Integrand size = 24, antiderivative size = 403 \[ \int \frac {x^3 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {6 x}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {6 \arctan (a x)}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {3 x \arctan (a x)^2}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^3}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^4 c^2}-\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {6 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {6 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}} \]
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Time = 0.38 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5084, 5050, 5010, 5008, 4266, 2611, 2320, 6724, 5018, 197} \[ \int \frac {x^3 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^4 c^2}-\frac {6 i \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^4 c \sqrt {a^2 c x^2+c}}+\frac {6 i \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^4 c \sqrt {a^2 c x^2+c}}+\frac {6 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^4 c \sqrt {a^2 c x^2+c}}-\frac {6 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^4 c \sqrt {a^2 c x^2+c}}+\frac {\arctan (a x)^3}{a^4 c \sqrt {a^2 c x^2+c}}+\frac {6 i \sqrt {a^2 x^2+1} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^4 c \sqrt {a^2 c x^2+c}}-\frac {6 \arctan (a x)}{a^4 c \sqrt {a^2 c x^2+c}}-\frac {3 x \arctan (a x)^2}{a^3 c \sqrt {a^2 c x^2+c}}+\frac {6 x}{a^3 c \sqrt {a^2 c x^2+c}} \]
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Rule 197
Rule 2320
Rule 2611
Rule 4266
Rule 5008
Rule 5010
Rule 5018
Rule 5050
Rule 5084
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^2}+\frac {\int \frac {x \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{a^2 c} \\ & = \frac {\arctan (a x)^3}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^4 c^2}-\frac {3 \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^3}-\frac {3 \int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{a^3 c} \\ & = -\frac {6 \arctan (a x)}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {3 x \arctan (a x)^2}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^3}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^4 c^2}+\frac {6 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^3}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{a^3 c \sqrt {c+a^2 c x^2}} \\ & = \frac {6 x}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {6 \arctan (a x)}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {3 x \arctan (a x)^2}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^3}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^4 c^2}-\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^4 c \sqrt {c+a^2 c x^2}} \\ & = \frac {6 x}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {6 \arctan (a x)}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {3 x \arctan (a x)^2}{a^3 c \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^4 c \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^3}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^4 c^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^4 c \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^4 c \sqrt {c+a^2 c x^2}} \\ & = \frac {6 x}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {6 \arctan (a x)}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {3 x \arctan (a x)^2}{a^3 c \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^4 c \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^3}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^4 c^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^4 c \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^4 c \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^4 c \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^4 c \sqrt {c+a^2 c x^2}} \\ & = \frac {6 x}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {6 \arctan (a x)}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {3 x \arctan (a x)^2}{a^3 c \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^4 c \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^3}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^4 c^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^4 c \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^4 c \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^4 c \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^4 c \sqrt {c+a^2 c x^2}} \\ & = \frac {6 x}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {6 \arctan (a x)}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {3 x \arctan (a x)^2}{a^3 c \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^4 c \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^3}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{a^4 c^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^4 c \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^4 c \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^4 c \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^4 c \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.76 (sec) , antiderivative size = 308, normalized size of antiderivative = 0.76 \[ \int \frac {x^3 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {1+a^2 x^2} \left (\frac {6 a x}{\sqrt {1+a^2 x^2}}-3 \sqrt {1+a^2 x^2} \arctan (a x)-\frac {3 a x \arctan (a x)^2}{\sqrt {1+a^2 x^2}}+\frac {3}{2} \sqrt {1+a^2 x^2} \arctan (a x)^3-3 \sqrt {1+a^2 x^2} \arctan (a x) \cos (2 \arctan (a x))+\frac {1}{2} \sqrt {1+a^2 x^2} \arctan (a x)^3 \cos (2 \arctan (a x))-3 \arctan (a x)^2 \log \left (1-i e^{i \arctan (a x)}\right )+3 \arctan (a x)^2 \log \left (1+i e^{i \arctan (a x)}\right )-6 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+6 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )+6 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )-6 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )}{a^4 c \sqrt {c \left (1+a^2 x^2\right )}} \]
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\[\int \frac {x^{3} \arctan \left (a x \right )^{3}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {x^3 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {x^{3} \arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^3 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x^{3} \operatorname {atan}^{3}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {x^3 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {x^{3} \arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {x^3 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^3 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x^3\,{\mathrm {atan}\left (a\,x\right )}^3}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
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