Integrand size = 22, antiderivative size = 229 \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{x} \, dx=\sqrt {c+a^2 c x^2} \arctan (a x)-\frac {2 c \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {i c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {i c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}} \]
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Time = 0.16 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {5066, 5078, 5074, 223, 212} \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{x} \, dx=-\frac {2 c \sqrt {a^2 x^2+1} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )+\frac {i c \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}-\frac {i c \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}} \]
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Rule 212
Rule 223
Rule 5066
Rule 5074
Rule 5078
Rubi steps \begin{align*} \text {integral}& = \sqrt {c+a^2 c x^2} \arctan (a x)+c \int \frac {\arctan (a x)}{x \sqrt {c+a^2 c x^2}} \, dx-(a c) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx \\ & = \sqrt {c+a^2 c x^2} \arctan (a x)-(a c) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )+\frac {\left (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}} \\ & = \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {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}}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {i 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 {i 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}} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{x} \, dx=\frac {\sqrt {c+a^2 c x^2} \left (\sqrt {1+a^2 x^2} \arctan (a x)+\arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )-\arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )+\log \left (\cos \left (\frac {1}{2} \arctan (a x)\right )-\sin \left (\frac {1}{2} \arctan (a x)\right )\right )-\log \left (\cos \left (\frac {1}{2} \arctan (a x)\right )+\sin \left (\frac {1}{2} \arctan (a x)\right )\right )+i \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )\right )}{\sqrt {1+a^2 x^2}} \]
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Time = 0.49 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.66
method | result | size |
default | \(\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (a x \right )-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (\arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-2 i \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )\right )}{\sqrt {a^{2} x^{2}+1}}\) | \(151\) |
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\[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{x} \, dx=\int { \frac {\sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )}{x} \,d x } \]
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\[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{x} \, dx=\int \frac {\sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}{\left (a x \right )}}{x}\, dx \]
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\[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{x} \, dx=\int { \frac {\sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )}{x} \,d x } \]
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Exception generated. \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{x} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{x} \, dx=\int \frac {\mathrm {atan}\left (a\,x\right )\,\sqrt {c\,a^2\,x^2+c}}{x} \,d x \]
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