Optimal. Leaf size=67 \[ \frac {i (1-i a x)^{3/2}}{5 a (1+i a x)^{5/2}}+\frac {i (1-i a x)^{3/2}}{15 a (1+i a x)^{3/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5181, 47, 37}
\begin {gather*} \frac {i (1-i a x)^{3/2}}{15 a (1+i a x)^{3/2}}+\frac {i (1-i a x)^{3/2}}{5 a (1+i a x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 5181
Rubi steps
\begin {align*} \int \frac {e^{-4 i \tan ^{-1}(a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx &=\int \frac {\sqrt {1-i a x}}{(1+i a x)^{7/2}} \, dx\\ &=\frac {i (1-i a x)^{3/2}}{5 a (1+i a x)^{5/2}}+\frac {1}{5} \int \frac {\sqrt {1-i a x}}{(1+i a x)^{5/2}} \, dx\\ &=\frac {i (1-i a x)^{3/2}}{5 a (1+i a x)^{5/2}}+\frac {i (1-i a x)^{3/2}}{15 a (1+i a x)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 47, normalized size = 0.70 \begin {gather*} \frac {(1-i a x)^{3/2} (-4 i+a x)}{15 a \sqrt {1+i a x} (-i+a x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 92, normalized size = 1.37
method | result | size |
default | \(\frac {\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{5 a \left (x -\frac {i}{a}\right )^{4}}-\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{15 \left (x -\frac {i}{a}\right )^{3}}}{a^{4}}\) | \(92\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 99 vs. \(2 (43) = 86\).
time = 0.26, size = 99, normalized size = 1.48 \begin {gather*} \frac {2 i \, \sqrt {a^{2} x^{2} + 1}}{-5 i \, a^{4} x^{3} - 15 \, a^{3} x^{2} + 15 i \, a^{2} x + 5 \, a} + \frac {i \, \sqrt {a^{2} x^{2} + 1}}{15 \, {\left (a^{3} x^{2} - 2 i \, a^{2} x - a\right )}} - \frac {i \, \sqrt {a^{2} x^{2} + 1}}{15 i \, a^{2} x + 15 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 7.37, size = 75, normalized size = 1.12 \begin {gather*} -\frac {a^{3} x^{3} - 3 i \, a^{2} x^{2} - 3 \, a x + {\left (a^{2} x^{2} - 3 i \, a x + 4\right )} \sqrt {a^{2} x^{2} + 1} + i}{15 \, {\left (a^{4} x^{3} - 3 i \, a^{3} x^{2} - 3 \, a^{2} x + i \, a\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 {\sqrt {a^{2} x^{2} + 1}}{\left (a x - i\right )^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 111 vs. \(2 (43) = 86\).
time = 0.45, size = 111, normalized size = 1.66 \begin {gather*} \frac {2 \, {\left (4 \, a^{4} - 25 \, a^{2} {\left (\sqrt {a^{2} + \frac {1}{x^{2}}} - \frac {1}{x}\right )}^{2} - 15 i \, a {\left (\sqrt {a^{2} + \frac {1}{x^{2}}} - \frac {1}{x}\right )}^{3} + 15 \, {\left (\sqrt {a^{2} + \frac {1}{x^{2}}} - \frac {1}{x}\right )}^{4} - 5 \, a^{3} {\left (-i \, \sqrt {a^{2} + \frac {1}{x^{2}}} + \frac {i}{x}\right )}\right )}}{15 \, {\left (-i \, a + \sqrt {a^{2} + \frac {1}{x^{2}}} - \frac {1}{x}\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.52, size = 40, normalized size = 0.60 \begin {gather*} \frac {\sqrt {a^2\,x^2+1}\,\left (a^2\,x^2-a\,x\,3{}\mathrm {i}+4\right )\,1{}\mathrm {i}}{15\,a\,{\left (1+a\,x\,1{}\mathrm {i}\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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