Optimal. Leaf size=46 \[ -\frac {(a x+1)^2 \sqrt {c-\frac {c}{a^2 x^2}}}{2 a x^2 \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rubi [A] time = 0.24, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6197, 6193, 37} \[ -\frac {(a x+1)^2 \sqrt {c-\frac {c}{a^2 x^2}}}{2 a x^2 \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 6193
Rule 6197
Rubi steps
\begin {align*} \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2} \, dx &=\frac {\sqrt {c-\frac {c}{a^2 x^2}} \int \frac {e^{\coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}}}{x^2} \, dx}{\sqrt {1-\frac {1}{a^2 x^2}}}\\ &=\frac {\sqrt {c-\frac {c}{a^2 x^2}} \int \frac {1+a x}{x^3} \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}}}\\ &=-\frac {\sqrt {c-\frac {c}{a^2 x^2}} (1+a x)^2}{2 a \sqrt {1-\frac {1}{a^2 x^2}} x^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 47, normalized size = 1.02 \[ \frac {\left (-\frac {a}{x}-\frac {1}{2 x^2}\right ) \sqrt {c-\frac {c}{a^2 x^2}}}{a \sqrt {1-\frac {1}{a^2 x^2}}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.48, size = 21, normalized size = 0.46 \[ -\frac {\sqrt {a^{2} c} {\left (2 \, a x + 1\right )}}{2 \, a^{2} x^{2}} \]
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 \[ \int \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}}}{x^{2} \sqrt {\frac {a x - 1}{a x + 1}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 53, normalized size = 1.15 \[ -\frac {\left (2 a x +1\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{2 x \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}} \]
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 \[ \int \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}}}{x^{2} \sqrt {\frac {a x - 1}{a x + 1}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.40, size = 63, normalized size = 1.37 \[ \frac {\left (x\,\sqrt {c-\frac {c}{a^2\,x^2}}+\frac {\sqrt {c-\frac {c}{a^2\,x^2}}}{2\,a}\right )\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{\frac {x}{a}-x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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