Integrand size = 22, antiderivative size = 242 \[ \int \frac {\arctan (a x)}{x^3 \sqrt {c+a^2 c x^2}} \, dx=-\frac {a \sqrt {c+a^2 c x^2}}{2 c x}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{2 c x^2}+\frac {a^2 \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {i a^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {i a^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}} \]
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Time = 0.16 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {5082, 270, 5078, 5074} \[ \int \frac {\arctan (a x)}{x^3 \sqrt {c+a^2 c x^2}} \, dx=\frac {a^2 \sqrt {a^2 x^2+1} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{2 c x^2}-\frac {i a^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {i a^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {a \sqrt {a^2 c x^2+c}}{2 c x} \]
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Rule 270
Rule 5074
Rule 5078
Rule 5082
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{2 c x^2}+\frac {1}{2} a \int \frac {1}{x^2 \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{2} a^2 \int \frac {\arctan (a x)}{x \sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {a \sqrt {c+a^2 c x^2}}{2 c x}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{2 c x^2}-\frac {\left (a^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{x \sqrt {1+a^2 x^2}} \, dx}{2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {a \sqrt {c+a^2 c x^2}}{2 c x}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{2 c x^2}+\frac {a^2 \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 {i a^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {i a^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.76 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.68 \[ \int \frac {\arctan (a x)}{x^3 \sqrt {c+a^2 c x^2}} \, dx=\frac {a^2 \sqrt {1+a^2 x^2} \left (-2 \cot \left (\frac {1}{2} \arctan (a x)\right )-\arctan (a x) \csc ^2\left (\frac {1}{2} \arctan (a x)\right )-4 \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )+4 \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )-4 i \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )+4 i \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )+\arctan (a x) \sec ^2\left (\frac {1}{2} \arctan (a x)\right )-2 \tan \left (\frac {1}{2} \arctan (a x)\right )\right )}{8 \sqrt {c \left (1+a^2 x^2\right )}} \]
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Time = 0.45 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.72
method | result | size |
default | \(-\frac {\left (a x +\arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 c \,x^{2}}-\frac {i a^{2} \left (i \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-i \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \sqrt {a^{2} x^{2}+1}\, c}\) | \(175\) |
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\[ \int \frac {\arctan (a x)}{x^3 \sqrt {c+a^2 c x^2}} \, dx=\int { \frac {\arctan \left (a x\right )}{\sqrt {a^{2} c x^{2} + c} x^{3}} \,d x } \]
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\[ \int \frac {\arctan (a x)}{x^3 \sqrt {c+a^2 c x^2}} \, dx=\int \frac {\operatorname {atan}{\left (a x \right )}}{x^{3} \sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]
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\[ \int \frac {\arctan (a x)}{x^3 \sqrt {c+a^2 c x^2}} \, dx=\int { \frac {\arctan \left (a x\right )}{\sqrt {a^{2} c x^{2} + c} x^{3}} \,d x } \]
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\[ \int \frac {\arctan (a x)}{x^3 \sqrt {c+a^2 c x^2}} \, dx=\int { \frac {\arctan \left (a x\right )}{\sqrt {a^{2} c x^{2} + c} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\arctan (a x)}{x^3 \sqrt {c+a^2 c x^2}} \, dx=\int \frac {\mathrm {atan}\left (a\,x\right )}{x^3\,\sqrt {c\,a^2\,x^2+c}} \,d x \]
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