Optimal. Leaf size=147 \[ \frac{c x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}} \]
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Rubi [A] time = 0.124496, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6197, 6193, 75} \[ \frac{c x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 6197
Rule 6193
Rule 75
Rubi steps
\begin{align*} \int e^{-\coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^{3/2} \, dx &=\frac{\left (c \sqrt{c-\frac{c}{a^2 x^2}}\right ) \int e^{-\coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right )^{3/2} \, dx}{\sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\left (c \sqrt{c-\frac{c}{a^2 x^2}}\right ) \int \frac{(-1+a x)^2 (1+a x)}{x^3} \, dx}{a^3 \sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\left (c \sqrt{c-\frac{c}{a^2 x^2}}\right ) \int \left (a^3+\frac{1}{x^3}-\frac{a}{x^2}-\frac{a^2}{x}\right ) \, dx}{a^3 \sqrt{1-\frac{1}{a^2 x^2}}}\\ &=-\frac{c \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^3 \sqrt{1-\frac{1}{a^2 x^2}} x^2}+\frac{c \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 \sqrt{1-\frac{1}{a^2 x^2}} x}+\frac{c \sqrt{c-\frac{c}{a^2 x^2}} x}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c \sqrt{c-\frac{c}{a^2 x^2}} \log (x)}{a \sqrt{1-\frac{1}{a^2 x^2}}}\\ \end{align*}
Mathematica [A] time = 0.0442541, size = 65, normalized size = 0.44 \[ \frac{\left (c-\frac{c}{a^2 x^2}\right )^{3/2} \left (a^3 x-a^2 \log (x)-\frac{3 a^2}{2}+\frac{a}{x}-\frac{1}{2 x^2}\right )}{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.226, size = 80, normalized size = 0.5 \begin{align*} -{\frac{x \left ( -2\,{x}^{3}{a}^{3}+2\,{a}^{2}\ln \left ( x \right ){x}^{2}-2\,ax+1 \right ) }{ \left ( 2\,ax-2 \right ) \left ({a}^{2}{x}^{2}-1 \right ) } \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}} \right ) ^{{\frac{3}{2}}}\sqrt{{\frac{ax-1}{ax+1}}}} \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{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{3}{2}} \sqrt{\frac{a x - 1}{a x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99713, size = 103, normalized size = 0.7 \begin{align*} \frac{{\left (2 \, a^{3} c x^{3} - 2 \, a^{2} c x^{2} \log \left (x\right ) + 2 \, a c x - c\right )} \sqrt{a^{2} c}}{2 \, a^{4} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{3}{2}} \sqrt{\frac{a x - 1}{a x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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