Integrand size = 14, antiderivative size = 137 \[ \int e^{-3 i \arctan (a x)} x^3 \, dx=\frac {(1-i a x)^3}{a^4 \sqrt {1+a^2 x^2}}+\frac {27 \sqrt {1+a^2 x^2}}{4 a^4}-\frac {x^2 \sqrt {1+a^2 x^2}}{a^2}+\frac {i x^3 \sqrt {1+a^2 x^2}}{4 a}-\frac {9 i (2 i+3 a x) \sqrt {1+a^2 x^2}}{8 a^4}+\frac {51 i \text {arcsinh}(a x)}{8 a^4} \]
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Time = 0.42 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {5168, 1647, 1607, 12, 866, 1649, 1829, 27, 757, 655, 221} \[ \int e^{-3 i \arctan (a x)} x^3 \, dx=\frac {51 i \text {arcsinh}(a x)}{8 a^4}-\frac {x^2 \sqrt {a^2 x^2+1}}{a^2}+\frac {i x^3 \sqrt {a^2 x^2+1}}{4 a}-\frac {9 i (3 a x+2 i) \sqrt {a^2 x^2+1}}{8 a^4}+\frac {27 \sqrt {a^2 x^2+1}}{4 a^4}+\frac {(1-i a x)^3}{a^4 \sqrt {a^2 x^2+1}} \]
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Rule 12
Rule 27
Rule 221
Rule 655
Rule 757
Rule 866
Rule 1607
Rule 1647
Rule 1649
Rule 1829
Rule 5168
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3 (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^3}{a}-x^4\right )}{(1+i a x)^2} \, dx \\ & = (i a) \int \frac {\left (-\frac {i}{a}-x\right ) x^3 \sqrt {1+a^2 x^2}}{(1+i a x)^2} \, dx \\ & = a^2 \int \frac {x^3 \left (1+a^2 x^2\right )^{3/2}}{a^2 (1+i a x)^3} \, dx \\ & = \int \frac {x^3 \left (1+a^2 x^2\right )^{3/2}}{(1+i a x)^3} \, dx \\ & = \int \frac {x^3 (1-i a x)^3}{\left (1+a^2 x^2\right )^{3/2}} \, dx \\ & = \frac {(1-i a x)^3}{a^4 \sqrt {1+a^2 x^2}}-\int \frac {(1-i a x)^2 \left (-\frac {3 i}{a^3}-\frac {x}{a^2}+\frac {i x^2}{a}\right )}{\sqrt {1+a^2 x^2}} \, dx \\ & = \frac {(1-i a x)^3}{a^4 \sqrt {1+a^2 x^2}}+\frac {i x^3 \sqrt {1+a^2 x^2}}{4 a}-\frac {\int \frac {-\frac {12 i}{a}-28 x+27 i a x^2+12 a^2 x^3}{\sqrt {1+a^2 x^2}} \, dx}{4 a^2} \\ & = \frac {(1-i a x)^3}{a^4 \sqrt {1+a^2 x^2}}-\frac {x^2 \sqrt {1+a^2 x^2}}{a^2}+\frac {i x^3 \sqrt {1+a^2 x^2}}{4 a}-\frac {\int \frac {-36 i a-108 a^2 x+81 i a^3 x^2}{\sqrt {1+a^2 x^2}} \, dx}{12 a^4} \\ & = \frac {(1-i a x)^3}{a^4 \sqrt {1+a^2 x^2}}-\frac {x^2 \sqrt {1+a^2 x^2}}{a^2}+\frac {i x^3 \sqrt {1+a^2 x^2}}{4 a}-\frac {\int \frac {9 i a (2 i+3 a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{12 a^4} \\ & = \frac {(1-i a x)^3}{a^4 \sqrt {1+a^2 x^2}}-\frac {x^2 \sqrt {1+a^2 x^2}}{a^2}+\frac {i x^3 \sqrt {1+a^2 x^2}}{4 a}-\frac {(3 i) \int \frac {(2 i+3 a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{4 a^3} \\ & = \frac {(1-i a x)^3}{a^4 \sqrt {1+a^2 x^2}}-\frac {x^2 \sqrt {1+a^2 x^2}}{a^2}+\frac {i x^3 \sqrt {1+a^2 x^2}}{4 a}-\frac {9 i (2 i+3 a x) \sqrt {1+a^2 x^2}}{8 a^4}-\frac {(3 i) \int \frac {-17 a^2+18 i a^3 x}{\sqrt {1+a^2 x^2}} \, dx}{8 a^5} \\ & = \frac {(1-i a x)^3}{a^4 \sqrt {1+a^2 x^2}}+\frac {27 \sqrt {1+a^2 x^2}}{4 a^4}-\frac {x^2 \sqrt {1+a^2 x^2}}{a^2}+\frac {i x^3 \sqrt {1+a^2 x^2}}{4 a}-\frac {9 i (2 i+3 a x) \sqrt {1+a^2 x^2}}{8 a^4}+\frac {(51 i) \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx}{8 a^3} \\ & = \frac {(1-i a x)^3}{a^4 \sqrt {1+a^2 x^2}}+\frac {27 \sqrt {1+a^2 x^2}}{4 a^4}-\frac {x^2 \sqrt {1+a^2 x^2}}{a^2}+\frac {i x^3 \sqrt {1+a^2 x^2}}{4 a}-\frac {9 i (2 i+3 a x) \sqrt {1+a^2 x^2}}{8 a^4}+\frac {51 i \text {arcsinh}(a x)}{8 a^4} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.58 \[ \int e^{-3 i \arctan (a x)} x^3 \, dx=\sqrt {1+a^2 x^2} \left (\frac {6}{a^4}-\frac {19 i x}{8 a^3}-\frac {x^2}{a^2}+\frac {i x^3}{4 a}-\frac {4 i}{a^4 (-i+a x)}\right )+\frac {51 i \text {arcsinh}(a x)}{8 a^4} \]
