Integrand size = 20, antiderivative size = 132 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x} \, dx=-\frac {11}{12} a c^3 x-\frac {7}{36} a^3 c^3 x^3-\frac {1}{30} a^5 c^3 x^5+\frac {11}{12} c^3 \arctan (a x)+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)+\frac {1}{2} i c^3 \operatorname {PolyLog}(2,-i a x)-\frac {1}{2} i c^3 \operatorname {PolyLog}(2,i a x) \]
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Time = 0.11 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {5068, 4940, 2438, 4946, 327, 209, 308} \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x} \, dx=\frac {1}{6} a^6 c^3 x^6 \arctan (a x)-\frac {1}{30} a^5 c^3 x^5+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)-\frac {7}{36} a^3 c^3 x^3+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)+\frac {11}{12} c^3 \arctan (a x)+\frac {1}{2} i c^3 \operatorname {PolyLog}(2,-i a x)-\frac {1}{2} i c^3 \operatorname {PolyLog}(2,i a x)-\frac {11}{12} a c^3 x \]
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Rule 209
Rule 308
Rule 327
Rule 2438
Rule 4940
Rule 4946
Rule 5068
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c^3 \arctan (a x)}{x}+3 a^2 c^3 x \arctan (a x)+3 a^4 c^3 x^3 \arctan (a x)+a^6 c^3 x^5 \arctan (a x)\right ) \, dx \\ & = c^3 \int \frac {\arctan (a x)}{x} \, dx+\left (3 a^2 c^3\right ) \int x \arctan (a x) \, dx+\left (3 a^4 c^3\right ) \int x^3 \arctan (a x) \, dx+\left (a^6 c^3\right ) \int x^5 \arctan (a x) \, dx \\ & = \frac {3}{2} a^2 c^3 x^2 \arctan (a x)+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)+\frac {1}{2} \left (i c^3\right ) \int \frac {\log (1-i a x)}{x} \, dx-\frac {1}{2} \left (i c^3\right ) \int \frac {\log (1+i a x)}{x} \, dx-\frac {1}{2} \left (3 a^3 c^3\right ) \int \frac {x^2}{1+a^2 x^2} \, dx-\frac {1}{4} \left (3 a^5 c^3\right ) \int \frac {x^4}{1+a^2 x^2} \, dx-\frac {1}{6} \left (a^7 c^3\right ) \int \frac {x^6}{1+a^2 x^2} \, dx \\ & = -\frac {3}{2} a c^3 x+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)+\frac {1}{2} i c^3 \operatorname {PolyLog}(2,-i a x)-\frac {1}{2} i c^3 \operatorname {PolyLog}(2,i a x)+\frac {1}{2} \left (3 a c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx-\frac {1}{4} \left (3 a^5 c^3\right ) \int \left (-\frac {1}{a^4}+\frac {x^2}{a^2}+\frac {1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx-\frac {1}{6} \left (a^7 c^3\right ) \int \left (\frac {1}{a^6}-\frac {x^2}{a^4}+\frac {x^4}{a^2}-\frac {1}{a^6 \left (1+a^2 x^2\right )}\right ) \, dx \\ & = -\frac {11}{12} a c^3 x-\frac {7}{36} a^3 c^3 x^3-\frac {1}{30} a^5 c^3 x^5+\frac {3}{2} c^3 \arctan (a x)+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)+\frac {1}{2} i c^3 \operatorname {PolyLog}(2,-i a x)-\frac {1}{2} i c^3 \operatorname {PolyLog}(2,i a x)+\frac {1}{6} \left (a c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx-\frac {1}{4} \left (3 a c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx \\ & = -\frac {11}{12} a c^3 x-\frac {7}{36} a^3 c^3 x^3-\frac {1}{30} a^5 c^3 x^5+\frac {11}{12} c^3 \arctan (a x)+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)+\frac {1}{2} i c^3 \operatorname {PolyLog}(2,-i a x)-\frac {1}{2} i c^3 \operatorname {PolyLog}(2,i a x) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x} \, dx=-\frac {11}{12} a c^3 x-\frac {7}{36} a^3 c^3 x^3-\frac {1}{30} a^5 c^3 x^5+\frac {11}{12} c^3 \arctan (a x)+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)+\frac {1}{2} i c^3 \operatorname {PolyLog}(2,-i a x)-\frac {1}{2} i c^3 \operatorname {PolyLog}(2,i a x) \]
