Integrand size = 14, antiderivative size = 75 \[ \int e^{-i \arctan (a x)} x^2 \, dx=\frac {i \sqrt {1+a^2 x^2}}{a^3}+\frac {x \sqrt {1+a^2 x^2}}{2 a^2}-\frac {i \left (1+a^2 x^2\right )^{3/2}}{3 a^3}-\frac {\text {arcsinh}(a x)}{2 a^3} \]
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Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5168, 811, 655, 201, 221} \[ \int e^{-i \arctan (a x)} x^2 \, dx=-\frac {\text {arcsinh}(a x)}{2 a^3}+\frac {x \sqrt {a^2 x^2+1}}{2 a^2}-\frac {i \left (a^2 x^2+1\right )^{3/2}}{3 a^3}+\frac {i \sqrt {a^2 x^2+1}}{a^3} \]
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Rule 201
Rule 221
Rule 655
Rule 811
Rule 5168
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^2 (1-i a x)}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {\int \frac {1-i a x}{\sqrt {1+a^2 x^2}} \, dx}{a^2}+\frac {\int (1-i a x) \sqrt {1+a^2 x^2} \, dx}{a^2} \\ & = \frac {i \sqrt {1+a^2 x^2}}{a^3}-\frac {i \left (1+a^2 x^2\right )^{3/2}}{3 a^3}-\frac {\int \frac {1}{\sqrt {1+a^2 x^2}} \, dx}{a^2}+\frac {\int \sqrt {1+a^2 x^2} \, dx}{a^2} \\ & = \frac {i \sqrt {1+a^2 x^2}}{a^3}+\frac {x \sqrt {1+a^2 x^2}}{2 a^2}-\frac {i \left (1+a^2 x^2\right )^{3/2}}{3 a^3}-\frac {\text {arcsinh}(a x)}{a^3}+\frac {\int \frac {1}{\sqrt {1+a^2 x^2}} \, dx}{2 a^2} \\ & = \frac {i \sqrt {1+a^2 x^2}}{a^3}+\frac {x \sqrt {1+a^2 x^2}}{2 a^2}-\frac {i \left (1+a^2 x^2\right )^{3/2}}{3 a^3}-\frac {\text {arcsinh}(a x)}{2 a^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.61 \[ \int e^{-i \arctan (a x)} x^2 \, dx=\frac {\left (4 i+3 a x-2 i a^2 x^2\right ) \sqrt {1+a^2 x^2}-3 \text {arcsinh}(a x)}{6 a^3} \]
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Time = 0.25 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.89
method | result | size |
risch | \(-\frac {i \left (2 a^{2} x^{2}+3 i a x -4\right ) \sqrt {a^{2} x^{2}+1}}{6 a^{3}}-\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{2 a^{2} \sqrt {a^{2}}}\) | \(67\) |
default | \(\frac {\frac {\sqrt {a^{2} x^{2}+1}\, x}{2}+\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{2 \sqrt {a^{2}}}}{a^{2}}-\frac {i \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3 a^{3}}+\frac {i \left (\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}+\frac {i a \ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{\sqrt {a^{2}}}\right )}{a^{3}}\) | \(167\) |
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Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.68 \[ \int e^{-i \arctan (a x)} x^2 \, dx=\frac {\sqrt {a^{2} x^{2} + 1} {\left (-2 i \, a^{2} x^{2} + 3 \, a x + 4 i\right )} + 3 \, \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right )}{6 \, a^{3}} \]
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\[ \int e^{-i \arctan (a x)} x^2 \, dx=- i \int \frac {x^{2} \sqrt {a^{2} x^{2} + 1}}{a x - i}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.79 \[ \int e^{-i \arctan (a x)} x^2 \, dx=\frac {\sqrt {a^{2} x^{2} + 1} x}{2 \, a^{2}} - \frac {i \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, a^{3}} - \frac {\operatorname {arsinh}\left (a x\right )}{2 \, a^{3}} + \frac {i \, \sqrt {a^{2} x^{2} + 1}}{a^{3}} \]
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Exception generated. \[ \int e^{-i \arctan (a x)} x^2 \, dx=\text {Exception raised: TypeError} \]
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Time = 0.46 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.95 \[ \int e^{-i \arctan (a x)} x^2 \, dx=\frac {\sqrt {a^2\,x^2+1}\,\left (\frac {x\,\sqrt {a^2}}{2\,a^2}+\frac {a\,2{}\mathrm {i}}{3\,{\left (a^2\right )}^{3/2}}-\frac {a^3\,x^2\,1{}\mathrm {i}}{3\,{\left (a^2\right )}^{3/2}}\right )}{\sqrt {a^2}}-\frac {\mathrm {asinh}\left (x\,\sqrt {a^2}\right )}{2\,a^2\,\sqrt {a^2}} \]
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