Optimal. Leaf size=82 \[ -\frac {2 i \sqrt [4]{a+i a x}}{3 a^2 (a-i a x)^{3/4}}+\frac {2 \left (1+x^2\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{3 a (a-i a x)^{3/4} (a+i a x)^{3/4}} \]
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Rubi [A]
time = 0.01, antiderivative size = 82, 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 = {53, 42, 239,
237} \begin {gather*} \frac {2 \left (x^2+1\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{3 a (a-i a x)^{3/4} (a+i a x)^{3/4}}-\frac {2 i \sqrt [4]{a+i a x}}{3 a^2 (a-i a x)^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 42
Rule 53
Rule 237
Rule 239
Rubi steps
\begin {align*} \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{3/4}} \, dx &=-\frac {2 i \sqrt [4]{a+i a x}}{3 a^2 (a-i a x)^{3/4}}+\frac {\int \frac {1}{(a-i a x)^{3/4} (a+i a x)^{3/4}} \, dx}{3 a}\\ &=-\frac {2 i \sqrt [4]{a+i a x}}{3 a^2 (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 (a-i a x)^{3/4} (a+i a x)^{3/4}}\\ &=-\frac {2 i \sqrt [4]{a+i a x}}{3 a^2 (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 (a-i a x)^{3/4} (a+i a x)^{3/4}}\\ &=-\frac {2 i \sqrt [4]{a+i a x}}{3 a^2 (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 (a-i a x)^{3/4} (a+i a x)^{3/4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.02, size = 70, normalized size = 0.85 \begin {gather*} -\frac {2 i \sqrt [4]{2} (1+i x)^{3/4} \, _2F_1\left (-\frac {3}{4},\frac {3}{4};\frac {1}{4};\frac {1}{2}-\frac {i x}{2}\right )}{3 a (a-i a x)^{3/4} (a+i a x)^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (-i a x +a \right )^{\frac {7}{4}} \left (i a x +a \right )^{\frac {3}{4}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.30, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (i a \left (x - i\right )\right )^{\frac {3}{4}} \left (- i a \left (x + i\right )\right )^{\frac {7}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{{\left (a-a\,x\,1{}\mathrm {i}\right )}^{7/4}\,{\left (a+a\,x\,1{}\mathrm {i}\right )}^{3/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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