Optimal. Leaf size=81 \[ \frac{x^{m+1} \sqrt{c-\frac{c}{a^2 x^2}}}{m \sqrt{1-a^2 x^2}}-\frac{a x^{m+2} \sqrt{c-\frac{c}{a^2 x^2}}}{(m+1) \sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.234089, antiderivative size = 81, normalized size of antiderivative = 1., 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, 43} \[ \frac{x^{m+1} \sqrt{c-\frac{c}{a^2 x^2}}}{m \sqrt{1-a^2 x^2}}-\frac{a x^{m+2} \sqrt{c-\frac{c}{a^2 x^2}}}{(m+1) \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 6160
Rule 6150
Rule 43
Rubi steps
\begin{align*} \int e^{-\tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x^m \, dx &=\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int e^{-\tanh ^{-1}(a x)} x^{-1+m} \sqrt{1-a^2 x^2} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int x^{-1+m} (1-a x) \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \left (x^{-1+m}-a x^m\right ) \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} x^{1+m}}{m \sqrt{1-a^2 x^2}}-\frac{a \sqrt{c-\frac{c}{a^2 x^2}} x^{2+m}}{(1+m) \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0387454, size = 52, normalized size = 0.64 \[ \frac{x \sqrt{c-\frac{c}{a^2 x^2}} \left (\frac{x^m}{m}-\frac{a x^{m+1}}{m+1}\right )}{\sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.08, size = 69, normalized size = 0.9 \begin{align*}{\frac{{x}^{1+m} \left ( axm-m-1 \right ) }{ \left ( 1+m \right ) m \left ( ax-1 \right ) \left ( ax+1 \right ) }\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}}\sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.14975, size = 74, normalized size = 0.91 \begin{align*} \frac{{\left (i \, a \sqrt{c} m x + \sqrt{c}{\left (-i \, m - i\right )}\right )}{\left (a x + 1\right )}{\left (a x - 1\right )} x^{m}}{{\left (m^{2} + m\right )} a^{3} x^{2} -{\left (m^{2} + m\right )} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.20564, size = 151, normalized size = 1.86 \begin{align*} \frac{\sqrt{-a^{2} x^{2} + 1}{\left (a m x^{2} -{\left (m + 1\right )} x\right )} x^{m} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{{\left (a^{2} m^{2} + a^{2} m\right )} x^{2} - m^{2} - m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m} \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )} \sqrt{- c \left (-1 + \frac{1}{a x}\right ) \left (1 + \frac{1}{a x}\right )}}{a x + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{c - \frac{c}{a^{2} x^{2}}} x^{m}}{a x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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