x^3+\frac {3}{4} a^2 c^3 \arctan (a x)-\frac {c^3 \arctan (a x)}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \arctan (a x)+\frac {1}{4} a^6 c^3 x^4 \arctan (a x)+\frac {3}{2} i a^2 c^3 \operatorname {PolyLog}(2,-i a x)-\frac {3}{2} i a^2 c^3 \operatorname {PolyLog}(2,i a x) \] Output:
-1/2*a*c^3/x-5/4*a^3*c^3*x-1/12*a^5*c^3*x^3+3/4*a^2*c^3*arctan(a*x)-1/2*c^ 3*arctan(a*x)/x^2+3/2*a^4*c^3*x^2*arctan(a*x)+1/4*a^6*c^3*x^4*arctan(a*x)+ 3/2*I*a^2*c^3*polylog(2,-I*a*x)-3/2*I*a^2*c^3*polylog(2,I*a*x)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.03 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.12 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x^3} \, dx=-\frac {5}{4} a^3 c^3 x-\frac {1}{12} a^5 c^3 x^3+\frac {5}{4} a^2 c^3 \arctan (a x)-\frac {c^3 \arctan (a x)}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \arctan (a x)+\frac {1}{4} a^6 c^3 x^4 \arctan (a x)-\frac {a c^3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-a^2 x^2\right )}{2 x}+\frac {3}{2} i a^2 c^3 \operatorname {PolyLog}(2,-i a x)-\frac {3}{2} i a^2 c^3 \operatorname {PolyLog}(2,i a x) \] Input:
Integrate[((c + a^2*c*x^2)^3*ArcTan[a*x])/x^3,x]
Output:
(-5*a^3*c^3*x)/4 - (a^5*c^3*x^3)/12 + (5*a^2*c^3*ArcTan[a*x])/4 - (c^3*Arc Tan[a*x])/(2*x^2) + (3*a^4*c^3*x^2*ArcTan[a*x])/2 + (a^6*c^3*x^4*ArcTan[a* x])/4 - (a*c^3*Hypergeometric2F1[-1/2, 1, 1/2, -(a^2*x^2)])/(2*x) + ((3*I) /2)*a^2*c^3*PolyLog[2, (-I)*a*x] - ((3*I)/2)*a^2*c^3*PolyLog[2, I*a*x]
Time = 0.35 (sec) , antiderivative size = 138, 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.
| \(\displaystyle \int \frac {\arctan (a x) \left (a^2 c x^2+c\right )^3}{x^3} \, dx\) |
| \(\Big \downarrow \) 5483 |
| \(\displaystyle \int \left (a^6 c^3 x^3 \arctan (a x)+3 a^4 c^3 x \arctan (a x)+\frac {3 a^2 c^3 \arctan (a x)}{x}+\frac {c^3 \arctan (a x)}{x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} a^6 c^3 x^4 \arctan (a x)-\frac {1}{12} a^5 c^3 x^3+\frac {3}{2} a^4 c^3 x^2 \arctan (a x)-\frac {5}{4} a^3 c^3 x+\frac {3}{4} a^2 c^3 \arctan (a x)+\frac {3}{2} i a^2 c^3 \operatorname {PolyLog}(2,-i a x)-\frac {3}{2} i a^2 c^3 \operatorname {PolyLog}(2,i a x)-\frac {c^3 \arctan (a x)}{2 x^2}-\frac {a c^3}{2 x}\) |
Input:
Int[((c + a^2*c*x^2)^3*ArcTan[a*x])/x^3,x]
Output:
-1/2*(a*c^3)/x - (5*a^3*c^3*x)/4 - (a^5*c^3*x^3)/12 + (3*a^2*c^3*ArcTan[a* x])/4 - (c^3*ArcTan[a*x])/(2*x^2) + (3*a^4*c^3*x^2*ArcTan[a*x])/2 + (a^6*c ^3*x^4*ArcTan[a*x])/4 + ((3*I)/2)*a^2*c^3*PolyLog[2, (-I)*a*x] - ((3*I)/2) *a^2*c^3*PolyLog[2, I*a*x]
