Optimal. Leaf size=140 \[ -a^2 \sqrt{c-\frac{c}{a^2 x^2}}+\frac{a (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}{3 x}-\frac{(1-a x)^2 \sqrt{c-\frac{c}{a^2 x^2}}}{3 x^2}+\frac{a^3 x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{\sqrt{1-a x} \sqrt{a x+1}} \]
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Rubi [A] time = 0.393428, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6159, 6129, 96, 94, 92, 208} \[ -a^2 \sqrt{c-\frac{c}{a^2 x^2}}+\frac{a (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}{3 x}-\frac{(1-a x)^2 \sqrt{c-\frac{c}{a^2 x^2}}}{3 x^2}+\frac{a^3 x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{\sqrt{1-a x} \sqrt{a x+1}} \]
Antiderivative was successfully verified.
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Rule 6159
Rule 6129
Rule 96
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^3} \, 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^4} \, 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^4 \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}{3 x^2}-\frac{\left (2 a \sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{(1-a x)^{3/2}}{x^3 \sqrt{1+a x}} \, dx}{3 \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{a \sqrt{c-\frac{c}{a^2 x^2}} (1-a x)}{3 x}-\frac{\sqrt{c-\frac{c}{a^2 x^2}} (1-a x)^2}{3 x^2}+\frac{\left (a^2 \sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{\sqrt{1-a x}}{x^2 \sqrt{1+a x}} \, dx}{\sqrt{1-a x} \sqrt{1+a x}}\\ &=-a^2 \sqrt{c-\frac{c}{a^2 x^2}}+\frac{a \sqrt{c-\frac{c}{a^2 x^2}} (1-a x)}{3 x}-\frac{\sqrt{c-\frac{c}{a^2 x^2}} (1-a x)^2}{3 x^2}-\frac{\left (a^3 \sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{1}{x \sqrt{1-a x} \sqrt{1+a x}} \, dx}{\sqrt{1-a x} \sqrt{1+a x}}\\ &=-a^2 \sqrt{c-\frac{c}{a^2 x^2}}+\frac{a \sqrt{c-\frac{c}{a^2 x^2}} (1-a x)}{3 x}-\frac{\sqrt{c-\frac{c}{a^2 x^2}} (1-a x)^2}{3 x^2}+\frac{\left (a^4 \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 )}{\sqrt{1-a x} \sqrt{1+a x}}\\ &=-a^2 \sqrt{c-\frac{c}{a^2 x^2}}+\frac{a \sqrt{c-\frac{c}{a^2 x^2}} (1-a x)}{3 x}-\frac{\sqrt{c-\frac{c}{a^2 x^2}} (1-a x)^2}{3 x^2}+\frac{a^3 \sqrt{c-\frac{c}{a^2 x^2}} x \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{1+a x}\right )}{\sqrt{1-a x} \sqrt{1+a x}}\\ \end{align*}
Mathematica [A] time = 0.0730487, size = 86, normalized size = 0.61 \[ -\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left (\sqrt{a^2 x^2-1} \left (5 a^2 x^2-3 a x+1\right )+3 a^3 x^3 \tan ^{-1}\left (\frac{1}{\sqrt{a^2 x^2-1}}\right )\right )}{3 x^2 \sqrt{a^2 x^2-1}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.129, size = 378, normalized size = 2.7 \begin{align*}{\frac{a}{3\,c{x}^{2}}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}} \left ( -6\,\sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}{x}^{4}{a}^{3}c+6\,\sqrt{-{\frac{c}{{a}^{2}}}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{3/2}{x}^{2}{a}^{3}+6\,{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}^{3}a-6\,{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}^{3}a+6\,\sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}}}{x}^{3}{a}^{2}c-3\,\sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}{x}^{3}{a}^{2}c-3\,\sqrt{-{\frac{c}{{a}^{2}}}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{3/2}x{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}^{3}{c}^{2}+a \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{{\frac{3}{2}}}\sqrt{-{\frac{c}{{a}^{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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.35167, size = 441, normalized size = 3.15 \begin{align*} \left [\frac{3 \, a^{2} \sqrt{-c} x^{2} \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 (5 \, a^{2} x^{2} - 3 \, a x + 1\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{6 \, x^{2}}, -\frac{3 \, a^{2} \sqrt{c} x^{2} \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 (5 \, a^{2} x^{2} - 3 \, a x + 1\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{3 \, x^{2}}\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^{4} + x^{3}}\, dx - \int \frac{a x \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x^{4} + x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.98771, size = 312, normalized size = 2.23 \begin{align*} \frac{2}{3} \,{\left (3 \, a \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{3 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{5} a c \mathrm{sgn}\left (x\right ) + 3 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{4} c^{\frac{3}{2}}{\left | a \right |} \mathrm{sgn}\left (x\right ) + 12 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{2} c^{\frac{5}{2}}{\left | a \right |} \mathrm{sgn}\left (x\right ) - 3 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )} a c^{3} \mathrm{sgn}\left (x\right ) + 5 \, c^{\frac{7}{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 )}^{3}}\right )}{\left | a \right |} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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