Optimal. Leaf size=107 \[ -\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}+\frac {3 a x \log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}-\frac {4 a x \sqrt {c-\frac {c}{a^2 x^2}} \log (1-a x)}{\sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.24, 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 = {6160, 6150, 88} \[ -\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}+\frac {3 a x \log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}-\frac {4 a x \sqrt {c-\frac {c}{a^2 x^2}} \log (1-a x)}{\sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
Rule 6160
Rubi steps
\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {e^{3 \tanh ^{-1}(a x)} \sqrt {1-a^2 x^2}}{x^2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {(1+a x)^2}{x^2 (1-a x)} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \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-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}+\frac {3 a \sqrt {c-\frac {c}{a^2 x^2}} x \log (x)}{\sqrt {1-a^2 x^2}}-\frac {4 a \sqrt {c-\frac {c}{a^2 x^2}} x \log (1-a x)}{\sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 49, normalized size = 0.46 \[ \frac {\sqrt {c-\frac {c}{a^2 x^2}} (3 a x \log (x)-4 a x \log (1-a x)-1)}{\sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left (a x + 1\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} x^{3} - 2 \, a x^{2} + x}, x\right ) \]
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 \[ \int \frac {{\left (a x + 1\right )}^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 63, normalized size = 0.59 \[ -\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \left (3 a \ln \relax (x ) x -4 \ln \left (a x -1\right ) x a -1\right ) \sqrt {-a^{2} x^{2}+1}}{a^{2} x^{2}-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.52, size = 144, normalized size = 1.35 \[ -\frac {1}{2} \, a^{3} {\left (-\frac {i \, \sqrt {c} \log \left (a x + 1\right )}{a^{3}} - \frac {i \, \sqrt {c} \log \left (a x - 1\right )}{a^{3}}\right )} - \frac {3}{2} \, a^{2} {\left (\frac {i \, \sqrt {c} \log \left (a x + 1\right )}{a^{2}} - \frac {i \, \sqrt {c} \log \left (a x - 1\right )}{a^{2}}\right )} - \frac {3}{2} \, a {\left (-\frac {i \, \sqrt {c} \log \left (a x + 1\right )}{a} - \frac {i \, \sqrt {c} \log \left (a x - 1\right )}{a} + \frac {2 i \, \sqrt {c} \log \relax (x)}{a}\right )} - \frac {1}{2} i \, \sqrt {c} \log \left (a x + 1\right ) + \frac {1}{2} i \, \sqrt {c} \log \left (a x - 1\right ) + \frac {i \, \sqrt {c}}{a x} \]
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 \[ \int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,{\left (a\,x+1\right )}^3}{x\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} \left (a x + 1\right )^{3}}{x \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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