Integrand size = 14, antiderivative size = 102 \[ \int e^{-3 i \arctan (a x)} x^2 \, dx=-\frac {i (1-i a x)^3}{a^3 \sqrt {1+a^2 x^2}}-\frac {i (3-i a x)^2 \sqrt {1+a^2 x^2}}{3 a^3}-\frac {(28 i+3 a x) \sqrt {1+a^2 x^2}}{6 a^3}+\frac {11 \text {arcsinh}(a x)}{2 a^3} \]
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Time = 0.39 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {5168, 1647, 1607, 12, 866, 1649, 1668, 794, 221} \[ \int e^{-3 i \arctan (a x)} x^2 \, dx=\frac {11 \text {arcsinh}(a x)}{2 a^3}-\frac {i (1-i a x)^3}{a^3 \sqrt {a^2 x^2+1}}-\frac {i (3-i a x)^2 \sqrt {a^2 x^2+1}}{3 a^3}-\frac {(3 a x+28 i) \sqrt {a^2 x^2+1}}{6 a^3} \]
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Rule 12
Rule 221
Rule 794
Rule 866
Rule 1607
Rule 1647
Rule 1649
Rule 1668
Rule 5168
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^2 (1-i a x)^2}{(1+i a x) \sqrt {1+a^2 x^2}} \, dx \\ & = (i a) \int \frac {\sqrt {1+a^2 x^2} \left (-\frac {i x^2}{a}-x^3\right )}{(1+i a x)^2} \, dx \\ & = (i a) \int \frac {\left (-\frac {i}{a}-x\right ) x^2 \sqrt {1+a^2 x^2}}{(1+i a x)^2} \, dx \\ & = a^2 \int \frac {x^2 \left (1+a^2 x^2\right )^{3/2}}{a^2 (1+i a x)^3} \, dx \\ & = \int \frac {x^2 \left (1+a^2 x^2\right )^{3/2}}{(1+i a x)^3} \, dx \\ & = \int \frac {x^2 (1-i a x)^3}{\left (1+a^2 x^2\right )^{3/2}} \, dx \\ & = -\frac {i (1-i a x)^3}{a^3 \sqrt {1+a^2 x^2}}-\int \frac {\left (-\frac {3}{a^2}+\frac {i x}{a}\right ) (1-i a x)^2}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {i (1-i a x)^3}{a^3 \sqrt {1+a^2 x^2}}-\frac {i (3-i a x)^2 \sqrt {1+a^2 x^2}}{3 a^3}+\frac {1}{3} \int \frac {\left (-\frac {3}{a^2}+\frac {i x}{a}\right ) (-5+3 i a x)}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {i (1-i a x)^3}{a^3 \sqrt {1+a^2 x^2}}-\frac {i (3-i a x)^2 \sqrt {1+a^2 x^2}}{3 a^3}-\frac {(28 i+3 a x) \sqrt {1+a^2 x^2}}{6 a^3}+\frac {11 \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx}{2 a^2} \\ & = -\frac {i (1-i a x)^3}{a^3 \sqrt {1+a^2 x^2}}-\frac {i (3-i a x)^2 \sqrt {1+a^2 x^2}}{3 a^3}-\frac {(28 i+3 a x) \sqrt {1+a^2 x^2}}{6 a^3}+\frac {11 \text {arcsinh}(a x)}{2 a^3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.62 \[ \int e^{-3 i \arctan (a x)} x^2 \, dx=\frac {\frac {\sqrt {1+a^2 x^2} \left (-52-19 i a x-7 a^2 x^2+2 i a^3 x^3\right )}{-i+a x}+33 \text {arcsinh}(a x)}{6 a^3} \]
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Time = 0.36 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.09
method | result | size |
risch | \(\frac {i \left (2 a^{2} x^{2}+9 i a x -28\right ) \sqrt {a^{2} x^{2}+1}}{6 a^{3}}+\frac {11 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{2 a^{2} \sqrt {a^{2}}}-\frac {4 \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{a^{4} \left (x -\frac {i}{a}\right )}\) | \(111\) |
