Integrand size = 24, antiderivative size = 602 \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x^3} \, dx=-\frac {3 a \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{2 x^2}-\frac {a^2 c \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^2 c \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 a^2 c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 a^2 c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \]
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Time = 0.82 (sec) , antiderivative size = 602, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {5070, 5082, 5064, 5078, 5074, 5076, 4268, 2611, 6744, 2320, 6724} \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x^3} \, dx=-\frac {a^2 c \sqrt {a^2 x^2+1} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {6 a^2 c \sqrt {a^2 x^2+1} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}+\frac {3 i a^2 c \sqrt {a^2 x^2+1} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {3 i a^2 c \sqrt {a^2 x^2+1} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {3 a^2 c \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {3 a^2 c \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {3 i a^2 c \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {3 i a^2 c \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (4,e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 x^2}-\frac {3 a \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 x}+\frac {3 i a^2 c \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}-\frac {3 i a^2 c \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}} \]
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Rule 2320
Rule 2611
Rule 4268
Rule 5064
Rule 5070
Rule 5074
Rule 5076
Rule 5078
Rule 5082
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = c \int \frac {\arctan (a x)^3}{x^3 \sqrt {c+a^2 c x^2}} \, dx+\left (a^2 c\right ) \int \frac {\arctan (a x)^3}{x \sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{2 x^2}+\frac {1}{2} (3 a c) \int \frac {\arctan (a x)^2}{x^2 \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{2} \left (a^2 c\right ) \int \frac {\arctan (a x)^3}{x \sqrt {c+a^2 c x^2}} \, dx+\frac {\left (a^2 c \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^3}{x \sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {3 a \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{2 x^2}+\left (3 a^2 c\right ) \int \frac {\arctan (a x)}{x \sqrt {c+a^2 c x^2}} \, dx-\frac {\left (a^2 c \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^3}{x \sqrt {1+a^2 x^2}} \, dx}{2 \sqrt {c+a^2 c x^2}}+\frac {\left (a^2 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^3 \csc (x) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {3 a \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{2 x^2}-\frac {2 a^2 c \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (a^2 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^3 \csc (x) \, dx,x,\arctan (a x)\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 a^2 c \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{x \sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}-\frac {\left (3 a^2 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (3 a^2 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1+e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {3 a \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{2 x^2}-\frac {a^2 c \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^2 c \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 i a^2 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 i a^2 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (3 a^2 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 a^2 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1+e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {3 a \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{2 x^2}-\frac {a^2 c \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^2 c \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^2 c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 a^2 c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (3 i a^2 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (3 i a^2 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 a^2 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 a^2 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {3 a \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{2 x^2}-\frac {a^2 c \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^2 c \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 a^2 c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 a^2 c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 i a^2 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 i a^2 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (3 a^2 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (3 a^2 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {3 a \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{2 x^2}-\frac {a^2 c \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^2 c \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 a^2 c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 a^2 c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 i a^2 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i a^2 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (3 i a^2 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (3 i a^2 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {3 a \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{2 x^2}-\frac {a^2 c \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^2 c \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 a^2 c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 a^2 c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 4.16 (sec) , antiderivative size = 345, normalized size of antiderivative = 0.57 \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x^3} \, dx=\frac {a^2 \sqrt {c \left (1+a^2 x^2\right )} \left (-i \pi ^4+2 i \arctan (a x)^4-12 \arctan (a x)^2 \cot \left (\frac {1}{2} \arctan (a x)\right )-2 \arctan (a x)^3 \csc ^2\left (\frac {1}{2} \arctan (a x)\right )+8 \arctan (a x)^3 \log \left (1-e^{-i \arctan (a x)}\right )+48 \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )-48 \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )-8 \arctan (a x)^3 \log \left (1+e^{i \arctan (a x)}\right )+24 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{-i \arctan (a x)}\right )+24 i \left (2+\arctan (a x)^2\right ) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-48 i \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )+48 \arctan (a x) \operatorname {PolyLog}\left (3,e^{-i \arctan (a x)}\right )-48 \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )-48 i \operatorname {PolyLog}\left (4,e^{-i \arctan (a x)}\right )-48 i \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )+2 \arctan (a x)^3 \sec ^2\left (\frac {1}{2} \arctan (a x)\right )-12 \arctan (a x)^2 \tan \left (\frac {1}{2} \arctan (a x)\right )\right )}{16 \sqrt {1+a^2 x^2}} \]
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Time = 4.33 (sec) , antiderivative size = 404, normalized size of antiderivative = 0.67
method | result | size |
default | \(-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (a x \right )^{2} \left (3 a x +\arctan \left (a x \right )\right )}{2 x^{2}}+\frac {i a^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-i \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )+6 i \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{2 \sqrt {a^{2} x^{2}+1}}\) | \(404\) |
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\[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x^3} \, dx=\int { \frac {\sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{3}}{x^{3}} \,d x } \]
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\[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x^3} \, dx=\int \frac {\sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{3}{\left (a x \right )}}{x^{3}}\, dx \]
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\[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x^3} \, dx=\int { \frac {\sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{3}}{x^{3}} \,d x } \]
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Exception generated. \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x^3} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x^3} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,\sqrt {c\,a^2\,x^2+c}}{x^3} \,d x \]
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