Integrand size = 22, antiderivative size = 283 \[ \int \frac {x \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\frac {6 i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^2 c}-\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^2 \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^2 \sqrt {c+a^2 c x^2}}+\frac {6 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^2 \sqrt {c+a^2 c x^2}}-\frac {6 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^2 \sqrt {c+a^2 c x^2}} \]
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Time = 0.18 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {5050, 5010, 5008, 4266, 2611, 2320, 6724} \[ \int \frac {x \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=-\frac {6 i \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^2 \sqrt {a^2 c x^2+c}}+\frac {6 i \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^2 \sqrt {a^2 c x^2+c}}+\frac {6 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^2 \sqrt {a^2 c x^2+c}}-\frac {6 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^2 \sqrt {a^2 c x^2+c}}+\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}+\frac {6 i \sqrt {a^2 x^2+1} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^2 \sqrt {a^2 c x^2+c}} \]
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Rule 2320
Rule 2611
Rule 4266
Rule 5008
Rule 5010
Rule 5050
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^2 c}-\frac {3 \int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{a} \\ & = \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^2 c}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{a \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^2 c}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\arctan (a x)\right )}{a^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {6 i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^2 c}+\frac {\left (6 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{a^2 \sqrt {c+a^2 c x^2}}-\frac {\left (6 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{a^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {6 i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^2 c}-\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^2 \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (6 i \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{a^2 \sqrt {c+a^2 c x^2}}-\frac {\left (6 i \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{a^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {6 i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^2 c}-\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^2 \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (6 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{a^2 \sqrt {c+a^2 c x^2}}-\frac {\left (6 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{a^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {6 i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^2 c}-\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^2 \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^2 \sqrt {c+a^2 c x^2}}+\frac {6 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^2 \sqrt {c+a^2 c x^2}}-\frac {6 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^2 \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.59 \[ \int \frac {x \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\sqrt {c \left (1+a^2 x^2\right )} \left (\arctan (a x)^3-\frac {3 \left (\arctan (a x)^2 \left (\log \left (1-i e^{i \arctan (a x)}\right )-\log \left (1+i e^{i \arctan (a x)}\right )\right )+2 i \arctan (a x) \left (\operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )\right )-2 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )+2 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )}{\sqrt {1+a^2 x^2}}\right )}{a^2 c} \]
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\[\int \frac {x \arctan \left (a x \right )^{3}}{\sqrt {a^{2} c \,x^{2}+c}}d x\]
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\[ \int \frac {x \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {x \arctan \left (a x\right )^{3}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]
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\[ \int \frac {x \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x \operatorname {atan}^{3}{\left (a x \right )}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]
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\[ \int \frac {x \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {x \arctan \left (a x\right )^{3}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]
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\[ \int \frac {x \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {x \arctan \left (a x\right )^{3}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]
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Timed out. \[ \int \frac {x \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x\,{\mathrm {atan}\left (a\,x\right )}^3}{\sqrt {c\,a^2\,x^2+c}} \,d x \]
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