Optimal. Leaf size=146 \[ -\frac{a^3 x^4 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}{\left (1-a^2 x^2\right )^{3/2}}-\frac{a x^2 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}{\left (1-a^2 x^2\right )^{3/2}}-\frac{x \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}{2 \left (1-a^2 x^2\right )^{3/2}}-\frac{a^2 x^3 \log (x) \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}{\left (1-a^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.167457, antiderivative size = 146, 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, 75} \[ -\frac{a^3 x^4 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}{\left (1-a^2 x^2\right )^{3/2}}-\frac{a x^2 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}{\left (1-a^2 x^2\right )^{3/2}}-\frac{x \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}{2 \left (1-a^2 x^2\right )^{3/2}}-\frac{a^2 x^3 \log (x) \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}{\left (1-a^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6160
Rule 6150
Rule 75
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^{3/2} \, dx &=\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac{e^{\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{3/2}}{x^3} \, dx}{\left (1-a^2 x^2\right )^{3/2}}\\ &=\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac{(1-a x) (1+a x)^2}{x^3} \, dx}{\left (1-a^2 x^2\right )^{3/2}}\\ &=\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \left (-a^3+\frac{1}{x^3}+\frac{a}{x^2}-\frac{a^2}{x}\right ) \, dx}{\left (1-a^2 x^2\right )^{3/2}}\\ &=-\frac{\left (c-\frac{c}{a^2 x^2}\right )^{3/2} x}{2 \left (1-a^2 x^2\right )^{3/2}}-\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^2}{\left (1-a^2 x^2\right )^{3/2}}-\frac{a^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^4}{\left (1-a^2 x^2\right )^{3/2}}-\frac{a^2 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} x^3 \log (x)}{\left (1-a^2 x^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.034949, size = 72, normalized size = 0.49 \[ \frac{c \sqrt{c-\frac{c}{a^2 x^2}} \left (2 a^3 x^3+3 a^2 x^2+2 a^2 x^2 \log (x)+2 a x+1\right )}{2 a^2 x \sqrt{1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.157, size = 70, 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 ) }{2\, \left ({a}^{2}{x}^{2}-1 \right ) ^{2}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}} \right ) ^{{\frac{3}{2}}}\sqrt{-{a}^{2}{x}^{2}+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 \frac{{\left (a x + 1\right )}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{3}{2}}}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.55127, size = 778, normalized size = 5.33 \begin{align*} \left [\frac{{\left (a^{3} c x^{3} - a c x\right )} \sqrt{-c} \log \left (\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} +{\left (a x^{5} - a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right ) -{\left (2 \, a^{3} c x^{3} -{\left (2 \, a^{3} + 2 \, a + 1\right )} c x^{2} + 2 \, a c x + c\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \,{\left (a^{4} x^{3} - a^{2} x\right )}}, -\frac{2 \,{\left (a^{3} c x^{3} - a c x\right )} \sqrt{c} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left (a x^{3} + a x\right )} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} -{\left (a^{2} + 1\right )} c x^{2} + c}\right ) +{\left (2 \, a^{3} c x^{3} -{\left (2 \, a^{3} + 2 \, a + 1\right )} c x^{2} + 2 \, a c x + c\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \,{\left (a^{4} x^{3} - a^{2} x\right )}}\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{\left (- c \left (-1 + \frac{1}{a x}\right ) \left (1 + \frac{1}{a x}\right )\right )^{\frac{3}{2}} \left (a x + 1\right )}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, 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{{\left (a x + 1\right )}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{3}{2}}}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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