Optimal. Leaf size=107 \[ -\frac {\sqrt {c-a^2 c x^2}}{x \sqrt {1-a^2 x^2}}-\frac {3 a \log (x) \sqrt {c-a^2 c x^2}}{\sqrt {1-a^2 x^2}}+\frac {4 a \sqrt {c-a^2 c x^2} \log (a x+1)}{\sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.20, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6153, 6150, 88} \[ -\frac {\sqrt {c-a^2 c x^2}}{x \sqrt {1-a^2 x^2}}-\frac {3 a \log (x) \sqrt {c-a^2 c x^2}}{\sqrt {1-a^2 x^2}}+\frac {4 a \sqrt {c-a^2 c x^2} \log (a x+1)}{\sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
Rule 6153
Rubi steps
\begin {align*} \int \frac {e^{-3 \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^{-3 \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)^2}{x^2 (1+a x)} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \left (\frac {1}{x^2}-\frac {3 a}{x}+\frac {4 a^2}{1+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 {3 a \sqrt {c-a^2 c x^2} \log (x)}{\sqrt {1-a^2 x^2}}+\frac {4 a \sqrt {c-a^2 c x^2} \log (1+a x)}{\sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 50, normalized size = 0.47 \[ \frac {\sqrt {c-a^2 c x^2} \left (-3 a \log (x)+4 a \log (a x+1)-\frac {1}{x}\right )}{\sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.30, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} {\left (a x - 1\right )}}{a^{2} x^{4} + 2 \, a x^{3} + x^{2}}, x\right ) \]
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 \[ \int \frac {\sqrt {-a^{2} c x^{2} + c} {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 60, normalized size = 0.56 \[ \frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \left (3 a \ln \relax (x ) x -4 a x \ln \left (a x +1\right )+1\right )}{\left (a^{2} x^{2}-1\right ) x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-a^{2} c x^{2} + c} {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {c-a^2\,c\,x^2}\,{\left (1-a^2\,x^2\right )}^{3/2}}{x^2\,{\left (a\,x+1\right )}^3} \,d x \]
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 \[ \int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}{x^{2} \left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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