Optimal. Leaf size=168 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{3 \sqrt{1-\frac{1}{a^2 x^2}}}{a c (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{2 a c (a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{3 \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{a c \sqrt{c-\frac{c}{a^2 x^2}}} \]
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Rubi [A] time = 0.140652, antiderivative size = 168, 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, 43} \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{3 \sqrt{1-\frac{1}{a^2 x^2}}}{a c (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{2 a c (a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{3 \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{a c \sqrt{c-\frac{c}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 6197
Rule 6193
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{-3 \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^{-3 \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)^3} \, 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}{a^3 (1+a x)^3}+\frac{3}{a^3 (1+a x)^2}-\frac{3}{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)^2}-\frac{3 \sqrt{1-\frac{1}{a^2 x^2}}}{a c \sqrt{c-\frac{c}{a^2 x^2}} (1+a x)}-\frac{3 \sqrt{1-\frac{1}{a^2 x^2}} \log (1+a x)}{a c \sqrt{c-\frac{c}{a^2 x^2}}}\\ \end{align*}
Mathematica [A] time = 0.0731578, size = 63, normalized size = 0.38 \[ \frac{\left (1-\frac{1}{a^2 x^2}\right )^{3/2} \left (2 a x+\frac{-6 a x-5}{(a x+1)^2}-6 \log (a x+1)\right )}{2 a \left (c-\frac{c}{a^2 x^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.232, size = 102, normalized size = 0.6 \begin{align*} -{\frac{ \left ( ax+1 \right ) \left ( -2\,{x}^{3}{a}^{3}+6\,\ln \left ( ax+1 \right ){a}^{2}{x}^{2}-4\,{a}^{2}{x}^{2}+12\,ax\ln \left ( ax+1 \right ) +4\,ax+6\,\ln \left ( ax+1 \right ) +5 \right ) }{2\,{a}^{4}{x}^{3}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}} \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{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75579, size = 176, normalized size = 1.05 \begin{align*} \frac{{\left (2 \, a^{3} x^{3} + 4 \, a^{2} x^{2} - 4 \, a x - 6 \,{\left (a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (a x + 1\right ) - 5\right )} \sqrt{a^{2} c}}{2 \,{\left (a^{4} c^{2} x^{2} + 2 \, 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{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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