Integrand size = 25, antiderivative size = 115 \[ \int \frac {1}{(a-i a x)^{13/4} (a+i a x)^{5/4}} \, dx=-\frac {2 i}{9 a^2 (a-i a x)^{9/4} \sqrt [4]{a+i a x}}-\frac {2 i}{9 a^3 (a-i a x)^{5/4} \sqrt [4]{a+i a x}}+\frac {2 \sqrt [4]{1+x^2} E\left (\left .\frac {\arctan (x)}{2}\right |2\right )}{3 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]
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Time = 0.02 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {53, 42, 203, 202} \[ \int \frac {1}{(a-i a x)^{13/4} (a+i a x)^{5/4}} \, dx=\frac {2 \sqrt [4]{x^2+1} E\left (\left .\frac {\arctan (x)}{2}\right |2\right )}{3 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {2 i}{9 a^3 (a-i a x)^{5/4} \sqrt [4]{a+i a x}}-\frac {2 i}{9 a^2 (a-i a x)^{9/4} \sqrt [4]{a+i a x}} \]
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Rule 42
Rule 53
Rule 202
Rule 203
Rubi steps \begin{align*} \text {integral}& = -\frac {2 i}{9 a^2 (a-i a x)^{9/4} \sqrt [4]{a+i a x}}+\frac {5 \int \frac {1}{(a-i a x)^{9/4} (a+i a x)^{5/4}} \, dx}{9 a} \\ & = -\frac {2 i}{9 a^2 (a-i a x)^{9/4} \sqrt [4]{a+i a x}}-\frac {2 i}{9 a^3 (a-i a x)^{5/4} \sqrt [4]{a+i a x}}+\frac {\int \frac {1}{(a-i a x)^{5/4} (a+i a x)^{5/4}} \, dx}{3 a^2} \\ & = -\frac {2 i}{9 a^2 (a-i a x)^{9/4} \sqrt [4]{a+i a x}}-\frac {2 i}{9 a^3 (a-i a x)^{5/4} \sqrt [4]{a+i a x}}+\frac {\sqrt [4]{a^2+a^2 x^2} \int \frac {1}{\left (a^2+a^2 x^2\right )^{5/4}} \, dx}{3 a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \\ & = -\frac {2 i}{9 a^2 (a-i a x)^{9/4} \sqrt [4]{a+i a x}}-\frac {2 i}{9 a^3 (a-i a x)^{5/4} \sqrt [4]{a+i a x}}+\frac {\sqrt [4]{1+x^2} \int \frac {1}{\left (1+x^2\right )^{5/4}} \, dx}{3 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \\ & = -\frac {2 i}{9 a^2 (a-i a x)^{9/4} \sqrt [4]{a+i a x}}-\frac {2 i}{9 a^3 (a-i a x)^{5/4} \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 )}{3 a^4 \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 = 0.61 \[ \int \frac {1}{(a-i a x)^{13/4} (a+i a x)^{5/4}} \, dx=-\frac {i 2^{3/4} \sqrt [4]{1+i x} \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},\frac {5}{4},-\frac {5}{4},\frac {1}{2}-\frac {i x}{2}\right )}{9 a^2 (a-i a x)^{9/4} \sqrt [4]{a+i a x}} \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 0.41 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.98
method | result | size |
risch | \(\frac {\frac {2}{3} x^{3}+\frac {4}{3} i x^{2}-\frac {4}{9} x +\frac {4}{9} i}{\left (x +i\right )^{2} a^{4} \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}-\frac {x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};-x^{2}\right ) \left (-a^{2} \left (i x -1\right ) \left (i x +1\right )\right )^{\frac {1}{4}}}{3 \left (a^{2}\right )^{\frac {1}{4}} a^{4} \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}\) | \(113\) |
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\[ \int \frac {1}{(a-i a x)^{13/4} (a+i a x)^{5/4}} \, dx=\int { \frac {1}{{\left (i \, a x + a\right )}^{\frac {5}{4}} {\left (-i \, a x + a\right )}^{\frac {13}{4}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a-i a x)^{13/4} (a+i a x)^{5/4}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {1}{(a-i a x)^{13/4} (a+i a x)^{5/4}} \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int \frac {1}{(a-i a x)^{13/4} (a+i a x)^{5/4}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {1}{(a-i a x)^{13/4} (a+i a x)^{5/4}} \, dx=\int \frac {1}{{\left (a-a\,x\,1{}\mathrm {i}\right )}^{13/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{5/4}} \,d x \]
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