Optimal. Leaf size=72 \[ 3 a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right )-\frac {c \sqrt {1-a^2 x^2}}{x \sqrt {c-a c x}} \]
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Rubi [A] time = 0.16, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6128, 879, 875, 208} \[ 3 a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right )-\frac {c \sqrt {1-a^2 x^2}}{x \sqrt {c-a c x}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 875
Rule 879
Rule 6128
Rubi steps
\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx &=\frac {\int \frac {(c-a c x)^{3/2}}{x^2 \sqrt {1-a^2 x^2}} \, dx}{c}\\ &=-\frac {c \sqrt {1-a^2 x^2}}{x \sqrt {c-a c x}}-\frac {1}{2} (3 a) \int \frac {\sqrt {c-a c x}}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c \sqrt {1-a^2 x^2}}{x \sqrt {c-a c x}}-\left (3 a^3 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{-a^2 c+a^2 c^2 x^2} \, dx,x,\frac {\sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right )\\ &=-\frac {c \sqrt {1-a^2 x^2}}{x \sqrt {c-a c x}}+3 a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 52, normalized size = 0.72 \[ \frac {\sqrt {1-a x} \left (3 a c \tanh ^{-1}\left (\sqrt {a x+1}\right )-\frac {c \sqrt {a x+1}}{x}\right )}{\sqrt {c-a c x}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.49, size = 207, normalized size = 2.88 \[ \left [\frac {3 \, {\left (a^{2} x^{2} - a x\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + a c x - 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{a x^{2} - x}\right ) + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{2 \, {\left (a x^{2} - x\right )}}, \frac {3 \, {\left (a^{2} x^{2} - a x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{a x^{2} - x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.02, size = 90, normalized size = 1.25 \[ -a {\left (\frac {3 \, \arctan \left (\frac {\sqrt {a c x + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} + \frac {\sqrt {a c x + c}}{a c x}\right )} {\left | c \right |} + \frac {3 \, a \sqrt {c} {\left | c \right |} \arctan \left (\frac {\sqrt {2} \sqrt {c}}{\sqrt {-c}}\right ) + \sqrt {2} a \sqrt {-c} {\left | c \right |}}{\sqrt {-c} \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 79, normalized size = 1.10 \[ \frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \left (-3 \arctanh \left (\frac {\sqrt {c \left (a x +1\right )}}{\sqrt {c}}\right ) x a c +\sqrt {c \left (a x +1\right )}\, \sqrt {c}\right )}{\left (a x -1\right ) \sqrt {c \left (a x +1\right )}\, x \sqrt {c}} \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{{\left (a x + 1\right )} x^{2}}\,{d x} \]
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 \[ \int \frac {\sqrt {1-a^2\,x^2}\,\sqrt {c-a\,c\,x}}{x^2\,\left (a\,x+1\right )} \,d x \]
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 \[ \int \frac {\sqrt {- c \left (a x - 1\right )} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{x^{2} \left (a x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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