Optimal. Leaf size=96 \[ \frac {\sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{x \sqrt {1-\frac {1}{a x}}}-\frac {3 \sqrt {a} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \sinh ^{-1}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\sqrt {1-\frac {1}{a x}}} \]
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Rubi [A] time = 0.20, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6176, 6181, 80, 54, 215} \[ \frac {\sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{x \sqrt {1-\frac {1}{a x}}}-\frac {3 \sqrt {a} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \sinh ^{-1}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\sqrt {1-\frac {1}{a x}}} \]
Antiderivative was successfully verified.
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Rule 54
Rule 80
Rule 215
Rule 6176
Rule 6181
Rubi steps
\begin {align*} \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx &=\frac {\sqrt {c-a c x} \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}}}{x^{3/2}} \, dx}{\sqrt {1-\frac {1}{a x}} \sqrt {x}}\\ &=-\frac {\left (\sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {x}{a}}{\sqrt {x} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} x}-\frac {\left (3 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 \sqrt {1-\frac {1}{a x}}}\\ &=\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} x}-\frac {\left (3 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{a}}} \, dx,x,\sqrt {\frac {1}{x}}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} x}-\frac {3 \sqrt {a} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \sinh ^{-1}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\sqrt {1-\frac {1}{a x}}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 78, normalized size = 0.81 \[ -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (3 \sqrt {a} \sinh ^{-1}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )-\sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}\right )}{\sqrt {1-\frac {1}{a x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 232, normalized size = 2.42 \[ \left [\frac {3 \, {\left (a^{2} x^{2} - a x\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + a c x + 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 2 \, c}{a x^{2} - x}\right ) + 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, {\left (a x^{2} - x\right )}}, -\frac {3 \, {\left (a^{2} x^{2} - a x\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x^{2} - x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 48, normalized size = 0.50 \[ a {\left (\frac {3 \, \arctan \left (\frac {\sqrt {-a c x - c}}{\sqrt {c}}\right )}{\sqrt {c}} - \frac {\sqrt {-a c x - c}}{a c x}\right )} {\left | c \right |} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 92, normalized size = 0.96 \[ -\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (3 \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}}{\sqrt {c}}\right ) x a c -\sqrt {-c \left (a x +1\right )}\, \sqrt {c}\right )}{\left (a x -1\right ) \sqrt {-c \left (a x +1\right )}\, x \sqrt {c}} \]
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 \[ \int \frac {\sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{x^{2}}\,{d x} \]
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 \[ \int \frac {\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{x^2} \,d x \]
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 \[ \int \frac {\sqrt {\frac {a x - 1}{a x + 1}} \sqrt {- c \left (a x - 1\right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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