Integrand size = 20, antiderivative size = 59 \[ \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^2 \sqrt {c}} \]
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Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {5050, 223, 212} \[ \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}} \]
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Rule 212
Rule 223
Rule 5050
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^2 c}-\frac {\int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{a} \\ & = \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^2 c}-\frac {\text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{a} \\ & = \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^2 \sqrt {c}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.02 \[ \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\sqrt {c+a^2 c x^2} \arctan (a x)-\sqrt {c} \log \left (a c x+\sqrt {c} \sqrt {c+a^2 c x^2}\right )}{a^2 c} \]
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Result contains complex when optimal does not.
Time = 0.40 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.69
method | result | size |
default | \(\frac {\left (\arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}-\ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+i\right )+\ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-i\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, a^{2} c}\) | \(100\) |
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none
Time = 0.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.08 \[ \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=\frac {2 \, \sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right ) + \sqrt {c} \log \left (-2 \, a^{2} c x^{2} + 2 \, \sqrt {a^{2} c x^{2} + c} a \sqrt {c} x - c\right )}{2 \, a^{2} c} \]
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Exception generated. \[ \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=\text {Exception raised: TypeError} \]
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none
Time = 0.33 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.03 \[ \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=\frac {2 \, \sqrt {a^{2} x^{2} + 1} \arctan \left (a x\right ) - \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) + \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right )}{2 \, a^{2} \sqrt {c}} \]
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\[ \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {x \arctan \left (a x\right )}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]
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Timed out. \[ \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x\,\mathrm {atan}\left (a\,x\right )}{\sqrt {c\,a^2\,x^2+c}} \,d x \]
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