Integrand size = 24, antiderivative size = 444 \[ \int \frac {x^4 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {2 x^3}{27 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {22 x}{9 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 x^2 \arctan (a x)}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {22 \arctan (a x)}{9 a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \arctan (a x)^2}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \arctan (a x)^2}{a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}} \]
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Time = 0.56 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5084, 5010, 5008, 4266, 2611, 2320, 6724, 5018, 197, 5064, 5058, 5050} \[ \int \frac {x^4 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=-\frac {x^3 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 x^3}{27 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 i \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {2 i \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}+\frac {2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {2 i \sqrt {a^2 x^2+1} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {22 \arctan (a x)}{9 a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {x \arctan (a x)^2}{a^4 c^2 \sqrt {a^2 c x^2+c}}+\frac {22 x}{9 a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2 x^2 \arctan (a x)}{9 a^3 c \left (a^2 c x^2+c\right )^{3/2}} \]
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Rule 197
Rule 2320
Rule 2611
Rule 4266
Rule 5008
Rule 5010
Rule 5018
Rule 5050
Rule 5058
Rule 5064
Rule 5084
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{a^2}+\frac {\int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^2 c} \\ & = -\frac {x^3 \arctan (a x)^2}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 \int \frac {x^3 \arctan (a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{3 a}+\frac {\int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{a^4 c^2}-\frac {\int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^4 c} \\ & = \frac {2 x^3}{27 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {2 x^2 \arctan (a x)}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {2 \arctan (a x)}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \arctan (a x)^2}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \arctan (a x)^2}{a^4 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^4 c}+\frac {4 \int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{9 a^3 c}+\frac {\sqrt {1+a^2 x^2} \int \frac {\arctan (a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{a^4 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {2 x^3}{27 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x}{a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 x^2 \arctan (a x)}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {22 \arctan (a x)}{9 a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \arctan (a x)^2}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \arctan (a x)^2}{a^4 c^2 \sqrt {c+a^2 c x^2}}+\frac {4 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{9 a^4 c}+\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\arctan (a x)\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {2 x^3}{27 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {22 x}{9 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 x^2 \arctan (a x)}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {22 \arctan (a x)}{9 a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \arctan (a x)^2}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \arctan (a x)^2}{a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (2 \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^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (2 \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^5 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {2 x^3}{27 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {22 x}{9 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 x^2 \arctan (a x)}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {22 \arctan (a x)}{9 a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \arctan (a x)^2}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \arctan (a x)^2}{a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (2 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^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (2 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^5 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {2 x^3}{27 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {22 x}{9 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 x^2 \arctan (a x)}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {22 \arctan (a x)}{9 a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \arctan (a x)^2}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \arctan (a x)^2}{a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (2 \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^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (2 \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^5 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {2 x^3}{27 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {22 x}{9 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 x^2 \arctan (a x)}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {22 \arctan (a x)}{9 a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \arctan (a x)^2}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \arctan (a x)^2}{a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.56 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.54 \[ \int \frac {x^4 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {\sqrt {c \left (1+a^2 x^2\right )} \left (-\frac {270 \arctan (a x)}{\sqrt {1+a^2 x^2}}-\frac {135 a x \left (-2+\arctan (a x)^2\right )}{\sqrt {1+a^2 x^2}}+6 \arctan (a x) \cos (3 \arctan (a x))+108 \arctan (a x)^2 \left (\log \left (1-i e^{i \arctan (a x)}\right )-\log \left (1+i e^{i \arctan (a x)}\right )\right )+216 i \arctan (a x) \left (\operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )\right )-216 \left (\operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )+\left (-2+9 \arctan (a x)^2\right ) \sin (3 \arctan (a x))\right )}{108 a^5 c^3 \sqrt {1+a^2 x^2}} \]
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\[\int \frac {x^{4} \arctan \left (a x \right )^{2}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}d x\]
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\[ \int \frac {x^4 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{4} \arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {x^4 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^{4} \operatorname {atan}^{2}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {x^4 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{4} \arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {x^4 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{4} \arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^4 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^4\,{\mathrm {atan}\left (a\,x\right )}^2}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
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