Optimal. Leaf size=149 \[ -\frac{104}{105} a^3 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}-\frac{104 a^3 c \sqrt{1-\frac{1}{a^2 x^2}}}{105 \sqrt{c-\frac{c}{a x}}}-\frac{26 a c \sqrt{1-\frac{1}{a^2 x^2}}}{35 x^2 \sqrt{c-\frac{c}{a x}}}+\frac{2 c \sqrt{1-\frac{1}{a^2 x^2}}}{7 x^3 \sqrt{c-\frac{c}{a x}}} \]
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Rubi [A] time = 0.302801, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {6178, 881, 871, 795, 649} \[ -\frac{104}{105} a^3 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}-\frac{104 a^3 c \sqrt{1-\frac{1}{a^2 x^2}}}{105 \sqrt{c-\frac{c}{a x}}}-\frac{26 a c \sqrt{1-\frac{1}{a^2 x^2}}}{35 x^2 \sqrt{c-\frac{c}{a x}}}+\frac{2 c \sqrt{1-\frac{1}{a^2 x^2}}}{7 x^3 \sqrt{c-\frac{c}{a x}}} \]
Antiderivative was successfully verified.
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Rule 6178
Rule 881
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 &=-\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (c-\frac{c x}{a}\right )^{3/2}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=\frac{2 c \sqrt{1-\frac{1}{a^2 x^2}}}{7 \sqrt{c-\frac{c}{a x}} x^3}-\frac{13}{7} \operatorname{Subst}\left (\int \frac{x^2 \sqrt{c-\frac{c x}{a}}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{2 c \sqrt{1-\frac{1}{a^2 x^2}}}{7 \sqrt{c-\frac{c}{a x}} x^3}-\frac{26 a c \sqrt{1-\frac{1}{a^2 x^2}}}{35 \sqrt{c-\frac{c}{a x}} x^2}+\frac{1}{35} (52 a) \operatorname{Subst}\left (\int \frac{x \sqrt{c-\frac{c x}{a}}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{104}{105} a^3 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+\frac{2 c \sqrt{1-\frac{1}{a^2 x^2}}}{7 \sqrt{c-\frac{c}{a x}} x^3}-\frac{26 a c \sqrt{1-\frac{1}{a^2 x^2}}}{35 \sqrt{c-\frac{c}{a x}} x^2}-\frac{1}{105} \left (52 a^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c-\frac{c x}{a}}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{104 a^3 c \sqrt{1-\frac{1}{a^2 x^2}}}{105 \sqrt{c-\frac{c}{a x}}}-\frac{104}{105} a^3 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+\frac{2 c \sqrt{1-\frac{1}{a^2 x^2}}}{7 \sqrt{c-\frac{c}{a x}} x^3}-\frac{26 a c \sqrt{1-\frac{1}{a^2 x^2}}}{35 \sqrt{c-\frac{c}{a x}} x^2}\\ \end{align*}
Mathematica [A] time = 0.123479, size = 66, normalized size = 0.44 \[ -\frac{2 a \sqrt{1-\frac{1}{a^2 x^2}} \left (104 a^3 x^3-52 a^2 x^2+39 a x-15\right ) \sqrt{c-\frac{c}{a x}}}{105 x^2 (a x-1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.112, size = 70, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,ax+2 \right ) \left ( 104\,{x}^{3}{a}^{3}-52\,{a}^{2}{x}^{2}+39\,ax-15 \right ) }{105\,{x}^{3} \left ( ax-1 \right ) }\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}\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}} \sqrt{\frac{a x - 1}{a x + 1}}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55549, size = 170, normalized size = 1.14 \begin{align*} -\frac{2 \,{\left (104 \, a^{4} x^{4} + 52 \, a^{3} x^{3} - 13 \, a^{2} x^{2} + 24 \, 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}} \sqrt{\frac{a x - 1}{a x + 1}}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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