Integrand size = 25, antiderivative size = 84 \[ \int \frac {e^{3 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {2 \sqrt {1+a^2 x^2}}{a (i+a x) \sqrt {c+a^2 c x^2}}-\frac {i \sqrt {1+a^2 x^2} \log (i+a x)}{a \sqrt {c+a^2 c x^2}} \]
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Time = 0.06 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {5184, 5181, 45} \[ \int \frac {e^{3 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {2 \sqrt {a^2 x^2+1}}{a (a x+i) \sqrt {a^2 c x^2+c}}-\frac {i \sqrt {a^2 x^2+1} \log (a x+i)}{a \sqrt {a^2 c x^2+c}} \]
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Rule 45
Rule 5181
Rule 5184
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+a^2 x^2} \int \frac {e^{3 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+i a x}{(1-i a x)^2} \, dx}{\sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {1+a^2 x^2} \int \left (-\frac {2}{(i+a x)^2}-\frac {i}{i+a x}\right ) \, dx}{\sqrt {c+a^2 c x^2}} \\ & = \frac {2 \sqrt {1+a^2 x^2}}{a (i+a x) \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 = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.65 \[ \int \frac {e^{3 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\sqrt {1+a^2 x^2} \left (\frac {2}{i+a x}-i \log (i+a x)\right )}{a \sqrt {c+a^2 c x^2}} \]
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Time = 0.25 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.73
method | result | size |
default | \(\frac {\left (-i \ln \left (a x +i\right ) a x +\ln \left (a x +i\right )+2\right ) \sqrt {c \left (a^{2} x^{2}+1\right )}}{\sqrt {a^{2} x^{2}+1}\, c a \left (a x +i\right )}\) | \(61\) |
risch | \(\frac {2 \sqrt {a^{2} x^{2}+1}}{\sqrt {c \left (a^{2} x^{2}+1\right )}\, a \left (a x +i\right )}-\frac {i \sqrt {a^{2} x^{2}+1}\, \ln \left (a x +i\right )}{\sqrt {c \left (a^{2} x^{2}+1\right )}\, a}\) | \(76\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (70) = 140\).
Time = 0.29 (sec) , antiderivative size = 357, normalized size of antiderivative = 4.25 \[ \int \frac {e^{3 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {{\left (-i \, a^{3} c x^{3} + a^{2} c x^{2} - i \, a c x + c\right )} \sqrt {\frac {1}{a^{2} c}} \log \left (\frac {{\left (i \, a^{6} x^{2} - 2 \, a^{5} x - 2 i \, a^{4}\right )} \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} + {\left (i \, a^{9} c x^{4} - 2 \, a^{8} c x^{3} + i \, a^{7} c x^{2} - 2 \, a^{6} c x\right )} \sqrt {\frac {1}{a^{2} c}}}{8 \, {\left (a^{3} x^{3} + i \, a^{2} x^{2} + a x + i\right )}}\right ) + {\left (i \, a^{3} c x^{3} - a^{2} c x^{2} + i \, a c x - c\right )} \sqrt {\frac {1}{a^{2} c}} \log \left (\frac {{\left (i \, a^{6} x^{2} - 2 \, a^{5} x - 2 i \, a^{4}\right )} \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} + {\left (-i \, a^{9} c x^{4} + 2 \, a^{8} c x^{3} - i \, a^{7} c x^{2} + 2 \, a^{6} c x\right )} \sqrt {\frac {1}{a^{2} c}}}{8 \, {\left (a^{3} x^{3} + i \, a^{2} x^{2} + a x + i\right )}}\right ) + 4 i \, \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} x}{2 \, {\left (a^{3} c x^{3} + i \, a^{2} c x^{2} + a c x + i \, c\right )}} \]
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\[ \int \frac {e^{3 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=- i \left (\int \frac {i}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\, dx + \int \left (- \frac {3 a x}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\right )\, dx + \int \frac {a^{3} x^{3}}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\, dx + \int \left (- \frac {3 i a^{2} x^{2}}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\right )\, dx\right ) \]
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\[ \int \frac {e^{3 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{3}}{\sqrt {a^{2} c x^{2} + c} {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {e^{3 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{3}}{\sqrt {a^{2} c x^{2} + c} {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {e^{3 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {{\left (1+a\,x\,1{}\mathrm {i}\right )}^3}{\sqrt {c\,a^2\,x^2+c}\,{\left (a^2\,x^2+1\right )}^{3/2}} \,d x \]
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