Optimal. Leaf size=46 \[ \frac{2 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]
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Rubi [A] time = 0.0085974, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {42, 197, 196} \[ \frac{2 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]
Antiderivative was successfully verified.
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Rule 42
Rule 197
Rule 196
Rubi steps
\begin{align*} \int \frac{1}{(a-i a x)^{5/4} (a+i a x)^{5/4}} \, dx &=\frac{\sqrt [4]{a^2+a^2 x^2} \int \frac{1}{\left (a^2+a^2 x^2\right )^{5/4}} \, dx}{\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}{a^2 \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 )}{a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ \end{align*}
Mathematica [C] time = 0.0229696, size = 68, normalized size = 1.48 \[ -\frac{i 2^{3/4} \sqrt [4]{1+i x} \, _2F_1\left (-\frac{1}{4},\frac{5}{4};\frac{3}{4};\frac{1}{2}-\frac{i x}{2}\right )}{a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.038, size = 91, normalized size = 2. \begin{align*} 2\,{\frac{x}{{a}^{2}\sqrt [4]{-a \left ( -1+ix \right ) }\sqrt [4]{a \left ( 1+ix \right ) }}}-{\frac{x}{{a}^{2}}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,-{x}^{2})}\sqrt [4]{-{a}^{2} \left ( -1+ix \right ) \left ( 1+ix \right ) }{\frac{1}{\sqrt [4]{{a}^{2}}}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{5}{4}}{\left (-i \, a x + a\right )}^{\frac{5}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}} x +{\left (a^{4} x^{2} + a^{4}\right )}{\rm integral}\left (-\frac{{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{a^{4} x^{2} + a^{4}}, x\right )}{a^{4} x^{2} + a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 54.0124, size = 97, normalized size = 2.11 \begin{align*} - \frac{i{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{5}{8}, \frac{9}{8}, 1 & \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \\\frac{5}{8}, \frac{3}{4}, \frac{9}{8}, \frac{5}{4}, \frac{7}{4} & 0 \end{matrix} \middle |{\frac{e^{- 3 i \pi }}{x^{2}}} \right )} e^{- \frac{3 i \pi }{4}}}{4 \pi a^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right )} + \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, 0, \frac{1}{8}, \frac{1}{2}, \frac{5}{8}, 1 & \\\frac{1}{8}, \frac{5}{8} & - \frac{1}{2}, 0, \frac{3}{4}, 0 \end{matrix} \middle |{\frac{e^{- i \pi }}{x^{2}}} \right )}}{4 \pi a^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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