Optimal. Leaf size=263 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{a c^2 (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{8 a c^2 (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{8 a c^2 (1-a x)^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{23 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{16 a c^2 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{7 \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{16 a c^2 \sqrt{c-\frac{c}{a^2 x^2}}} \]
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Rubi [A] time = 0.159471, antiderivative size = 263, 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} \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{a c^2 (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{8 a c^2 (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{8 a c^2 (1-a x)^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{23 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{16 a c^2 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{7 \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{16 a c^2 \sqrt{c-\frac{c}{a^2 x^2}}} \]
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 )^{5/2}} \, 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 )^{5/2}} \, dx}{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}\\ &=\frac{\left (a^5 \sqrt{1-\frac{1}{a^2 x^2}}\right ) \int \frac{x^5}{(-1+a x)^3 (1+a x)^2} \, dx}{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}\\ &=\frac{\left (a^5 \sqrt{1-\frac{1}{a^2 x^2}}\right ) \int \left (\frac{1}{a^5}+\frac{1}{4 a^5 (-1+a x)^3}+\frac{1}{a^5 (-1+a x)^2}+\frac{23}{16 a^5 (-1+a x)}+\frac{1}{8 a^5 (1+a x)^2}-\frac{7}{16 a^5 (1+a x)}\right ) \, dx}{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}\\ &=\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{8 a c^2 \sqrt{c-\frac{c}{a^2 x^2}} (1-a x)^2}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{a c^2 \sqrt{c-\frac{c}{a^2 x^2}} (1-a x)}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{8 a c^2 \sqrt{c-\frac{c}{a^2 x^2}} (1+a x)}+\frac{23 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{16 a c^2 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{7 \sqrt{1-\frac{1}{a^2 x^2}} \log (1+a x)}{16 a c^2 \sqrt{c-\frac{c}{a^2 x^2}}}\\ \end{align*}
Mathematica [A] time = 0.107094, size = 86, normalized size = 0.33 \[ \frac{\left (1-\frac{1}{a^2 x^2}\right )^{5/2} \left (16 a x+\frac{16}{1-a x}-\frac{2}{a x+1}-\frac{2}{(a x-1)^2}+23 \log (1-a x)-7 \log (a x+1)\right )}{16 a \left (c-\frac{c}{a^2 x^2}\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.234, size = 175, normalized size = 0.7 \begin{align*} -{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) \left ( -16\,{x}^{4}{a}^{4}+7\,{a}^{3}{x}^{3}\ln \left ( ax+1 \right ) -23\,\ln \left ( ax-1 \right ){x}^{3}{a}^{3}+16\,{x}^{3}{a}^{3}-7\,\ln \left ( ax+1 \right ){a}^{2}{x}^{2}+23\,\ln \left ( ax-1 \right ){a}^{2}{x}^{2}+34\,{a}^{2}{x}^{2}-7\,ax\ln \left ( ax+1 \right ) +23\,\ln \left ( ax-1 \right ) xa-18\,ax+7\,\ln \left ( ax+1 \right ) -23\,\ln \left ( ax-1 \right ) -12 \right ) }{16\,{a}^{6}{x}^{5}}{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}} \right ) ^{-{\frac{5}{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{5}{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 = 1.46809, size = 292, normalized size = 1.11 \begin{align*} \frac{{\left (16 \, a^{4} x^{4} - 16 \, a^{3} x^{3} - 34 \, a^{2} x^{2} + 18 \, a x - 7 \,{\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x + 1\right ) + 23 \,{\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x - 1\right ) + 12\right )} \sqrt{a^{2} c}}{16 \,{\left (a^{5} c^{3} x^{3} - a^{4} c^{3} x^{2} - a^{3} c^{3} x + a^{2} c^{3}\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{5}{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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