Integrand size = 28, antiderivative size = 132 \[ \int \frac {\arctan (a+b x)}{\sqrt {1+a^2+2 a b x+b^2 x^2}} \, dx=-\frac {2 i \arctan (a+b x) \arctan \left (\frac {\sqrt {1+i (a+b x)}}{\sqrt {1-i (a+b x)}}\right )}{b}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i (a+b x)}}{\sqrt {1-i (a+b x)}}\right )}{b}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i (a+b x)}}{\sqrt {1-i (a+b x)}}\right )}{b} \]
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Time = 0.07 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {5163, 5006} \[ \int \frac {\arctan (a+b x)}{\sqrt {1+a^2+2 a b x+b^2 x^2}} \, dx=-\frac {2 i \arctan (a+b x) \arctan \left (\frac {\sqrt {1+i (a+b x)}}{\sqrt {1-i (a+b x)}}\right )}{b}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i (a+b x)+1}}{\sqrt {1-i (a+b x)}}\right )}{b}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i (a+b x)+1}}{\sqrt {1-i (a+b x)}}\right )}{b} \]
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Rule 5006
Rule 5163
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\arctan (x)}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{b} \\ & = -\frac {2 i \arctan (a+b x) \arctan \left (\frac {\sqrt {1+i (a+b x)}}{\sqrt {1-i (a+b x)}}\right )}{b}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i (a+b x)}}{\sqrt {1-i (a+b x)}}\right )}{b}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i (a+b x)}}{\sqrt {1-i (a+b x)}}\right )}{b} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.51 \[ \int \frac {\arctan (a+b x)}{\sqrt {1+a^2+2 a b x+b^2 x^2}} \, dx=-\frac {i \left (2 \arctan \left (e^{i \arctan (a+b x)}\right ) \arctan (a+b x)-\operatorname {PolyLog}\left (2,-i e^{i \arctan (a+b x)}\right )+\operatorname {PolyLog}\left (2,i e^{i \arctan (a+b x)}\right )\right )}{b} \]
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Time = 0.43 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {-\arctan \left (b x +a \right ) \ln \left (1+\frac {i \left (1+i \left (b x +a \right )\right )}{\sqrt {1+\left (b x +a \right )^{2}}}\right )+\arctan \left (b x +a \right ) \ln \left (1-\frac {i \left (1+i \left (b x +a \right )\right )}{\sqrt {1+\left (b x +a \right )^{2}}}\right )+i \operatorname {dilog}\left (1+\frac {i \left (1+i \left (b x +a \right )\right )}{\sqrt {1+\left (b x +a \right )^{2}}}\right )-i \operatorname {dilog}\left (1-\frac {i \left (1+i \left (b x +a \right )\right )}{\sqrt {1+\left (b x +a \right )^{2}}}\right )}{b}\) | \(135\) |
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\[ \int \frac {\arctan (a+b x)}{\sqrt {1+a^2+2 a b x+b^2 x^2}} \, dx=\int { \frac {\arctan \left (b x + a\right )}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} \,d x } \]
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\[ \int \frac {\arctan (a+b x)}{\sqrt {1+a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {\operatorname {atan}{\left (a + b x \right )}}{\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx \]
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\[ \int \frac {\arctan (a+b x)}{\sqrt {1+a^2+2 a b x+b^2 x^2}} \, dx=\int { \frac {\arctan \left (b x + a\right )}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} \,d x } \]
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\[ \int \frac {\arctan (a+b x)}{\sqrt {1+a^2+2 a b x+b^2 x^2}} \, dx=\int { \frac {\arctan \left (b x + a\right )}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} \,d x } \]
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Timed out. \[ \int \frac {\arctan (a+b x)}{\sqrt {1+a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {\mathrm {atan}\left (a+b\,x\right )}{\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}} \,d x \]
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