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 = 1.06 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.07
| method | result | size |
| derivativedivides | \(a^{2} \left (\frac {a^{4} c^{3} x^{4} \arctan \left (a x \right )}{4}+\frac {3 a^{2} c^{3} x^{2} \arctan \left (a x \right )}{2}+3 c^{3} \arctan \left (a x \right ) \ln \left (a x \right )-\frac {c^{3} \arctan \left (a x \right )}{2 a^{2} x^{2}}-\frac {c^{3} \left (\frac {a^{3} x^{3}}{3}+5 a x +\frac {2}{a x}-3 \arctan \left (a x \right )-6 i \ln \left (a x \right ) \ln \left (i a x +1\right )+6 i \ln \left (a x \right ) \ln \left (-i a x +1\right )-6 i \operatorname {dilog}\left (i a x +1\right )+6 i \operatorname {dilog}\left (-i a x +1\right )\right )}{4}\right )\) | \(148\) |
| default | \(a^{2} \left (\frac {a^{4} c^{3} x^{4} \arctan \left (a x \right )}{4}+\frac {3 a^{2} c^{3} x^{2} \arctan \left (a x \right )}{2}+3 c^{3} \arctan \left (a x \right ) \ln \left (a x \right )-\frac {c^{3} \arctan \left (a x \right )}{2 a^{2} x^{2}}-\frac {c^{3} \left (\frac {a^{3} x^{3}}{3}+5 a x +\frac {2}{a x}-3 \arctan \left (a x \right )-6 i \ln \left (a x \right ) \ln \left (i a x +1\right )+6 i \ln \left (a x \right ) \ln \left (-i a x +1\right )-6 i \operatorname {dilog}\left (i a x +1\right )+6 i \operatorname {dilog}\left (-i a x +1\right )\right )}{4}\right )\) | \(148\) |
| parts | \(\frac {a^{6} c^{3} x^{4} \arctan \left (a x \right )}{4}+\frac {3 a^{4} c^{3} x^{2} \arctan \left (a x \right )}{2}-\frac {c^{3} \arctan \left (a x \right )}{2 x^{2}}+3 c^{3} \arctan \left (a x \right ) a^{2} \ln \left (x \right )-\frac {a \,c^{3} \left (\frac {x^{3} a^{4}}{3}+5 a^{2} x -3 a \arctan \left (a x \right )+\frac {2}{x}+12 a^{2} \left (-\frac {i \ln \left (x \right ) \left (-\ln \left (-i a x +1\right )+\ln \left (i a x +1\right )\right )}{2 a}-\frac {i \left (\operatorname {dilog}\left (i a x +1\right )-\operatorname {dilog}\left (-i a x +1\right )\right )}{2 a}\right )\right )}{4}\) | \(149\) |
| meijerg | \(\frac {a^{2} c^{3} \left (\frac {a x \left (-5 a^{2} x^{2}+15\right )}{15}-\frac {a x \left (-5 a^{4} x^{4}+5\right ) \arctan \left (\sqrt {a^{2} x^{2}}\right )}{5 \sqrt {a^{2} x^{2}}}\right )}{4}+\frac {3 a^{2} c^{3} \left (-2 a x +\frac {2 \left (3 a^{2} x^{2}+3\right ) \arctan \left (a x \right )}{3}\right )}{4}+\frac {3 a^{2} c^{3} \left (-\frac {2 i a x \operatorname {polylog}\left (2, i \sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}+\frac {2 i a x \operatorname {polylog}\left (2, -i \sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}\right )}{4}+\frac {a^{2} c^{3} \left (-\frac {2}{a x}-\frac {2 \left (a^{2} x^{2}+1\right ) \arctan \left (a x \right )}{a^{2} x^{2}}\right )}{4}\) | \(190\) |