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Time = 0.37 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.91
method | result | size |
risch | \(\frac {i \left (2 a^{3} x^{3}+8 i a^{2} x^{2}-19 a x -48 i\right ) \sqrt {a^{2} x^{2}+1}}{8 a^{4}}+\frac {i \left (\frac {51 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{\sqrt {a^{2}}}-\frac {32 \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{a^{2} \left (x -\frac {i}{a}\right )}\right )}{8 a^{3}}\) | \(124\) |
default | \(\frac {i \left (\frac {x \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4}+\frac {3 \sqrt {a^{2} x^{2}+1}\, x}{8}+\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{8 \sqrt {a^{2}}}\right )}{a^{3}}+\frac {\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 )}{a^{6}}-\frac {3 \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^{4}}-\frac {3 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 )^{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^{5}}\) | \(682\) |
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Time = 0.25 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.64 \[ \int e^{-3 i \arctan (a x)} x^3 \, dx=\frac {-32 i \, a x - 51 \, {\left (i \, a x + 1\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) + {\left (2 i \, a^{4} x^{4} - 6 \, a^{3} x^{3} - 11 i \, a^{2} x^{2} + 29 \, a x - 80 i\right )} \sqrt {a^{2} x^{2} + 1} - 32}{8 \, {\left (a^{5} x - i \, a^{4}\right )}} \]
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\[ \int e^{-3 i \arctan (a x)} x^3 \, dx=i \left (\int \frac {x^{3} \sqrt {a^{2} x^{2} + 1}}{a^{3} x^{3} - 3 i a^{2} x^{2} - 3 a x + i}\, dx + \int \frac {a^{2} x^{5} \sqrt {a^{2} x^{2} + 1}}{a^{3} x^{3} - 3 i a^{2} x^{2} - 3 a x + i}\, dx\right ) \]
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Time = 0.28 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.58 \[ \int e^{-3 i \arctan (a x)} x^3 \, dx=\frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{6} x^{2} - 2 i \, a^{5} x - a^{4}} + \frac {3 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{2 i \, a^{5} x + 2 \, a^{4}} + \frac {6 \, \sqrt {a^{2} x^{2} + 1}}{i \, a^{5} x + a^{4}} + \frac {i \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{4 \, a^{3}} + \frac {3 i \, \sqrt {a^{2} x^{2} + 1} x}{8 \, a^{3}} - \frac {3 i \, \sqrt {-a^{2} x^{2} + 4 i \, a x + 3} x}{2 \, a^{3}} - \frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{4}} + \frac {3 i \, \arcsin \left (i \, a x + 2\right )}{2 \, a^{4}} + \frac {63 i \, \operatorname {arsinh}\left (a x\right )}{8 \, a^{4}} + \frac {9 \, \sqrt {a^{2} x^{2} + 1}}{2 \, a^{4}} - \frac {3 \, \sqrt {-a^{2} x^{2} + 4 i \, a x + 3}}{a^{4}} \]
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Exception generated. \[ \int e^{-3 i \arctan (a x)} x^3 \, dx=\text {Exception raised: TypeError} \]
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Time = 0.51 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.01 \[ \int e^{-3 i \arctan (a x)} x^3 \, dx=\frac {\sqrt {a^2\,x^2+1}\,\left (\frac {4}{{\left (a^2\right )}^{3/2}}+\frac {2\,\sqrt {a^2}}{a^4}-\frac {x^2\,\sqrt {a^2}}{a^2}+\frac {x^3\,{\left (a^2\right )}^{3/2}\,1{}\mathrm {i}}{4\,a^3}-\frac {x\,\sqrt {a^2}\,19{}\mathrm {i}}{8\,a^3}\right )}{\sqrt {a^2}}+\frac {\mathrm {asinh}\left (x\,\sqrt {a^2}\right )\,51{}\mathrm {i}}{8\,a^3\,\sqrt {a^2}}+\frac {\sqrt {a^2\,x^2+1}\,4{}\mathrm {i}}{a^3\,\left (-x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \]
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