Optimal. Leaf size=175 \[ \frac{\left (1-a^2 x^2\right )^{3/2}}{2 a^4 x^3 (1-a x) \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}+\frac{\left (1-a^2 x^2\right )^{3/2}}{a^3 x^2 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}+\frac{5 \left (1-a^2 x^2\right )^{3/2} \log (1-a x)}{4 a^4 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}-\frac{\left (1-a^2 x^2\right )^{3/2} \log (a x+1)}{4 a^4 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}} \]
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Rubi [A] time = 0.184058, antiderivative size = 175, 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 = {6160, 6150, 88} \[ \frac{\left (1-a^2 x^2\right )^{3/2}}{2 a^4 x^3 (1-a x) \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}+\frac{\left (1-a^2 x^2\right )^{3/2}}{a^3 x^2 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}+\frac{5 \left (1-a^2 x^2\right )^{3/2} \log (1-a x)}{4 a^4 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}-\frac{\left (1-a^2 x^2\right )^{3/2} \log (a x+1)}{4 a^4 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6160
Rule 6150
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^{3/2}} \, dx &=\frac{\left (1-a^2 x^2\right )^{3/2} \int \frac{e^{\tanh ^{-1}(a x)} x^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{\left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac{\left (1-a^2 x^2\right )^{3/2} \int \frac{x^3}{(1-a x)^2 (1+a x)} \, dx}{\left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac{\left (1-a^2 x^2\right )^{3/2} \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}{\left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac{\left (1-a^2 x^2\right )^{3/2}}{a^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^2}+\frac{\left (1-a^2 x^2\right )^{3/2}}{2 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3 (1-a x)}+\frac{5 \left (1-a^2 x^2\right )^{3/2} \log (1-a x)}{4 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3}-\frac{\left (1-a^2 x^2\right )^{3/2} \log (1+a x)}{4 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3}\\ \end{align*}
Mathematica [A] time = 0.0516773, size = 91, normalized size = 0.52 \[ -\frac{\sqrt{1-a^2 x^2} \left (a^2 x^2-1\right ) \left (\frac{x}{a^3}+\frac{1}{2 a^4 (1-a x)}+\frac{5 \log (1-a x)}{4 a^4}-\frac{\log (a x+1)}{4 a^4}\right )}{x^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.174, size = 94, normalized size = 0.5 \begin{align*}{\frac{ \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}}\sqrt{-{a}^{2}{x}^{2}+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{a x + 1}{\sqrt{-a^{2} x^{2} + 1}{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a^{2} x^{2} + 1} a^{4} x^{4} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{5} c^{2} x^{5} - a^{4} c^{2} x^{4} - 2 \, a^{3} c^{2} x^{3} + 2 \, a^{2} c^{2} x^{2} + a c^{2} x - c^{2}}, x\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{a x + 1}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (-1 + \frac{1}{a x}\right ) \left (1 + \frac{1}{a x}\right )\right )^{\frac{3}{2}}}\, dx \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{a x + 1}{\sqrt{-a^{2} x^{2} + 1}{\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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