Optimal. Leaf size=82 \[ -\frac {2 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-4 a \sqrt {c-\frac {c}{a x}}+4 \sqrt {2} a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \]
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Rubi [A] time = 0.38, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6167, 6133, 25, 514, 444, 50, 63, 208} \[ -\frac {2 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-4 a \sqrt {c-\frac {c}{a x}}+4 \sqrt {2} a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \]
Antiderivative was successfully verified.
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Rule 25
Rule 50
Rule 63
Rule 208
Rule 444
Rule 514
Rule 6133
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx &=-\int \frac {e^{-2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx\\ &=-\int \frac {\sqrt {c-\frac {c}{a x}} (1-a x)}{x^2 (1+a x)} \, dx\\ &=\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2}}{x (1+a x)} \, dx}{c}\\ &=\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2}}{\left (a+\frac {1}{x}\right ) x^2} \, dx}{c}\\ &=-\frac {a \operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{3/2}}{a+x} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {2 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-(2 a) \operatorname {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{a+x} \, dx,x,\frac {1}{x}\right )\\ &=-4 a \sqrt {c-\frac {c}{a x}}-\frac {2 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-(4 a c) \operatorname {Subst}\left (\int \frac {1}{(a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-4 a \sqrt {c-\frac {c}{a x}}-\frac {2 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}+\left (8 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )\\ &=-4 a \sqrt {c-\frac {c}{a x}}-\frac {2 a \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}+4 \sqrt {2} a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )\\ \end {align*}
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Mathematica [A] time = 0.07, size = 69, normalized size = 0.84 \[ \frac {2 (1-7 a x) \sqrt {c-\frac {c}{a x}}}{3 x}+4 \sqrt {2} a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 161, normalized size = 1.96 \[ \left [\frac {2 \, {\left (3 \, \sqrt {2} a \sqrt {c} x \log \left (-\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + 3 \, a c x - c}{a x + 1}\right ) - {\left (7 \, a x - 1\right )} \sqrt {\frac {a c x - c}{a x}}\right )}}{3 \, x}, -\frac {2 \, {\left (6 \, \sqrt {2} a \sqrt {-c} x \arctan \left (\frac {\sqrt {2} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{2 \, c}\right ) + {\left (7 \, a x - 1\right )} \sqrt {\frac {a c x - c}{a x}}\right )}}{3 \, x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 254, normalized size = 3.10 \[ \frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (-18 \sqrt {a \,x^{2}-x}\, a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, x^{3}+6 a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, x^{3}+12 a^{\frac {3}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x \sqrt {\frac {1}{a}}+9 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, x^{3} a^{2}-6 a^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) x^{3}-9 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, x^{3} a^{2}-2 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\, \sqrt {\frac {1}{a}}\right )}{3 x^{2} \sqrt {\left (a x -1\right ) x}\, \sqrt {a}\, \sqrt {\frac {1}{a}}} \]
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 \[ \int \frac {{\left (a x - 1\right )} \sqrt {c - \frac {c}{a x}}}{{\left (a x + 1\right )} x^{2}}\,{d x} \]
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 \[ \int \frac {\sqrt {c-\frac {c}{a\,x}}\,\left (a\,x-1\right )}{x^2\,\left (a\,x+1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x - 1\right )}{x^{2} \left (a x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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