Optimal. Leaf size=187 \[ -\frac {4 a \sqrt {c-\frac {c}{a^2 x^2}}}{x \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 \sqrt {c-\frac {c}{a^2 x^2}}}{2 x^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 a^2 \log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 a^2 \sqrt {c-\frac {c}{a^2 x^2}} \log (a x+1)}{\sqrt {1-\frac {1}{a^2 x^2}}}-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{3 a x^3 \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rubi [A] time = 0.28, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6197, 6193, 88} \[ -\frac {4 a \sqrt {c-\frac {c}{a^2 x^2}}}{x \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 \sqrt {c-\frac {c}{a^2 x^2}}}{2 x^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{3 a x^3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 a^2 \log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 a^2 \sqrt {c-\frac {c}{a^2 x^2}} \log (a x+1)}{\sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6193
Rule 6197
Rubi steps
\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^3} \, dx &=\frac {\sqrt {c-\frac {c}{a^2 x^2}} \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}}}{x^3} \, dx}{\sqrt {1-\frac {1}{a^2 x^2}}}\\ &=\frac {\sqrt {c-\frac {c}{a^2 x^2}} \int \frac {(-1+a x)^2}{x^4 (1+a x)} \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}}}\\ &=\frac {\sqrt {c-\frac {c}{a^2 x^2}} \int \left (\frac {1}{x^4}-\frac {3 a}{x^3}+\frac {4 a^2}{x^2}-\frac {4 a^3}{x}+\frac {4 a^4}{1+a x}\right ) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}}}\\ &=-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{3 a \sqrt {1-\frac {1}{a^2 x^2}} x^3}+\frac {3 \sqrt {c-\frac {c}{a^2 x^2}}}{2 \sqrt {1-\frac {1}{a^2 x^2}} x^2}-\frac {4 a \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}} x}-\frac {4 a^2 \sqrt {c-\frac {c}{a^2 x^2}} \log (x)}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 a^2 \sqrt {c-\frac {c}{a^2 x^2}} \log (1+a x)}{\sqrt {1-\frac {1}{a^2 x^2}}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 75, normalized size = 0.40 \[ \frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (-4 a^3 \log (x)+4 a^3 \log (a x+1)-\frac {4 a^2}{x}+\frac {3 a}{2 x^2}-\frac {1}{3 x^3}\right )}{a \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 91, normalized size = 0.49 \[ \frac {24 \, a^{4} \sqrt {c} x^{3} \log \left (\frac {2 \, a^{3} c x^{2} + 2 \, a^{2} c x + \sqrt {a^{2} c} {\left (2 \, a x + 1\right )} \sqrt {c} + a c}{a x^{2} + x}\right ) - {\left (24 \, a^{2} x^{2} - 9 \, a x + 2\right )} \sqrt {a^{2} c}}{6 \, a^{2} x^{3}} \]
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 \[ \int \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 90, normalized size = 0.48 \[ -\frac {\left (24 a^{3} \ln \relax (x ) x^{3}-24 a^{3} x^{3} \ln \left (a x +1\right )+24 a^{2} x^{2}-9 a x +2\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{6 \left (a x -1\right )^{2} x^{2}} \]
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 \[ \int \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{x^3} \,d x \]
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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