Optimal. Leaf size=113 \[ \frac {2 a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{3/2}}{5 c}+\frac {2}{5} a^2 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}}+\frac {8 a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}{5 \sqrt {c-\frac {c}{a x}}} \]
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Rubi [A] time = 0.22, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6178, 795, 657, 649} \[ \frac {2 a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{3/2}}{5 c}+\frac {2}{5} a^2 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}}+\frac {8 a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}{5 \sqrt {c-\frac {c}{a x}}} \]
Antiderivative was successfully verified.
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Rule 649
Rule 657
Rule 795
Rule 6178
Rubi steps
\begin {align*} \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {x \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 a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{3/2}}{5 c}+\frac {(3 a) \operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{3/2}}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{5 c}\\ &=\frac {2}{5} a^2 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}}+\frac {2 a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{3/2}}{5 c}+\frac {1}{5} (4 a) \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 {8 a^2 c \sqrt {1-\frac {1}{a^2 x^2}}}{5 \sqrt {c-\frac {c}{a x}}}+\frac {2}{5} a^2 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}}+\frac {2 a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{3/2}}{5 c}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 58, normalized size = 0.51 \[ \frac {2 a \sqrt {1-\frac {1}{a^2 x^2}} \left (6 a^2 x^2-3 a x+1\right ) \sqrt {c-\frac {c}{a x}}}{5 x (a x-1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 69, normalized size = 0.61 \[ \frac {2 \, {\left (6 \, a^{3} x^{3} + 3 \, a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{5 \, {\left (a x^{3} - x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c - \frac {c}{a x}} \sqrt {\frac {a x - 1}{a x + 1}}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 62, normalized size = 0.55 \[ \frac {2 \left (a x +1\right ) \left (6 a^{2} x^{2}-3 a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {a x -1}{a x +1}}}{5 x^{2} \left (a x -1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c - \frac {c}{a x}} \sqrt {\frac {a x - 1}{a x + 1}}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.35, size = 62, normalized size = 0.55 \[ \frac {2\,\sqrt {c-\frac {c}{a\,x}}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (6\,a^3\,x^3+3\,a^2\,x^2-2\,a\,x+1\right )}{5\,x^2\,\left (a\,x-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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