Optimal. Leaf size=16 \[ \frac {1}{2} i \sin ^{-1}\left (1-\frac {8 i x}{3}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {633, 221}
\begin {gather*} \frac {1}{2} i \text {ArcSin}\left (1-\frac {8 i x}{3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 633
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3 i x+4 x^2}} \, dx &=\frac {1}{6} \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{9}}} \, dx,x,3 i+8 x\right )\\ &=\frac {1}{2} i \sin ^{-1}\left (1-\frac {8 i x}{3}\right )\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(51\) vs. \(2(16)=32\).
time = 0.03, size = 51, normalized size = 3.19 \begin {gather*} -\frac {\sqrt {x} \sqrt {3 i+4 x} \log \left (-2 \sqrt {x}+\sqrt {3 i+4 x}\right )}{\sqrt {x (3 i+4 x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.43, size = 10, normalized size = 0.62
method | result | size |
default | \(\frac {\arcsinh \left (i+\frac {8 x}{3}\right )}{2}\) | \(10\) |
trager | \(-\frac {\ln \left (-440 x -144-165 i-192 i \sqrt {4 x^{2}+3 i x}+384 i x +220 \sqrt {4 x^{2}+3 i x}\right )}{2}\) | \(44\) |
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. 21 vs. \(2 (8) = 16\).
time = 0.49, size = 21, normalized size = 1.31 \begin {gather*} \frac {1}{2} \, \log \left (8 \, x + 4 \, \sqrt {4 \, x^{2} + 3 i \, x} + 3 i\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 19 vs. \(2 (8) = 16\).
time = 1.35, size = 19, normalized size = 1.19 \begin {gather*} -\frac {1}{2} \, \log \left (-2 \, x + \sqrt {4 \, x^{2} + 3 i \, x} - \frac {3}{4} i\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}{\sqrt {4 x^{2} + 3 i x}}\, 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. 110 vs. \(2 (8) = 16\).
time = 1.58, size = 110, normalized size = 6.88 \begin {gather*} \frac {1}{32} \, \sqrt {8 \, x^{2} + 2 \, \sqrt {16 \, x^{2} + 9} x} {\left (8 \, x + 3 i\right )} {\left (\frac {3 i \, x}{4 \, x^{2} + \sqrt {16 \, x^{4} + 9 \, x^{2}}} + 1\right )} - \frac {9}{64} \, \log \left (2 \, \sqrt {8 \, x^{2} + 2 \, \sqrt {16 \, x^{2} + 9} x} {\left (\frac {3 i \, x}{4 \, x^{2} + \sqrt {16 \, x^{4} + 9 \, x^{2}}} + 1\right )} - 8 \, x - 3 i\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.28, size = 19, normalized size = 1.19 \begin {gather*} \frac {\ln \left (x+\frac {\sqrt {x\,\left (4\,x+3{}\mathrm {i}\right )}}{2}+\frac {3}{8}{}\mathrm {i}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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