Integrand size = 16, antiderivative size = 741 \[ \int e^{\frac {1}{4} i \arctan (a x)} x^2 \, dx=-\frac {11 i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{32 a^3}-\frac {i (1-i a x)^{7/8} (1+i a x)^{9/8}}{24 a^3}+\frac {x (1-i a x)^{7/8} (1+i a x)^{9/8}}{3 a^2}+\frac {11 i \sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )}{128 a^3}+\frac {11 i \sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )}{128 a^3}-\frac {11 i \sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )}{128 a^3}-\frac {11 i \sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )}{128 a^3}-\frac {11 i \sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{256 a^3}+\frac {11 i \sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{256 a^3}-\frac {11 i \sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{256 a^3}+\frac {11 i \sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{256 a^3} \]
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Time = 0.53 (sec) , antiderivative size = 741, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.812, Rules used = {5170, 92, 81, 52, 65, 338, 305, 1136, 1183, 648, 632, 210, 642} \[ \int e^{\frac {1}{4} i \arctan (a x)} x^2 \, dx=\frac {11 i \sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )}{128 a^3}+\frac {11 i \sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )}{128 a^3}-\frac {11 i \sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )}{128 a^3}-\frac {11 i \sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )}{128 a^3}-\frac {i (1-i a x)^{7/8} (1+i a x)^{9/8}}{24 a^3}-\frac {11 i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{32 a^3}-\frac {11 i \sqrt {2-\sqrt {2}} \log \left (\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{256 a^3}+\frac {11 i \sqrt {2-\sqrt {2}} \log \left (\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{256 a^3}-\frac {11 i \sqrt {2+\sqrt {2}} \log \left (\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{256 a^3}+\frac {11 i \sqrt {2+\sqrt {2}} \log \left (\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{256 a^3}+\frac {x (1-i a x)^{7/8} (1+i a x)^{9/8}}{3 a^2} \]
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Rule 52
Rule 65
Rule 81
Rule 92
Rule 210
Rule 305
Rule 338
Rule 632
Rule 642
Rule 648
Rule 1136
Rule 1183
Rule 5170
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^2 \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}} \, dx \\ & = \frac {x (1-i a x)^{7/8} (1+i a x)^{9/8}}{3 a^2}+\frac {\int \frac {\left (-1-\frac {i a x}{4}\right ) \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}} \, dx}{3 a^2} \\ & = -\frac {i (1-i a x)^{7/8} (1+i a x)^{9/8}}{24 a^3}+\frac {x (1-i a x)^{7/8} (1+i a x)^{9/8}}{3 a^2}-\frac {11 \int \frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}} \, dx}{32 a^2} \\ & = -\frac {11 i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{32 a^3}-\frac {i (1-i a x)^{7/8} (1+i a x)^{9/8}}{24 a^3}+\frac {x (1-i a x)^{7/8} (1+i a x)^{9/8}}{3 a^2}-\frac {11 \int \frac {1}{\sqrt [8]{1-i a x} (1+i a x)^{7/8}} \, dx}{128 a^2} \\ & = -\frac {11 i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{32 a^3}-\frac {i (1-i a x)^{7/8} (1+i a x)^{9/8}}{24 a^3}+\frac {x (1-i a x)^{7/8} (1+i a x)^{9/8}}{3 a^2}-\frac {(11 i) \text {Subst}\left (\int \frac {x^6}{\left (2-x^8\right )^{7/8}} \, dx,x,\sqrt [8]{1-i a x}\right )}{16 a^3} \\ & = -\frac {11 i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{32 a^3}-\frac {i (1-i a x)^{7/8} (1+i a x)^{9/8}}{24 a^3}+\frac {x (1-i a x)^{7/8} (1+i a x)^{9/8}}{3 a^2}-\frac {(11 i) \text {Subst}\left (\int \frac {x^6}{1+x^8} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{16 a^3} \\ & = -\frac {11 i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{32 a^3}-\frac {i (1-i a x)^{7/8} (1+i a x)^{9/8}}{24 a^3}+\frac {x (1-i a x)^{7/8} (1+i a x)^{9/8}}{3 a^2}-\frac {(11 i) \text {Subst}\left (\int \frac {x^4}{1-\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{32 \sqrt {2} a^3}+\frac {(11 i) \text {Subst}\left (\int \frac {x^4}{1+\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{32 \sqrt {2} a^3} \\ & = -\frac {11 i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{32 a^3}-\frac {i (1-i a x)^{7/8} (1+i a x)^{9/8}}{24 a^3}+\frac {x (1-i a x)^{7/8} (1+i a x)^{9/8}}{3 a^2}+\frac {(11 i) \text {Subst}\left (\int \frac {1-\sqrt {2} x^2}{1-\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{32 \sqrt {2} a^3}-\frac {(11 i) \text {Subst}\left (\int \frac {1+\sqrt {2} x^2}{1+\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{32 \sqrt {2} a^3} \\ & = -\frac {11 i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{32 a^3}-\frac {i (1-i a x)^{7/8} (1+i a x)^{9/8}}{24 a^3}+\frac {x (1-i a x)^{7/8} (1+i a x)^{9/8}}{3 a^2}-\frac {(11 i) \text {Subst}\left (\int \frac {\sqrt {2-\sqrt {2}}-\left (1-\sqrt {2}\right ) x}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 \sqrt {2 \left (2-\sqrt {2}\right )} a^3}-\frac {(11 i) \text {Subst}\left (\int \frac {\sqrt {2-\sqrt {2}}+\left (1-\sqrt {2}\right ) x}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 \sqrt {2 \left (2-\sqrt {2}\right )} a^3}+\frac {(11 i) \text {Subst}\left (\int \frac {\sqrt {2+\sqrt {2}}-\left (1+\sqrt {2}\right ) x}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 \sqrt {2 \left (2+\sqrt {2}\right )} a^3}+\frac {(11 i) \text {Subst}\left (\int \frac {\sqrt {2+\sqrt {2}}+\left (1+\sqrt {2}\right ) x}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 \sqrt {2 \left (2+\sqrt {2}\right )} a^3} \\ & = -\frac {11 i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{32 a^3}-\frac {i (1-i a x)^{7/8} (1+i a x)^{9/8}}{24 a^3}+\frac {x (1-i a x)^{7/8} (1+i a x)^{9/8}}{3 a^2}-\frac {\left (11 i \sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{128 a^3}-\frac {\left (11 i \sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{128 a^3}-\frac {\left (11 i \sqrt {2-\sqrt {2}}\right ) \text {Subst}\left (\int \frac {-\sqrt {2-\sqrt {2}}+2 x}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{256 a^3}+\frac {\left (11 i \sqrt {2-\sqrt {2}}\right ) \text {Subst}\left (\int \frac {\sqrt {2-\sqrt {2}}+2 x}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{256 a^3}-\frac {\left (11 i \sqrt {2+\sqrt {2}}\right ) \text {Subst}\left (\int \frac {-\sqrt {2+\sqrt {2}}+2 x}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{256 a^3}+\frac {\left (11 i \sqrt {2+\sqrt {2}}\right ) \text {Subst}\left (\int \frac {\sqrt {2+\sqrt {2}}+2 x}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{256 a^3}-\frac {\left (11 i \sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{128 a^3}-\frac {\left (11 i \sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{128 a^3} \\ & = -\frac {11 i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{32 a^3}-\frac {i (1-i a x)^{7/8} (1+i a x)^{9/8}}{24 a^3}+\frac {x (1-i a x)^{7/8} (1+i a x)^{9/8}}{3 a^2}-\frac {11 i \sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{256 a^3}+\frac {11 i \sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{256 a^3}-\frac {11 i \sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{256 a^3}+\frac {11 i \sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{256 a^3}+\frac {\left (11 i \sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )}\right ) \text {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,-\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^3}+\frac {\left (11 i \sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )}\right ) \text {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^3}+\frac {\left (11 i \sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )}\right ) \text {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,-\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^3}+\frac {\left (11 i \sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )}\right ) \text {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^3} \\ & = -\frac {11 i (1-i a x)^{7/8} \sqrt [8]{1+i a x}}{32 a^3}-\frac {i (1-i a x)^{7/8} (1+i a x)^{9/8}}{24 a^3}+\frac {x (1-i a x)^{7/8} (1+i