Optimal. Leaf size=43 \[ -\frac {\sqrt {c-\frac {c}{a^2 x^2}} (1+a x)^2}{2 x \sqrt {1-a^2 x^2}} \]
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Rubi [A]
time = 0.15, 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 = {6295, 6285, 37}
\begin {gather*} -\frac {(a x+1)^2 \sqrt {c-\frac {c}{a^2 x^2}}}{2 x \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 6285
Rule 6295
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 = 0.03, size = 42, normalized size = 0.98 \begin {gather*} -\frac {\sqrt {c-\frac {c}{a^2 x^2}} (1+2 a x)}{2 x \sqrt {1-a^2 x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.04, size = 54, normalized size = 1.26
method | result | size |
gosper | \(-\frac {\left (2 a x +1\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{2 x \sqrt {-a^{2} x^{2}+1}}\) | \(43\) |
default | \(\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \sqrt {-a^{2} x^{2}+1}\, \left (2 a x +1\right )}{2 x \left (a^{2} x^{2}-1\right )}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.30, size = 20, normalized size = 0.47 \begin {gather*} \frac {i \, \sqrt {c}}{x} + \frac {i \, \sqrt {c}}{2 \, a x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 63, normalized size = 1.47 \begin {gather*} -\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 )}} \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 (-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 \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 [B]
time = 1.01, size = 35, normalized size = 0.81 \begin {gather*} -\frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a\,x+\frac {1}{2}\right )}{x\,\sqrt {1-a^2\,x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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