Integrand size = 20, antiderivative size = 90 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)}{x^3} \, dx=-\frac {a c^2}{2 x}-\frac {1}{2} a^3 c^2 x-\frac {c^2 \arctan (a x)}{2 x^2}+\frac {1}{2} a^4 c^2 x^2 \arctan (a x)+i a^2 c^2 \operatorname {PolyLog}(2,-i a x)-i a^2 c^2 \operatorname {PolyLog}(2,i a x) \]
-1/2*a*c^2/x-1/2*a^3*c^2*x-1/2*c^2*arctan(a*x)/x^2+1/2*a^4*c^2*x^2*arctan( a*x)+I*a^2*c^2*polylog(2,-I*a*x)-I*a^2*c^2*polylog(2,I*a*x)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.04 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.33 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)}{x^3} \, dx=-\frac {1}{2} a^3 c^2 x+\frac {1}{2} a^2 c^2 \arctan (a x)-\frac {c^2 \arctan (a x)}{2 x^2}+\frac {1}{2} a^4 c^2 x^2 \arctan (a x)-\frac {a c^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-a^2 x^2\right )}{2 x}+i a^2 c^2 \operatorname {PolyLog}(2,-i a x)-i a^2 c^2 \operatorname {PolyLog}(2,i a x) \]
-1/2*(a^3*c^2*x) + (a^2*c^2*ArcTan[a*x])/2 - (c^2*ArcTan[a*x])/(2*x^2) + ( a^4*c^2*x^2*ArcTan[a*x])/2 - (a*c^2*Hypergeometric2F1[-1/2, 1, 1/2, -(a^2* x^2)])/(2*x) + I*a^2*c^2*PolyLog[2, (-I)*a*x] - I*a^2*c^2*PolyLog[2, I*a*x ]
Time = 0.29 (sec) , antiderivative size = 90, 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.
\(\displaystyle \int \frac {\arctan (a x) \left (a^2 c x^2+c\right )^2}{x^3} \, dx\) |
\(\Big \downarrow \) 5483 |
\(\displaystyle \int \left (a^4 c^2 x \arctan (a x)+\frac {2 a^2 c^2 \arctan (a x)}{x}+\frac {c^2 \arctan (a x)}{x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} a^4 c^2 x^2 \arctan (a x)-\frac {1}{2} a^3 c^2 x+i a^2 c^2 \operatorname {PolyLog}(2,-i a x)-i a^2 c^2 \operatorname {PolyLog}(2,i a x)-\frac {c^2 \arctan (a x)}{2 x^2}-\frac {a c^2}{2 x}\) |
-1/2*(a*c^2)/x - (a^3*c^2*x)/2 - (c^2*ArcTan[a*x])/(2*x^2) + (a^4*c^2*x^2* ArcTan[a*x])/2 + I*a^2*c^2*PolyLog[2, (-I)*a*x] - I*a^2*c^2*PolyLog[2, I*a *x]
3.2.63.3.1 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 = 0.48 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.29
method | result | size |
parts | \(\frac {a^{4} c^{2} x^{2} \arctan \left (a x \right )}{2}+2 c^{2} \arctan \left (a x \right ) a^{2} \ln \left (x \right )-\frac {c^{2} \arctan \left (a x \right )}{2 x^{2}}-\frac {c^{2} a \left (a^{2} x +\frac {1}{x}+4 a^{2} \left (-\frac {i \ln \left (x \right ) \left (\ln \left (i a x +1\right )-\ln \left (-i a x +1\right )\right )}{2 a}-\frac {i \left (\operatorname {dilog}\left (i a x +1\right )-\operatorname {dilog}\left (-i a x +1\right )\right )}{2 a}\right )\right )}{2}\) | \(116\) |
derivativedivides | \(a^{2} \left (\frac {a^{2} c^{2} x^{2} \arctan \left (a x \right )}{2}-\frac {c^{2} \arctan \left (a x \right )}{2 a^{2} x^{2}}+2 c^{2} \arctan \left (a x \right ) \ln \left (a x \right )-\frac {c^{2} \left (a x +\frac {1}{a x}-2 i \ln \left (a x \right ) \ln \left (i a x +1\right )+2 i \ln \left (a x \right ) \ln \left (-i a x +1\right )-2 i \operatorname {dilog}\left (i a x +1\right )+2 i \operatorname {dilog}\left (-i a x +1\right )\right )}{2}\right )\) | \(117\) |
default | \(a^{2} \left (\frac {a^{2} c^{2} x^{2} \arctan \left (a x \right )}{2}-\frac {c^{2} \arctan \left (a x \right )}{2 a^{2} x^{2}}+2 c^{2} \arctan \left (a x \right ) \ln \left (a x \right )-\frac {c^{2} \left (a x +\frac {1}{a x}-2 i \ln \left (a x \right ) \ln \left (i a x +1\right )+2 i \ln \left (a x \right ) \ln \left (-i a x +1\right )-2 i \operatorname {dilog}\left (i a x +1\right )+2 i \operatorname {dilog}\left (-i a x +1\right )\right )}{2}\right )\) | \(117\) |
