Integrand size = 25, antiderivative size = 71 \[ \int \frac {1}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \, dx=\frac {2 x}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {2 \sqrt [4]{1+x^2} E\left (\left .\frac {\arctan (x)}{2}\right |2\right )}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]
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Time = 0.01 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {42, 235, 233, 202} \[ \int \frac {1}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \, dx=\frac {2 x}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {2 \sqrt [4]{x^2+1} E\left (\left .\frac {\arctan (x)}{2}\right |2\right )}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]
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Rule 42
Rule 202
Rule 233
Rule 235
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{a^2+a^2 x^2} \int \frac {1}{\sqrt [4]{a^2+a^2 x^2}} \, dx}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \\ & = \frac {\sqrt [4]{1+x^2} \int \frac {1}{\sqrt [4]{1+x^2}} \, dx}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \\ & = \frac {2 x}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {\sqrt [4]{1+x^2} \int \frac {1}{\left (1+x^2\right )^{5/4}} \, dx}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \\ & = \frac {2 x}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {2 \sqrt [4]{1+x^2} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \, dx=\frac {2 i 2^{3/4} \sqrt [4]{1+i x} (a-i a x)^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {7}{4},\frac {1}{2}-\frac {i x}{2}\right )}{3 a \sqrt [4]{a+i a x}} \]
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\[\int \frac {1}{\left (-i a x +a \right )^{\frac {1}{4}} \left (i a x +a \right )^{\frac {1}{4}}}d x\]
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\[ \int \frac {1}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \, dx=\int { \frac {1}{{\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}} \,d x } \]
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Time = 2.39 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.44 \[ \int \frac {1}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \, dx=- \frac {i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{8}, \frac {5}{8}, 1 & \frac {1}{4}, \frac {1}{2}, \frac {3}{4} \\- \frac {1}{4}, \frac {1}{8}, \frac {1}{4}, \frac {5}{8}, \frac {3}{4} & 0 \end {matrix} \middle | {\frac {e^{- 3 i \pi }}{x^{2}}} \right )} e^{\frac {i \pi }{4}}}{4 \pi \sqrt {a} \Gamma \left (\frac {1}{4}\right )} + \frac {i {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, - \frac {3}{8}, 0, \frac {1}{8}, \frac {1}{2}, 1 & \\- \frac {3}{8}, \frac {1}{8} & - \frac {1}{2}, - \frac {1}{4}, 0, 0 \end {matrix} \middle | {\frac {e^{- i \pi }}{x^{2}}} \right )}}{4 \pi \sqrt {a} \Gamma \left (\frac {1}{4}\right )} \]
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\[ \int \frac {1}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \, dx=\int { \frac {1}{{\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}} \,d x } \]
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Exception generated. \[ \int \frac {1}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {1}{\sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \, dx=\int \frac {1}{{\left (a-a\,x\,1{}\mathrm {i}\right )}^{1/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{1/4}} \,d x \]
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