Integrand size = 20, antiderivative size = 90 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)}{x^3} \, dx=-\frac {a c^2}{2 x}-\frac {1}{2} a^3 c^2 x-\frac {c^2 \arctan (a x)}{2 x^2}+\frac {1}{2} a^4 c^2 x^2 \arctan (a x)+i a^2 c^2 \operatorname {PolyLog}(2,-i a x)-i a^2 c^2 \operatorname {PolyLog}(2,i a x) \]
[Out]
Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {5068, 4946, 331, 209, 4940, 2438, 327} \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)}{x^3} \, dx=\frac {1}{2} a^4 c^2 x^2 \arctan (a x)-\frac {1}{2} a^3 c^2 x+i a^2 c^2 \operatorname {PolyLog}(2,-i a x)-i a^2 c^2 \operatorname {PolyLog}(2,i a x)-\frac {c^2 \arctan (a x)}{2 x^2}-\frac {a c^2}{2 x} \]
[In]
[Out]
Rule 209
Rule 327
Rule 331
Rule 2438
Rule 4940
Rule 4946
Rule 5068
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c^2 \arctan (a x)}{x^3}+\frac {2 a^2 c^2 \arctan (a x)}{x}+a^4 c^2 x \arctan (a x)\right ) \, dx \\ & = c^2 \int \frac {\arctan (a x)}{x^3} \, dx+\left (2 a^2 c^2\right ) \int \frac {\arctan (a x)}{x} \, dx+\left (a^4 c^2\right ) \int x \arctan (a x) \, dx \\ & = -\frac {c^2 \arctan (a x)}{2 x^2}+\frac {1}{2} a^4 c^2 x^2 \arctan (a x)+\frac {1}{2} \left (a c^2\right ) \int \frac {1}{x^2 \left (1+a^2 x^2\right )} \, dx+\left (i a^2 c^2\right ) \int \frac {\log (1-i a x)}{x} \, dx-\left (i a^2 c^2\right ) \int \frac {\log (1+i a x)}{x} \, dx-\frac {1}{2} \left (a^5 c^2\right ) \int \frac {x^2}{1+a^2 x^2} \, 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) \\ \end{align*}
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 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)}{x^3} \, dx=-\frac {1}{2} a^3 c^2 x+\frac {1}{2} a^2 c^2 \arctan (a x)-\frac {c^2 \arctan (a x)}{2 x^2}+\frac {1}{2} a^4 c^2 x^2 \arctan (a x)-\frac {a c^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-a^2 x^2\right )}{2 x}+i a^2 c^2 \operatorname {PolyLog}(2,-i a x)-i a^2 c^2 \operatorname {PolyLog}(2,i a x) \]
[In]
[Out]
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 | \(\frac {i c^{2} a^{4} \ln \left (-i a x +1\right ) x^{2}}{4}-\frac {a^{3} c^{2} x}{2}+\frac {i c^{2} a^{2} \ln \left (-i a x \right )}{4}-\frac {a \,c^{2}}{2 x}-\frac {i c^{2} \ln \left (-i a x +1\right )}{4 x^{2}}-i c^{2} a^{2} \operatorname {dilog}\left (-i a x +1\right )-\frac {i c^{2} a^{4} \ln \left (i a x +1\right ) x^{2}}{4}-\frac {i c^{2} a^{2} \ln \left (i a x \right )}{4}+\frac {i c^{2} \ln \left (i a x +1\right )}{4 x^{2}}+i c^{2} a^{2} \operatorname {dilog}\left (i a x +1\right )\) | \(158\) |
[In]
[Out]
\[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )}{x^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)}{x^3} \, dx=c^{2} \left (\int \frac {\operatorname {atan}{\left (a x \right )}}{x^{3}}\, dx + \int \frac {2 a^{2} \operatorname {atan}{\left (a x \right )}}{x}\, dx + \int a^{4} x \operatorname {atan}{\left (a x \right )}\, dx\right ) \]
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.33 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)}{x^3} \, dx=-\frac {a^{3} c^{2} x^{3} + \pi a^{2} c^{2} x^{2} \log \left (a^{2} x^{2} + 1\right ) - 4 \, a^{2} c^{2} x^{2} \arctan \left (a x\right ) \log \left (a x\right ) + 2 i \, a^{2} c^{2} x^{2} {\rm Li}_2\left (i \, a x + 1\right ) - 2 i \, a^{2} c^{2} x^{2} {\rm Li}_2\left (-i \, a x + 1\right ) + a c^{2} x - {\left (a^{4} c^{2} x^{4} - c^{2}\right )} \arctan \left (a x\right )}{2 \, x^{2}} \]
[In]
[Out]
\[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )}{x^{3}} \,d x } \]
[In]
[Out]
Time = 0.57 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.22 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)}{x^3} \, dx=\left \{\begin {array}{cl} 0 & \text {\ if\ \ }a=0\\ a^4\,c^2\,\mathrm {atan}\left (a\,x\right )\,\left (\frac {1}{2\,a^2}+\frac {x^2}{2}\right )-\frac {c^2\,\mathrm {atan}\left (a\,x\right )}{2\,x^2}-\frac {c^2\,\left (a^3\,\mathrm {atan}\left (a\,x\right )+\frac {a^2}{x}\right )}{2\,a}-\frac {a^3\,c^2\,x}{2}-a^2\,c^2\,{\mathrm {Li}}_{\mathrm {2}}\left (1-a\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}+a^2\,c^2\,{\mathrm {Li}}_{\mathrm {2}}\left (1+a\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i} & \text {\ if\ \ }a\neq 0 \end {array}\right . \]
[In]
[Out]