Optimal. Leaf size=41 \[ \frac {c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{a}-\frac {c \sqrt {1-a^2 x^2}}{a} \]
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Rubi [A] time = 0.07, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6131, 6128, 266, 50, 63, 208} \[ \frac {c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{a}-\frac {c \sqrt {1-a^2 x^2}}{a} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 208
Rule 266
Rule 6128
Rule 6131
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx &=-\frac {c \int \frac {e^{\tanh ^{-1}(a x)} (1-a x)}{x} \, dx}{a}\\ &=-\frac {c \int \frac {\sqrt {1-a^2 x^2}}{x} \, dx}{a}\\ &=-\frac {c \operatorname {Subst}\left (\int \frac {\sqrt {1-a^2 x}}{x} \, dx,x,x^2\right )}{2 a}\\ &=-\frac {c \sqrt {1-a^2 x^2}}{a}-\frac {c \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{2 a}\\ &=-\frac {c \sqrt {1-a^2 x^2}}{a}+\frac {c \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{a^3}\\ &=-\frac {c \sqrt {1-a^2 x^2}}{a}+\frac {c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 42, normalized size = 1.02 \[ -\frac {c \left (\sqrt {1-a^2 x^2}-\log \left (\sqrt {1-a^2 x^2}+1\right )+\log (x)\right )}{a} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.43, size = 41, normalized size = 1.00 \[ -\frac {c \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + \sqrt {-a^{2} x^{2} + 1} c}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 58, normalized size = 1.41 \[ \frac {c \log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right ) - c \log \left (-\sqrt {-a^{2} x^{2} + 1} + 1\right ) - 2 \, \sqrt {-a^{2} x^{2} + 1} c}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 34, normalized size = 0.83 \[ \frac {c \left (-\sqrt {-a^{2} x^{2}+1}+\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 50, normalized size = 1.22 \[ \frac {c \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right )}{a} - \frac {\sqrt {-a^{2} x^{2} + 1} c}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.82, size = 37, normalized size = 0.90 \[ \frac {c\,\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right )}{a}-\frac {c\,\sqrt {1-a^2\,x^2}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 20.41, size = 61, normalized size = 1.49 \[ \begin {cases} \frac {- c \sqrt {- a^{2} x^{2} + 1} + \frac {c \left (- \log {\left (-1 + \frac {1}{\sqrt {- a^{2} x^{2} + 1}} \right )} + \log {\left (1 + \frac {1}{\sqrt {- a^{2} x^{2} + 1}} \right )}\right )}{2}}{a} & \text {for}\: a \neq 0 \\c x + \tilde {\infty } c \log {\relax (x )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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