Optimal. Leaf size=73 \[ -\frac {\sqrt {c-a^2 c x^2}}{a \sqrt {1-\frac {1}{a^2 x^2}} x^2}+\frac {\sqrt {c-a^2 c x^2} \log (x)}{\sqrt {1-\frac {1}{a^2 x^2}} x} \]
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Rubi [A]
time = 0.13, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6327, 6328, 45}
\begin {gather*} \frac {\log (x) \sqrt {c-a^2 c x^2}}{x \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {\sqrt {c-a^2 c x^2}}{a x^2 \sqrt {1-\frac {1}{a^2 x^2}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6327
Rule 6328
Rubi steps
\begin {align*} \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^2} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int \frac {e^{\coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}}}{x} \, dx}{\sqrt {1-\frac {1}{a^2 x^2}} x}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \frac {1+a x}{x^2} \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \left (\frac {1}{x^2}+\frac {a}{x}\right ) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x}\\ &=-\frac {\sqrt {c-a^2 c x^2}}{a \sqrt {1-\frac {1}{a^2 x^2}} x^2}+\frac {\sqrt {c-a^2 c x^2} \log (x)}{\sqrt {1-\frac {1}{a^2 x^2}} x}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 43, normalized size = 0.59 \begin {gather*} \frac {\sqrt {c-a^2 c x^2} (-1+a x \log (x))}{a \sqrt {1-\frac {1}{a^2 x^2}} x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 48, normalized size = 0.66
method | result | size |
default | \(\frac {\left (a \ln \left (x \right ) x -1\right ) \sqrt {-c \left (a^{2} x^{2}-1\right )}}{x \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(48\) |
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 [A]
time = 0.36, size = 22, normalized size = 0.30 \begin {gather*} \frac {\sqrt {-a^{2} c} {\left (a x \log \left (x\right ) - 1\right )}}{a x} \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 {- c \left (a x - 1\right ) \left (a x + 1\right )}}{x^{2} \sqrt {\frac {a x - 1}{a x + 1}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
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 {\sqrt {c-a^2\,c\,x^2}}{x^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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