Integrand size = 16, antiderivative size = 371 \[ \int e^{\frac {5}{2} i \arctan (a x)} x^2 \, dx=\frac {55 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{8 a^3}+\frac {11 i (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^3}+\frac {2 i (1+i a x)^{9/4}}{a^3 \sqrt [4]{1-i a x}}+\frac {i (1-i a x)^{3/4} (1+i a x)^{9/4}}{3 a^3}-\frac {55 i \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt {2} a^3}+\frac {55 i \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt {2} a^3}+\frac {55 i \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{16 \sqrt {2} a^3}-\frac {55 i \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{16 \sqrt {2} a^3} \]
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Time = 0.18 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5170, 91, 81, 52, 65, 338, 303, 1176, 631, 210, 1179, 642} \[ \int e^{\frac {5}{2} i \arctan (a x)} x^2 \, dx=-\frac {55 i \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt {2} a^3}+\frac {55 i \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt {2} a^3}+\frac {i (1-i a x)^{3/4} (1+i a x)^{9/4}}{3 a^3}+\frac {2 i (1+i a x)^{9/4}}{a^3 \sqrt [4]{1-i a x}}+\frac {11 i (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^3}+\frac {55 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{8 a^3}+\frac {55 i \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{16 \sqrt {2} a^3}-\frac {55 i \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{16 \sqrt {2} a^3} \]
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Rule 52
Rule 65
Rule 81
Rule 91
Rule 210
Rule 303
Rule 338
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 5170
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^2 (1+i a x)^{5/4}}{(1-i a x)^{5/4}} \, dx \\ & = \frac {2 i (1+i a x)^{9/4}}{a^3 \sqrt [4]{1-i a x}}-\frac {(2 i) \int \frac {(1+i a x)^{5/4} \left (\frac {5 i a}{2}-\frac {a^2 x}{2}\right )}{\sqrt [4]{1-i a x}} \, dx}{a^3} \\ & = \frac {2 i (1+i a x)^{9/4}}{a^3 \sqrt [4]{1-i a x}}+\frac {i (1-i a x)^{3/4} (1+i a x)^{9/4}}{3 a^3}+\frac {11 \int \frac {(1+i a x)^{5/4}}{\sqrt [4]{1-i a x}} \, dx}{2 a^2} \\ & = \frac {11 i (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^3}+\frac {2 i (1+i a x)^{9/4}}{a^3 \sqrt [4]{1-i a x}}+\frac {i (1-i a x)^{3/4} (1+i a x)^{9/4}}{3 a^3}+\frac {55 \int \frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}} \, dx}{8 a^2} \\ & = \frac {55 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{8 a^3}+\frac {11 i (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^3}+\frac {2 i (1+i a x)^{9/4}}{a^3 \sqrt [4]{1-i a x}}+\frac {i (1-i a x)^{3/4} (1+i a x)^{9/4}}{3 a^3}+\frac {55 \int \frac {1}{\sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx}{16 a^2} \\ & = \frac {55 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{8 a^3}+\frac {11 i (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^3}+\frac {2 i (1+i a x)^{9/4}}{a^3 \sqrt [4]{1-i a x}}+\frac {i (1-i a x)^{3/4} (1+i a x)^{9/4}}{3 a^3}+\frac {(55 i) \text {Subst}\left (\int \frac {x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-i a x}\right )}{4 a^3} \\ & = \frac {55 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{8 a^3}+\frac {11 i (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^3}+\frac {2 i (1+i a x)^{9/4}}{a^3 \sqrt [4]{1-i a x}}+\frac {i (1-i a x)^{3/4} (1+i a x)^{9/4}}{3 a^3}+\frac {(55 i) \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 a^3} \\ & = \frac {55 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{8 a^3}+\frac {11 i (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^3}+\frac {2 i (1+i a x)^{9/4}}{a^3 \sqrt [4]{1-i a x}}+\frac {i (1-i a x)^{3/4} (1+i a x)^{9/4}}{3 a^3}-\frac {(55 i) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 a^3}+\frac {(55 i) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 a^3} \\ & = \frac {55 