Optimal. Leaf size=68 \[ \frac {a \log (x) \sqrt {c-a^2 c x^2}}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {c-a^2 c x^2}}{x \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.18, antiderivative size = 68, 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 = {6153, 6150, 43} \[ \frac {a \log (x) \sqrt {c-a^2 c x^2}}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {c-a^2 c x^2}}{x \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6150
Rule 6153
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^2} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int \frac {e^{\tanh ^{-1}(a x)} \sqrt {1-a^2 x^2}}{x^2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \frac {1+a x}{x^2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \left (\frac {1}{x^2}+\frac {a}{x}\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {\sqrt {c-a^2 c x^2}}{x \sqrt {1-a^2 x^2}}+\frac {a \sqrt {c-a^2 c x^2} \log (x)}{\sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 40, normalized size = 0.59 \[ \frac {\sqrt {c-a^2 c x^2} \left (a \log (x)-\frac {1}{x}\right )}{\sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.75, size = 264, normalized size = 3.88 \[ \left [\frac {{\left (a^{3} x^{3} - a x\right )} \sqrt {c} \log \left (\frac {a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} - \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} {\left (x^{4} - 1\right )} \sqrt {c} - c}{a^{2} x^{4} - x^{2}}\right ) - 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} {\left (x - 1\right )}}{2 \, {\left (a^{2} x^{3} - x\right )}}, \frac {{\left (a^{3} x^{3} - a x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} {\left (x^{2} + 1\right )} \sqrt {-c}}{a^{2} c x^{4} - {\left (a^{2} + 1\right )} c x^{2} + c}\right ) - \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} {\left (x - 1\right )}}{a^{2} x^{3} - x}\right ] \]
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 \[ \int \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x + 1\right )}}{\sqrt {-a^{2} x^{2} + 1} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 50, normalized size = 0.74 \[ \frac {\left (-a \ln \relax (x ) x +1\right ) \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}}{\left (a^{2} x^{2}-1\right ) x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 60, normalized size = 0.88 \[ -\frac {1}{2} \, {\left (\left (-1\right )^{-2 \, a^{2} c x^{2} + 2 \, c} \sqrt {c} \log \left (-2 \, a^{2} c + \frac {2 \, c}{x^{2}}\right ) + \sqrt {c} \log \left (x^{2} - \frac {1}{a^{2}}\right )\right )} a - \frac {\sqrt {c}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {c-a^2\,c\,x^2}\,\left (a\,x+1\right )}{x^2\,\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}{x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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