| risch | \(-\frac {i c^{3} a^{6} \ln \left (i a x +1\right ) x^{4}}{8}+\frac {3 i c^{3} a^{4} \ln \left (-i a x +1\right ) x^{2}}{4}-\frac {3 i c^{3} a^{4} \ln \left (i a x +1\right ) x^{2}}{4}+\frac {3 a^{2} c^{3} \arctan \left (a x \right )}{4}-\frac {a^{5} c^{3} x^{3}}{12}-\frac {5 a^{3} c^{3} x}{4}-\frac {a \,c^{3}}{2 x}+\frac {i c^{3} a^{2} \ln \left (-i a x \right )}{4}+\frac {i c^{3} \ln \left (i a x +1\right )}{4 x^{2}}+\frac {i c^{3} a^{6} \ln \left (-i a x +1\right ) x^{4}}{8}-\frac {3 i c^{3} a^{2} \operatorname {dilog}\left (-i a x +1\right )}{2}-\frac {i c^{3} \ln \left (-i a x +1\right )}{4 x^{2}}+\frac {3 i c^{3} a^{2} \operatorname {dilog}\left (i a x +1\right )}{2}-\frac {i c^{3} a^{2} \ln \left (i a x \right )}{4}\) | \(221\) |
Input:
int((a^2*c*x^2+c)^3*arctan(a*x)/x^3,x,method=_RETURNVERBOSE)
Output:
a^2*(1/4*a^4*c^3*x^4*arctan(a*x)+3/2*a^2*c^3*x^2*arctan(a*x)+3*c^3*arctan( a*x)*ln(a*x)-1/2*c^3*arctan(a*x)/a^2/x^2-1/4*c^3*(1/3*a^3*x^3+5*a*x+2/a/x- 3*arctan(a*x)-6*I*ln(a*x)*ln(1+I*a*x)+6*I*ln(a*x)*ln(1-I*a*x)-6*I*dilog(1+ I*a*x)+6*I*dilog(1-I*a*x)))
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )}{x^{3}} \,d x } \] Input:
integrate((a^2*c*x^2+c)^3*arctan(a*x)/x^3,x, algorithm="fricas")
Output:
integral((a^6*c^3*x^6 + 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 + c^3)*arctan(a*x)/x ^3, x)
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x^3} \, dx=c^{3} \left (\int \frac {\operatorname {atan}{\left (a x \right )}}{x^{3}}\, dx + \int \frac {3 a^{2} \operatorname {atan}{\left (a x \right )}}{x}\, dx + \int 3 a^{4} x \operatorname {atan}{\left (a x \right )}\, dx + \int a^{6} x^{3} \operatorname {atan}{\left (a x \right )}\, dx\right ) \] Input:
integrate((a**2*c*x**2+c)**3*atan(a*x)/x**3,x)
Output:
c**3*(Integral(atan(a*x)/x**3, x) + Integral(3*a**2*atan(a*x)/x, x) + Inte gral(3*a**4*x*atan(a*x), x) + Integral(a**6*x**3*atan(a*x), x))
Time = 0.16 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.12 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x^3} \, dx=-\frac {a^{5} c^{3} x^{5} + 15 \, a^{3} c^{3} x^{3} + 9 \, \pi a^{2} c^{3} x^{2} \log \left (a^{2} x^{2} + 1\right ) - 36 \, a^{2} c^{3} x^{2} \arctan \left (a x\right ) \log \left (a x\right ) + 18 i \, a^{2} c^{3} x^{2} {\rm Li}_2\left (i \, a x + 1\right ) - 18 i \, a^{2} c^{3} x^{2} {\rm Li}_2\left (-i \, a x + 1\right ) + 6 \, a c^{3} x - 3 \, {\left (a^{6} c^{3} x^{6} + 6 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - 2 \, c^{3}\right )} \arctan \left (a x\right )}{12 \, x^{2}} \] Input:
