Optimal. Leaf size=100 \[ -\frac {2 i}{7 a^2 (a-i a x)^{7/4} \sqrt [4]{a+i a x}}-\frac {8 i}{21 a^3 (a-i a x)^{3/4} \sqrt [4]{a+i a x}}+\frac {16 i \sqrt [4]{a-i a x}}{21 a^4 \sqrt [4]{a+i a x}} \]
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Rubi [A]
time = 0.01, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {47, 37}
\begin {gather*} \frac {16 i \sqrt [4]{a-i a x}}{21 a^4 \sqrt [4]{a+i a x}}-\frac {8 i}{21 a^3 \sqrt [4]{a+i a x} (a-i a x)^{3/4}}-\frac {2 i}{7 a^2 \sqrt [4]{a+i a x} (a-i a x)^{7/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{(a-i a x)^{11/4} (a+i a x)^{5/4}} \, dx &=-\frac {2 i}{7 a^2 (a-i a x)^{7/4} \sqrt [4]{a+i a x}}+\frac {4 \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{5/4}} \, dx}{7 a}\\ &=-\frac {2 i}{7 a^2 (a-i a x)^{7/4} \sqrt [4]{a+i a x}}-\frac {8 i}{21 a^3 (a-i a x)^{3/4} \sqrt [4]{a+i a x}}+\frac {8 \int \frac {1}{(a-i a x)^{3/4} (a+i a x)^{5/4}} \, dx}{21 a^2}\\ &=-\frac {2 i}{7 a^2 (a-i a x)^{7/4} \sqrt [4]{a+i a x}}-\frac {8 i}{21 a^3 (a-i a x)^{3/4} \sqrt [4]{a+i a x}}+\frac {16 i \sqrt [4]{a-i a x}}{21 a^4 \sqrt [4]{a+i a x}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 52, normalized size = 0.52 \begin {gather*} \frac {2 (a+i a x)^{3/4} \left (i+12 x-8 i x^2\right )}{21 a^4 (a-i a x)^{3/4} \left (1+x^2\right )} \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 [A]
time = 0.15, size = 44, normalized size = 0.44
method | result | size |
risch | \(\frac {\frac {16}{21} x^{2}+\frac {8}{7} i x -\frac {2}{21}}{a^{3} \left (-a \left (i x -1\right )\right )^{\frac {3}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}} \left (x +i\right )}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 56, normalized size = 0.56 \begin {gather*} \frac {2 \, {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}} {\left (8 \, x^{2} + 12 i \, x - 1\right )}}{21 \, {\left (a^{5} x^{3} + i \, a^{5} x^{2} + a^{5} x + i \, a^{5}\right )}} \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 {5}{4}} \left (- i a \left (x + i\right )\right )^{\frac {11}{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 [B]
time = 0.76, size = 46, normalized size = 0.46 \begin {gather*} -\frac {{\left (-a\,\left (-1+x\,1{}\mathrm {i}\right )\right )}^{1/4}\,\left (8\,x^2+x\,12{}\mathrm {i}-1\right )\,2{}\mathrm {i}}{21\,a^4\,{\left (-1+x\,1{}\mathrm {i}\right )}^2\,{\left (a\,\left (1+x\,1{}\mathrm {i}\right )\right )}^{1/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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