Optimal. Leaf size=111 \[ \frac{(1-a x)^2}{a^2 x \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{2 (a x+1) (1-a x)}{a^2 x \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{2 \sqrt{a x+1} \sqrt{1-a x} \sin ^{-1}(a x)}{a^2 x \sqrt{c-\frac{c}{a^2 x^2}}} \]
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Rubi [A] time = 0.24115, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6159, 6129, 78, 50, 41, 216} \[ \frac{(1-a x)^2}{a^2 x \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{2 (a x+1) (1-a x)}{a^2 x \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{2 \sqrt{a x+1} \sqrt{1-a x} \sin ^{-1}(a x)}{a^2 x \sqrt{c-\frac{c}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 6159
Rule 6129
Rule 78
Rule 50
Rule 41
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{-2 \tanh ^{-1}(a x)}}{\sqrt{c-\frac{c}{a^2 x^2}}} \, dx &=\frac{\left (\sqrt{1-a x} \sqrt{1+a x}\right ) \int \frac{e^{-2 \tanh ^{-1}(a x)} x}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{\sqrt{c-\frac{c}{a^2 x^2}} x}\\ &=\frac{\left (\sqrt{1-a x} \sqrt{1+a x}\right ) \int \frac{x \sqrt{1-a x}}{(1+a x)^{3/2}} \, dx}{\sqrt{c-\frac{c}{a^2 x^2}} x}\\ &=\frac{(1-a x)^2}{a^2 \sqrt{c-\frac{c}{a^2 x^2}} x}+\frac{\left (2 \sqrt{1-a x} \sqrt{1+a x}\right ) \int \frac{\sqrt{1-a x}}{\sqrt{1+a x}} \, dx}{a \sqrt{c-\frac{c}{a^2 x^2}} x}\\ &=\frac{(1-a x)^2}{a^2 \sqrt{c-\frac{c}{a^2 x^2}} x}+\frac{2 (1-a x) (1+a x)}{a^2 \sqrt{c-\frac{c}{a^2 x^2}} x}+\frac{\left (2 \sqrt{1-a x} \sqrt{1+a x}\right ) \int \frac{1}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{a \sqrt{c-\frac{c}{a^2 x^2}} x}\\ &=\frac{(1-a x)^2}{a^2 \sqrt{c-\frac{c}{a^2 x^2}} x}+\frac{2 (1-a x) (1+a x)}{a^2 \sqrt{c-\frac{c}{a^2 x^2}} x}+\frac{\left (2 \sqrt{1-a x} \sqrt{1+a x}\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{a \sqrt{c-\frac{c}{a^2 x^2}} x}\\ &=\frac{(1-a x)^2}{a^2 \sqrt{c-\frac{c}{a^2 x^2}} x}+\frac{2 (1-a x) (1+a x)}{a^2 \sqrt{c-\frac{c}{a^2 x^2}} x}+\frac{2 \sqrt{1-a x} \sqrt{1+a x} \sin ^{-1}(a x)}{a^2 \sqrt{c-\frac{c}{a^2 x^2}} x}\\ \end{align*}
Mathematica [A] time = 0.0692871, size = 69, normalized size = 0.62 \[ \frac{-a^2 x^2+2 \sqrt{a^2 x^2-1} \log \left (\sqrt{a^2 x^2-1}+a x\right )-2 a x+3}{a^2 x \sqrt{c-\frac{c}{a^2 x^2}}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.145, size = 177, normalized size = 1.6 \begin{align*} -{\frac{1}{ \left ( ax+1 \right ) ax}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}} \left ( \sqrt{c}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}x{a}^{2}-2\,\ln \left ( x\sqrt{c}+\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}} \right ) xac+\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}a\sqrt{c}+2\,a\sqrt{{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}}}\sqrt{c}-2\,\ln \left ( x\sqrt{c}+\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}} \right ) c \right ){\frac{1}{\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}}}}{c}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{a^{2} x^{2} - 1}{{\left (a x + 1\right )}^{2} \sqrt{c - \frac{c}{a^{2} x^{2}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97899, size = 448, normalized size = 4.04 \begin{align*} \left [\frac{{\left (a x + 1\right )} \sqrt{c} \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) -{\left (a^{2} x^{2} + 3 \, a x\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x + a c}, -\frac{2 \,{\left (a x + 1\right )} \sqrt{-c} \arctan \left (\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) +{\left (a^{2} x^{2} + 3 \, a x\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x + a c}\right ] \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{a x}{a x \sqrt{c - \frac{c}{a^{2} x^{2}}} + \sqrt{c - \frac{c}{a^{2} x^{2}}}}\, dx - \int - \frac{1}{a x \sqrt{c - \frac{c}{a^{2} x^{2}}} + \sqrt{c - \frac{c}{a^{2} x^{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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