Optimal. Leaf size=20 \[ \frac {\log \left (a+\sqrt {-a} x\right )}{\sqrt {-a}} \]
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Rubi [A]
time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {31}
\begin {gather*} \frac {\log \left (\sqrt {-a} x+a\right )}{\sqrt {-a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rubi steps
\begin {align*} \int \frac {1}{a+\sqrt {-a} x} \, dx &=\frac {\log \left (a+\sqrt {-a} x\right )}{\sqrt {-a}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 20, normalized size = 1.00 \begin {gather*} \frac {\log \left (a+\sqrt {-a} x\right )}{\sqrt {-a}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.64, size = 16, normalized size = 0.80 \begin {gather*} \frac {\text {Log}\left [a+x \sqrt {-a}\right ]}{\sqrt {-a}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 17, normalized size = 0.85
method | result | size |
default | \(\frac {\ln \left (a +x \sqrt {-a}\right )}{\sqrt {-a}}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 16, normalized size = 0.80 \begin {gather*} \frac {\log \left (\sqrt {-a} x + a\right )}{\sqrt {-a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 20, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {-a} \log \left (x - \sqrt {-a}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 17, normalized size = 0.85 \begin {gather*} \frac {\log {\left (a + x \sqrt {- a} \right )}}{\sqrt {- a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 21, normalized size = 1.05 \begin {gather*} -\frac {\sqrt {-a} \ln \left |x-\sqrt {-a}\right |}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 16, normalized size = 0.80 \begin {gather*} \frac {\ln \left (x-\sqrt {-a}\right )}{\sqrt {-a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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