Optimal. Leaf size=187 \[ -\frac{4 a^2 \sqrt{c-a^2 c x^2}}{x \sqrt{1-a^2 x^2}}+\frac{3 a \sqrt{c-a^2 c x^2}}{2 x^2 \sqrt{1-a^2 x^2}}-\frac{\sqrt{c-a^2 c x^2}}{3 x^3 \sqrt{1-a^2 x^2}}-\frac{4 a^3 \log (x) \sqrt{c-a^2 c x^2}}{\sqrt{1-a^2 x^2}}+\frac{4 a^3 \sqrt{c-a^2 c x^2} \log (a x+1)}{\sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.207641, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6153, 6150, 88} \[ -\frac{4 a^2 \sqrt{c-a^2 c x^2}}{x \sqrt{1-a^2 x^2}}+\frac{3 a \sqrt{c-a^2 c x^2}}{2 x^2 \sqrt{1-a^2 x^2}}-\frac{\sqrt{c-a^2 c x^2}}{3 x^3 \sqrt{1-a^2 x^2}}-\frac{4 a^3 \log (x) \sqrt{c-a^2 c x^2}}{\sqrt{1-a^2 x^2}}+\frac{4 a^3 \sqrt{c-a^2 c x^2} \log (a x+1)}{\sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 6153
Rule 6150
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{-3 \tanh ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^4} \, dx &=\frac{\sqrt{c-a^2 c x^2} \int \frac{e^{-3 \tanh ^{-1}(a x)} \sqrt{1-a^2 x^2}}{x^4} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\sqrt{c-a^2 c x^2} \int \frac{(1-a x)^2}{x^4 (1+a x)} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\sqrt{c-a^2 c x^2} \int \left (\frac{1}{x^4}-\frac{3 a}{x^3}+\frac{4 a^2}{x^2}-\frac{4 a^3}{x}+\frac{4 a^4}{1+a x}\right ) \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{\sqrt{c-a^2 c x^2}}{3 x^3 \sqrt{1-a^2 x^2}}+\frac{3 a \sqrt{c-a^2 c x^2}}{2 x^2 \sqrt{1-a^2 x^2}}-\frac{4 a^2 \sqrt{c-a^2 c x^2}}{x \sqrt{1-a^2 x^2}}-\frac{4 a^3 \sqrt{c-a^2 c x^2} \log (x)}{\sqrt{1-a^2 x^2}}+\frac{4 a^3 \sqrt{c-a^2 c x^2} \log (1+a x)}{\sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0426764, size = 72, normalized size = 0.39 \[ \frac{\sqrt{c-a^2 c x^2} \left (-\frac{4 a^2}{x}-4 a^3 \log (x)+4 a^3 \log (a x+1)+\frac{3 a}{2 x^2}-\frac{1}{3 x^3}\right )}{\sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.094, size = 81, normalized size = 0.4 \begin{align*}{\frac{24\,{a}^{3}\ln \left ( x \right ){x}^{3}-24\,{a}^{3}{x}^{3}\ln \left ( ax+1 \right ) +24\,{a}^{2}{x}^{2}-9\,ax+2}{ \left ( 6\,{a}^{2}{x}^{2}-6 \right ){x}^{3}}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }\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{\sqrt{-a^{2} c x^{2} + c}{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (a x + 1\right )}^{3} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.21139, size = 1022, normalized size = 5.47 \begin{align*} \left [\frac{12 \,{\left (a^{5} x^{5} - a^{3} x^{3}\right )} \sqrt{c} \log \left (\frac{4 \, a^{5} c x^{5} +{\left (2 \, a^{6} + 4 \, a^{5} + 6 \, a^{4} + 4 \, a^{3} + a^{2}\right )} c x^{6} +{\left (4 \, a^{4} - 4 \, a^{3} - 6 \, a^{2} - 4 \, a - 1\right )} c x^{4} - 5 \, a^{2} c x^{2} - 4 \, a c x -{\left (4 \, a^{3} x^{3} -{\left (4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} x^{4} + 6 \, a^{2} x^{2} + 4 \, a x + 1\right )} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - c}{a^{4} x^{6} + 2 \, a^{3} x^{5} - 2 \, a x^{3} - x^{2}}\right ) + \sqrt{-a^{2} c x^{2} + c}{\left (24 \, a^{2} x^{2} -{\left (24 \, a^{2} - 9 \, a + 2\right )} x^{3} - 9 \, a x + 2\right )} \sqrt{-a^{2} x^{2} + 1}}{6 \,{\left (a^{2} x^{5} - x^{3}\right )}}, -\frac{24 \,{\left (a^{5} x^{5} - a^{3} x^{3}\right )} \sqrt{-c} \arctan \left (-\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}{\left ({\left (2 \, a^{2} + 2 \, a + 1\right )} x^{2} + 2 \, a x + 1\right )} \sqrt{-c}}{2 \, a^{3} c x^{3} -{\left (2 \, a^{3} + a^{2}\right )} c x^{4} +{\left (a^{2} + 2 \, a + 1\right )} c x^{2} - 2 \, a c x - c}\right ) - \sqrt{-a^{2} c x^{2} + c}{\left (24 \, a^{2} x^{2} -{\left (24 \, a^{2} - 9 \, a + 2\right )} x^{3} - 9 \, a x + 2\right )} \sqrt{-a^{2} x^{2} + 1}}{6 \,{\left (a^{2} x^{5} - x^{3}\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 (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}} \sqrt{- c \left (a x - 1\right ) \left (a x + 1\right )}}{x^{4} \left (a x + 1\right )^{3}}\, 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{\sqrt{-a^{2} c x^{2} + c}{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (a x + 1\right )}^{3} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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