Optimal. Leaf size=107 \[ -\frac {\sqrt {c-a^2 c x^2}}{x \sqrt {1-a^2 x^2}}-\frac {3 a \sqrt {c-a^2 c x^2} \log (x)}{\sqrt {1-a^2 x^2}}+\frac {4 a \sqrt {c-a^2 c x^2} \log (1+a x)}{\sqrt {1-a^2 x^2}} \]
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Rubi [A]
time = 0.14, 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 = {6288, 6285, 90}
\begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 6285
Rule 6288
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 = 0.02, size = 50, normalized size = 0.47 \begin {gather*} \frac {\sqrt {c-a^2 c x^2} \left (-\frac {1}{x}-3 a \log (x)+4 a \log (1+a x)\right )}{\sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 60, normalized size = 0.56
method | result | size |
default | \(\frac {\left (-4 \ln \left (a x +1\right ) a x +3 a \ln \left (x \right ) x +1\right ) \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}}{\left (a^{2} x^{2}-1\right ) x}\) | \(60\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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