Integrand size = 25, antiderivative size = 42 \[ \int \frac {e^{i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {i \sqrt {1+a^2 x^2} \log (i+a x)}{a \sqrt {c+a^2 c x^2}} \]
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Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {5184, 5181, 31} \[ \int \frac {e^{i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {i \sqrt {a^2 x^2+1} \log (a x+i)}{a \sqrt {a^2 c x^2+c}} \]
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Rule 31
Rule 5181
Rule 5184
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+a^2 x^2} \int \frac {e^{i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {1+a^2 x^2} \int \frac {1}{1-i a x} \, dx}{\sqrt {c+a^2 c x^2}} \\ & = \frac {i \sqrt {1+a^2 x^2} \log (i+a x)}{a \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \frac {e^{i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {i \sqrt {1+a^2 x^2} \log (i+a x)}{a \sqrt {c+a^2 c x^2}} \]
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Time = 0.23 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.90
method | result | size |
risch | \(\frac {i \sqrt {a^{2} x^{2}+1}\, \ln \left (a x +i\right )}{\sqrt {c \left (a^{2} x^{2}+1\right )}\, a}\) | \(38\) |
default | \(\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (i \ln \left (a^{2} x^{2}+1\right )+2 \arctan \left (a x \right )\right )}{2 \sqrt {a^{2} x^{2}+1}\, c a}\) | \(53\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (34) = 68\).
Time = 0.28 (sec) , antiderivative size = 253, normalized size of antiderivative = 6.02 \[ \int \frac {e^{i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {1}{2} i \, \sqrt {\frac {1}{a^{2} c}} \log \left (\frac {{\left (i \, a^{6} x^{2} - 2 \, a^{5} x - 2 i \, a^{4}\right )} \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} + {\left (i \, a^{9} c x^{4} - 2 \, a^{8} c x^{3} + i \, a^{7} c x^{2} - 2 \, a^{6} c x\right )} \sqrt {\frac {1}{a^{2} c}}}{8 \, {\left (a^{3} x^{3} + i \, a^{2} x^{2} + a x + i\right )}}\right ) - \frac {1}{2} i \, \sqrt {\frac {1}{a^{2} c}} \log \left (\frac {{\left (i \, a^{6} x^{2} - 2 \, a^{5} x - 2 i \, a^{4}\right )} \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} + {\left (-i \, a^{9} c x^{4} + 2 \, a^{8} c x^{3} - i \, a^{7} c x^{2} + 2 \, a^{6} c x\right )} \sqrt {\frac {1}{a^{2} c}}}{8 \, {\left (a^{3} x^{3} + i \, a^{2} x^{2} + a x + i\right )}}\right ) \]
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\[ \int \frac {e^{i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=i \left (\int \left (- \frac {i}{\sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\right )\, dx + \int \frac {a x}{\sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\, dx\right ) \]
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Exception generated. \[ \int \frac {e^{i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {e^{i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {i \, a x + 1}{\sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1}} \,d x } \]
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Timed out. \[ \int \frac {e^{i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {c\,a^2\,x^2+c}\,\sqrt {a^2\,x^2+1}} \,d x \]
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