Optimal. Leaf size=81 \[ \frac{2 \left (x^2+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \tan ^{-1}(x),2\right )}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}}+\frac{2 x}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}} \]
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Rubi [A] time = 0.0148509, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {42, 199, 233, 231} \[ \frac{2 \left (x^2+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}}+\frac{2 x}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}} \]
Antiderivative was successfully verified.
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Rule 42
Rule 199
Rule 233
Rule 231
Rubi steps
\begin{align*} \int \frac{1}{(a-i a x)^{7/4} (a+i a x)^{7/4}} \, dx &=\frac{\left (a^2+a^2 x^2\right )^{3/4} \int \frac{1}{\left (a^2+a^2 x^2\right )^{7/4}} \, dx}{(a-i a x)^{3/4} (a+i a x)^{3/4}}\\ &=\frac{2 x}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}}+\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}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}}\\ &=\frac{2 x}{3 a^2 (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}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}}\\ &=\frac{2 x}{3 a^2 (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 )}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0219649, size = 70, normalized size = 0.86 \[ -\frac{i \sqrt [4]{2} (1+i x)^{3/4} \, _2F_1\left (-\frac{3}{4},\frac{7}{4};\frac{1}{4};\frac{1}{2}-\frac{i x}{2}\right )}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.041, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a-iax \right ) ^{-{\frac{7}{4}}} \left ( a+iax \right ) ^{-{\frac{7}{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{7}{4}}{\left (-i \, a x + a\right )}^{\frac{7}{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{3 \,{\left (a^{4} x^{2} + a^{4}\right )}{\rm integral}\left (\frac{{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{3 \,{\left (a^{4} x^{2} + a^{4}\right )}}, x\right ) + 2 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}} x}{3 \,{\left (a^{4} x^{2} + a^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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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