Integrand size = 22, antiderivative size = 250 \[ \int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=-\frac {\sqrt {c+a^2 c x^2}}{2 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)}{2 a^2 c}+\frac {i \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a^3 \sqrt {c+a^2 c x^2}} \]
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Time = 0.11 (sec) , antiderivative size = 250, 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 = {5072, 267, 5010, 5006} \[ \int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=\frac {x \arctan (a x) \sqrt {a^2 c x^2+c}}{2 a^2 c}+\frac {i \sqrt {a^2 x^2+1} \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right ) \arctan (a x)}{a^3 \sqrt {a^2 c x^2+c}}-\frac {i \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {i \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{2 a^3 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c} \]
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Rule 267
Rule 5006
Rule 5010
Rule 5072
Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt {c+a^2 c x^2} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{2 a^2}-\frac {\int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{2 a} \\ & = -\frac {\sqrt {c+a^2 c x^2}}{2 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {1+a^2 x^2} \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{2 a^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {\sqrt {c+a^2 c x^2}}{2 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)}{2 a^2 c}+\frac {i \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a^3 \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.62 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.63 \[ \int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=-\frac {\sqrt {c \left (1+a^2 x^2\right )} \left (\sqrt {1+a^2 x^2}-a x \sqrt {1+a^2 x^2} \arctan (a x)+\arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )-\arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )+i \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-i \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )\right )}{2 a^3 c \sqrt {1+a^2 x^2}} \]
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Time = 0.46 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.74
method | result | size |
default | \(\frac {\left (x \arctan \left (a x \right ) a -1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 c \,a^{3}}+\frac {\left (\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \sqrt {a^{2} x^{2}+1}\, a^{3} c}\) | \(184\) |
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\[ \int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {x^{2} \arctan \left (a x\right )}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]
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\[ \int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x^{2} \operatorname {atan}{\left (a x \right )}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]
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\[ \int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {x^{2} \arctan \left (a x\right )}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]
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\[ \int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {x^{2} \arctan \left (a x\right )}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]
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Timed out. \[ \int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x^2\,\mathrm {atan}\left (a\,x\right )}{\sqrt {c\,a^2\,x^2+c}} \,d x \]
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