Integrand size = 12, antiderivative size = 268 \[ \int e^{\frac {1}{2} i \arctan (a x)} \, dx=\frac {i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}-\frac {i \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2} a}+\frac {i \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2} a}+\frac {i \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{2 \sqrt {2} a}-\frac {i \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{2 \sqrt {2} a} \]
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Time = 0.10 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5169, 52, 65, 338, 303, 1176, 631, 210, 1179, 642} \[ \int e^{\frac {1}{2} i \arctan (a x)} \, dx=-\frac {i \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2} a}+\frac {i \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2} a}+\frac {i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}+\frac {i \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{2 \sqrt {2} a}-\frac {i \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{2 \sqrt {2} a} \]
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Rule 52
Rule 65
Rule 210
Rule 303
Rule 338
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 5169
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}} \, dx \\ & = \frac {i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}+\frac {1}{2} \int \frac {1}{\sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx \\ & = \frac {i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}+\frac {(2 i) \text {Subst}\left (\int \frac {x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-i a x}\right )}{a} \\ & = \frac {i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}+\frac {(2 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 )}{a} \\ & = \frac {i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}-\frac {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 )}{a}+\frac {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 )}{a} \\ & = \frac {i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}+\frac {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 )}{2 a}+\frac {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 )}{2 a}+\frac {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 )}{2 \sqrt {2} a}+\frac {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 )}{2 \sqrt {2} a} \\ & = \frac {i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}+\frac {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 )}{2 \sqrt {2} a}-\frac {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 )}{2 \sqrt {2} a}+\frac {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 )}{\sqrt {2} a}-\frac {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 )}{\sqrt {2} a} \\ & = \frac {i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}-\frac {i \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2} a}+\frac {i \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2} a}+\frac {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 )}{2 \sqrt {2} a}-\frac {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 )}{2 \sqrt {2} a} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.15 \[ \int e^{\frac {1}{2} i \arctan (a x)} \, dx=-\frac {8 i e^{\frac {5}{2} i \arctan (a x)} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},2,\frac {9}{4},-e^{2 i \arctan (a x)}\right )}{5 a} \]
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\[\int \sqrt {\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}}d x\]
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none
Time = 0.26 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.78 \[ \int e^{\frac {1}{2} i \arctan (a x)} \, dx=\frac {a \sqrt {\frac {i}{a^{2}}} \log \left (i \, a \sqrt {\frac {i}{a^{2}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - a \sqrt {\frac {i}{a^{2}}} \log \left (-i \, a \sqrt {\frac {i}{a^{2}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) + a \sqrt {-\frac {i}{a^{2}}} \log \left (i \, a \sqrt {-\frac {i}{a^{2}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - a \sqrt {-\frac {i}{a^{2}}} \log \left (-i \, a \sqrt {-\frac {i}{a^{2}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) + 2 \, {\left (a x + i\right )} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{2 \, a} \]
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\[ \int e^{\frac {1}{2} i \arctan (a x)} \, dx=\int \sqrt {\frac {i a x + 1}{\sqrt {a^{2} x^{2} + 1}}}\, dx \]
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\[ \int e^{\frac {1}{2} i \arctan (a x)} \, dx=\int { \sqrt {\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}} \,d x } \]
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Exception generated. \[ \int e^{\frac {1}{2} i \arctan (a x)} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int e^{\frac {1}{2} i \arctan (a x)} \, dx=\int \sqrt {\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}} \,d x \]
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