Optimal. Leaf size=173 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{2 a c (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{5 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{4 a c \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{4 a c \sqrt{c-\frac{c}{a^2 x^2}}} \]
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Rubi [A] time = 0.133429, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6197, 6193, 88} \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{2 a c (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{5 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{4 a c \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{4 a c \sqrt{c-\frac{c}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 6197
Rule 6193
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^{3/2}} \, dx &=\frac{\sqrt{1-\frac{1}{a^2 x^2}} \int \frac{e^{\coth ^{-1}(a x)}}{\left (1-\frac{1}{a^2 x^2}\right )^{3/2}} \, dx}{c \sqrt{c-\frac{c}{a^2 x^2}}}\\ &=\frac{\left (a^3 \sqrt{1-\frac{1}{a^2 x^2}}\right ) \int \frac{x^3}{(-1+a x)^2 (1+a x)} \, dx}{c \sqrt{c-\frac{c}{a^2 x^2}}}\\ &=\frac{\left (a^3 \sqrt{1-\frac{1}{a^2 x^2}}\right ) \int \left (\frac{1}{a^3}+\frac{1}{2 a^3 (-1+a x)^2}+\frac{5}{4 a^3 (-1+a x)}-\frac{1}{4 a^3 (1+a x)}\right ) \, dx}{c \sqrt{c-\frac{c}{a^2 x^2}}}\\ &=\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{2 a c \sqrt{c-\frac{c}{a^2 x^2}} (1-a x)}+\frac{5 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{4 a c \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}} \log (1+a x)}{4 a c \sqrt{c-\frac{c}{a^2 x^2}}}\\ \end{align*}
Mathematica [A] time = 0.0726939, size = 68, normalized size = 0.39 \[ \frac{\left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (4 a x+\frac{2}{1-a x}+5 \log (1-a x)-\log (a x+1)\right )}{4 a \left (c-\frac{c}{a^2 x^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.228, size = 102, normalized size = 0.6 \begin{align*} -{\frac{ \left ( ax-1 \right ) \left ( -4\,{a}^{2}{x}^{2}+ax\ln \left ( ax+1 \right ) -5\,\ln \left ( ax-1 \right ) xa+4\,ax-\ln \left ( ax+1 \right ) +5\,\ln \left ( ax-1 \right ) +2 \right ) \left ( ax+1 \right ) }{4\,{a}^{4}{x}^{3}}{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}} \right ) ^{-{\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{1}{{\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 = 1.42893, size = 155, normalized size = 0.9 \begin{align*} \frac{{\left (4 \, a^{2} x^{2} - 4 \, a x -{\left (a x - 1\right )} \log \left (a x + 1\right ) + 5 \,{\left (a x - 1\right )} \log \left (a x - 1\right ) - 2\right )} \sqrt{a^{2} c}}{4 \,{\left (a^{3} c^{2} x - a^{2} c^{2}\right )}} \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 \frac{1}{{\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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