Optimal. Leaf size=43 \[ -\frac {(a x+1)^2 \sqrt {c-\frac {c}{a^2 x^2}}}{2 x \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.21, antiderivative size = 43, 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 = {6160, 6150, 37} \[ -\frac {(a x+1)^2 \sqrt {c-\frac {c}{a^2 x^2}}}{2 x \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 6150
Rule 6160
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2} \, dx &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {e^{\tanh ^{-1}(a x)} \sqrt {1-a^2 x^2}}{x^3} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1+a x}{x^3} \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {\sqrt {c-\frac {c}{a^2 x^2}} (1+a x)^2}{2 x \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 42, normalized size = 0.98 \[ -\frac {(2 a x+1) \sqrt {c-\frac {c}{a^2 x^2}}}{2 x \sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.56, size = 63, normalized size = 1.47 \[ -\frac {\sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, a + 1\right )} x^{2} - 2 \, a x - 1\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left (a^{2} x^{3} - x\right )}} \]
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 \[ \int \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{\sqrt {-a^{2} x^{2} + 1} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 43, normalized size = 1.00 \[ -\frac {\left (2 a x +1\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{2 x \sqrt {-a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.38, size = 20, normalized size = 0.47 \[ \frac {i \, \sqrt {c}}{x} + \frac {i \, \sqrt {c}}{2 \, a x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.01, size = 35, normalized size = 0.81 \[ -\frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a\,x+\frac {1}{2}\right )}{x\,\sqrt {1-a^2\,x^2}} \]
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 \[ \int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} \left (a x + 1\right )}{x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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