meijerg | \(\frac {a^{2} c^{2} \left (-2 a x +\frac {2 \left (3 a^{2} x^{2}+3\right ) \arctan \left (a x \right )}{3}\right )}{4}+\frac {a^{2} c^{2} \left (-\frac {2 i a x \operatorname {polylog}\left (2, i \sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}+\frac {2 i a x \operatorname {polylog}\left (2, -i \sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}\right )}{2}+\frac {a^{2} c^{2} \left (-\frac {2}{a x}-\frac {2 \left (a^{2} x^{2}+1\right ) \arctan \left (a x \right )}{a^{2} x^{2}}\right )}{4}\) | \(134\) |
risch | \(\frac {i c^{2} a^{4} \ln \left (-i a x +1\right ) x^{2}}{4}-\frac {a^{3} c^{2} x}{2}+\frac {i c^{2} a^{2} \ln \left (-i a x \right )}{4}-\frac {a \,c^{2}}{2 x}-\frac {i c^{2} \ln \left (-i a x +1\right )}{4 x^{2}}-i c^{2} a^{2} \operatorname {dilog}\left (-i a x +1\right )-\frac {i c^{2} a^{4} \ln \left (i a x +1\right ) x^{2}}{4}-\frac {i c^{2} a^{2} \ln \left (i a x \right )}{4}+\frac {i c^{2} \ln \left (i a x +1\right )}{4 x^{2}}+i c^{2} a^{2} \operatorname {dilog}\left (i a x +1\right )\) | \(158\) |
1/2*a^4*c^2*x^2*arctan(a*x)+2*c^2*arctan(a*x)*a^2*ln(x)-1/2*c^2*arctan(a*x )/x^2-1/2*c^2*a*(a^2*x+1/x+4*a^2*(-1/2*I*ln(x)*(ln(1+I*a*x)-ln(1-I*a*x))/a -1/2*I*(dilog(1+I*a*x)-dilog(1-I*a*x))/a))
\[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )}{x^{3}} \,d x } \]
\[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)}{x^3} \, dx=c^{2} \left (\int \frac {\operatorname {atan}{\left (a x \right )}}{x^{3}}\, dx + \int \frac {2 a^{2} \operatorname {atan}{\left (a x \right )}}{x}\, dx + \int a^{4} x \operatorname {atan}{\left (a x \right )}\, dx\right ) \]
c**2*(Integral(atan(a*x)/x**3, x) + Integral(2*a**2*atan(a*x)/x, x) + Inte gral(a**4*x*atan(a*x), x))
Time = 0.33 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.33 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)}{x^3} \, dx=-\frac {a^{3} c^{2} x^{3} + \pi a^{2} c^{2} x^{2} \log \left (a^{2} x^{2} + 1\right ) - 4 \, a^{2} c^{2} x^{2} \arctan \left (a x\right ) \log \left (a x\right ) + 2 i \, a^{2} c^{2} x^{2} {\rm Li}_2\left (i \, a x + 1\right ) - 2 i \, a^{2} c^{2} x^{2} {\rm Li}_2\left (-i \, a x + 1\right ) + a c^{2} x - {\left (a^{4} c^{2} x^{4} - c^{2}\right )} \arctan \left (a x\right )}{2 \, x^{2}} \]
-1/2*(a^3*c^2*x^3 + pi*a^2*c^2*x^2*log(a^2*x^2 + 1) - 4*a^2*c^2*x^2*arctan (a*x)*log(a*x) + 2*I*a^2*c^2*x^2*dilog(I*a*x + 1) - 2*I*a^2*c^2*x^2*dilog( -I*a*x + 1) + a*c^2*x - (a^4*c^2*x^4 - c^2)*arctan(a*x))/x^2
\[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )}{x^{3}} \,d x } \]
Time = 0.57 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.22 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)}{x^3} \, dx=\left \{\begin {array}{cl} 0 & \text {\ if\ \ }a=0\\ a^4\,c^2\,\mathrm {atan}\left (a\,x\right )\,\left (\frac {1}{2\,a^2}+\frac {x^2}{2}\right )-\frac {c^2\,\mathrm {atan}\left (a\,x\right )}{2\,x^2}-\frac {c^2\,\left (a^3\,\mathrm {atan}\left (a\,x\right )+\frac {a^2}{x}\right )}{2\,a}-\frac {a^3\,c^2\,x}{2}-a^2\,c^2\,{\mathrm {Li}}_{\mathrm {2}}\left (1-a\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}+a^2\,c^2\,{\mathrm {Li}}_{\mathrm {2}}\left (1+a\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i} & \text {\ if\ \ }a\neq 0 \end {array}\right . \]