Optimal. Leaf size=75 \[ \sqrt {c-a^2 c x^2}+2 \sqrt {c} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )+\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \]
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Rubi [A] time = 0.35, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6167, 6152, 1809, 844, 217, 203, 266, 63, 208} \[ \sqrt {c-a^2 c x^2}+2 \sqrt {c} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )+\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 208
Rule 217
Rule 266
Rule 844
Rule 1809
Rule 6152
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx &=-\int \frac {e^{-2 \tanh ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x} \, dx\\ &=-\left (c \int \frac {(1-a x)^2}{x \sqrt {c-a^2 c x^2}} \, dx\right )\\ &=\sqrt {c-a^2 c x^2}+\frac {\int \frac {-a^2 c+2 a^3 c x}{x \sqrt {c-a^2 c x^2}} \, dx}{a^2}\\ &=\sqrt {c-a^2 c x^2}-c \int \frac {1}{x \sqrt {c-a^2 c x^2}} \, dx+(2 a c) \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx\\ &=\sqrt {c-a^2 c x^2}-\frac {1}{2} c \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-a^2 c x}} \, dx,x,x^2\right )+(2 a c) \operatorname {Subst}\left (\int \frac {1}{1+a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c-a^2 c x^2}}\right )\\ &=\sqrt {c-a^2 c x^2}+2 \sqrt {c} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )+\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2 c}} \, dx,x,\sqrt {c-a^2 c x^2}\right )}{a^2}\\ &=\sqrt {c-a^2 c x^2}+2 \sqrt {c} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )+\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )\\ \end {align*}
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Mathematica [A] time = 0.09, size = 97, normalized size = 1.29 \[ \sqrt {c-a^2 c x^2}+\sqrt {c} \log \left (\sqrt {c} \sqrt {c-a^2 c x^2}+c\right )-2 \sqrt {c} \tan ^{-1}\left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (a^2 x^2-1\right )}\right )-\sqrt {c} \log (x) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.89, size = 191, normalized size = 2.55 \[ \left [-2 \, \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) + \frac {1}{2} \, \sqrt {c} \log \left (-\frac {a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) + \sqrt {-a^{2} c x^{2} + c}, \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + \sqrt {-c} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) + \sqrt {-a^{2} c x^{2} + c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 95, normalized size = 1.27 \[ -\frac {2 \, c \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} + \frac {2 \, a \sqrt {-c} \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{{\left | a \right |}} + \sqrt {-a^{2} c x^{2} + c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 121, normalized size = 1.61 \[ -\sqrt {-a^{2} c \,x^{2}+c}+\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )+2 \sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 \left (x +\frac {1}{a}\right ) a c}+\frac {2 a c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 \left (x +\frac {1}{a}\right ) a c}}\right )}{\sqrt {a^{2} c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 86, normalized size = 1.15 \[ a^{2} {\left (\frac {\sqrt {c} \arcsin \left (a x\right )}{a^{2}} + \frac {\sqrt {-a^{2} c x^{2} + c}}{a^{2}}\right )} + a {\left (\frac {\sqrt {c} \arcsin \left (a x\right )}{a} + \frac {\sqrt {c} \log \left (\frac {2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c}}{{\left | x \right |}} + \frac {2 \, c}{{\left | x \right |}}\right )}{a}\right )} \]
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 \[ \int \frac {\sqrt {c-a^2\,c\,x^2}\,\left (a\,x-1\right )}{x\,\left (a\,x+1\right )} \,d x \]
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 \[ \int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x - 1\right )}{x \left (a x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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