a x)^{9/8}}{3 a^2}+\frac {11 i \sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )}{128 a^3}+\frac {11 i \sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )}{128 a^3}-\frac {11 i \sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )}{128 a^3}-\frac {11 i \sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )}{128 a^3}-\frac {11 i \sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{256 a^3}+\frac {11 i \sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{256 a^3}-\frac {11 i \sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{256 a^3}+\frac {11 i \sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{256 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 = 83, normalized size of antiderivative = 0.11 \[ \int e^{\frac {1}{4} i \arctan (a x)} x^2 \, dx=\frac {(1-i a x)^{7/8} \left (7 \sqrt [8]{1+i a x} \left (-i+9 a x+8 i a^2 x^2\right )-66 i \sqrt [8]{2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{8},\frac {7}{8},\frac {15}{8},\frac {1}{2} (1-i a x)\right )\right )}{168 a^3} \]
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\[\int {\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}^{\frac {1}{4}} x^{2}d x\]
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none
Time = 0.28 (sec) , antiderivative size = 435, normalized size of antiderivative = 0.59 \[ \int e^{\frac {1}{4} i \arctan (a x)} x^2 \, dx=\frac {96 i \, a^{3} \left (\frac {14641 i}{268435456 \, a^{12}}\right )^{\frac {1}{4}} \log \left (\frac {128}{11} \, a^{3} \left (\frac {14641 i}{268435456 \, a^{12}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - 96 \, a^{3} \left (\frac {14641 i}{268435456 \, a^{12}}\right )^{\frac {1}{4}} \log \left (\frac {128}{11} i \, a^{3} \left (\frac {14641 i}{268435456 \, a^{12}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + 96 \, a^{3} \left (\frac {14641 i}{268435456 \, a^{12}}\right )^{\frac {1}{4}} \log \left (-\frac {128}{11} i \, a^{3} \left (\frac {14641 i}{268435456 \, a^{12}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - 96 i \, a^{3} \left (\frac {14641 i}{268435456 \, a^{12}}\right )^{\frac {1}{4}} \log \left (-\frac {128}{11} \, a^{3} \left (\frac {14641 i}{268435456 \, a^{12}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + 96 i \, a^{3} \left (-\frac {14641 i}{268435456 \, a^{12}}\right )^{\frac {1}{4}} \log \left (\frac {128}{11} \, a^{3} \left (-\frac {14641 i}{268435456 \, a^{12}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - 96 \, a^{3} \left (-\frac {14641 i}{268435456 \, a^{12}}\right )^{\frac {1}{4}} \log \left (\frac {128}{11} i \, a^{3} \left (-\frac {14641 i}{268435456 \, a^{12}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + 96 \, a^{3} \left (-\frac {14641 i}{268435456 \, a^{12}}\right )^{\frac {1}{4}} \log \left (-\frac {128}{11} i \, a^{3} \left (-\frac {14641 i}{268435456 \, a^{12}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - 96 i \, a^{3} \left (-\frac {14641 i}{268435456 \, a^{12}}\right )^{\frac {1}{4}} \log \left (-\frac {128}{11} \, a^{3} \left (-\frac {14641 i}{268435456 \, a^{12}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + {\left (32 \, a^{3} x^{3} - 4 i \, a^{2} x^{2} - a x - 37 i\right )} \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}}{96 \, a^{3}} \]
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\[ \int e^{\frac {1}{4} i \arctan (a x)} x^2 \, dx=\int x^{2} \sqrt [4]{\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}}\, dx \]
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\[ \int e^{\frac {1}{4} i \arctan (a x)} x^2 \, dx=\int { x^{2} \left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {1}{4}} \,d x } \]
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Exception generated. \[ \int e^{\frac {1}{4} i \arctan (a x)} x^2 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int e^{\frac {1}{4} i \arctan (a x)} x^2 \, dx=\int x^2\,{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{1/4} \,d x \]
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