Optimal. Leaf size=117 \[ \frac{8 a^3 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{105 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{2 a c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{7 x^2 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{8 a^3 c \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{35 \sqrt{c-\frac{c}{a x}}} \]
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Rubi [A] time = 0.246082, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {6178, 871, 795, 649} \[ \frac{8 a^3 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{105 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{2 a c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{7 x^2 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{8 a^3 c \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{35 \sqrt{c-\frac{c}{a x}}} \]
Antiderivative was successfully verified.
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Rule 6178
Rule 871
Rule 795
Rule 649
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^4} \, dx &=-\left (c \operatorname{Subst}\left (\int \frac{x^2 \sqrt{1-\frac{x^2}{a^2}}}{\sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )\right )\\ &=-\frac{2 a c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{7 \left (c-\frac{c}{a x}\right )^{3/2} x^2}+\frac{1}{7} (4 a c) \operatorname{Subst}\left (\int \frac{x \sqrt{1-\frac{x^2}{a^2}}}{\sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{8 a^3 c \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{35 \sqrt{c-\frac{c}{a x}}}-\frac{2 a c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{7 \left (c-\frac{c}{a x}\right )^{3/2} x^2}+\frac{1}{35} \left (4 a^2 c\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x^2}{a^2}}}{\sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{8 a^3 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{105 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{8 a^3 c \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{35 \sqrt{c-\frac{c}{a x}}}-\frac{2 a c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{7 \left (c-\frac{c}{a x}\right )^{3/2} x^2}\\ \end{align*}
Mathematica [A] time = 0.108857, size = 66, normalized size = 0.56 \[ -\frac{2 a \sqrt{1-\frac{1}{a^2 x^2}} \left (8 a^3 x^3-4 a^2 x^2+3 a x+15\right ) \sqrt{c-\frac{c}{a x}}}{105 x^2 (a x-1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.119, size = 55, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,ax+2 \right ) \left ( 8\,{a}^{2}{x}^{2}-12\,ax+15 \right ) }{105\,{x}^{3}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}{\frac{1}{\sqrt{{\frac{ax-1}{ax+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{c - \frac{c}{a x}}}{x^{4} \sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58065, size = 162, normalized size = 1.38 \begin{align*} -\frac{2 \,{\left (8 \, a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} + 18 \, a x + 15\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{105 \,{\left (a x^{4} - x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{c - \frac{c}{a x}}}{x^{4} \sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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