Optimal. Leaf size=68 \[ \frac {a x^2 \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}+\frac {x \log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.14, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6160, 6150, 43} \[ \frac {a x^2 \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}+\frac {x \log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6150
Rule 6160
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a^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} \, 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} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \left (a+\frac {1}{x}\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {a \sqrt {c-\frac {c}{a^2 x^2}} x^2}{\sqrt {1-a^2 x^2}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x \log (x)}{\sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 37, normalized size = 0.54 \[ \frac {x \sqrt {c-\frac {c}{a^2 x^2}} (a x+\log (x))}{\sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.69, size = 320, normalized size = 4.71 \[ \left [\frac {{\left (a^{2} x^{2} - 1\right )} \sqrt {-c} \log \left (\frac {a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} + {\left (a x^{5} - a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right ) - 2 \, {\left (a^{2} x^{2} - a^{2} x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left (a^{3} x^{2} - a\right )}}, -\frac {{\left (a^{2} x^{2} - 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left (a x^{3} + a x\right )} \sqrt {c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} - {\left (a^{2} + 1\right )} c x^{2} + c}\right ) + {\left (a^{2} x^{2} - a^{2} x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} x^{2} - a}\right ] \]
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 \[ \int \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 52, normalized size = 0.76 \[ -\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \left (a x +\ln \relax (x )\right ) \sqrt {-a^{2} x^{2}+1}}{a^{2} x^{2}-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.39, size = 17, normalized size = 0.25 \[ -i \, \sqrt {c} x - \frac {i \, \sqrt {c} \log \relax (x)}{a} \]
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 \[ \int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a\,x+1\right )}{\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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