Integrand size = 20, antiderivative size = 276 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x} \, dx=-\frac {3}{2} i c \arctan (a x)^2-\frac {3}{2} a c x \arctan (a x)^2+\frac {1}{2} c \arctan (a x)^3+\frac {1}{2} a^2 c x^2 \arctan (a x)^3+2 c \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-3 c \arctan (a x) \log \left (\frac {2}{1+i a x}\right )-\frac {3}{2} i c \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} c \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right ) \]
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Time = 0.39 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5070, 4942, 5108, 5004, 5114, 5118, 6745, 4946, 5036, 4930, 5040, 4964, 2449, 2352} \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x} \, dx=\frac {1}{2} a^2 c x^2 \arctan (a x)^3+2 c \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )+\frac {3}{2} i c \arctan (a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{i a x+1}-1\right )-\frac {3}{2} c \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )+\frac {3}{2} c \arctan (a x) \operatorname {PolyLog}\left (3,\frac {2}{i a x+1}-1\right )+\frac {1}{2} c \arctan (a x)^3-\frac {3}{2} i c \arctan (a x)^2-\frac {3}{2} a c x \arctan (a x)^2-3 c \arctan (a x) \log \left (\frac {2}{1+i a x}\right )-\frac {3}{2} i c \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )+\frac {3}{4} i c \operatorname {PolyLog}\left (4,1-\frac {2}{i a x+1}\right )-\frac {3}{4} i c \operatorname {PolyLog}\left (4,\frac {2}{i a x+1}-1\right ) \]
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Rule 2352
Rule 2449
Rule 4930
Rule 4942
Rule 4946
Rule 4964
Rule 5004
Rule 5036
Rule 5040
Rule 5070
Rule 5108
Rule 5114
Rule 5118
Rule 6745
Rubi steps \begin{align*} \text {integral}& = c \int \frac {\arctan (a x)^3}{x} \, dx+\left (a^2 c\right ) \int x \arctan (a x)^3 \, dx \\ & = \frac {1}{2} a^2 c x^2 \arctan (a x)^3+2 c \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-(6 a c) \int \frac {\arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{2} \left (3 a^3 c\right ) \int \frac {x^2 \arctan (a x)^2}{1+a^2 x^2} \, dx \\ & = \frac {1}{2} a^2 c x^2 \arctan (a x)^3+2 c \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-\frac {1}{2} (3 a c) \int \arctan (a x)^2 \, dx+\frac {1}{2} (3 a c) \int \frac {\arctan (a x)^2}{1+a^2 x^2} \, dx+(3 a c) \int \frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-(3 a c) \int \frac {\arctan (a x)^2 \log \left (2-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx \\ & = -\frac {3}{2} a c x \arctan (a x)^2+\frac {1}{2} c \arctan (a x)^3+\frac {1}{2} a^2 c x^2 \arctan (a x)^3+2 c \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )+(3 i a c) \int \frac {\arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-(3 i a c) \int \frac {\arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\left (3 a^2 c\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx \\ & = -\frac {3}{2} i c \arctan (a x)^2-\frac {3}{2} a c x \arctan (a x)^2+\frac {1}{2} c \arctan (a x)^3+\frac {1}{2} a^2 c x^2 \arctan (a x)^3+2 c \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} c \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {1}{2} (3 a c) \int \frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{2} (3 a c) \int \frac {\operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-(3 a c) \int \frac {\arctan (a x)}{i-a x} \, dx \\ & = -\frac {3}{2} i c \arctan (a x)^2-\frac {3}{2} a c x \arctan (a x)^2+\frac {1}{2} c \arctan (a x)^3+\frac {1}{2} a^2 c x^2 \arctan (a x)^3+2 c \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-3 c \arctan (a x) \log \left (\frac {2}{1+i a x}\right )-\frac {3}{2} i c \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} c \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right )+(3 a c) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx \\ & = -\frac {3}{2} i c \arctan (a x)^2-\frac {3}{2} a c x \arctan (a x)^2+\frac {1}{2} c \arctan (a x)^3+\frac {1}{2} a^2 c x^2 \arctan (a x)^3+2 c \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-3 c \arctan (a x) \log \left (\frac {2}{1+i a x}\right )-\frac {3}{2} i c \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} c \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right )-(3 i c) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right ) \\ & = -\frac {3}{2} i c \arctan (a x)^2-\frac {3}{2} a c x \arctan (a x)^2+\frac {1}{2} c \arctan (a x)^3+\frac {1}{2} a^2 c x^2 \arctan (a x)^3+2 c \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-3 c \arctan (a x) \log \left (\frac {2}{1+i a x}\right )-\frac {3}{2} i c \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} c \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right ) \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.96 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x} \, dx=\frac {1}{2} c \left (1+a^2 x^2\right ) \arctan (a x)^3-\frac {3}{2} c \left (-i \arctan (a x)^2+a x \arctan (a x)^2+2 \arctan (a x) \log \left (1+e^{2 i \arctan (a x)}\right )-i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )\right )-\frac {1}{64} i c \left (\pi ^4-32 \arctan (a x)^4+64 i \arctan (a x)^3 \log \left (1-e^{-2 i \arctan (a x)}\right )-64 i \arctan (a x)^3 \log \left (1+e^{2 i \arctan (a x)}\right )-96 \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )-96 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+96 i \arctan (a x) \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )-96 i \arctan (a x) \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )+48 \operatorname {PolyLog}\left (4,e^{-2 i \arctan (a x)}\right )+48 \operatorname {PolyLog}\left (4,-e^{2 i \arctan (a x)}\right )\right ) \]
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Time = 19.54 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.67
method | result | size |
derivativedivides | \(\frac {c \arctan \left (a x \right )^{2} \left (-i \arctan \left (a x \right )+x \arctan \left (a x \right ) a -3\right ) \left (a x +i\right )}{2}+\frac {3 i c \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}-3 c \arctan \left (a x \right ) \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )+3 i c \arctan \left (a x \right )^{2}+c \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-3 i c \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 c \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i c \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-c \arctan \left (a x \right )^{3} \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )+6 i c \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {3 c \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+\frac {3 i c \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+c \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {3 i c \operatorname {polylog}\left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{4}+6 c \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i c \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\) | \(460\) |
default | \(\frac {c \arctan \left (a x \right )^{2} \left (-i \arctan \left (a x \right )+x \arctan \left (a x \right ) a -3\right ) \left (a x +i\right )}{2}+\frac {3 i c \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}-3 c \arctan \left (a x \right ) \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )+3 i c \arctan \left (a x \right )^{2}+c \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-3 i c \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 c \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i c \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-c \arctan \left (a x \right )^{3} \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )+6 i c \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {3 c \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+\frac {3 i c \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+c \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {3 i c \operatorname {polylog}\left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{4}+6 c \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i c \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\) | \(460\) |
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\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}}{x} \,d x } \]
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\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x} \, dx=c \left (\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{x}\, dx + \int a^{2} x \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]
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\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}}{x} \,d x } \]
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\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}}{x} \,d x } \]
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Timed out. \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,\left (c\,a^2\,x^2+c\right )}{x} \,d x \]
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