Optimal. Leaf size=43 \[ \frac{2 \left (x^2+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \tan ^{-1}(x),2\right )}{(a-i a x)^{3/4} (a+i a x)^{3/4}} \]
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Rubi [A] time = 0.0081492, antiderivative size = 43, 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, 233, 231} \[ \frac{2 \left (x^2+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{(a-i a x)^{3/4} (a+i a x)^{3/4}} \]
Antiderivative was successfully verified.
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Rule 42
Rule 233
Rule 231
Rubi steps
\begin{align*} \int \frac{1}{(a-i a x)^{3/4} (a+i a x)^{3/4}} \, dx &=\frac{\left (a^2+a^2 x^2\right )^{3/4} \int \frac{1}{\left (a^2+a^2 x^2\right )^{3/4}} \, dx}{(a-i a x)^{3/4} (a+i a x)^{3/4}}\\ &=\frac{\left (1+x^2\right )^{3/4} \int \frac{1}{\left (1+x^2\right )^{3/4}} \, dx}{(a-i a x)^{3/4} (a+i a x)^{3/4}}\\ &=\frac{2 \left (1+x^2\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{(a-i a x)^{3/4} (a+i a x)^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0198353, size = 68, normalized size = 1.58 \[ \frac{2 i \sqrt [4]{2} (1+i x)^{3/4} \sqrt [4]{a-i a x} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{1}{2}-\frac{i x}{2}\right )}{a (a+i a x)^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.039, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a-iax \right ) ^{-{\frac{3}{4}}} \left ( a+iax \right ) ^{-{\frac{3}{4}}}}\, dx \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{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{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*}{\rm integral}\left (\frac{{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{a^{2} x^{2} + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.2738, size = 100, normalized size = 2.33 \begin{align*} - \frac{i{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{8}, \frac{7}{8}, 1 & \frac{1}{2}, \frac{3}{4}, \frac{5}{4} \\\frac{1}{4}, \frac{3}{8}, \frac{3}{4}, \frac{7}{8}, \frac{5}{4} & 0 \end{matrix} \middle |{\frac{e^{- 3 i \pi }}{x^{2}}} \right )} e^{\frac{3 i \pi }{4}}}{4 \pi a^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right )} + \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{8}, 0, \frac{3}{8}, \frac{1}{2}, 1 & \\- \frac{1}{8}, \frac{3}{8} & - \frac{1}{2}, 0, \frac{1}{4}, 0 \end{matrix} \middle |{\frac{e^{- i \pi }}{x^{2}}} \right )}}{4 \pi a^{\frac{3}{2}} \Gamma \left (\frac{3}{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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