Optimal. Leaf size=112 \[ \frac{3}{2} a \sqrt{c-\frac{c}{a^2 x^2}}-\frac{(1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}{2 x}-\frac{3 a^2 x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{2 \sqrt{1-a x} \sqrt{a x+1}} \]
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Rubi [A] time = 0.390893, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {6159, 6129, 94, 92, 208} \[ \frac{3}{2} a \sqrt{c-\frac{c}{a^2 x^2}}-\frac{(1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}{2 x}-\frac{3 a^2 x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{2 \sqrt{1-a x} \sqrt{a x+1}} \]
Antiderivative was successfully verified.
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Rule 6159
Rule 6129
Rule 94
Rule 92
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-2 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^2} \, dx &=\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{e^{-2 \tanh ^{-1}(a x)} \sqrt{1-a x} \sqrt{1+a x}}{x^3} \, dx}{\sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{(1-a x)^{3/2}}{x^3 \sqrt{1+a x}} \, dx}{\sqrt{1-a x} \sqrt{1+a x}}\\ &=-\frac{\sqrt{c-\frac{c}{a^2 x^2}} (1-a x)}{2 x}-\frac{\left (3 a \sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{\sqrt{1-a x}}{x^2 \sqrt{1+a x}} \, dx}{2 \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{3}{2} a \sqrt{c-\frac{c}{a^2 x^2}}-\frac{\sqrt{c-\frac{c}{a^2 x^2}} (1-a x)}{2 x}+\frac{\left (3 a^2 \sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{1}{x \sqrt{1-a x} \sqrt{1+a x}} \, dx}{2 \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{3}{2} a \sqrt{c-\frac{c}{a^2 x^2}}-\frac{\sqrt{c-\frac{c}{a^2 x^2}} (1-a x)}{2 x}-\frac{\left (3 a^3 \sqrt{c-\frac{c}{a^2 x^2}} x\right ) \operatorname{Subst}\left (\int \frac{1}{a-a x^2} \, dx,x,\sqrt{1-a x} \sqrt{1+a x}\right )}{2 \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{3}{2} a \sqrt{c-\frac{c}{a^2 x^2}}-\frac{\sqrt{c-\frac{c}{a^2 x^2}} (1-a x)}{2 x}-\frac{3 a^2 \sqrt{c-\frac{c}{a^2 x^2}} x \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{1+a x}\right )}{2 \sqrt{1-a x} \sqrt{1+a x}}\\ \end{align*}
Mathematica [A] time = 0.0652253, size = 78, normalized size = 0.7 \[ \frac{\sqrt{c-\frac{c}{a^2 x^2}} \left ((4 a x-1) \sqrt{a^2 x^2-1}+3 a^2 x^2 \tan ^{-1}\left (\frac{1}{\sqrt{a^2 x^2-1}}\right )\right )}{2 x \sqrt{a^2 x^2-1}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.124, size = 348, normalized size = 3.1 \begin{align*} -{\frac{1}{2\,cx}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}} \left ( -4\,\sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}{x}^{3}{a}^{3}c+4\,\sqrt{-{\frac{c}{{a}^{2}}}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{3/2}x{a}^{3}+4\,{c}^{3/2}\ln \left ( x\sqrt{c}+\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}} \right ) \sqrt{-{\frac{c}{{a}^{2}}}}{x}^{2}a-4\,{c}^{3/2}\sqrt{-{\frac{c}{{a}^{2}}}}\ln \left ({\frac{1}{\sqrt{c}} \left ( \sqrt{{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}}}\sqrt{c}+cx \right ) } \right ){x}^{2}a+4\,\sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}}}{x}^{2}{a}^{2}c-3\,\sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}{x}^{2}{a}^{2}c-{a}^{2} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{{\frac{3}{2}}}\sqrt{-{\frac{c}{{a}^{2}}}}-3\,\ln \left ( 2\,{\frac{1}{{a}^{2}x} \left ( \sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}{a}^{2}-c \right ) } \right ){x}^{2}{c}^{2} \right ){\frac{1}{\sqrt{-{\frac{c}{{a}^{2}}}}}}{\frac{1}{\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{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{{\left (a^{2} x^{2} - 1\right )} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{{\left (a x + 1\right )}^{2} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.22041, size = 392, normalized size = 3.5 \begin{align*} \left [\frac{3 \, a \sqrt{-c} x \log \left (-\frac{a^{2} c x^{2} - 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) + 2 \,{\left (4 \, a x - 1\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, x}, \frac{3 \, a \sqrt{c} x \arctan \left (\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) +{\left (4 \, a x - 1\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, x}\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{\sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x^{3} + x^{2}}\, dx - \int \frac{a x \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x^{3} + x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.60445, size = 263, normalized size = 2.35 \begin{align*} -{\left (3 \, \sqrt{c} \arctan \left (-\frac{\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}}{\sqrt{c}}\right ) \mathrm{sgn}\left (x\right ) - \frac{{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{3} a c \mathrm{sgn}\left (x\right ) + 4 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{2} c^{\frac{3}{2}}{\left | a \right |} \mathrm{sgn}\left (x\right ) -{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )} a c^{2} \mathrm{sgn}\left (x\right ) + 4 \, c^{\frac{5}{2}}{\left | a \right |} \mathrm{sgn}\left (x\right )}{{\left ({\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{2} a}\right )}{\left | a \right |} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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