Optimal. Leaf size=39 \[ -2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {875, 208} \[ -2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right ) \]
Antiderivative was successfully verified.
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Rule 208
Rule 875
Rubi steps
\begin {align*} \int \frac {\sqrt {c-a c x}}{x \sqrt {1-a^2 x^2}} \, dx &=\left (2 a^2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{-a^2 c+a^2 c^2 x^2} \, dx,x,\frac {\sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right )\\ &=-2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 67, normalized size = 1.72 \[ -\frac {2 \sqrt {c} \sqrt {\frac {a x}{c}+\frac {1}{c}} \sqrt {c-a c x} \tanh ^{-1}\left (\sqrt {c} \sqrt {\frac {a x}{c}+\frac {1}{c}}\right )}{\sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 110, normalized size = 2.82 \[ \left [\sqrt {c} \log \left (-\frac {a^{2} c x^{2} + a c x + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{a x^{2} - x}\right ), -2 \, \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right )\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 57, normalized size = 1.46 \[ -\frac {2 \, c^{3} {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c}}{\sqrt {-c}}\right )}{\sqrt {-c} c} - \frac {\arctan \left (\frac {\sqrt {a c x + c}}{\sqrt {-c}}\right )}{\sqrt {-c} c}\right )}}{{\left | c \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 58, normalized size = 1.49 \[ \frac {2 \sqrt {-\left (a x -1\right ) c}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {c}\, \arctanh \left (\frac {\sqrt {\left (a x +1\right ) c}}{\sqrt {c}}\right )}{\left (a x -1\right ) \sqrt {\left (a x +1\right ) c}} \]
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 \[ \int \frac {\sqrt {-a c x + c}}{\sqrt {-a^{2} x^{2} + 1} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\sqrt {c-a\,c\,x}}{x\,\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {\sqrt {- c \left (a x - 1\right )}}{x \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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