Optimal. Leaf size=101 \[ \frac{5 a^2 \sqrt{c-a^2 c x^2}}{3 x}-\frac{a \sqrt{c-a^2 c x^2}}{x^2}+\frac{\sqrt{c-a^2 c x^2}}{3 x^3}+a^3 \left (-\sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right ) \]
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Rubi [A] time = 0.368385, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {6167, 6152, 1807, 835, 807, 266, 63, 208} \[ \frac{5 a^2 \sqrt{c-a^2 c x^2}}{3 x}-\frac{a \sqrt{c-a^2 c x^2}}{x^2}+\frac{\sqrt{c-a^2 c x^2}}{3 x^3}+a^3 \left (-\sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right ) \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6152
Rule 1807
Rule 835
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^4} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^4} \, dx\\ &=-\left (c \int \frac{(1-a x)^2}{x^4 \sqrt{c-a^2 c x^2}} \, dx\right )\\ &=\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 = 0.11434, size = 82, normalized size = 0.81 \[ \frac{\left (5 a^2 x^2-3 a x+1\right ) \sqrt{c-a^2 c x^2}}{3 x^3}-a^3 \sqrt{c} \log \left (\sqrt{c} \sqrt{c-a^2 c x^2}+c\right )+a^3 \sqrt{c} \log (x) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.059, size = 254, normalized size = 2.5 \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{-{a}^{2}c \left ( x+{a}^{-1} \right ) ^{2}+2\, \left ( x+{a}^{-1} \right ) ac}-2\,{\frac{{a}^{4}c}{\sqrt{{a}^{2}c}}\arctan \left ({\frac{\sqrt{{a}^{2}c}x}{\sqrt{-{a}^{2}c \left ( x+{a}^{-1} \right ) ^{2}+2\, \left ( x+{a}^{-1} \right ) ac}}} \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 x - 1\right )}}{{\left (a x + 1\right )} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72065, size = 371, normalized size = 3.67 \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{- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x - 1\right )}{x^{4} \left (a x + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14053, size = 338, normalized size = 3.35 \begin{align*} \frac{2 \, a^{3} c \arctan \left (-\frac{\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} - \frac{2 \,{\left (3 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{5} a^{3} c + 3 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{4} a^{2} \sqrt{-c} c{\left | a \right |} - 12 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{2} a^{2} \sqrt{-c} c^{2}{\left | a \right |} - 3 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )} a^{3} c^{3} + 5 \, a^{2} \sqrt{-c} c^{3}{\left | a \right |}\right )}}{3 \,{\left ({\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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