Optimal. Leaf size=152 \[ \frac {3 \sqrt {c-a^2 c x^2}}{x^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 a \log (x) \sqrt {c-a^2 c x^2}}{x \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 a \sqrt {c-a^2 c x^2} \log (a x+1)}{x \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {\sqrt {c-a^2 c x^2}}{2 a x^3 \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rubi [A] time = 0.24, antiderivative size = 152, 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 = {6192, 6193, 88} \[ \frac {3 \sqrt {c-a^2 c x^2}}{x^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {\sqrt {c-a^2 c x^2}}{2 a x^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 a \log (x) \sqrt {c-a^2 c x^2}}{x \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 a \sqrt {c-a^2 c x^2} \log (a x+1)}{x \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6192
Rule 6193
Rubi steps
\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}}}{x^2} \, dx}{\sqrt {1-\frac {1}{a^2 x^2}} x}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \frac {(-1+a x)^2}{x^3 (1+a x)} \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \left (\frac {1}{x^3}-\frac {3 a}{x^2}+\frac {4 a^2}{x}-\frac {4 a^3}{1+a x}\right ) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x}\\ &=-\frac {\sqrt {c-a^2 c x^2}}{2 a \sqrt {1-\frac {1}{a^2 x^2}} x^3}+\frac {3 \sqrt {c-a^2 c x^2}}{\sqrt {1-\frac {1}{a^2 x^2}} x^2}+\frac {4 a \sqrt {c-a^2 c x^2} \log (x)}{\sqrt {1-\frac {1}{a^2 x^2}} x}-\frac {4 a \sqrt {c-a^2 c x^2} \log (1+a x)}{\sqrt {1-\frac {1}{a^2 x^2}} x}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 68, normalized size = 0.45 \[ \frac {\sqrt {c-a^2 c x^2} \left (4 a^2 \log (x)-4 a^2 \log (a x+1)+\frac {3 a}{x}-\frac {1}{2 x^2}\right )}{a x \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 88, normalized size = 0.58 \[ \frac {8 \, a^{3} \sqrt {-c} x^{2} \log \left (\frac {2 \, a^{3} c x^{2} + 2 \, a^{2} c x + \sqrt {-a^{2} c} {\left (2 \, a x + 1\right )} \sqrt {-c} + a c}{a x^{2} + x}\right ) + \sqrt {-a^{2} c} {\left (6 \, a x - 1\right )}}{2 \, a x^{2}} \]
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 \[ \int \frac {\sqrt {-a^{2} c x^{2} + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 77, normalized size = 0.51 \[ \frac {\left (8 a^{2} \ln \relax (x ) x^{2}-8 \ln \left (a x +1\right ) x^{2} a^{2}+6 a x -1\right ) \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{2 x^{2} \left (a x -1\right )^{2}} \]
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 \[ \int \frac {\sqrt {-a^{2} c x^{2} + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{3}}\,{d x} \]
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 \[ \int \frac {\sqrt {c-a^2\,c\,x^2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{x^3} \,d x \]
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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