default | \(-\frac {2 \left (-\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{2}}+3 i a \left (\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{3}+i a \left (\frac {\left (2 \left (x -\frac {i}{a}\right ) a^{2}+2 i a \right ) \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{4 a^{2}}+\frac {\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 )}{2 \sqrt {a^{2}}}\right )\right )\right )}{a^{4}}+\frac {i \left (\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{3}+i a \left (\frac {\left (2 \left (x -\frac {i}{a}\right ) a^{2}+2 i a \right ) \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{4 a^{2}}+\frac {\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 )}{2 \sqrt {a^{2}}}\right )\right )}{a^{3}}-\frac {i \left (\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{3}}-2 i a \left (-\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{2}}+3 i a \left (\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{3}+i a \left (\frac {\left (2 \left (x -\frac {i}{a}\right ) a^{2}+2 i a \right ) \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{4 a^{2}}+\frac {\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 )}{2 \sqrt {a^{2}}}\right )\right )\right )\right )}{a^{5}}\) | \(619\) |
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Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.78 \[ \int e^{-3 i \arctan (a x)} x^2 \, dx=-\frac {24 \, a x + 33 \, {\left (a x - i\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) - {\left (2 i \, a^{3} x^{3} - 7 \, a^{2} x^{2} - 19 i \, a x - 52\right )} \sqrt {a^{2} x^{2} + 1} - 24 i}{6 \, {\left (a^{4} x - i \, a^{3}\right )}} \]
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\[ \int e^{-3 i \arctan (a x)} x^2 \, dx=i \left (\int \frac {x^{2} \sqrt {a^{2} x^{2} + 1}}{a^{3} x^{3} - 3 i a^{2} x^{2} - 3 a x + i}\, dx + \int \frac {a^{2} x^{4} \sqrt {a^{2} x^{2} + 1}}{a^{3} x^{3} - 3 i a^{2} x^{2} - 3 a x + i}\, dx\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 181 vs. \(2 (80) = 160\).
Time = 0.30 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.77 \[ \int e^{-3 i \arctan (a x)} x^2 \, dx=-\frac {i \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{5} x^{2} - 2 i \, a^{4} x - a^{3}} - \frac {i \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{i \, a^{4} x + a^{3}} - \frac {6 i \, \sqrt {a^{2} x^{2} + 1}}{i \, a^{4} x + a^{3}} - \frac {\sqrt {-a^{2} x^{2} + 4 i \, a x + 3} x}{2 \, a^{2}} + \frac {i \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, a^{3}} + \frac {\arcsin \left (i \, a x + 2\right )}{2 \, a^{3}} + \frac {6 \, \operatorname {arsinh}\left (a x\right )}{a^{3}} - \frac {3 i \, \sqrt {a^{2} x^{2} + 1}}{a^{3}} + \frac {i \, \sqrt {-a^{2} x^{2} + 4 i \, a x + 3}}{a^{3}} \]
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\[ \int e^{-3 i \arctan (a x)} x^2 \, dx=\int { \frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}}{{\left (i \, a x + 1\right )}^{3}} \,d x } \]
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Time = 0.07 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.13 \[ \int e^{-3 i \arctan (a x)} x^2 \, dx=\frac {11\,\mathrm {asinh}\left (x\,\sqrt {a^2}\right )}{2\,a^2\,\sqrt {a^2}}-\frac {\sqrt {a^2\,x^2+1}\,\left (\frac {3\,x\,\sqrt {a^2}}{2\,a^2}+\frac {a\,14{}\mathrm {i}}{3\,{\left (a^2\right )}^{3/2}}-\frac {a^3\,x^2\,1{}\mathrm {i}}{3\,{\left (a^2\right )}^{3/2}}\right )}{\sqrt {a^2}}+\frac {4\,\sqrt {a^2\,x^2+1}}{a^2\,\left (-x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \]
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