Integrand size = 14, antiderivative size = 90 \[ \int e^{i \arctan (a x)} x^3 \, dx=\frac {x^2 \sqrt {1+a^2 x^2}}{3 a^2}+\frac {i x^3 \sqrt {1+a^2 x^2}}{4 a}-\frac {(16+9 i a x) \sqrt {1+a^2 x^2}}{24 a^4}+\frac {3 i \text {arcsinh}(a x)}{8 a^4} \]
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Time = 0.05 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5168, 847, 794, 221} \[ \int e^{i \arctan (a x)} x^3 \, dx=\frac {3 i \text {arcsinh}(a x)}{8 a^4}+\frac {x^2 \sqrt {a^2 x^2+1}}{3 a^2}+\frac {i x^3 \sqrt {a^2 x^2+1}}{4 a}-\frac {(16+9 i a x) \sqrt {a^2 x^2+1}}{24 a^4} \]
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Rule 221
Rule 794
Rule 847
Rule 5168
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3 (1+i a x)}{\sqrt {1+a^2 x^2}} \, dx \\ & = \frac {i x^3 \sqrt {1+a^2 x^2}}{4 a}+\frac {\int \frac {x^2 \left (-3 i a+4 a^2 x\right )}{\sqrt {1+a^2 x^2}} \, dx}{4 a^2} \\ & = \frac {x^2 \sqrt {1+a^2 x^2}}{3 a^2}+\frac {i x^3 \sqrt {1+a^2 x^2}}{4 a}+\frac {\int \frac {x \left (-8 a^2-9 i a^3 x\right )}{\sqrt {1+a^2 x^2}} \, dx}{12 a^4} \\ & = \frac {x^2 \sqrt {1+a^2 x^2}}{3 a^2}+\frac {i x^3 \sqrt {1+a^2 x^2}}{4 a}-\frac {(16+9 i a x) \sqrt {1+a^2 x^2}}{24 a^4}+\frac {(3 i) \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx}{8 a^3} \\ & = \frac {x^2 \sqrt {1+a^2 x^2}}{3 a^2}+\frac {i x^3 \sqrt {1+a^2 x^2}}{4 a}-\frac {(16+9 i a x) \sqrt {1+a^2 x^2}}{24 a^4}+\frac {3 i \text {arcsinh}(a x)}{8 a^4} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.62 \[ \int e^{i \arctan (a x)} x^3 \, dx=\frac {\sqrt {1+a^2 x^2} \left (-16-9 i a x+8 a^2 x^2+6 i a^3 x^3\right )+9 i \text {arcsinh}(a x)}{24 a^4} \]
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Time = 0.25 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.86
method | result | size |
risch | \(\frac {i \left (6 a^{3} x^{3}-8 i a^{2} x^{2}-9 a x +16 i\right ) \sqrt {a^{2} x^{2}+1}}{24 a^{4}}+\frac {3 i \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{8 a^{3} \sqrt {a^{2}}}\) | \(77\) |
meijerg | \(\frac {\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right ) \sqrt {a^{2} x^{2}+1}}{6}}{2 a^{4} \sqrt {\pi }}+\frac {i \left (-\frac {\sqrt {\pi }\, x \left (a^{2}\right )^{\frac {5}{2}} \left (-10 a^{2} x^{2}+15\right ) \sqrt {a^{2} x^{2}+1}}{20 a^{4}}+\frac {3 \sqrt {\pi }\, \left (a^{2}\right )^{\frac {5}{2}} \operatorname {arcsinh}\left (a x \right )}{4 a^{5}}\right )}{2 a^{3} \sqrt {\pi }\, \sqrt {a^{2}}}\) | \(109\) |
default | \(\frac {x^{2} \sqrt {a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {a^{2} x^{2}+1}}{3 a^{4}}+i a \left (\frac {x^{3} \sqrt {a^{2} x^{2}+1}}{4 a^{2}}-\frac {3 \left (\frac {x \sqrt {a^{2} x^{2}+1}}{2 a^{2}}-\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{2 a^{2} \sqrt {a^{2}}}\right )}{4 a^{2}}\right )\) | \(117\) |
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Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.66 \[ \int e^{i \arctan (a x)} x^3 \, dx=\frac {{\left (6 i \, a^{3} x^{3} + 8 \, a^{2} x^{2} - 9 i \, a x - 16\right )} \sqrt {a^{2} x^{2} + 1} - 9 i \, \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right )}{24 \, a^{4}} \]
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Time = 0.56 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.16 \[ \int e^{i \arctan (a x)} x^3 \, dx=\begin {cases} \sqrt {a^{2} x^{2} + 1} \left (\frac {i x^{3}}{4 a} + \frac {x^{2}}{3 a^{2}} - \frac {3 i x}{8 a^{3}} - \frac {2}{3 a^{4}}\right ) + \frac {3 i \log {\left (2 a^{2} x + 2 \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2}} \right )}}{8 a^{3} \sqrt {a^{2}}} & \text {for}\: a^{2} \neq 0 \\\frac {i a x^{5}}{5} + \frac {x^{4}}{4} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.90 \[ \int e^{i \arctan (a x)} x^3 \, dx=\frac {i \, \sqrt {a^{2} x^{2} + 1} x^{3}}{4 \, a} + \frac {\sqrt {a^{2} x^{2} + 1} x^{2}}{3 \, a^{2}} - \frac {3 i \, \sqrt {a^{2} x^{2} + 1} x}{8 \, a^{3}} + \frac {3 i \, \operatorname {arsinh}\left (a x\right )}{8 \, a^{4}} - \frac {2 \, \sqrt {a^{2} x^{2} + 1}}{3 \, a^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.78 \[ \int e^{i \arctan (a x)} x^3 \, dx=-\frac {1}{24} \, \sqrt {a^{2} x^{2} + 1} {\left ({\left (2 \, x {\left (-\frac {3 i \, x}{a} - \frac {4}{a^{2}}\right )} + \frac {9 i}{a^{3}}\right )} x + \frac {16}{a^{4}}\right )} - \frac {3 i \, \log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right )}{8 \, a^{3} {\left | a \right |}} \]
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Time = 0.50 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.94 \[ \int e^{i \arctan (a x)} x^3 \, dx=\frac {\mathrm {asinh}\left (x\,\sqrt {a^2}\right )\,3{}\mathrm {i}}{8\,a^3\,\sqrt {a^2}}-\frac {\sqrt {a^2\,x^2+1}\,\left (\frac {2}{3\,{\left (a^2\right )}^{3/2}}-\frac {a^2\,x^2}{3\,{\left (a^2\right )}^{3/2}}-\frac {x^3\,{\left (a^2\right )}^{3/2}\,1{}\mathrm {i}}{4\,a^3}+\frac {x\,\sqrt {a^2}\,3{}\mathrm {i}}{8\,a^3}\right )}{\sqrt {a^2}} \]
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