Optimal. Leaf size=99 \[ -\frac{5 a^2 \sqrt{c-a^2 c x^2}}{3 x}+\frac{a \sqrt{c-a^2 c x^2}}{x^2}-\frac{\sqrt{c-a^2 c x^2}}{3 x^3}+a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right ) \]
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Rubi [A] time = 0.274834, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {6152, 1807, 835, 807, 266, 63, 208} \[ -\frac{5 a^2 \sqrt{c-a^2 c x^2}}{3 x}+\frac{a \sqrt{c-a^2 c x^2}}{x^2}-\frac{\sqrt{c-a^2 c x^2}}{3 x^3}+a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right ) \]
Antiderivative was successfully verified.
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Rule 6152
Rule 1807
Rule 835
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-2 \tanh ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^4} \, dx &=c \int \frac{(1-a x)^2}{x^4 \sqrt{c-a^2 c x^2}} \, dx\\ &=-\frac{\sqrt{c-a^2 c x^2}}{3 x^3}-\frac{1}{3} \int \frac{6 a c-5 a^2 c x}{x^3 \sqrt{c-a^2 c x^2}} \, dx\\ &=-\frac{\sqrt{c-a^2 c x^2}}{3 x^3}+\frac{a \sqrt{c-a^2 c x^2}}{x^2}+\frac{\int \frac{10 a^2 c^2-6 a^3 c^2 x}{x^2 \sqrt{c-a^2 c x^2}} \, dx}{6 c}\\ &=-\frac{\sqrt{c-a^2 c x^2}}{3 x^3}+\frac{a \sqrt{c-a^2 c x^2}}{x^2}-\frac{5 a^2 \sqrt{c-a^2 c x^2}}{3 x}-\left (a^3 c\right ) \int \frac{1}{x \sqrt{c-a^2 c x^2}} \, dx\\ &=-\frac{\sqrt{c-a^2 c x^2}}{3 x^3}+\frac{a \sqrt{c-a^2 c x^2}}{x^2}-\frac{5 a^2 \sqrt{c-a^2 c x^2}}{3 x}-\frac{1}{2} \left (a^3 c\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-a^2 c x}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{c-a^2 c x^2}}{3 x^3}+\frac{a \sqrt{c-a^2 c x^2}}{x^2}-\frac{5 a^2 \sqrt{c-a^2 c x^2}}{3 x}+a \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2 c}} \, dx,x,\sqrt{c-a^2 c x^2}\right )\\ &=-\frac{\sqrt{c-a^2 c x^2}}{3 x^3}+\frac{a \sqrt{c-a^2 c x^2}}{x^2}-\frac{5 a^2 \sqrt{c-a^2 c x^2}}{3 x}+a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )\\ \end{align*}
Mathematica [A] time = 0.127972, size = 82, normalized size = 0.83 \[ \frac{\left (-5 a^2 x^2+3 a x-1\right ) \sqrt{c-a^2 c x^2}}{3 x^3}+a^3 \sqrt{c} \log \left (\sqrt{c} \sqrt{c-a^2 c x^2}+c\right )+a^3 \left (-\sqrt{c}\right ) \log (x) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.047, size = 253, normalized size = 2.6 \begin{align*} -2\,{\frac{{a}^{2} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{3/2}}{cx}}-2\,{a}^{4}x\sqrt{-{a}^{2}c{x}^{2}+c}-2\,{\frac{{a}^{4}c}{\sqrt{{a}^{2}c}}\arctan \left ({\frac{\sqrt{{a}^{2}c}x}{\sqrt{-{a}^{2}c{x}^{2}+c}}} \right ) }+\sqrt{c}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{-{a}^{2}c{x}^{2}+c} \right ) } \right ){a}^{3}-\sqrt{-{a}^{2}c{x}^{2}+c}{a}^{3}+2\,{a}^{3}\sqrt{-c{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,ac \left ( x+{a}^{-1} \right ) }+2\,{\frac{{a}^{4}c}{\sqrt{{a}^{2}c}}\arctan \left ({\frac{\sqrt{{a}^{2}c}x}{\sqrt{-c{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,ac \left ( x+{a}^{-1} \right ) }}} \right ) }+{\frac{a}{c{x}^{2}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{1}{3\,c{x}^{3}} \left ( -{a}^{2}c{x}^{2}+c \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{\sqrt{-a^{2} c x^{2} + c}{\left (a^{2} x^{2} - 1\right )}}{{\left (a x + 1\right )}^{2} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.4601, size = 370, normalized size = 3.74 \begin{align*} \left [\frac{3 \, a^{3} \sqrt{c} x^{3} \log \left (-\frac{a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right ) - 2 \, \sqrt{-a^{2} c x^{2} + c}{\left (5 \, a^{2} x^{2} - 3 \, a x + 1\right )}}{6 \, x^{3}}, \frac{3 \, a^{3} \sqrt{-c} x^{3} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) - \sqrt{-a^{2} c x^{2} + c}{\left (5 \, a^{2} x^{2} - 3 \, a x + 1\right )}}{3 \, x^{3}}\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{\sqrt{- a^{2} c x^{2} + c}}{a x^{5} + x^{4}}\, dx - \int \frac{a x \sqrt{- a^{2} c x^{2} + c}}{a x^{5} + x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21167, size = 284, normalized size = 2.87 \begin{align*} -\frac{1}{12} \,{\left (\frac{24 \, a^{2} c \arctan \left (\frac{\sqrt{-c + \frac{2 \, c}{a x + 1}}}{\sqrt{-c}}\right ) \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right )}{\sqrt{-c}} - \frac{2 \,{\left (3 \, \pi a^{2} c - 10 \, a^{2} c\right )} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right )}{\sqrt{-c}} + \frac{9 \, a^{2}{\left (c - \frac{2 \, c}{a x + 1}\right )}^{2} c \sqrt{-c + \frac{2 \, c}{a x + 1}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) + 3 \, a^{2} c^{3} \sqrt{-c + \frac{2 \, c}{a x + 1}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) - 8 \, a^{2} c^{2}{\left (-c + \frac{2 \, c}{a x + 1}\right )}^{\frac{3}{2}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right )}{{\left (c - \frac{c}{a x + 1}\right )}^{3}}\right )}{\left | a \right |} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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