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Time = 0.47 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.08
method | result | size |
derivativedivides | \(\frac {a^{6} c^{3} x^{6} \arctan \left (a x \right )}{6}+\frac {3 a^{4} c^{3} x^{4} \arctan \left (a x \right )}{4}+\frac {3 a^{2} c^{3} x^{2} \arctan \left (a x \right )}{2}+c^{3} \arctan \left (a x \right ) \ln \left (a x \right )-\frac {c^{3} \left (\frac {2 a^{5} x^{5}}{5}+\frac {7 a^{3} x^{3}}{3}+11 a x -11 \arctan \left (a x \right )-6 i \ln \left (a x \right ) \ln \left (i a x +1\right )+6 i \ln \left (a x \right ) \ln \left (-i a x +1\right )-6 i \operatorname {dilog}\left (i a x +1\right )+6 i \operatorname {dilog}\left (-i a x +1\right )\right )}{12}\) | \(143\) |
default | \(\frac {a^{6} c^{3} x^{6} \arctan \left (a x \right )}{6}+\frac {3 a^{4} c^{3} x^{4} \arctan \left (a x \right )}{4}+\frac {3 a^{2} c^{3} x^{2} \arctan \left (a x \right )}{2}+c^{3} \arctan \left (a x \right ) \ln \left (a x \right )-\frac {c^{3} \left (\frac {2 a^{5} x^{5}}{5}+\frac {7 a^{3} x^{3}}{3}+11 a x -11 \arctan \left (a x \right )-6 i \ln \left (a x \right ) \ln \left (i a x +1\right )+6 i \ln \left (a x \right ) \ln \left (-i a x +1\right )-6 i \operatorname {dilog}\left (i a x +1\right )+6 i \operatorname {dilog}\left (-i a x +1\right )\right )}{12}\) | \(143\) |
parts | \(\frac {a^{6} c^{3} x^{6} \arctan \left (a x \right )}{6}+\frac {3 a^{4} c^{3} x^{4} \arctan \left (a x \right )}{4}+\frac {3 a^{2} c^{3} x^{2} \arctan \left (a x \right )}{2}+c^{3} \arctan \left (a x \right ) \ln \left (x \right )-\frac {c^{3} a \left (\frac {2 a^{4} x^{5}}{5}+\frac {7 a^{2} x^{3}}{3}+11 x -\frac {11 \arctan \left (a x \right )}{a}-\frac {6 i \ln \left (x \right ) \left (\ln \left (i a x +1\right )-\ln \left (-i a x +1\right )\right )}{a}-\frac {6 i \left (\operatorname {dilog}\left (i a x +1\right )-\operatorname {dilog}\left (-i a x +1\right )\right )}{a}\right )}{12}\) | \(144\) |
risch | \(-\frac {11 a \,c^{3} x}{12}+\frac {3 i c^{3} \ln \left (-i a x +1\right ) x^{4} a^{4}}{8}+\frac {3 i c^{3} \ln \left (-i a x +1\right ) x^{2} a^{2}}{4}+\frac {i c^{3} \ln \left (-i a x +1\right ) x^{6} a^{6}}{12}-\frac {i c^{3} \operatorname {dilog}\left (-i a x +1\right )}{2}+\frac {11 c^{3} \arctan \left (a x \right )}{12}-\frac {a^{5} c^{3} x^{5}}{30}-\frac {7 a^{3} c^{3} x^{3}}{36}-\frac {3 i c^{3} \ln \left (i a x +1\right ) x^{4} a^{4}}{8}-\frac {3 i c^{3} \ln \left (i a x +1\right ) x^{2} a^{2}}{4}-\frac {i c^{3} \ln \left (i a x +1\right ) x^{6} a^{6}}{12}+\frac {i c^{3} \operatorname {dilog}\left (i a x +1\right )}{2}\) | \(188\) |
