Optimal. Leaf size=81 \[ \frac {2 \left (x^2+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \tan ^{-1}(x),2\right )}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}}+\frac {2 x}{3 a^2 (a-i a x)^{3/4} (a+i a x)^{3/4}} \]
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Rubi [A] time = 0.01, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, 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 231
Rule 233
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*}
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Mathematica [C] time = 0.03, 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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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ \frac {3 \, {\left (a^{4} x^{2} + a^{4}\right )} {\rm integral}\left (\frac {{\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{3 \, {\left (a^{4} x^{2} + a^{4}\right )}}, x\right ) + 2 \, {\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}} x}{3 \, {\left (a^{4} x^{2} + a^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (i \, a x + a\right )}^{\frac {7}{4}} {\left (-i \, a x + a\right )}^{\frac {7}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-i a x +a \right )^{\frac {7}{4}} \left (i a x +a \right )^{\frac {7}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (i \, a x + a\right )}^{\frac {7}{4}} {\left (-i \, a x + a\right )}^{\frac {7}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a-a\,x\,1{}\mathrm {i}\right )}^{7/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{7/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 36.70, size = 95, normalized size = 1.17 \[ - \frac {i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {7}{8}, \frac {11}{8}, 1 & \frac {1}{2}, \frac {7}{4}, \frac {9}{4} \\\frac {7}{8}, \frac {5}{4}, \frac {11}{8}, \frac {7}{4}, \frac {9}{4} & 0 \end {matrix} \middle | {\frac {e^{- 3 i \pi }}{x^{2}}} \right )} e^{- \frac {i \pi }{4}}}{4 \pi a^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {i {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, 0, \frac {3}{8}, \frac {1}{2}, \frac {7}{8}, 1 & \\\frac {3}{8}, \frac {7}{8} & - \frac {1}{2}, 0, \frac {5}{4}, 0 \end {matrix} \middle | {\frac {e^{- i \pi }}{x^{2}}} \right )}}{4 \pi a^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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