integrate((a^2*c*x^2+c)^3*arctan(a*x)/x^3,x, algorithm="maxima")
Output:
-1/12*(a^5*c^3*x^5 + 15*a^3*c^3*x^3 + 9*pi*a^2*c^3*x^2*log(a^2*x^2 + 1) - 36*a^2*c^3*x^2*arctan(a*x)*log(a*x) + 18*I*a^2*c^3*x^2*dilog(I*a*x + 1) - 18*I*a^2*c^3*x^2*dilog(-I*a*x + 1) + 6*a*c^3*x - 3*(a^6*c^3*x^6 + 6*a^4*c^ 3*x^4 + 3*a^2*c^3*x^2 - 2*c^3)*arctan(a*x))/x^2
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )}{x^{3}} \,d x } \] Input:
integrate((a^2*c*x^2+c)^3*arctan(a*x)/x^3,x, algorithm="giac")
Output:
integrate((a^2*c*x^2 + c)^3*arctan(a*x)/x^3, x)
Time = 0.83 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.10 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x^3} \, dx=\left \{\begin {array}{cl} 0 & \text {\ if\ \ }a=0\\ 3\,a^4\,c^3\,\mathrm {atan}\left (a\,x\right )\,\left (\frac {1}{2\,a^2}+\frac {x^2}{2}\right )-\frac {a^2\,c^3\,\left (3\,\mathrm {atan}\left (a\,x\right )-3\,a\,x+a^3\,x^3\right )}{12}-\frac {c^3\,\mathrm {atan}\left (a\,x\right )}{2\,x^2}-\frac {c^3\,\left (a^3\,\mathrm {atan}\left (a\,x\right )+\frac {a^2}{x}\right )}{2\,a}-\frac {3\,a^3\,c^3\,x}{2}+\frac {a^6\,c^3\,x^4\,\mathrm {atan}\left (a\,x\right )}{4}-\frac {a^2\,c^3\,{\mathrm {Li}}_{\mathrm {2}}\left (1-a\,x\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2}+\frac {a^2\,c^3\,{\mathrm {Li}}_{\mathrm {2}}\left (1+a\,x\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2} & \text {\ if\ \ }a\neq 0 \end {array}\right . \] Input:
int((atan(a*x)*(c + a^2*c*x^2)^3)/x^3,x)
Output:
piecewise(a == 0, 0, a ~= 0, - (3*a^3*c^3*x)/2 - (a^2*c^3*(3*atan(a*x) - 3 *a*x + a^3*x^3))/12 - (c^3*atan(a*x))/(2*x^2) - (a^2*c^3*dilog(- a*x*1i + 1)*3i)/2 + (a^2*c^3*dilog(a*x*1i + 1)*3i)/2 - (c^3*(a^3*atan(a*x) + a^2/x) )/(2*a) + 3*a^4*c^3*atan(a*x)*(1/(2*a^2) + x^2/2) + (a^6*c^3*x^4*atan(a*x) )/4)
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x^3} \, dx=\frac {c^{3} \left (3 \mathit {atan} \left (a x \right ) a^{6} x^{6}+18 \mathit {atan} \left (a x \right ) a^{4} x^{4}+9 \mathit {atan} \left (a x \right ) a^{2} x^{2}-6 \mathit {atan} \left (a x \right )+36 \left (\int \frac {\mathit {atan} \left (a x \right )}{x}d x \right ) a^{2} x^{2}-a^{5} x^{5}-15 a^{3} x^{3}-6 a x \right )}{12 x^{2}} \] Input:
int((a^2*c*x^2+c)^3*atan(a*x)/x^3,x)
Output:
(c**3*(3*atan(a*x)*a**6*x**6 + 18*atan(a*x)*a**4*x**4 + 9*atan(a*x)*a**2*x **2 - 6*atan(a*x) + 36*int(atan(a*x)/x,x)*a**2*x**2 - a**5*x**5 - 15*a**3* x**3 - 6*a*x))/(12*x**2)