meijerg | \(\frac {c^{3} \left (-\frac {2 a x \left (21 a^{4} x^{4}-35 a^{2} x^{2}+105\right )}{315}+\frac {2 a x \left (7 a^{6} x^{6}+7\right ) \arctan \left (\sqrt {a^{2} x^{2}}\right )}{21 \sqrt {a^{2} x^{2}}}\right )}{4}+\frac {3 c^{3} \left (\frac {a x \left (-5 a^{2} x^{2}+15\right )}{15}-\frac {a x \left (-5 a^{4} x^{4}+5\right ) \arctan \left (\sqrt {a^{2} x^{2}}\right )}{5 \sqrt {a^{2} x^{2}}}\right )}{4}+\frac {3 c^{3} \left (-2 a x +\frac {2 \left (3 a^{2} x^{2}+3\right ) \arctan \left (a x \right )}{3}\right )}{4}+\frac {c^{3} \left (-\frac {2 i a x \operatorname {polylog}\left (2, i \sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}+\frac {2 i a x \operatorname {polylog}\left (2, -i \sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}\right )}{4}\) | \(204\) |
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\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )}{x} \,d x } \]
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\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x} \, dx=c^{3} \left (\int \frac {\operatorname {atan}{\left (a x \right )}}{x}\, dx + \int 3 a^{2} x \operatorname {atan}{\left (a x \right )}\, dx + \int 3 a^{4} x^{3} \operatorname {atan}{\left (a x \right )}\, dx + \int a^{6} x^{5} \operatorname {atan}{\left (a x \right )}\, dx\right ) \]
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Time = 0.34 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.96 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x} \, dx=-\frac {1}{30} \, a^{5} c^{3} x^{5} - \frac {7}{36} \, a^{3} c^{3} x^{3} - \frac {11}{12} \, a c^{3} x - \frac {1}{4} \, \pi c^{3} \log \left (a^{2} x^{2} + 1\right ) + c^{3} \arctan \left (a x\right ) \log \left (a x\right ) - \frac {1}{2} i \, c^{3} {\rm Li}_2\left (i \, a x + 1\right ) + \frac {1}{2} i \, c^{3} {\rm Li}_2\left (-i \, a x + 1\right ) + \frac {1}{12} \, {\left (2 \, a^{6} c^{3} x^{6} + 9 \, a^{4} c^{3} x^{4} + 18 \, a^{2} c^{3} x^{2} + 11 \, c^{3}\right )} \arctan \left (a x\right ) \]
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\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )}{x} \,d x } \]
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Time = 0.77 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.18 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x} \, dx=\left \{\begin {array}{cl} 0 & \text {\ if\ \ }a=0\\ 3\,a^2\,c^3\,\mathrm {atan}\left (a\,x\right )\,\left (\frac {1}{2\,a^2}+\frac {x^2}{2}\right )-\frac {a^5\,c^3\,\left (\frac {x}{a^4}-\frac {\mathrm {atan}\left (a\,x\right )}{a^5}+\frac {x^5}{5}-\frac {x^3}{3\,a^2}\right )}{6}-\frac {3\,a\,c^3\,x}{2}-\frac {c^3\,\left (3\,\mathrm {atan}\left (a\,x\right )-3\,a\,x+a^3\,x^3\right )}{4}+\frac {3\,a^4\,c^3\,x^4\,\mathrm {atan}\left (a\,x\right )}{4}+\frac {a^6\,c^3\,x^6\,\mathrm {atan}\left (a\,x\right )}{6}-\frac {c^3\,{\mathrm {Li}}_{\mathrm {2}}\left (1-a\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\frac {c^3\,{\mathrm {Li}}_{\mathrm {2}}\left (1+a\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} & \text {\ if\ \ }a\neq 0 \end {array}\right . \]
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