Integrand size = 24, antiderivative size = 625 \[ \int \frac {x^2 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=-\frac {3 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^3}{2 a^2 c}+\frac {i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{a^3 \sqrt {c+a^2 c x^2}}-\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}} \]
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Time = 0.36 (sec) , antiderivative size = 625, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5072, 5050, 5010, 5006, 5008, 4266, 2611, 6744, 2320, 6724} \[ \int \frac {x^2 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {3 i \sqrt {a^2 x^2+1} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {3 i \sqrt {a^2 x^2+1} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {3 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^3 \sqrt {a^2 c x^2+c}}-\frac {3 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^3 \sqrt {a^2 c x^2+c}}+\frac {3 i \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )}{a^3 \sqrt {a^2 c x^2+c}}-\frac {3 i \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )}{a^3 \sqrt {a^2 c x^2+c}}+\frac {i \sqrt {a^2 x^2+1} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{a^3 \sqrt {a^2 c x^2+c}}-\frac {3 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a^3 c}-\frac {6 i \sqrt {a^2 x^2+1} \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right ) \arctan (a x)}{a^3 \sqrt {a^2 c x^2+c}}+\frac {3 i \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {a^2 c x^2+c}}-\frac {3 i \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {a^2 c x^2+c}} \]
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Rule 2320
Rule 2611
Rule 4266
Rule 5006
Rule 5008
Rule 5010
Rule 5050
Rule 5072
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^3}{2 a^2 c}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{2 a^2}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{2 a} \\ & = -\frac {3 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^3}{2 a^2 c}+\frac {3 \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a^2}-\frac {\sqrt {1+a^2 x^2} \int \frac {\arctan (a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{2 a^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {3 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^3}{2 a^2 c}-\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int x^3 \sec (x) \, dx,x,\arctan (a x)\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{a^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {3 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^3}{2 a^2 c}+\frac {i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{a^3 \sqrt {c+a^2 c x^2}}-\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{2 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{2 a^3 \sqrt {c+a^2 c x^2}} \\ & = -\frac {3 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^3}{2 a^2 c}+\frac {i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{a^3 \sqrt {c+a^2 c x^2}}-\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 i \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 i \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{a^3 \sqrt {c+a^2 c x^2}} \\ & = -\frac {3 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^3}{2 a^2 c}+\frac {i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{a^3 \sqrt {c+a^2 c x^2}}-\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{a^3 \sqrt {c+a^2 c x^2}} \\ & = -\frac {3 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^3}{2 a^2 c}+\frac {i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{a^3 \sqrt {c+a^2 c x^2}}-\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 i \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 i \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}} \\ & = -\frac {3 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^3}{2 a^2 c}+\frac {i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{a^3 \sqrt {c+a^2 c x^2}}-\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 5.66 (sec) , antiderivative size = 812, normalized size of antiderivative = 1.30 \[ \int \frac {x^2 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\sqrt {c \left (1+a^2 x^2\right )} \left (\frac {7 i \pi ^4}{32}+\frac {1}{4} i \pi ^3 \arctan (a x)-6 \arctan (a x)^2-\frac {3}{4} i \pi ^2 \arctan (a x)^2+i \pi \arctan (a x)^3-\frac {1}{2} i \arctan (a x)^4-\frac {3}{2} \pi ^2 \arctan (a x) \log \left (1-i e^{-i \arctan (a x)}\right )+3 \pi \arctan (a x)^2 \log \left (1-i e^{-i \arctan (a x)}\right )+\frac {1}{4} \pi ^3 \log \left (1+i e^{-i \arctan (a x)}\right )-2 \arctan (a x)^3 \log \left (1+i e^{-i \arctan (a x)}\right )+12 \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )-\frac {1}{4} \pi ^3 \log \left (1+i e^{i \arctan (a x)}\right )-12 \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )+\frac {3}{2} \pi ^2 \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )-3 \pi \arctan (a x)^2 \log \left (1+i e^{i \arctan (a x)}\right )+2 \arctan (a x)^3 \log \left (1+i e^{i \arctan (a x)}\right )-\frac {1}{4} \pi ^3 \log \left (\tan \left (\frac {1}{4} (\pi +2 \arctan (a x))\right )\right )-6 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{-i \arctan (a x)}\right )-\frac {3}{2} i \pi (\pi -4 \arctan (a x)) \operatorname {PolyLog}\left (2,i e^{-i \arctan (a x)}\right )+12 i \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\frac {3}{2} i \pi ^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+6 i \pi \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-6 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-12 i \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-12 \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{-i \arctan (a x)}\right )+6 \pi \operatorname {PolyLog}\left (3,i e^{-i \arctan (a x)}\right )-6 \pi \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )+12 \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )+12 i \operatorname {PolyLog}\left (4,-i e^{-i \arctan (a x)}\right )+12 i \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )+\frac {\arctan (a x)^3}{\left (\cos \left (\frac {1}{2} \arctan (a x)\right )-\sin \left (\frac {1}{2} \arctan (a x)\right )\right )^2}-\frac {6 \arctan (a x)^2 \sin \left (\frac {1}{2} \arctan (a x)\right )}{\cos \left (\frac {1}{2} \arctan (a x)\right )-\sin \left (\frac {1}{2} \arctan (a x)\right )}-\frac {\arctan (a x)^3}{\left (\cos \left (\frac {1}{2} \arctan (a x)\right )+\sin \left (\frac {1}{2} \arctan (a x)\right )\right )^2}+\frac {6 \arctan (a x)^2 \sin \left (\frac {1}{2} \arctan (a x)\right )}{\cos \left (\frac {1}{2} \arctan (a x)\right )+\sin \left (\frac {1}{2} \arctan (a x)\right )}\right )}{4 a^3 c \sqrt {1+a^2 x^2}} \]
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Time = 3.89 (sec) , antiderivative size = 428, normalized size of antiderivative = 0.68
method | result | size |
default | \(\frac {\left (x \arctan \left (a x \right ) a -3\right ) \arctan \left (a x \right )^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 c \,a^{3}}+\frac {\left (\arctan \left (a x \right )^{3} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right )^{3} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \operatorname {polylog}\left (4, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \operatorname {polylog}\left (4, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \sqrt {a^{2} x^{2}+1}\, a^{3} c}\) | \(428\) |
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\[ \int \frac {x^2 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {x^{2} \arctan \left (a x\right )^{3}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]
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\[ \int \frac {x^2 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x^{2} \operatorname {atan}^{3}{\left (a x \right )}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]
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\[ \int \frac {x^2 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {x^{2} \arctan \left (a x\right )^{3}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]
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\[ \int \frac {x^2 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {x^{2} \arctan \left (a x\right )^{3}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]
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Timed out. \[ \int \frac {x^2 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x^2\,{\mathrm {atan}\left (a\,x\right )}^3}{\sqrt {c\,a^2\,x^2+c}} \,d x \]
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