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{8 a^3}+\frac {11 i (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^3}+\frac {2 i (1+i a x)^{9/4}}{a^3 \sqrt [4]{1-i a x}}+\frac {i (1-i a x)^{3/4} (1+i a x)^{9/4}}{3 a^3}+\frac {(55 i) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{16 a^3}+\frac {(55 i) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{16 a^3}+\frac {(55 i) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{16 \sqrt {2} a^3}+\frac {(55 i) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{16 \sqrt {2} a^3} \\ & = \frac {55 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{8 a^3}+\frac {11 i (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^3}+\frac {2 i (1+i a x)^{9/4}}{a^3 \sqrt [4]{1-i a x}}+\frac {i (1-i a x)^{3/4} (1+i a x)^{9/4}}{3 a^3}+\frac {55 i \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{16 \sqrt {2} a^3}-\frac {55 i \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{16 \sqrt {2} a^3}+\frac {(55 i) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt {2} a^3}-\frac {(55 i) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt {2} a^3} \\ & = \frac {55 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{8 a^3}+\frac {11 i (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^3}+\frac {2 i (1+i a x)^{9/4}}{a^3 \sqrt [4]{1-i a x}}+\frac {i (1-i a x)^{3/4} (1+i a x)^{9/4}}{3 a^3}-\frac {55 i \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt {2} a^3}+\frac {55 i \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt {2} a^3}+\frac {55 i \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{16 \sqrt {2} a^3}-\frac {55 i \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{16 \sqrt {2} a^3} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.23 \[ \int e^{\frac {5}{2} i \arctan (a x)} x^2 \, dx=\frac {-\sqrt [4]{1+i a x} (-i+a x)^2 (7 i+a x)+44 \sqrt [4]{2} (i+a x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {3}{4},\frac {7}{4},\frac {1}{2} (1-i a x)\right )}{3 a^3 \sqrt [4]{1-i a x}} \]
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\[\int {\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}^{\frac {5}{2}} x^{2}d x\]
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none
Time = 0.27 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.66 \[ \int e^{\frac {5}{2} i \arctan (a x)} x^2 \, dx=\frac {12 \, a^{3} \sqrt {\frac {3025 i}{64 \, a^{6}}} \log \left (\frac {8}{55} i \, a^{3} \sqrt {\frac {3025 i}{64 \, a^{6}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - 12 \, a^{3} \sqrt {\frac {3025 i}{64 \, a^{6}}} \log \left (-\frac {8}{55} i \, a^{3} \sqrt {\frac {3025 i}{64 \, a^{6}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) + 12 \, a^{3} \sqrt {-\frac {3025 i}{64 \, a^{6}}} \log \left (\frac {8}{55} i \, a^{3} \sqrt {-\frac {3025 i}{64 \, a^{6}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - 12 \, a^{3} \sqrt {-\frac {3025 i}{64 \, a^{6}}} \log \left (-\frac {8}{55} i \, a^{3} \sqrt {-\frac {3025 i}{64 \, a^{6}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - {\left (8 \, a^{3} x^{3} - 26 i \, a^{2} x^{2} - 61 \, a x - 287 i\right )} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{24 \, a^{3}} \]
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Timed out. \[ \int e^{\frac {5}{2} i \arctan (a x)} x^2 \, dx=\text {Timed out} \]
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\[ \int e^{\frac {5}{2} i \arctan (a x)} x^2 \, dx=\int { x^{2} \left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {5}{2}} \,d x } \]
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Exception generated. \[ \int e^{\frac {5}{2} i \arctan (a x)} x^2 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int e^{\frac {5}{2} i \arctan (a x)} x^2 \, dx=\int x^2\,{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{5/2} \,d x \]
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